Mechano-chemical interface processes
with application to rock weathering
Anja Røyne
Thesis submitted for the degree of Philosophia Doctor
Department of Physics, University of Oslo, Norway
January 2011
til Tord og Synne
Preface
The subject of this thesis is coupled mechanical and chemical processes that
take place at interfaces in rocks during weathering, and which have an effect
on the chemical and physical transformation of the rock. It is put in the
context of rock weathering, but is also relevant to other research areas such
as material science.
The research is at the cross-section between physics, geology and physical chemistry. The methods used include field work, theory and laboratory
experiments. Coming from primarily an experimental physics background, I
feel very lucky to have had the opportunity to expand my knowledge in these
different directions.
In geological studies, we are faced with the end result of millions of years of
coupled processes which have taken place under largely unknown conditions.
Laboratory studies offer a chance to study isolated processes under controlled
conditions. Theoretical studies allow us to explore the consequences of what
we know, and of what we think we know, in beautiful, simple geometries with
well defined boundary conditions. I think that in order to fully understand
the how the Earth works, it is necessary to combine these three approaches.
This thesis takes one tiny step towards that goal.
I have had three excellent supervisors who have helped me out along the
way. I’d like to thank Bjørn Jamtveit particularly for welcoming me to PGP
with open arms, and for taking me on field trips to Svalbard and South
Africa. Bjørn has also spent countless hours with me at the microscope and
the microprobe, trying to make sense of these extrordinarily difficult and
complicated rocks and to do something about my ignorance on elementary
geology and chemistry.
Anders Malthe-Sørenssen has been my theoretical support, and since he
knows just about everything, it always pays off to stop by his office for help
and advice.
My principal supervisor has been Dag Kristian Dysthe. He has also been
a great co-worker in my laboratory studies. Experimental work is something
which has to be learned by doing, and I have been fortunate to learn from
ii
Preface
Dag’s wealth of experience, and to benefit from his contagious enthusiasm
and creativity.
There are some other people without whom my lab activities would have
been much more troublesome. Olav Gundersen has very patiently helped me
with large and small practical issues and has had to deal with my frequent
frustrations over things not working. Big thanks also go to Finn, Steinar,
Øyvind, Helge and the other guys at the workshop who have made my clumsy
constructions become working equipment. Lab work would also have been
much more frustrating and less fun without the good company and enormous
amount of help I’ve received from my fellow experimentalists, including Jan
Bisschop, Anders Nermoen, Christophe Raufaste, Matthieu Angeli, Delphine
Croizé, Julien Scheibert, Olivier Galland and Simon De Villiers.
Then there are some people at PGP who know ”everything” and are
always willing to take the time to share their wisdom. I would particularly
like to thank Paul Meakin, Francois Renard, Ray Fletcher, Jens Feder and
Håkon Austrheim for everything they have taught me.
Thanks to past and present office mates - Hans, Marta, Andreas and
Harry - for the good compay. Jan Ludvig, Ingrid, Jaqueline, Maya, Luiza,
Nina, Filip, Kirsten, Marcin&Marcin, Espen, Karen, Karin, Trine-Lise and
all you other PGPans, you are a fantastic bunch of talented and friendly
people.
During the course of my PhD studies a number of things have happened in
my life. I have lost two people dear to me, my mother and my grandmother,
but I also got married and had two wonderful children. Although these
events have sometimes severely distracted me from my work, I have never
had anything but support from my colleagues and supervisors, and to that I
am grateful.
I would also like to acknowledge the support from friends and family
outside of work. You’re such great people, and just the fact that you’re
reading this means you are actually making an effort at understanding this
project which I can never seem to explain in a very clear manner.
Asbjørn, my love, my life would have been much poorer without you.
Thanks for doing all of the Christmas preparations while I was finishing my
thesis.
Tord and Synne, I dedicate the thesis to you. When you’re old enough
to read this, I want you to know that the best part of every day during this
time was coming home to see you.
Oslo, January 2011
Anja Røyne
Contents
Preface
i
1 Introduction
1
2 Generation of stresses due to crystal growth
2.1 Thermodynamics of stressed crystal surfaces in contact with
their solution . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Effect of surface tension, and a possibility of sustaining
large stresses at equilibrium . . . . . . . . . . . . . .
2.1.2 Calculation of solution activity . . . . . . . . . . . .
2.2 Confined fluid films . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Surface energies and disjoining pressure . . . . . . . .
2.2.2 Surface forces . . . . . . . . . . . . . . . . . . . . . .
2.3 Thesis results . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Interface structure . . . . . . . . . . . . . . . . . . .
2.3.2 Surface energy of calcite . . . . . . . . . . . . . . . .
2.3.3 Stress evolution through time: Effect of boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Fracture mechanics and kinetic fracture processes
3.1 Basic fracture mechanics: Griffith theory . . . . . .
3.2 Effect of the environment . . . . . . . . . . . . . . .
3.3 Time effects . . . . . . . . . . . . . . . . . . . . . .
3.4 Experimental crack velocity data . . . . . . . . . .
3.5 Proposed mechanisms for subcritical crack growth .
3.6 Reaction rate theory of slow crack growth . . . . .
3.7 Thesis results: Subcritical crack growth in calcite .
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4 Effects of stress generation in pores during weathering
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4.1 Replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Microcrack propagation . . . . . . . . . . . . . . . . . . . . . . 39
iv
CONTENTS
4.2.1
4.3
Thesis results: Coupling subcritical crack propagation
and stress generation due to crystallization . . . . . . . 39
Build-up of elastic strain energy and macroscopic expansion . 41
4.3.1 Thesis results: Hierarchical fracturing . . . . . . . . . . 44
5 Summary and outlook
47
Bibliography
51
Scientific Papers
Paper I - Controls on rock weathering rates by reaction-induced hierarchical fracturing . . . . . . . . . . . . . . . . . . . . . .
Paper II - Growth rims and interface structure of a single crystal face growing from solution in the direction of compressive
normal stress . . . . . . . . . . . . . . . . . . . . . . . . . .
Paper III - Experimental investigation of surface energy and subcritical crack growth in calcite . . . . . . . . . . . . . . . . .
Paper IV - Subcritical crack propagation driven by crystal growth .
63
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Chapter 1
Introduction
Most rocks are formed at pressures and temperatures exceeding those at the
Earth’s surface (Figure 1.1). When such rocks are brought to the surface
through uplifting process, they become thermodynamically unstable with respect to the prevailing conditions. Weathering refers to the in situ breakdown
and transformation of rocks to equilibrate with the conditions at or near the
surface of the Earth. The principle agent of rock weathering is water, and
the driving force for weathering is solar energy.
Weathering is an important part of the rock cycle together with erosion,
which are the processes by which rock debris is transported from the weathering site by agents such as ice, water and wind. When weathering products
are transported by erosion and subsequently deposited on riverbeds, lakes
and oceans, they are called sediment, which may transform to sedimentary
rock through burial and cementation.
Even slight variations in rock properties and climatic conditions may lead
to significant differences in weathering rates. This leads to the formation of
topological features, both on small and large scales, and thereby directly
affects the shape of the world we live in.
When organic matter becomes part of the sediment and is transformed
into sedimentary rock, nutrients and building-blocks which are essential for
life becomes locked in the rock cycle. Weathering is the process by which
these nutrients are released back into the biosphere, and it is therefore critical
for the existence of life on the planet [1]. Weathering also affects the long
term global climate because precipitation of carbonates during weathering
binds CO2 from the atmosphere [1, 2]. On shorter timescales, weathering
has a direct effect on the human society through its destructive effect on
concrete and building stones [3].
Weathering also produces some stunningly beautiful and intriguing patterns that we observe around us (e.g. Figure 1.2c-e). Patterns trigger our
2
Introduction
Solar energy
Rock at
surface
interaction with
biosphere
uplift
weathering
erosion
Igneous
rock
uplift
Sediment
uplift
metamorphism
burial
cementation
melting
Sedimentary
rock
Metamorphic
rock
metamorphism
Geothermal energy
Figure 1.1: The rock cycle.
3
curiosity and offer a pathway for gaining a deeper understanding of processes in nature. Understanding the formation of a particular weathering
morphology requires knowledge of the principal mechanisms involved in the
weathering process. By understanding the process behind the weathering
patterns on Earth, we may infer something about processes and conditions
in distant places such as Mars and other planets [4, 5].
Weathering takes place through chemical, mechanical and biological processes (Figure 1.2). Chemical weathering includes the dissolution of unstable
minerals and precipitation of stable phases, and takes place when mineral
surfaces are exposed to water containing dissolved chemical species.
During mechanical weathering, intact rock is broken into smaller pieces.
Perhaps the most important physical weathering processes is abrasion, which
is the frictional scraping of intact rock by moving particles during erosion
due to ice, water, wind or gravity. Fracturing and physical breakdown may
also take place due to tectonic processes, thermal stresses, expansion due to
unloading and the hydraulic action of waves, or when stresses build up in
pores due to the growth of ice, salt crystals or other minerals.
Biology affects rock weathering through both chemical and mechanical
processes. Dissolution and precipitation can be controlled by organic acids
excreted by microorganisms, fungi, lichen or plant roots. Tree roots may
expand to wedge open preexisting fractures in rocks (Figure 1.2b). The
extent to which microfractures can be opened by plant roots or fungal hyphae
is not yet known [6, 7].
Mechanical and chemical weathering are coupled processes. When rocks
are broken, new surfaces are formed, which may be exposed to water and
allow for increased chemical weathering. Fractures also act as efficient pathways for transport of water and dissolved species. In addition, chemical
reactions may change the stress state in the rock in such a way that new
fractures are formed (see Figure 1.3).
Because weathering rates are important for the evolution of global climate, considerable effort is being spent on models that describe the different
stages of weathering as well as the propagation rate of the weathering front
(the zone above the intact bedrock where weathering reactions are active)
and the erosion of the top surface of the Earth. These models need to include a basic understanding of transport, kinetic processes and the coupling
between mechanical and chemical processes [8–12].
During weathering, processes on the scale of outcrops and landscapes
may be controlled by processes that operate on the pore scale or below.
For instance, overall weathering rates can be controlled by the kinetics of
the mineral-fluid interface reactions [13]. In other cases, the rate limiting
step is transport, which takes place in large scale fractures as well as in
4
Introduction
a)
c)
b)
d)
e)
Figure 1.2: Weathering morphologies. a) This face on Angkor Wat (Cambodia) has been affected by chemical (colour changes, dissolutional rounding of
sharp angles) and mechanical (flaking, and possibly some granular disintegration leading to rounding of edges) processes. b) An extreme example of
biological weathering at Angkor Wat. c) Tafoni structures, Argentina (picture courtesy of Bjørn Jamtveit). d) Spalling of corners during spheroidal
weathering, Puerto Rico (picture courtesy of Ray Fletcher) e) Spheroidal
weathering patterns, Australia (picture courtesy of Andrew Putnis).
5
changed stress state
Mechanical
fracturing
effect on
dissolution
and growth rates
Chemical
dissolution
growth
increased reactive surface area
increased transport
Figure 1.3: Feedback between mechanical and chemical weathering processes.
Fracturing may increase the reactive surface area and transport of reactive
species, thereby accelerating the chemical processes. Dissolution and growth
may change the stress state of the rock in such a way that fractures form.
There is also a feedback between the state of stress of a solid surface and the
dissolution or growth of that surface.
6
Introduction
confined fluids at grain boundaries. The rate limiting step can vary across
the weathering profile depending on the local permeability and chemistry
[13, 14]. Laboratory studies are performed to investigate the dissolution
rates of rock forming minerals, and progress is being made on how to reconcile
field and laboratory data [15, 16]. On the outcrop scale, several field studies
indicate that the velocity of the weathering front is constant in time [17–19].
Constraints on large scale rates of erosion and weathering front propagation
are obtained through isotope measurements [20–22].
In this thesis, the focus has been on stress generation due to chemical
weathering processes and how this is coupled to fracture propagation both
on the pore and outcrop scale. Particular emphasis has been placed on the
interplay between kinetic processes at the crack tip and on the crystal surface.
The basic physics of stress generation due to crystallization and slow fracture propagation are introduced in Chapters 2 and 3. The coupling of these
processes is discussed in Chapter 4. In Chapter 5 the principal findings are
summarized and some reflections are made on issues which need to be addressed in future studies. The main results of this study consist of the four
scientific papers included at the end of the thesis. In each of the introductory chapters, the relevant results from the papers are presented and placed
in context. Figure 1.4 gives an overview of the processes and mechanisms
discussed in this thesis.
7
a)
c)
b)
d)
Chapter 2
Chapter 2
Paper II
Paper IV
Chapter 3
Chapter 4
Paper III
Paper IV
Chapter 4
Chapter 2
Chapter 3
Paper III
e)
Chapter 4
Paper I
Figure 1.4: Overview of processes which are discussed in this thesis. a)
Stresses generated due to crystal growth from a supersaturated solution in
a pore. b) Close-up on the crystal-pore wall interface, where important processes and properties are crystal growth in the direction of normal stress
(Chapter 2, Paper II, Paper IV), liquid film where normal stresses are sustained by the disjoining pressure (Chapter 2) and the energy of a solid-liquid
interface (Chapter 2, Chapter 3, Paper III). c) Crack propagation (Chapter 3,
Chapter 4, Paper III, Paper IV). d) Replacement (Chapter 4). e) Large-scale
build up of elastic strain energy (Chapter 4, Paper I).
8
Introduction
Chapter 2
Generation of stresses due to
crystal growth
Crystallization refers to the precipitation of a structured solid from a melt
or a solution. The focus of this chapter is crystallization from solutions
because this is the most common situation in weathering processes, with the
exception of frost damage.
The driving force for crystallization depends on a number of parameters
including subcooling (for melts) or supersaturation (for solutions), temperature, hydrostatic pressure and surface normal stress. Given the appropriate
conditions, it may be energetically favourable for a crystal to grow on a face
which is in close contact with a confining surface, and thereby perform mechanical work by ”pushing” on the surrounding matrix. The result is that
crystals may ”wedge” themselves into rocks to propagate fractures and produce damage through frost heave, salt weathering and the like. The stresses
generated in this context are often referred to as ”the force of crystallization”
or ”crystallization pressure”.
Crystallization from solutions and from melts have much in common, but
are also different in some respects. The main difference is that transport of
heat and fluid advection are important in the latter case, while diffusional
transport of dissolved species tends to govern the former. Stress generation
due to ice growth in pores is discussed in abundant literature [23–30].
Work generated by growing crystals was first reported by Lavalle in 1853
[31], who noted that crystals growing from solution were able to push themselves upwards. Later Becker and Dey [32] (see Figure 2.1) and Taber [33]
demonstrated that growing crystals could lift considerable weights. The only
quantitative measurements were made by Correns and Steinborn [34], who
found an agreement with their derived equation for equilibrium crystallization pressure. It has, however, been pointed out that the equation used by
10
Generation of stresses due to crystal growth
1 kg
GLASS
GLASS
Figure 2.1: Experimental setup used by Becker and Dey [32]. A 1 cm wide
crystal of alum placed between two glass plates is loaded with a 1 kg weight.
The crystal is immersed in a solution of alum which is supersaturated due to
evaporation.
Correns and Steinborn is off by almost a factor two due to neglecting the
number of ions in the dissolved salt (see Section 2.1.2) and the effects of
non-ideality. This factor is larger than the uncertainty in the reported data,
something which casts some doubts on the interpretation of his quantitative
results [35].
It was postulated by Correns [34], and has also later been assumed, that
in order for crystallization to take place in the direction of normal load, there
must be a liquid film present between the loading surface and the crystal.
Without this film, there would be no material transport to the growing surface (solid diffusion is orders of magnitudes too slow for the situations we
are considering), and the crystal could not grow. The existence of this liquid
film is discussed in more detail in Section 2.2.
2.1
Thermodynamics of stressed crystal surfaces in contact with their solution
The thermodynamic equilibrium of a non-hydrostatically stressed solid with
its solution was first considered by Gibbs [36] and later elaborated by, among
others, Paterson [37], whose derivations are followed in this section.
The solubility of a solid is controlled by a number of parameters, such
as temperature, pressure, surface stress and presence of other species in the
2.1 Thermodynamics of stressed crystal surfaces
11
Fluid
pressure p
A σn
Solid
stress σij
Figure 2.2: Schematic representation of closed system containing a solid
under stress σij and a fluid under pressure p in which the solid can dissolve.
The stress is applied on the solid by a permeable loading frame (hatched).
Figure after Paterson [37].
liquid. It can be quantified by considering the chemical potential µ of the
surface of the solid as a function of these parameters, defined by the Gibbs
d
(U − T S + pV ) where U is internal energy, T is temperrelation µi = dN
i
ature, S is entropy, p is pressure and V is volume. According to the laws
of thermodynamics, a system will always evolve towards a state of equilibrium, where the chemical potentials of all components and at all positions
are equal. However, different components of the system will evolve at different rates. In a situation involving the chemical equilibrium between a solid
and a fluid, it is enough to consider the chemical potential of the surface, as
the interior of the solid is dependent on solid diffusion and will take a much
longer time to equilibrate. As liquid diffusion is much faster, all of the fluid
in the system needs to be considered.
The equilibrium of a solid subject to a surface normal stress which is
different from the fluid pressure can be investigated in the model system
shown in Figure 2.2. The system consists of a container with a solid immersed
in a fluid of constant pressure p. Stress is applied on the solid by a permeable
loading frame. The solid is soluble in the fluid, and the solid component in
the fluid is initially in equilibrium with the solid state. A small reversible
change is then made which consists of the following steps:
1. m units of solid is detached from the surface of the solid at site A, where
the component of stress normal to the surface is σn , and removed from
the system. This changes the internal energy of the system by −uσ m,
12
Generation of stresses due to crystal growth
where uσ is the specific internal energy of the stressed solid.
2. The loading frame is adjusted to regain contact with the surface of the
solid. The change in potential energy of the loading frame (which is
part of the internal energy of the system), is pdV = −(σn − p)vσ m,
where vσ is the specific volume of the solid in its stressed state.
3. m units of the component of the solid is introduced to the system in the
same state as it occurs in the solution, and dispersed in the solution in
the vicinity of A. The change in internal energy of the system is uL m,
where uL is the partial specific internal energy of the component of the
solid in the solution.
4. A different volume is occupied by m units of solid and m units in the
solution. To accommodate this, the walls of the container have moved.
Moving the walls back to their original position requires a work of
magnitude pdV = p(vL − vσ )m where vL is the specific volume of the
solid component in solution. This ensures that no external work is done
on the system, since the boundaries remain stationary.
5. Since the entropy of the system has changed by ∆S = −sσ m + sL m
where sL and sσ are specific entropies of the stressed solid and solid
in solution, we must add a quantity of heat equal to −T (sL − sσ )m to
ensure that the entropy of the system is unchanged.
As there is no change in entropy, and no work is done on this closed system,
the change in internal energy is zero. This gives the following equilibrium
condition:
−uσ − (σn − p)σ + uL + p(vL − vσ ) − T (sL − sσ ) = 0,
(2.1)
Rearranging to get solid and liquid properties on either side of the equality,
this becomes
uL − T sL + pvL = uσ − T sσ + σn vσ .
(2.2)
It follows from the Gibbs relation that the expression on the left hand side
equals the chemical potential of the component of the solid in solution, µL .
The equilibrium condition for a stressed solid and its solution in the vicinity
of the stressed surface is therefore
µL = uσ − T sσ + σn vσ .
(2.3)
From the theory of solutions, the chemical potential of a solute in solution
is given as
µ ≡ µ∗ (p, T ) + kT ln a
(2.4)
2.1 Thermodynamics of stressed crystal surfaces
13
where k is the Boltzmann constant. µ∗ can be taken as a constant in our
case, since it is a function of pressure and temperature only, and a is defined
as the activity of the solute. This gives
µ∗ (p, T ) + kT ln aσ = uσ − T sσ + σn vσ
(2.5)
where aσ is the activity of the component in solution in equilibrium with the
solid under normal stress σn . In a hydrostatic condition, where σn = p, this
would give
µ∗ (p, T ) + kT ln ap = up − T sp + pvp .
(2.6)
Combining these equations gives us the following equilibrium condition:
kT ln
aσ
− [(uσ − T sσ ) − (up − T sp )] − σn (vσ − vp ) + (σn − p)vp = 0. (2.7)
ap
The u − T s terms represent the difference in elastic strain energy between
the solid under stress and at hydrostatic pressure, while the σn (vσ − vp ) term
is the contribution from the differences in specific volumes, which is also a
consequence of elastic strain. These terms will normally be much smaller
than the last term, unless σ ≈ p, which can be the case for the free face
of a uniaxially stressed crystal. We will therefore for now turn our interest
to the case where σn p. To simplify, we introduce the deviatoric stress
σc = σn − p. We will also assume that the solute activity and specific volume
at hydrostatic pressure p are very close to those of some reference state, so
that we can replace ap and vp with a0 and v0 to get
kT ln
aσ
− σc v0 = 0.
a0
(2.8)
This equation is a condition for equilibrium. It tells us that in every solidliquid system, the solid will neither grow nor dissolve when the deviatoric
surface normal stress equals σc , provided that the solute activity of the fluid
in the immediate vicinity of the stressed solid is aσ . If the equality is replaced
with >, the crystal will tend to grow; in the opposite case, it will tend to
dissolve.
2.1.1
Effect of surface tension, and a possibility of sustaining large stresses at equilibrium
For very small crystals (. 0.1 µm), an extra term is needed in the equilibrium
condition to account for the energy cost of creating new surface area [38]:
kT ln
dA
aσ
− σc v0 − γcl v0
= 0,
a0
dV
(2.9)
14
Generation of stresses due to crystal growth
where γcl is the surface energy of the crystal in the liquid. As this last
term is larger for smaller crystals, they are in equilibrium with solutions
of higher solute activity than larger crystals. This corresponds to a higher
solubility (higher equilibrium fluid concentration) in small pores, which has
been demonstrated experimentally [39].
If a large crystal is loaded on two faces, while the other faces remain free,
the chemical potential will be higher on the loaded faces than on the free
faces. Growth can take place on all faces given that the solution activity
is sufficiently high, but if the system is closed, the activity of the solution
will become lower as material is removed from solution to be incorporated
in the crystal. When the free faces reach equilibrium with the bulk solution,
the chemical potential of the loaded faces is such that they will tend to
dissolve. It is therefore not easy to envisage an equilibrium situation where
a high crystallization pressure is maintained through time. If the system is
completely saturated with fluid, a possible solution is to consider cylindrical
pores, or circular pores with small entrances, where the geometry of the
pore can be such that the free faces of the crystal will attain an equilibrium
curvature which is higher than the curvature of the loaded faces, and hence
be in equilibrium is reached with a fluid which on the loaded faces has a high
enough activity to allow for some sustained stress [26, 40].
2.1.2
Calculation of solution activity
The degree of supersaturation of the fluid is described in terms of the activity
ratio as in Equation (2.8), but depending on the nonideality of the fluid the
calculation of the exact value of the activity can be quite complex. For
nonhydrous salts of low solubility, it is sufficiently accurate to neglect the
non-ideal behavior of water and the water component of hydrated salts, which
yields the following approximation [40]:
kT ln
aσ
= nkT ln Ω,
a0
(2.10)
where n is the total number of ions released upon complete dissociation of the
salt, and Ω = c/c0 is the solution supersaturation. More accurate expressions
are given by Steiger [40].
2.2
Confined fluid films
In order for a liquid film to exist between two solid surfaces subject to a
deviatoric surface normal stress σ, force balance requires the pressure pf in
2.2 Confined fluid films
γwl
h
γcl
15
γwl
γcl
P(h)
h
γcw
Figure 2.3: Energy of a film between two solid surfaces. Left: In a thick film
the associated energy is the sum of the two solid-liquid interfacial energies
γcl and γwl . Middle: For thin films there is an additional energy term P (h)
which depends on the separation h. Right: When the film thickness goes to
zero, the energy is that of the dry solid-solid interface γcl .
the liquid film to be pf = σ and consequently higher than the hydrostatic
pressure p of the bulk fluid. This elevated pressure is often referred to as the
disjoining pressure of the thin film.
The disjoining pressure can be derived from a macroscopic continuum
picture by considering the interfacial energies of solid-liquid and solid-solid
interfaces or a microscopic continuum model where the interaction forces
between two surfaces separated by a liquid are considered; both of these
methods yield consistent results. The theory, as described in the following
subsections, is well established for static interfaces. If one interface is moving,
as may be the case when a crystal is growing against a load, there may be
additional effects which come into play.
2.2.1
Surface energies and disjoining pressure
This section follows the derivations by de Gennes et al. [41].
Consider a crystal growing in a pore, where the crystal surface is separated
from the pore wall surface by a thick (thickness h 100 nm) film of liquid.
The energy of the thick film is given by the sum of the crystal-liquid and
wall-liquid interfacial energies, U∞ = γcl + γwl (where the energy U in this
case is defined per unit area). If the surfaces are moved progressively closer
to each other so that the film becomes very thin (h → 0), the energy must
become that of the dry solid-solid interface, U0 = γcw (Figure 2.3).
For intermediate film thicknesses, a thickness dependent energy U (h) can
be assigned by introducing some function P (h) to get
U (h) = γcl + γwl + P (h),
(2.11)
16
Generation of stresses due to crystal growth
with the boundary conditions P (∞) = 0 and P (0) = γcw − (γcl + γwl ).
If the thickness of the film is increased by an amount dh by adding dN
molecules, then dN = dh/v0 where v0 is the specific volume of the molecule
in the liquid phase. The energy of the film is changed by an amount µf dN ,
where µf is the chemical potential of the liquid in the film, which includes
an additional term to the reference chemical potential µ∗ of the bulk liquid
to account for the change in P (h):
µf dN = µ∗ dN +
dP
dh.
dh
(2.12)
We can then define the disjoining pressure Π(h) by
Π(h) ≡ −
dP
.
dh
(2.13)
Combining these two equations reveals that the disjoining pressure is related
to the chemical potential through
µf = µ∗ − v0 Π(h).
(2.14)
This shows that if the disjoining pressure is positive, the chemical potential of
the film becomes lower than that of the bulk liquid, which will act as a driving
force for fluid migration against the positive pressure gradient and into the
thin film. Equilibrium with the bulk fluid is reached when dP/dh = 0.
2.2.2
Surface forces
The interaction energy between surfaces separated by a liquid by more than
a few molecular diameters is determined by long range van der Waals and
electrostatic forces. The non-retarded van der Waals interaction energy between two planar surfaces separated by a liquid film of thickness h follows
the dependence [42]
AH
UvdW = −
,
(2.15)
12πh2
where the Hamaker constant AH describes the interaction of the particular
materials involved. For interactions across a water film, the Hamaker constant is typically positive, which means that the resulting van der Waals
forces are attractive and increase with decreasing separation.
Electrostatic forces arise between surfaces in a liquid due to the build-up
of so-called electric double layers (DL) due to the balancing of surface charges
2.2 Confined fluid films
17
by ions in solution. For planar surfaces the interaction energy related to this
process depends exponentially on the separation distance, [42]
UDL ∝
1 −h/λD
e
,
λD
(2.16)
where the length scale is given by the Debye screening length λD , which is
a function of the electrolyte concentration of the fluid. In ultra pure water,
λD is close to 1 µm, while in concentrated solutions it is on the order of a
few Å.
The so-called DLVO theory [42], named after Derjaguin and Landau [43]
and Verwey and Overbeek [44] which approximates the net force acting between the surfaces to be the sum of these two forces, provides a good qualitative understanding of how the interaction energy varies with separation
and electrolyte concentration (Figure 2.4). At very short separations, the
attractive forces always dominate (provided that the Hamaker constant is
positive) implying that if surfaces are brought sufficiently close together,
they will form a strong, dry adhesive contact. Since the van der Waals interactions are largely insensitive to electrolyte concentration while the range
and magnitude of the electrostatic interactions are strongly dependent on
electrolyte concentration, the qualitative result of the combination of these
two forces depends on the electrolyte concentration. For low concentrations,
repulsive forces dominate up to very large separations. As the concentration
is increased, repulsive forces only dominate for some intermediate range of
separations, before at very high concentrations the repulsive contribution becomes negligible. In this case, the solid surfaces will be forced into contact
and no disjoining pressure develops in the film. In the cases where there
is a repulsive region, there exists a repulsive energy barrier which must be
overcome in order to bring the dry surfaces into contact. The force needed
to overcome this barrier defines the maximum disjoining pressure which can
develop in the thin film. If the normal stress across the two surfaces is larger
than the maximum disjoining pressure, then the liquid film will be displaced.
While the DLVO theory is a good qualitative explanation for many phenomena in colloid and surface science, it is often not sufficient for making
quantitative predictions. Experimental measurements are necessary for obtaining quantitative data on the forces between surfaces of different materials
and in different liquids. Measurements are typically made using the surface
forces apparatus (SFA) [45] or by atomic force microscopy (AFM). Some
measurements for geologically relevant systems are described by references
[46–48].
For surface separations less than about 2 nm, the interaction energy is
dominated by effects related to the discrete molecular nature of the surfaces,
18
Generation of stresses due to crystal growth
Interaction energy (U = UvdW + UDL)
Repulsive barrier
region
Long-range attractive region
UDL
UDL + UvdW
UvdW
Repulsive barrier height
Film thickness (h)
Attractive secondary minimum
Short range repulsive region
Deep primary minimum
Figure 2.4: Conceptual figure of the interaction energy of two surfaces separated by a fluid film of thickness h as described by DLVO theory (Figure
courtesy of Paul Meakin). The blue line shows the repulsive screened electrostatic interactions, which decay exponentially with film thickness. The
attractive van der Waals interactions, which follow an inverse power dependence of film thickness, are shown as the red line. In DLVO theory, the total
interaction energy is given by the sum of these two interactions. As a guide
to the eye, the pink line shows the negative of the van der Waals interaction
energy. In the situation shown, attractive forces dominate at long and very
short separations, while there is an intermediate range where the repulsive
forces dominate. A repulsive barrier must be overcome in order to bring the
two surfaces into contact at the deep primary minimum.
2.3 Thesis results
19
solvent and ions [45, 46, 49, 50], which are not accounted for in the continuum DLVO theory. At least in the case of mica surfaces (which have been
most commonly used in SFA experiments), ”steric-hydration” forces between
surfaces with a strongly bound layer of water molecules give rise to disjoining pressures of more than 100 MPa for concentrated solutions, while DLVO
theory can account for pressures on the order of 1 MPa [46].
2.3
2.3.1
Thesis results
Interface structure
Consider a crystal in contact with a supersaturated solution and loaded by a
heavy weight on the top face such as in the setup by Becker and Dey (Figure
2.1). If the crystal face in contact with the glass plate is initially rough, then
the stress distribution on the crystal face will range from very high in the areas closest to the glass plate, to essentially zero in portions which are further
away. From a continuum perspective, because the equilibrium supersaturation is higher for surfaces subject to a large normal stress (Equation (2.8)),
we would expect portions of the surface subject to a high normal stress to
grow much more slowly than unstressed portions of the surface, or even to
dissolve if the local stress is higher than σc . With time, this would cause
the surface to become smooth and the stress on the crystal face to become
evenly distributed (assuming that the loading surface is a perfectly smooth
plane).
The growth rate of a crystal surface may be limited by growth kinetics,
transport, or both. If a crystal face is separated from the loading surface
by a liquid film of only a few nanometer thickness (which is required for the
disjoining pressure to be sufficient to transmit the normal stress between the
surfaces [47]), it is reasonable to assume that growth is limited by transport,
since the diffusion current, which is proportional to the film thickness, will
be very small in this case. Weyl [51] analyzed this situation for a cylindrical
crystal with a normal stress applied to its circular face, and calculated the
rate of growth in the direction of load as a function of stress and bulk supersaturation. Weyl’s study is one of very few dynamic models of this sort of
process, and is extensively cited in papers dealing with stress generated by
crystallization.
In addition to calculating growth rates, the model also predicts that for
mean applied stresses up to half of the equilibrium stress σc , growth should
only take place on a rim surrounding a hollow core. Growth rims are commonly found on crystals grown from solution while resting on the bottom of
20
Generation of stresses due to crystal growth
0.2
δ /X
0.15
0.1
0.05
0
-6
-5
-4
-3
-2
-1
0
log10(s/sc)
Figure 2.5: Measured growth rim widths as a function of applied normal
stress (see Paper II). The prediction from Weyl’s model is shown as a solid
line.
a dish, and they were also reported and discussed by Becker and Dey [32,
see also figure 2.1] and Taber [33].
Paper II describes experiments which have been performed in order to
test the growth rim predictions by Weyl. The initial motivation was to use
these results to infer how the thickness of the liquid film was related to
applied stress and solution saturation. However, surprisingly, the measured
growth rim widths showed none of the dependencies given in Weyl’s model
(Figure 2.5). The reason for this became evident upon examining the surface
morphologies of the growth rims. They are in fact very rough with topological
features which are several µm in height (Figure 2.6). The contact between
the rim surface and glass cannot therefore be assumed to be smooth, which
violates the basic assumption of Weyl’s model.
It was found that the rims on the loaded faces most likely develop due
to a step bunching instability due to much more rapid growth along the
edges than in the central portion of the confined face. This happens due to
a strong concentration gradient parallel to the confined face which develops
as a consequence of the balance between diffusional transport from the bulk
2.3 Thesis results
21
a)
b)
0
h (µm)
h (µm)
0
-5
-10
0
-5
-10
1000
2000
3000
4000
position along rim (µm)
5000
6000
0
2000
4000
6000
8000
position along rim (µm)
10000
12000
Figure 2.6: Topographies of two loaded crystal faces measured at the end of
the experiment. The crystal in a) has been growing under its own weight
while that in b) was loaded by a 200g calibration weight. The line plots show
a 2D transect along the rim; different sides are marked by different colors.
The h scale shows the inferred distance from an imagined glass plate.
solution and removal of ions due to surface growth.
In the context of pressure solution, which can be viewed as the inverse
process of the force of crystallization [51], the structure and transport properties of the interface have been investigated in some detail. Since observed
rates of solution compaction are much larger than what can be explained by
diffusion through a smooth thin film [46], various models for rough interfaces
have been proposed, where diffusion through regions of thicker films allows
for increased transport. Some degree of interfacial roughness has been confirmed by experimental studies [52–54]. The results from Paper II suggest
that these considerations may also be valid in the context of the force of
crystallization.
2.3.2
Surface energy of calcite
An equilibrium thermodynamic requirement for a liquid film to exist between
a crystal face and a confining wall is that the sum of the solid-fluid interfacial
energies for the crystal and wall does not exceed the energy of a dry crystalwall interface; otherwise it may be energetically favourable for the system to
expel the liquid to produce a dry contact. This can be expressed as
γcw > γcl + γwl .
(2.17)
22
Generation of stresses due to crystal growth
If a crystal is growing in a pore where the above requirement is not fulfilled,
it would be expected to grow into dry contact with the pore wall rather than
maintaining a liquid film, and hence no stress would be exerted on the pore
wall. While this is useful as a conceptual criterion for whether stresses can
be generated during crystallization, it is less useful in practice due to the
limited availability of high quality data on interfacial energies.
The energies of mineral-fluid interfaces are typically found from numerical simulations or from precipitation studies. An alternative and less used
method is to infer the surface energies from the lower energy threshold for
subcritical crack propagation. In Paper III, this method has been used to
measure the surface energy of the {101̄4} cleavage plane of calcite as a function of water concentration (Figure 2.7). Our results agree well with previous
experiments in dry and water saturated conditions. As was also found in
molecular dynamics simulations, the effect of increased water concentration
is an almost linear decrease in surface energy.
Damage due to salt crystallization in calcite bearing building stones such
as limestone is an important engineering problem. For future studies, it
would be useful to investigate how the surface energy of calcite is affected by
the concentration of some common salts. This would have implications both
for the stress exerted by salt crystals on calcite surfaces, and also on possible
opening and propagation of fractures in calcite, which is further described in
Chapter 2.
2.3.3
Stress evolution through time: Effect of boundary conditions
Equilibrium thermodynamics can be used to predict the maximum normal
stress which can be sustained by a crystal surface before it will start to dissolve. It is often assumed that when a supersaturated solution is introduced
to a porous medium, the mean pressure which is generated in large pores can
be found from Equation (2.8) or that it is given by the pore geometry (see
Section 2.1.1).
However, it may take a long time before equilibrium is reached, and fractures may initiate and propagate while the system is still far from equilibrium. The time evolution of crystallization stresses may be coupled to
kinetic fracture propagation (See Chapter 3) and to transport within the
porous medium. Knowledge about the dynamics of stress generation due to
crystallization may therefore be important for understanding how rocks are
damaged due to crystallization stresses.
The stress on the pore wall is generated by the growing crystal and is
2.3 Thesis results
23
0.9
0.8
0.7
γes (J/m2)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
cw
0.6
0.8
1
Figure 2.7: Surface energies γse of calcite as function of water concentration
(mole fraction or surface cover fraction). Filled circles are estimates from
experiments on calcite (this study). The measured γse are plotted with asymmetric error bars: The upper shows the experimental uncertainty, while the
lower is undefined, indicating that our values represent upper bounds. Open
squares are surface energies for calcite surfaces with partial coverage of water,
calculated by atomistic simulations [55]. Ellipses show the range of numerical
results from literature for dry and wet calcite surfaces, while rectangles show
the range of experimental results from literature (see Paper III).
24
Generation of stresses due to crystal growth
initially zero before the crystal comes into contact with the pore wall. As
the crystal is growing into contact with the wall, the growth rate of the
crystal decreases with increasing stress until it reaches zero when σ = σc .
The rate of stress generation is determined by the compliance of the pore,
which describes the magnitude of stress needed for a given displacement of
the pore wall.
In Paper IV, the evolution of normal stress σ for a crystal growing in
the aperture of a penny shaped crack (Figure 4.2b) has been analyzed. One
outcome of this model is shown in Figure 2.8. Two results from the analysis
are worth mentioning in the context of this chapter. The first is that the
stress on the crystal face only asymptotically approaches σc , with a time
scale determined by the pore compliance and the kinetic constant and specific
volume of the crystal. This is a general observation which should hold for any
pore geometry, provided that the supersaturation of the solution is constant.
The second observation is that if fractures develop, the fracture velocity has
a direct control on the stress on the crystal face due to the change in fracture
compliance.
The discussion of fracture evolution in this model is presented in Chapter
4.
2.3 Thesis results
25
1
0.8
σ*, p*
II
0.6
III
I
0.4
0.2
0
0
5
10
15
20
t*
Figure 2.8: Evolution of normal stress (σ ∗ = σ/σc ) on the confined crystal
face in a penny crack (see Figure 4.2b) as a function of nondimensional
time (see Paper IV) for a crystal growing from a constant supersaturation.
The dashed blue line shows the normalized stress on the crystal face. The
effective pressure p∗ , which denotes the fluid pressure which would be required
to maintain the observed energy release rate of the crack, is shown as the
solid blue line. The red dashed line shows the asymptotic increase in stress
which is predicted for a stationary, filled crack. The crystal nucleates in the
center of the fracture with initial opening close to zero. The stress on the
crystal face which comes into contact with the crack wall is initially larger
than the effective pressure, but both the effective pressure and the crystal
stress approaches the asymptotic solution as the crystal grows to fill more
of the crack. At point I, the crystal fills the entire fracture (a = c). Crack
propagation is initiated at point II, and because the fracture compliance
increases with crack length, σ ∗ and p∗ decrease as the fracture grows. From
III the velocity of the crack tip is faster than the growth rate of the free
crystal face, so that a < c. As the crystal area supporting the load becomes
smaller relative to the fracture size, the stress on the crystal face increases.
For further details and a full description of the model, see Paper IV. The
dimensionless parameters used to generate this plot are A = B = C = 1 and
c∗0 = 2.
26
Generation of stresses due to crystal growth
Chapter 3
Fracture mechanics and kinetic
fracture processes
3.1
Basic fracture mechanics: Griffith theory
The basis of today’s fracture mechanics theories were established in the 1921
paper by Griffith [56]. At that time, the prevailing notion was that rupture
should be controlled by the maximum elastic stress or strain which could be
sustained by a material. On the other hand, experiments showed a clear relationship between scratch sizes and strength in rupture tests. In the search
for a theory to explain these observations, Griffith chose the thermodynamics
approach to a fractured system. He made use of the derivations of Inglis [57],
who had shown that the strain energy associated with a crack subject to a
remote stress is proportional to the square of the crack length. A thermodynamic rupture criterion would imply that for rupture to have taken place,
the system must have passed from the unbroken to the broken state through
a continuous decrease in the total energy of the system, which also includes
the energy of the two newly surfaces. In order to consider crack surfaces of
sufficient separation to ignore cohesive forces between the crack surfaces, and
to assess incremental crack advancements, Griffith’s starting scenario was a
body with a preexisting crack of a size fulfilling these requirements. The
equilibrium condition for a crack is then given as
δ
(UM − US ) = 0,
δc
(3.1)
which means that for an incremental change of the crack length c away from
equilibrium, the change in mechanical energy UM exactly matches the change
in surface energy US . This very powerful statement defines how fracture
propagation is understood today.
28
Fracture mechanics and kinetic fracture processes
The mechanical energy of the system can be described by the mechanical
energy release rate G (this is not a rate defined with respect to time, but
rather a mechanical energy defined per unit length of the crack front), defined
as [58]
dUM
G≡−
(3.2)
dc
The criterion for crack propagation then becomes
G>
dUS
.
dc
The mechanical energy release rate G is theoretically defined for a range of
idealized crack geometries. It is related to the stress intensity factors KI , KII
and KIII , which are commonly used in engineering, through
G=
2
KIII
(1 + ν)
KI2 KII2
+
+
,
E0
E0
E
(3.3)
where E 0 = E for plane stress and E 0 = E/(1 − ν 2 ) for plane strain, and
ν is Poisson’s ratio. The subscripts I, II and III refer, respectively, to pure
tension, in-plane shear and anti-plane shear loading.
The change in surface energy US with crack extension can be related to
the intrinsic surface energy γ of the material through [59]
2γs =
dUS
.
dc
(3.4)
This is a measure of the energy required to break bonds in order to create
two new surfaces as the crack extends. γs is defined as energy per unit area.
Within this framework, we would expect the crack to extend if G > 2γs , and
to heal if G < 2γs .
3.2
Effect of the environment
After the establishment of the Griffith theory, considerable effort was put
into trying to resolve observations which seemed to be in disagreement with
the theory. One of these was fatigue in glass, which, if loaded for long times,
was observed to fail at approximately one third of the short-time breaking
stress. Further allusion to a chemical effect was made in 1930 by Obreimoff’s
experiments on the slow fracture of mica in a controlled environment [60] .
These observations were reconciled with the Griffith theory in the 1944
paper by Orowan [61]. The change in force from vacuum to air can be related
to an adsorbed film of air or moisture which lowers the effective surface energy
3.3 Time effects
29
γs to a value γse which depends on the chemistry of the environment. This
implies that the air molecules must be able to penetrate to the crack tip. If
the crack is propagating very fast, the adsorbed film would not be able to
keep up with the motion of the crack tip. The new surfaces would in effect
open in vacuum, and so the vacuum surface energy γs should be used. If on
the other hand the crack opens very slowly, the adsorbed film has time to
diffuse to the crack tip, and the newly created surfaces have energy γse . This
explains the apparent lowering of strength from vacuum to air.
3.3
Time effects
Both in the fatigue observations and in Obreimoff’s experiment, there had
been a clear indication of a time effect on the fracture process. The Griffith
treatment of cracks was made only within the framework of the first law of
thermodynamics, and could as such not deal with effects away from equilibrium. Rice [62] generalized the Griffith theory by considering irreversible
thermodynamics. When a crack velocity v is introduced, he showed that
positive entropy production with time is only maintained if
(G − 2γse )v ≥ 0.
(3.5)
This implies that there should exist a well defined crack velocity function v(G)
defined for some deviation of G from 2γse , where v is positive (crack opening)
for G > 2γse , and negative (crack healing) for G < 2γse . For large departures
from equilibrium, crack propagation may become unstable. This happens as
the crack velocity approaches the Rayleigh wave speed in the medium, and
this dynamic fracture propagation has been studied extensively [63].
3.4
Experimental crack velocity data
Due to a number of devoted experimentalists since the middle of the 20th
century, crack velocity functions v(G) have been demonstrated for a range of
materials such as mica [60, 64, 65], glass [61, 66–71], quartz [72–78] and other
rock forming minerals including calcite [77], as well as polycrystalline rocks
[79–83] and ceramic materials [84]. The crack velocity is often reported as
a function of the stress intensity factor KI (Equation 3.3). The results can
generally be summarized as in Figure 3.1, where the subcritical crack growth
can be grouped into three regimes I, II and III. In regime I, the velocity fits
well to a power law of KI , as what is now refered to as the Charles’ law [85]
v = AKIn ,
(3.6)
Fracture mechanics and kinetic fracture processes
log(v)
30
vC
III
II
I
G0
GC
G
Figure 3.1: Schematic diagram of subcritical crack propagation (after Maugis
[59]). The curve to the right corresponds to propagation in vacuum. As water
is introduced, the threshold G0 becomes lower, and the whole curve is shifted
left. Region I crack propagation velocity is thus highly dependent on G and
on the environmental conditions. At some velocity, transport of chemically
active species to the crack tip may become rate limiting, and the rate of
crack growth becomes approximately constant (region I) until the vacuum
curve is reached. Propagation near Gc is highly stress dependent (region III).
Above Gc and vc , the dynamic fracture regime is entered, where the fracture
velocity does not depend on the environment because the crack tip is under
vacuum conditions.
3.5 Proposed mechanisms for subcritical crack growth
31
where A is some empirical prefactor and n is an exponent which is typically
in the range 10-90. When very low velocities can be measured, the curve
has been found to bend towards a threshold value which would represent the
Griffith equilibrium at G0 [59, 65, 86] (see Figure 3.2). KI gives a good fit to
a power law of KI with the exponent n in the range 10-90 [80]. While the
Charles law is a useful empirical relation, no firm theories have been posed to
explain the power law relationship, which is really indistinguishable from an
exponential relation for the small range of KI which can be observed (more
about this in Section 3.6).
Region II is more often observed in gaseous environments, and is thought
to be related to crack growth limited by the transport of environmental
species to the crack tip. Crack growth in this region is relatively insensitive
to G. The G dependence in this region can be attributed to the increased
aperture of the crack, which enhances transport.
Region III tends to fit a power law with a much higher exponent than
region I, and is thought to be controlled mainly by intrinsic crack tip processes under vacuum conditions, since chemical species can no longer keep up
with the propagating crack. Region III crack growth has been found to be
relatively insensitive to environmental conditions, but not completely [69].
In region I, increasing the concentration or activity of water tends to give
higher velocities for a given G. The degree of influence of reagent chemistry
on crack velocity is in general found to be most pronounced at low values of
G [75]. Raising the temperature gives, in most cases, increased crack velocity
[76, 80]. Changing the pH of the environment affects crack growth in a nontrivial way. For example, Atkinson [75] found that raising the pH increased
crack velocity and decreased the slope of the velocity curves for quartz, so
that the difference became less pronounced at higher G-values. Dunning [77]
found some weakening at high pH and strengthening at low pH in quartz,
but also found no simple relationship.
3.5
Proposed mechanisms for subcritical crack
growth
Much effort has gone into identifying possible mechanisms for subcritical
crack growth. Sometimes this involves some confusion about the Griffith
criterion, where the onset of dynamic fracture is thought to be the Griffith equilibrium point and subcritical fracture is explained as some sort of
anomalous fracture which has to be explained by a different mechanism than
regular crack propagation [59, 87].
32
Fracture mechanics and kinetic fracture processes
Subcritical cracking in glass is perhaps the most extensively studied system in this context, and it is where the most thorough theory for the underlying mechanism is established. Charles [66] was the first to suggest a
corrosion mechanism at the crack tip of glass where the strong Si-O-Si bond
is replaced by a much weaker OH+ -Si bond through a reaction with water
molecules. The weakening of bonds would then allow for crack growth at
smaller stresses, with a velocity controlled by the reaction rate. Michalske
[88, 89] developed this model by considering the bond rupture process as consisting of a three-step chemical reaction between the strained Si-O-Si bond
and a water molecule. This form of stress-enhanced reaction, which may
differ in glasses of different compositions [90], is now the prevailing theory
for subcritical fracture of glass [91] and quartz [92].
In some experiments, a velocity plateau has appeared at low crack velocities (well below region II behaviour). It has been suggested that this shows
crack growth through preferential dissolution at the crack tip, such that the
crack propagation rate is controlled by the dissolution rate of the material
[79, 80, 93, 94]. However dissolution and removal of material is related to
blunting of the crack tip, which is in contradiction to Griffith theory where
an atomically sharp crack tip is postulated [95]. AFM studies of crack surfaces have also indicated that although material may be removed behind the
crack tip, the tip itself remains sharp [96].
There has been some ongoing debate about whether nanoscale plastic
processes take place at the advancing crack tip in amorphous materials such
as glass [97, 98], but the most current agreement is that crack propagation
remains brittle, and that brittle nanocracks may open in a process zone ahead
of the main crack tip [99, 100].
Surface forces or surface energy effects have been found to be relevant in
many aspects. As discussed in Chapter 2, repulsive forces may be present
in liquid films confined between solid surfaces at small separations. When
fluids are present near the crack tip, the disjoining pressure in the fluid may
effectively enhance the stress at the crack tip, thus reducing the applied mechanical energy release rate necessary for crack propagation [58]. Wiederhorn
and Fuller [101] showed that surface forces associated with water trapped in a
small (< 1nm) gap between the crack walls could have an important effect on
subcritical fracture in glass.The charge densities of mica fracture surfaces has
been shown to depend on the rate of fracture propagation [64]. In addition,
the ζ-potential of the system, which is the electrostatic potential between the
adsorbed monolayer on the surface of the material and a distant point in the
fluid, may [102] or may not [103, 104] have an effect on crack propagation.
3.6 Reaction rate theory of slow crack growth
3.6
33
Reaction rate theory of slow crack growth
A system containing a crack for which G 6= G0 may evolve towards equilibrium through crack propagation or crack healing. The rate at which the
system evolves depends on the height of energy barriers which must be surpassed by some thermally activated process in order to reach an energetically
lower state. Macroscopic crack advance takes place if the energy barrier for
crack propagation, ∆F+ , is larger than that for crack healing, ∆F− . The
asymmetry in the forwards and backwards energy barriers is caused by the
departure from mechanical equilibrium given by G − G0 .
When the departure from equilibrium is small, we can express the energy
barriers ∆F± as a simple Taylor expansion of the quiescent energy barrier
∆F ∗ through [65]
∆F± = ∆F ∗ ∓ α(G − 2γse ) + ..., (G − 2γse 2γse )
(3.7)
where α is an activation area and 2γse is two times the surface energy of the
solid in the given environment for region I crack propagation. From classical
reaction rate theory, it follows that the crack velocities in region I is given
by [65]
vI = 2v0 a0 exp(−∆F ∗ /kT ) × sinh [α(G − 2γse )/kT ] ,
(3.8)
where v0 = kT /h is a fundamental lattice vibration frequency and a0 is some
characteristic atomic spacing between successive barriers.
The velocity function for region II will depend on transport processes,
and if free molecular flow is rate limiting, it will be given as
vII = λpG,
(3.9)
where λ is a material-geometry coefficient and p is the partial pressure of the
active species [65].
In region III, the functional dependency on G is the same as in region I
but with the vacuum surface energy instead of the environment dependent
value γse and possibly with a different activation area and quiescent energy
threshold.
This model gives a good fit to experimental data for a range of materials
and crack velocities [105]. It has also been established that reaction rate
treatment of atomistic and continuum of crack models yield consistent results
[106].
34
Fracture mechanics and kinetic fracture processes
-2
-3
log(v) (m/s)
-4
cw = 1.0
cw = 0.7
cw = 0.5
cw = 0.3
cw = 0.1
cw < 0.03
sinh fit
-5
-6
-7
-8
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
G - G0 (J/m2)
Figure 3.2: Measured fracture velocity v as a function of excess mechanical
energy release rate G − G0 . The different symbols denote measurements
at different water concentrations cw . The solid line shows the results from
Equation (3.8) using a0 = 5 Å as a characteristic barrier spacing and with
fitting parameters α = 2.0 × 10−20 m2 and ∆F ∗ = 7.6 × 10−20 J.
3.7
Thesis results: Subcritical crack growth
in calcite
In Paper III, measurements on subcritical cleavage crack propagation of single calcite crystals are presented. To the authors’ knowledge this is the
second (see [77]) experimental study of subcritical crack propagation in single calcite crystals and the first addressing the particular influence of water
concentration.
Measured crack velocities were quite variable, particularly at low water
concentrations and/or high G, something which may be related to low temperature plastic behaviour in calcite. The highest velocities measured at any
given G were assumed to correspond to the most ”true” brittle crack behaviour. Within the experimental range of energy release rates and fracture
velocities, the ”brittle” crack velocities were found to be well described by
the reaction rate model given by Equation (3.8) (see Figure 3.2).
The equilibrium mechanical energy release rate, G0 , was found to decrease
with water concentration, corresponding to a reduction in the solid-fluid in-
3.7 Thesis results: Subcritical crack growth in calcite
35
terfacial energy (see Figure 2.7). Within the experimental error, no systematic change in the fitting parameters α and ∆F ∗ was detected with changing
water concentration. This implies that the physical mechanism responsible
for crack propagation in calcite is not dependent on the water concentration.
As pointed out by Maugis [59], no particular chemical reaction with the
strained bonds at the crack tip is necessary to explain subcritical fracture
propagation near G0 . The stress corrosion mechanism for glasses is well
established, but in the case of mica it has been argued that the sharpness of
the crack makes direct access of environmental species to the strained bonds
at the crack tip unlikely. In this case, the most probable rate limited process
in region I may be activated interfacial diffusion, which would still yield the
functional dependence of Equation (3.8) [58]. More experimental work would
be needed in order to resolve whether calcite is more comparable with glass
or mica, or whether crack propagation in calcite is controlled by some other,
as yet unknown mechanism.
36
Fracture mechanics and kinetic fracture processes
Chapter 4
Effects of stress generation in
pores during weathering
In Figure 1.4, three possible effect of stress generation due to crystal growth
in pores were presented: Replacement, microcrack propagation and large
scale build-up of elastic strain energy. This list is not meant to be extensive.
Also, since weathering takes place at low temperatures and confining stresses,
plastic deformation and viscous relaxation have been disregarded. This may
be an oversimplification in some cases, such as for calcite which behaves
plastically even at room temperature (see Paper III). In this chapter the
chosen three examples are discussed in more detail.
4.1
Replacement
A very common feature in weathering profiles is the isovolumetric, shape
preserving replacement of one mineral by another [107–110]. The shape
preservation during replacement is thought to arise due to a stress mediated
coupling between the dissolution rate of one mineral and the growth rate of
the other mineral. Since this important mechano-chemical coupling has been
extensively researched it has not been a focus of the present study, but since
it belongs in this context a brief outline of the replacement mechanism is
described below.
Imagine a mineral A separated from mineral B by a thin fluid film (Figure
4.1). The fluid film is supersaturated with respect to mineral A and saturated
with respect to mineral B, which is soluble under these conditions. The film
is assumed to be connected to a larger reservoir so that the concentration
of species A and B remains constant. Mineral A is growing due to the
supersaturation and, in doing so, the separation between A and B becomes
38
Effects of stress generation in pores
A
A
ΩA > 1, ΩB = 1
ΩA > 1, ΩB = 1
B
B
Figure 4.1: Mineral A separated from mineral B by a thin liquid film (blue),
which is supersaturated with respect to A and saturated with respect to B.
The thickness of the film is greatly exaggerated for clarity. The growth of A
results in a normal stress exerted on the interface between A and B which
causes B to dissolve. With time, mineral B is replaced by mineral A with no
change in volume.
smaller. The decrease in film thickness sets up a disjoining pressure in the
fluid film which is equivalent to a normal stress exerted on the surfaces of
A and B. This increases the equilibrium concentration of B, causing B to
dissolve increasingly faster with increasing decreasing surface separation. At
some point, the rate of growth of A (which decreases with increasing normal
stress) equals the growth rate of B and the system self-adjusts at a steady
state normal stress, the magnitude of which depends on the kinetic constants
and specific volumes of A and B as well as the value of ΩA , but is always
smaller than σc [111].
Since in this model all of the work exerted by the growing crystal A goes
into dissolution of B, the result is isovolumetric replacement of mineral B by
A.
As is often the case (and is also illustrated by Paper II), continuum models
do not reflect all of the details of the process. When a replacement front is
moving into mineral B from all free surfaces, so that there is no obvious
connection between the interface between A and B and some larger liquid
reservoir, one would expect the liquid film to become saturated with respect
to both minerals causing the reaction to stop, since solid state diffusion for
most minerals is negligibly slow under weathering conditions. Experimental
studies [112–114] have revealed that replacement processes are, in fact, always
associated with porosity generation in the replacing mineral. This allows for
sustained transport to the interface between A and B through a percolating
4.2 Microcrack propagation
39
network of solution filled pores or fractures. The evolution of this porosity
is probably related to local crystal growth kinetics [115].
4.2
Microcrack propagation
If a crystal is growing in a pore with insoluble pore walls, stresses may become
large enough to enlarge the pore by fracture propagation. The magnitude of
stress required for crack propagation depends on the pore geometry and can
be characterized by the mechanical energy release rate G which is analytically
defined for a range of idealized geometries (see Chapter 3).
Microcrack propagation may lead to a variety of weathering morphologies
depending on the boundary conditions. Microcracking near the surface of the
rock leads to granular disintegration and flaking, which is often observed in
experiments on salt crystallization in rocks [116]. A spectacular result of
localized granular disintegration is the formation of tafoni structures (Figure
1.2c) [117].
The formation of veins due to crystal growth and propagation of cracks at
a fixed energy release rate or stress value has been analyzed by Fletcher and
Merino [111], and subcritical crack growth due to ice growth in a temperature
gradient, coupled to water transport through a partially frozen matrix, was
modelled by Walder and Hallet [118].
4.2.1
Thesis results: Coupling subcritical crack propagation and stress generation due to crystallization
One may envisage a number of geometries in which fractures are propagated
directly by the stress generated due to crystal growth in a pore. Two possible geometries are shown in Figure 4.2. The top panel shows radial cracks
propagating from a circular pore. These cracks form due to the tangential
component of stress which is present in a hole subject to a uniform internal pressure [119]. The bottom panel shows a penny-shaped crack which is
pushed open by a crystal growing in its aperture.
While the first geometry might be more intuitively appealing, the second
geometry was chosen for the model presented in Paper IV. The reason is that
analytical expressions relating the fracture opening and radius to the crystal
radius and stress makes it possible to obtain direct relations for the feedback
from crack propagation to normal stress and therefore crystal growth rate
through the change in crack compliance.
40
Effects of stress generation in pores
a)
b)
σ
2w
c
a
Figure 4.2: Crack propagation due to growth of a crystal (grey) in pore filled
with a supersaturated solution (hatched). a) Radial cracks growing from a
circular pore. b) Crystal growing in the aperture of a penny-shaped crack
with radius c and maximum opening 2w. The crystal is located in the center
of the crack and has radius a and thickness 2w which matches the aperture of
the crack (the fluid film thickness is assumed to be negligible). The growing
crystal exerts a normal stress σ on the crack walls.
4.3 Build-up of elastic strain energy and macroscopic expansion41
Some results for a stationary crack have already been presented in Chapter 2. It turns out that two dimensionless parameters, one which relates crack
growth kinetics to crystal growth kinetics and another which characterizes
the degree of supersaturation, controls the evolution of fractures. In particular, there is a crossover from dynamic cracking (where the energy release
rate G increases sharply with increasing crack length) to subcritical cracking
(where G is locked at a value close to G0 for long times) which is given by the
two dimensionless parameters. This crossover is illustrated in Figure 4.3.
4.3
Build-up of elastic strain energy and macroscopic expansion
If the solubility of neighbouring minerals is too low for replacement to take
place, and the geometry of the pore is such that the threshold for crack
propagation is not reached, then crystals may continue to grow in pores until
the maximum crystallization stress σc is exerted on all pore walls. When
crystals exert an isotropic stress in a large number of pores in a homogeneous
rock, the result is a bulk volume increase analogous to thermal expansion. If
the rock is confined in some directions, the result of the expansion is a buildup of elastic strain energy which may become large enough to propagate large
scale fractures.
Spheroidal weathering (Figure 4.4), which is a striking weathering morphology found in many lithologies and in a wide range of geographical settings, is an example of a pattern formed by fractures propagated due to expansion during weathering. This is the prevalent explanation for this pattern
[10, 120–123], although alternative hypotheses have been proposed including
inherent stress patterns in the rock [124, 125] and chemical processes with
no mechanical component [126, 127].
The origin of the bulk expansion resulting in elastic strain and fractures
may vary for different lithologies, and it is difficult to identify the exact
stress generating reaction by petrological examination of the weathered rock.
Buss et al. [128] found that the first weathering reaction to take place in a
spheroidally weathered quartz diorite in Puerto Rico was an oxidation of
biotite which caused an expansion in d(001) from 10.0Å to 10.5Å, which is
probably sufficient to produce the observed fractures. In an andesite of relatively high initial porosity, Jamtveit et al. [in prep.] found that precipitation
of iron oxides in pores from a supersaturated solution was the most probable
stress generating mechanism.
The strain resulting from a mean isotropic stress σc in a large number
42
Effects of stress generation in pores
≥2
2
1.9
1.5
1.8
1.7
1
log(C)
1.6
0.5
1.5
1.4
0
1.3
1.2
-0.5
1.1
-1
-1
-0.5
0
0.5
1
1.5
2
2.5
log(AeB)
Figure 4.3: The relative value of the mechanical energy release rate, G/G0 ,
which is reached in fractures after long times as a function of the dimensionless parameters AeB (which relates crack kinetics and crystal growth kinetics)
and C (describing the level of supersaturation). The range of parameters has
been chosen to illustrate the crossover in behaviour. If G/G0 > 2, then the
crack is accelerating quickly and the system enters the dynamic fracturing
regime. G/G0 close to one imply that fractures accelerate extremely slowly
and that subcritical fracture propagation is dominating. A perfectly constant
crack velocity is never reached which means that at long enough times, the
system will become unstable. For a full description of the model, see Paper
IV.
4.3 Build-up of elastic strain energy
43
Figure 4.4: Road section showing a spheroidally weathered dolerite sill in
Karoo, South Africa (see also Paper I).
of pores in a rock which is completely saturated with fluid is given by [129]
=
bSc σc
,
K
(4.1)
where K is the bulk modulus of the rock, b is the Biot coefficient of the rock
(b = 1 − K/KS where KS is the bulk modulus of the solid phase of the rock)
and Sc is the volume fraction of pores filled with crystals.
With a crystallization front moving from the free surface and into the
rock, the rock is free to expand in the direction normal to the free surface, but
constrained in the surface parallel direction. Disregarding surface curvature,
this confinement gives rise to a surface parallel stress given by [119]
σ = 3K.
(4.2)
When the elastic energy built up in the outer layer of the rock is sufficiently large, it can produce spalls by driving fractures parallel to the surface.
Assuming a uniform surface parallel on a flat surface, the elastic strain energy is large enough to drive surface parallel cracks when the thickness of the
layer is given by [130]
2ΓE
(4.3)
∆h = 2 ,
σ
where ∆h is the thickness of the layer and Γ is the surface energy of the
rock. The effect of surface curvature can also be included [119, 131], but
the expressions become more complicated. This model shows how spalling
can be produced by in-pore crystallization, and when realistic parameters
are used the length scale given by Equation (4.3) agrees well with observed
spall thicknesses [Jamtveit et al., in prep.].
44
Effects of stress generation in pores
Figure 4.5: Details showing how initially spheroidally weathered blocks have
become subdivided into smaller units which have subsequently been rounded
by spheroidal weathering.
4.3.1
Thesis results: Hierarchical fracturing
In spheroidal weathering profiles, the rounded blocks or corestones are sometimes observed to become divided into smaller units after some extent of
spheroidal weathering of the original block has taken place (Figure 4.5). This
is evident from some figures in literature [132, fig. 7],[121, fig. 5],[122, fig.
2], but has not been much discussed, although Sarracino and Prasad [127]
suggested that subdivision of corestones during the weathering process was
caused by some large scale tectonic event.
In Paper I, we show that the subdivision of corestones during spheroidal
weathering can be caused by the same elastic strain energy which is causing
the spalling. A conceptual explanation is that expansion of the outer layer
of a solid body also generates a tensile stress in the middle of the body [119],
which may be sufficient to propagate fractures depending on the geometry.
A numerical model demonstrates that subdivision of an initially intact solid
due can result from an expansive reaction (Figure 4.6).
The subdivision process can be viewed as a hierarchical process where
domains get progressively subdivided with new fractures forming at normal
angles to preexisting fractures. The effect of this domain divining process is
to accelerate large scale weathering rates since more surface area and more
efficient fluid pathways are created through fracturing.
4.3 Build-up of elastic strain energy
45
Figure 4.6: Result of a numerical simulation where an initially intact rectangular block reacts with water and expands along the outside. Fracturing
causes the block to become progressively subdivided and rounded. The color
scale shows the local extent of reaction.
46
Effects of stress generation in pores
Chapter 5
Summary and outlook
The principal agent in rock weathering is water, and it is the manifestations
of the properties of water at different scales which unify the topics adressed
in this thesis.
On the large scale, water acts as a transport medium for chemical species.
As water penetrates into unweathered rock it allows reactions to take place
through dissolution of unstable minerals and precipitation of more stable
minerals. The weathering front is an open system where species can be
supplied and removed through the aqueous phase.
During spheroidal weathering and the associated hierarchical fracturing
of rock, weathering is accelerated due to fracturing because the fractures
that open enables faster water transport, and also because more unreacted
rock surface area becomes exposed to water. This coupling between mechanical and chemical processes during weathering has been explored in Paper
I, where it was shown that stresses generated during weathering can cause
fracturing on a larger scale than what was previously recognized. This may
have significant implications for the rate of advancement of the weathering
front.
On the small scale, stresses are generated in pores when crystals precipitate from a thin film of water at a grain contact. The presence of water at
this contact is determined by the long range forces which act between surfaces separated by a thin liquid film. An alternative way of explaining the
presence of a thin film of water which transmits a normal stress between two
surfaces is to consider the effect of water on the surface energy of the solids.
If the sum of the solid-liquid interfacial energies is higher than the energy of
the dry solid-solid contact, it may be energetically favourable for the system
to sustain a liquid film at an elevated pressure. The complex dynamics of
the interface between a growing crystal and a confining wall is illustrated by
the experiments presented in Paper II. Contrary to what has been assumed,
48
Summary and outlook
there is no tendency for the loaded crystal face to approach a steady state
where it is in smooth contact with the confining surface.
If water is present in the aperture of a fracture, the energy of the fracture
surfaces are lower than they would have been if the fracture had opened up in
vacuum. Since fracture propagation requires the applied mechanical energy
to be higher than the energy of the two newly created surfaces, it may require
less energy to open a water filled crack than one which is empty. The effect of
water on fracture propagation can also be explained in terms of surface forces.
The crack tip in brittle materials is thought to be very sharp and to approach
molecular dimensions near the tip. If a film of water is present in the crack
aperture, then in the region near the crack tip, where the crack opening is
very small, long range forces will be acting between the fracture surfaces.
When these forces are repulsive, their effect is to increase the effective stress
at the crack tip and thus to lower the applied mechanical energy required
for crack propagation. The effect of water on the surface energy and crack
propagation in calcite is described in Paper III, which shows that increasing
the water concentration leads to a monotonic decrease in surface energy of
calcite.
Fracture propagation and stress generation due to crystal growth, which
are two of the principal processes responsible for damage in rocks during
weathering, are thus linked by the physical properties of thin liquid films.
They are also both time dependent processes, and coupled through their
effect on and response to the stress field around pores and fractures. The
model presented in Paper IV shows that the coupling of fracture kinetics
and crystallization kinetics may control how fractures evolve when they are
driven by crystallization stresses.
During the work of this thesis, some questions have been answered, and
many new questions have been raised.
The mechanisms responsible for subcritical crack growth in calcite, and
in many other geological materials, are still not known. As any thermally
activated process is expected to produce the same type of crack velocity
curves, identifying the exact physical or chemical mechanisms would require
carefully planned experiments in a range of chemical environment. Theoretical considerations and simulations may also help to understand the relative
importance of surface forces and crack-tip reactions such as stress corrosion
and dissolution.
The generation of stresses during crystal growth has been explained theoretically by considering the equilibrium between the bulk solution at hydrostatic pressure and the solid subject to a normal stress. In reality, the liquid
in intimate contact with the stressed crystal surface is not at hydrostatic
pressure. The effect of the disjoining pressure on the chemical potential of
49
the dissolved material in the thin film has not been fully worked out. This
is also the case for the effect of moving a surface on the surface forces transmitted through thin liquid films, something which may be important both
for growing crystals and for propagating fractures.
When diffusion takes place under the influence of external forces, it may
in some instances result in ions being transported in the direction of greater
ion concentration. This may be the case when large pressure gradients are
present, such as when the disjoining pressure is localized at areas of contact.
It is not known what effect this may have on transport and growth of crystals
under normal stress, and on transport of reactive species to the crack tip.
In order to advance the understanding of the process by which crystals
perform mechanical work on their surroundings through growth, a more thorough theoretical treatment of the process, including the effects of surface
forces, ion diffusion and the nucleation and propagation of steps on a faceted
crystal. In situ observations of the evolution of the loaded interface would
also be valuable.
In porous materials, transport has large scale effects on whether damage
takes place, and on the resulting weathering morphologies (see Figure 5.1).
Upscaling from the pore scale to the scale of rock samples requires an understanding of transport processes and how they couple to the kinetics of crystal
growth and fracture propagation.
The large scale degradation of rock does not take place by isolated fractures but rather by the interplay between a large number of microfractures.
The analysis presented in this thesis on the propagation of a single fracture
due to crystal growth can be extended to consider the interactions between
fractures and the evolution of fracture populations through time, which may
provide useful insight on how rocks are broken due to crystallization stresses.
It is not yet known to what extent biological agents such as plant roots
and fungal hyphae actively cause fracture propagation in rocks. The work
in this thesis has highlighted the intimate connection between the crack tip
chemistry and the stress required to propagate a crack. Since the chemical
environment near a root tip is most certainly affected by the plant, it may
be relevant to consider whether biological agents are able to cause fracture
propagation through excerting a pressure while affecting the crack tip chemistry in order to lower the pressure required for crack growth. This sort of
analysis may advance the current understanding of biological weathering.
50
Summary and outlook
Figure 5.1: This rock art on Angkor Wat has been damaged due to salt
crystallization. The white substance on the surface of the rock is so-called
efflorescence, where salts have crystallized on the outside of the rock. Crystallization is more damaging if it takes place in pores within the rock, so-called
subflorescence, and this has resulted in fracturing and scaling. Whether efflorescence or subflorescence takes place is largely controlled by transport
processes.
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Scientific Papers
Paper I
Controls on rock weathering rates by reaction-induced hierarchical fracturing
Anja Røyne, Bjørn Jamtveit and Anders Malthe-Sørenssen (2008)
Earth and Planetary Science Letters, 275, 364-369
64
Scientific Papers
Earth and Planetary Science Letters 275 (2008) 364–369
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Controls on rock weathering rates by reaction-induced hierarchical fracturing
Anja Røyne ⁎, Bjørn Jamtveit, Joachim Mathiesen, Anders Malthe-Sørenssen
Physics of Geological Processes (PGP), University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
a r t i c l e
i n f o
Article history:
Received 26 May 2008
Received in revised form 25 August 2008
Accepted 26 August 2008
Available online 14 October 2008
Editor: P. DeMenocal
Keywords:
weathering
weathering rates
spheroidal weathering
basalt
hierarchical fracturing
fracture patterns
a b s t r a c t
Weathering of rocks takes place through complex processes involving an intimate coupling between
mechanical disintegration of the rocks and interactions between the exposed rock surface, the hydrosphere
and the biosphere. It is still poorly understood how this coupling affects weathering rates and patterns
produced by weathering. Here, we show that the observed spheroidal weathering patterns in basaltic
intrusions (dolerites) in the Karoo Basin in South Africa are created through volume expanding reactions
which set up stresses that not only lead to spalling of spheroidal layers of rock, but also drive a large scale
hierarchical fracturing process by which the dolerite is continuously undergoing domain division. The
fracture processes generates new reactive surface area in a self-accelerating manner, thus providing a firstorder control on the total weathering rate.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Studies of rock weathering often distinguish between chemical,
physical and biological weathering (Bland and Rolls, 1998). However,
there is reason to believe that the coupling between chemical and
physical processes exerts a crucial control on the weathering process
(Anderson et al., 2004). The first quantitative model that relates
chemical and physical weathering was recently published by Fletcher
et al. (2006) who, using a 1-D model, showed how diffusioncontrolled chemical weathering generates elastic stresses that may
drive the surface parallel fracturing observed during spheroidal
weathering. Spheroidal weathering has been observed on a wide
range of rock types and climates. The coupling of chemical and
physical processes during this type of weathering may therefore
provide important constraints on weathering processes in general.
Dana (1896) was the first to suggest that spheroidal patterns are
caused by chemical changes during weathering. Blackwelder (1925)
found that hydration of the alumina and iron of complex silicate
minerals would cause a greater increase in volume than what is offset
by the removal of dissolved material, causing expansion of the reacted
outer shell of the boulder. This expansion produces stresses which are
large enough to fracture the rock. Larsen (1948) attributed spheroidal
weathering in granite to a slight expansion of biotite and other
minerals caused by hydration. Chapman and Greenfield (1949)
concluded that the spheroidal scaling of granites and basalts probably
⁎ Corresponding author. Tel.: +47 22 85 69 26; fax: +47 22 85 51 01.
E-mail address: anja.royne@fys.uio.no (A. Røyne).
0012-821X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2008.08.035
resulted from the oxidation and hydration of silicate minerals, based
on microscopic observations. Spheroidal patterns in argillaceous
sandstone were attributed to the formation and/or subsequent
expansion of dioctahedral vermuculite by Simpson (1964), and to
expansion of phyllosilicates by Heald et al. (1979). Spheroidal
weathering is most prominent in rocks dominated by plagioclase,
olivine and pyroxenes, and is normally not found in quartz-dominated
sandstones. Through geometrical arguments, Folk and Patton (1982)
showed that only very small expansions are needed to cause large
amounts of arching which must be accommodated by fracturing. The
total expansion in the numerical model which Fletcher et al. (2006)
used to explain spheroidal weathering is 0.25%.
We have made observations of spheroidal weathering of basaltic
intrusions (dolerites) on the South-facing slopes of the Hogsback and
Katberg mountains (26°48.6′E, 32°28.6′S) in the Karoo basin in South
Africa. The Karoo dolerites are particularly well suited for a study of
evolving fractures due to the general lack of pre-existing, tectonically
produced fractures. Most dolerites appear rather ‘flawless’ and their
resistance to weathering is demonstrated by their common occurrence as flat tops of ‘table mountains’. The Karoo Basin covers an area
of approximately 140,000 km2 (Johnson et al., 1997). Sedimentary
sequences dominated by shales and sandstone were intruded by large
volumes of basaltic melts about 183 Ma ago (Duncan et al., 1997;
Svensen et al., 2007). These subvolcanic doleritic intrusives locally
form up to 70% of the basin volume (Roswell and De Swardt, 1976). The
climate in this region is temperate, and the annual rainfall is 600–
800 mm/yr. Geochemical effects of dolerite weathering in this area
was previously described by Marsh (1991) who described large
kernels of fresh dolerite separated from soft, “buff-colored” clay-
A. Røyne et al. / Earth and Planetary Science Letters 275 (2008) 364–369
dominated weathering products by rinds of progressively more
altered dolerite, i.e. spheroidal weathering.
2. Field observations
Fig. 1a shows a road section from this area which displays typical spheroidal weathering patterns. Rounded cores with a wide
range of sizes are surrounded by spherical shells and separated by
regions of highly weathered rock. The remnants of the fracture
planes along which the weathering originated can be traced on the
road section. Some of these fractures are long and seem to divide
the entire visible domain. These may be joints produced during cooling of the sill or during subsequent exposure. In addition to these
original fractures, a succession of shorter fractures can be identified, each dividing a subdomain and joining older fractures nearly
at right angles. Fig. 1b shows these fractures labeled as a succession
of generations 1–6.
The hierarchical structure of the fracture pattern seen in Fig. 1b
indicates that the fractures formed sequentially. A recent study of
fracture patterns in drying films (Bohn et al., 2005a,b) attributed
hierarchical structures to the sequential formation of fractures, and
argued that sequentially formed patterns are dominated by four-sided
365
domains and T-shaped fracture junctions, where new fractures meet
pre-existing fractures at nearly right angles. Many of the temporal and
geometric characteristics of these quasi-2D systems are also found in
planar sections through 3D systems, and can be clearly distinguished
from patterns formed by simultaneous fracture growth, which
typically are dominated by triple junctions with 120° angles and sixsided domains (Bohn et al., 2005a; Toga and Alaca, 2006). Similar
hierarchical fracture patterns in ultramafic rocks undergoing expansion connected to serpentinization reactions have recently been
described by Iyer et al. (2008).
A statistical analysis of the fracture set found in Fig. 1b is shown in
Fig. 1c to compare the patterns we observe in the field with
hierarchical drying patterns. Corresponding analyses of areas with a
large number of identifiable fractures from two other localities
characterized by spheroidal weathering are presented as supplementary material in the Appendix. The patterns in all three cases were
found to be dominated by T-shaped fracture junctions and four-sided
domains, matching the expected results for hierarchical fracturing.
The observed fracture patterns are therefore consistent with a
sequential formation history and hierarchical domain division.
While the first generation of fractures may have been generated thermally or by tectonic forces, the hierarchical nature of the
Fig. 1. Spheroidal weathering and hierarchical domain division observed in a road section at Nico Malan Pass, South Africa (S32.50780°, E026.80417°). The scale is given by the people
in the right hand side of the picture. Note the very ‘patchy’ distribution of weathering on the road section scale. In part b, the fractures are marked according to generation following
the method of Bohn et al. (2005a). Long fractures whose origins are difficult to determine are labeled as generation 1. Fractures that join only those of generation 1 at approximately
right angles are labeled generation 2, and so forth until generation 6. Each higher generation fracture subdivides one domain into two smaller ones, which creates more surface area
available for chemical weathering. Part c shows the statistical analysis of the fracture pattern. The number of sides for each domain of the fracture pattern shows clear peak at foursided domains, which is characteristic of hierarchical fracture patterns (Bohn et al., 2005a). Triple junctions were found to comprise as much as 89.0% of the total junctions in the
network. The majority of the large angles at triple junctions are found at around 170°, indicating that these are T-shaped junctions consisting of a straight fracture intercepted by a
younger fracture. The distribution of the smallest angles at the triple junctions peaks close to 90°, indicating that new fractures meet old fractures nearly at right angles, which is
expected for hierarchical fracture patterns. The distribution includes a small number of very small angles. This is not surprising since, unlike 2D drying patterns (Bohn et al., 2005a),
the angles are measured in a 2D cross-section through a 3D fracture network. Fractures and domains at the outer edges of the image were excluded from the statistical analysis, since
the presented measures cannot be determined for edge domains.
366
A. Røyne et al. / Earth and Planetary Science Letters 275 (2008) 364–369
subsequent domain divisions indicate that they have been caused by
locally generated stresses. The most likely source of these stresses is
volume changes caused by chemical weathering, analogous to the
expansion driving the spheroidal patterns. Very similar reactiondriven fracture patterns have recently been produced experimentally,
but at much smaller scales, during mineral replacement processes
(Jamtveit et al., in press).
A detailed study of the road sections demonstrates how the
spheroidal and hierarchical weathering patterns are intimately
connected. As shown in Fig. 2a–d, a fresh block can weather
spheroidally for some time, spalling off a number of shells, before
the block is subdivided by radial fractures connecting to the innermost
shell. Fig. 2a and b also show that the newly formed smaller blocks can
continue to weather spheroidally, and it is reasonable to expect that
these smaller blocks may again be divided by a new generation of
fractures. These observations clearly demonstrate that division of
blocks takes place simultaneously with spheroidal weathering. Thus,
the observed pattern is not, as previously thought (Ollier, 1971; Turner
et al., 2003), controlled by pre-existing fractures only. Another image
that convincingly illustrates the simultaneous nature of the two
modes of cracking, this time in an andesitic sill from Argentina, is
shown in supplementary Fig. 2 in the Appendix. Patterns similar to
those in Fig. 2 can be seen in the figures from several previous studies
of spheroidal weathering (Bisdom, 1967; Blackwelder, 1925; Ollier,
1971), but are not commented on in the papers. Other studies (Larsen,
1948; Ollier, 1967; Pera and Sorriso-Valvo, 2000) also mention radial
or sequential fracturing in some form, but the focus of spheroidal
weathering studies has always been the formation of the spherical
shells.
3. Numerical model
A coupled diffusion–reaction–deformation model previously used
to study shrinkage-induced fractures was modified to include volume
expansion reactions, and applied to the weathering processes
(Jamtveit et al., 2000; Malthe-Sørenssen et al., 2006). The diffusion–
reaction process is described by:
Acf
¼ Dð/Þj2 cf −Q
At An
Vm
¼
Q
At
f0
/
ð1Þ
where the rate of consumption of the fluid is Q = kSρf0(1 − ξ)cf, cf is the
local fluid concentration, ξ is the extent of reaction, ϕ is the porosity,
D(ϕ) is the diffusion constant, k is the kinetic constant, S is the specific
surface, ρ is the density, f0 is the initial volume fraction of reactable
material in the solid, and Vm the molar volume of the solvent. Fluid
initially enters along the outer boundaries of a ‘block’ of fresh rock
(Fig. 3a) described using Neumann boundary condition, and diffuses
inwards. The reaction produces a local volume change proportional to
the extent of reaction, ξ. The local volume change produces a local,
inelastic strain which induces stresses in the elastic rock matrix. The
rock matrix is modeled using a discrete element model (Cundall and
Strack, 1979), and the local inelastic strain is represented by a local
change in the equilibrium distance between the discrete elements.
Given the stresses induced by the local volume change, the
equilibrium configuration of the elastic matrix is found using over-
Fig. 2. Details of fracture patterns created by spheroidal and hierarchical weathering.
Part a shows how one block with an initial right angled corner became progressively
rounded until it fractured into four smaller pieces, which were subsequently rounded
off by spheroidal weathering. The continuous spheroidal shells of the original block are
marked in red. Part b shows a similar pattern, where one initial block after a period of
rounding due to spheroidal weathering, was subdivided into five angular parts by
fracturing. These have subsequently been rounded by a new period of spheroidal
weathering. The scale bar is 0.10 m. Parts c and d show how fresh radial fractures
through an unweathered core meet the spheroidal shells of the original block. If
weathering had continued, the individual domains of the freshly fractured block would
have become rounded by spheroidal weathering. Part e shows that spalled layers are
thicker in the corners, and that spalling tends to round corners to form a more-or-less
spherical shape. Part f shows a corner with a less altered core. Scales are given by a
camera lens cap in c, a hammer in a and d and a tape measure in e and f.
A. Røyne et al. / Earth and Planetary Science Letters 275 (2008) 364–369
367
Fig. 3. Mechanical model for spalling and hierarchical fracturing. Parts a–f show six successive stages in a simulations of weathering due to a diffusion–reaction-induced volume
expansion process using a discrete element model coupled to a diffusion–reaction model (see Malthe-Sørenssen et al., 2006 for details). Fluid diffuses in from the outer boundaries,
reacting with the rock matrix, resulting in a local expansion. Expansion in the y-direction is limited by an elastic wall. The color-scale shows the normalized fluid concentration.
Fractures are illustrated by red lines. The outer corners are rounded by spalling. Four generations of subdomains are formed, and in each generation, a new fracture divides a cell into
two smaller cells. The fluid infiltrates the fracture as it becomes connected to the exterior. Therefore the boundary conditions for the reaction–diffusion process changes gradually.
The outer corners are sheared off early in the process, and shear bands due to the external boundaries are visible in the final fracture pattern.
relaxation methods. If the elastic stress in an element exceeds the local
yield strength of the material, a fracture occurs, implemented by
removing the attractive interaction between two elements. If a
fracture extends to the outer boundary of a block, fluid infiltrates
the fracture, and the fracture surfaces become new boundaries for the
diffusion–reaction process. The chemical transformations in the
spheroidal weathering processes that we investigated are transport
limited. The stress dependence of the transport processes is weak, and
it was neglected in the model. The model is scaled to represent
realistic weathering systems by matching relevant dimensionless
numbers (Malthe-Sørenssen et al., 2006) in model and nature. The
ratio of the maximum stresses set up by the reaction to the tensile
strength of the matrix, σc, is X = (f0ΔV/V) (E/σc). For weathering rocks,
E/σc is 102–104, spanning from granites to sandstones, and f0ΔV/V is in
the order 10− 4–10− 2, giving X = 10− 2–102. The corresponding value in
the simulations was X = 0.1 × 50 = 5 which is within the realistic range.
The simulation results (Fig. 3) show that, initially, layer by layer
spalling due to expansion produces a spheroidal shape by removing
sharp corners from the initially rectangular block. This process
matches what we see in the field (Fig. 2e). The simulation also
indicates that corners may be sheared off (Fig. 3b), resulting in large
corner blocks with unaltered cores, something which is also observed
in the field (Fig. 2f). Further shells are spalled off as the reaction front
continues to advance into the block. At the same time, the expanding
boundary generates a tensile stress in the block, which eventually
leads to the formation of a tensile fracture that divides the block in two
(c.f. Fig. 3c). As fluid penetrates into the newly formed fracture, the
diffusion–reaction process proceeds from the surfaces of the two new
blocks, and the entire process is repeated on a smaller scale. Four
generations of subdomain formation was observed in the simulations.
This domain-dividing fracture mechanism results in a hierarchical
system of fractures and unfractured blocks. Layer-by-layer spalling
may occur in each generation, and sharp corners generated by
domain-dividing fractures are rounded off by spalling, gradually
producing a spheroidal shape (Fig. 3).
4. Implications for weathering rates
We see that fracturing accelerates the reaction process. Fig. 4
shows a plot of the extent of reaction, Ξ, for a simulation with
fracturing (Fig. 3) and one without fracturing, as a function of time. For
small times, simulations with and without fracturing display nearly
identical behavior. However, approximately from the time where
the first domain-dividing fracture occurs, the total extent of reaction
for the fracturing system starts to grow significantly faster than for
the non-fracturing system. Eventually, the effect of finite system size
limits the growth, and both systems tend towards the same asymptotic value (Ξ = 1).
368
A. Røyne et al. / Earth and Planetary Science Letters 275 (2008) 364–369
stresses induced by the reaction process increases with the width Δw.
A domain division occurs after a time Δt ∝ (Δwc)1/q, which is when the
tensile stress reaches the material strength. How Δwc is related to
other parameters in the system depends on the mechanism causing
domain division. For example, from arguments used for thermal
stresses (Timoshenko and Goodier, 1970), we expect the maximum
tensile stress in the center of a block of thickness w to be proportional
to Δw/w and, consequently, we find that Δwc ∝ w. Subdivision due to
such a mechanism would lead to a subdivision time Δt ∝ Δwc1/q ∝ w1/q.
However, there may be several other mechanisms in the weathering
system that lead to subdomain formation, such as shear-band
formation due to boundary effects or the propagation of wedges
caused by morphological instabilities in the spalling front. Here, we
use a conservative, upper limit estimate for the subdivision time
consistent with observations from the simulations, and assume that
the critical width Δwc and the subdivision time Δt are constants that
does not depend strongly on the block size.
Based on the hierarchical model illustrated in Fig. 4c and the
assumption that the time between the formation of each subsequent
generation of fractures is constant, we estimate the reacted volume.
After a time t, the reaction front has reached a distance Δw(t −ti) inwards
from a fracture surface formed in generation i at the time ti. The reacted
volume is therefore VR = ∑(Si −Si − 1) Δw(t −ti), where Si −Si − 1 is the new
surface formed at ti. The total extent of reaction is found from Ξ =VR /V0,
where V0 is the initial volume. We have illustrated the prediction of the
hierarchical model in Fig. 4a and b. Here, we have used a transport model
Δw =ctm with m = 1.25, which is an approximation for the observed
behavior of the non fracture case for intermediate times. We notice that
this coarse theoretical prediction fits well with the observed behavior
at short times, but that it deviates from the observation for long times
as it does not include information about the finite volume that may
react.
The accelerated subdivision process will, however, be limited by
several factors not included in this simplified model. The fracturing
process will only continue down to a smallest characteristic lengthscale, l0, corresponding to the size of the smallest block that will
fracture before it is completely consumed by chemical processes.
Furthermore, the diffusion–reaction process driving the subdivision
process may be transport limited and controlled by the rate at which
fluid can be transported to the reacting surfaces along the evolving
fracture network. As the subdivision process continues the path from a
fluid reservoir to the reaction front becomes longer. Consequently, we
expect the real process to become gradually more transport limited,
which limits the acceleration.
5. Conclusion
Fig. 4. Time evolution of reacted volume on log–log scale (part a) and linear scale (part
b). The reacted volume is shown both in the case when the process is accelerated by
hierarchical fracturing, that is when the system fractures and fluid enters newly
generated fractures, and non-accelerated, where the system does not fracture and fluid
only enters from the initial external boundaries. The circles along the accelerated
simulation curve correspond to the times for the images in Fig. 3. Part c shows a cartoon
illustration of the first few generations in a hierarchically fracturing system. The gray
areas indicate how far the reaction has reached. For illustrations purposes, the reaction
front is assumed to grow inwards with approximately the square root of time,
corresponding to diffusional transport.
The domain-division by fracturing is caused by the diffusion–
reaction process. For a rapid reaction, the reaction profile is
approximately step-shaped. The width, Δw, of the step increases in
time according to the dominant transport mechanism. We use a
generalized model, Δw = atq, which by choice of q may describe
various transport mechanism, including diffusion dominated transport (q = 1/2) and fracturing-accelerated transport (q = 1). The tensile
The field observations presented in this report clearly demonstrate
the intimate coupling between chemical and physical weathering
through the generation of reactive surface area by reaction-induced
hierarchical fracturing. During this process, layer-by-layer spalling and
domain-dividing fracturing operates simultaneously to produce the
familiar spheroidal weathering patterns. Our simple mechanical
model demonstrates that the hierarchical process can lead to a rapid
acceleration of the local production of reactive surface area. Depending on the access of water or other fluids, this can be expected to lead
to a very ‘patchy’ distribution of weathering products, because
domains that start to weather early could be almost completely
weathered by the time the late starters advance beyond the spallation
stage. This is indeed what is observed in the Karoo road sections
where the extent of weathering is ‘patchy’ rather than controlled by
the distance to another dolerite body. Reaction-induced hierarchical
fracturing thus appears to represent a first-order control on weathering rates at a scale much larger than previously thought. Moreover,
in contrast to previous assumptions, this rate grows in a strongly
nonlinear way with time.
A. Røyne et al. / Earth and Planetary Science Letters 275 (2008) 364–369
Acknowledgements
Discussions with Dag Kristian Dysthe, Jens Feder and Karthik Iyer
(PGP) have significantly contributed to this paper. Paul Meakin has
contributed significantly through discussions and by his critical
review of the manuscript. Logistic support by Goonie Marsh (Rhodes
University, South Africa) is gratefully acknowledged. This study was
supported by a Center of Excellence grant to PGP from the Norwegian
Research Council.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.epsl.2008.08.035.
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Paper II
Growth rims and interface structure of a crystal face growing from solution
while exerting a force of crystallization
Anja Røyne and Dag Kristian Dysthe
to be submitted
Growth rims and morphology of a single crystal face growing from solution in the direction
of compressive normal stress
Anja Røyne and Dag Kristian Dysthe
When a crystal face is forced into contact with a confining surface by an applied normal stress, and
the crystal is growing from a supersaturated solution, the loaded crystal face is often observed to
develop a growth rim surrounding a hollow core. This has been thought to arise due to the balance
between surface growth and diffusional transport in a thin liquid film through the way in which the
local equilibrium concentration of the liquid is affected by the distribution of surface normal stress.
In this paper, we present experimental results which are in disagreement this hypothesis, and suggest
that the rim forms as a consequence of the steep concentration gradient that develops parallel to
the confined surface, as an analogue to the development of hopper crystal faces. The loaded crystal
surfaces display a high degree of roughness, indicating that steady state is not reached within the
time span of these measurements, and that there may be some mechanism which favours growth on
the most confined portions of the crystal surface.
I.
INTRODUCTION
The so-called force of crystallization, or crystallization pressure, has been an intriguing topic in geology
and geophysics for the last 150 years. Field observations have indicated that under some conditions, growing crystals are able to exert considerable force on their
environment, which may lead to fracture or deformation
of the surrounding matrix [1]. The force of crystallization is important in engineering, because of the deteriorating effects of salt weathering [2, 3] and delayed ettringite formation in cement [4]; and in geophysics, where
it is the mechanism behind pseudomorphic replacement
[5, 6] and can drive vein formation [6, 7]. The force resulting from the controlled growth of a nanocrystal may
also be utilized to drive a linear nanomotor [8].
The first experimental raport on the force of crystallization was made by Lavalle in 1853 [9] who noticed
that crystals were able to push themselves upwards
through growth on their lower surface. In 1905, Becker
and Dey [10] showed that centimeter sized alum crystals growing from a supersaturated solution could "raise
a weight of a kilogram through a distance of several
tens of a millimeter". Later Taber [11] and Correns [12]
followed up with experiments demonstrating the same
phenomenon in several different salts.
In the experiments of Becker and Dey [10] and Taber
[11], quantitative measurements of surface stress were
not feasible due to the appearance of growth rims on
the loaded crystal faces. Becker and Dey reported that
"a terraced cup forms below the crystal so that the bearing surface remains a mere edge throughout its growth"
(see Figure 1). Such growth rims are also commonly observed on the bottom face of crystals grown from solution in a glass dish. Taber [11] showed that the growth
rims were formed by growth on the loaded face of the
crystal as opposed to dissolution of the interior, and concluded that "no material is deposited on the under side
of the crystal except along the advancing outer edge,
and that the cavities (...) are due to the downward
growth of the outer edge". In both of these studies the
1 kg
GLASS
GLASS
FIG. 1: Experimental setup used in the study of Becker and
Dey [10], showing growth rims on the lower face of the crystal.
width of the rim was postulated (but not measured) increase with increasing normal stresses.
When a normal stress is applied to a solid surface in
contact with its solution, the chemical potential of the
solid at the stressed surface is increased with respect to
an unstressed surface. The effect is for the stressed crystal surface to be in equilibrium with a solution of higher
solute activity than an unstressed surface [13–15]. The
reduction in driving force for growth of a flat crystal
face, given by the excess chemical potential ∆µ, due to
the surface normal stress σ is given as [14]
∆µ = kT ln
aσ
− σ v0 ,
a0
(1)
where k is the Boltzmann constant, T is temperature, v0
is the specific volume of the solid and a and a0 are activities of the solution and of a solution saturated with
respect to a large unstressed crystal at temperature T.
The equilibrium normal stress σc , corresponding to zero
crystal growth, can for non-hydrous salts of low solubility be approximated as [15]
σc =
nkT
ln Ω.
v0
(2)
2
where n is the total number of ions released upon complete dissociation of the salt and the supersaturation Ω
is defined as Ω = c/c0 , where c and c0 solution and
equilibrium concentrations.
In order for a crystal to perform mechanical work by
displacing a confining wall in the direction of normal
load, there must be a liquid film present between the
crystal face and the confining surface through which
ions are supplied to the growth site. This was first recognized by Correns [12]. The normal stress between the
crystal face and the confining wall is transmitted by the
disjoining pressure which builds up in the liquid film
[16] when the thickness of the film is less than about 10
nm [17].
In the first order approximation, the normal growth
velocity v y of the crystal surface can be assumed to be
a linear function of the excess chemical potential ∆µ,
v y ∝ ∆µ. If the loaded crystal surface is initially rough,
then the normal stress will be high in in the contact areas
[18, 19] and close to zero otherwise. Since ∆µ is lower
on parts of the surface which are subject to a high normal stress (Equation (1)), unloaded parts of the crystal
face grow faster than the contact areas, which will make
"valleys" on the crystal surface grow faster than "hills"
until a perfectly smooth contact is made with the confining surface (provided that the confining surface is not
rough).
Growth on the confined crystal face must be sustained
by transport of ions from the bulk solution through diffusion in the thin film. In his classical paper on pressure solution and the force of crystallization, Weyl [20]
considered this situation for a cylindrical geometry. At
steady state, the normal growth velocity of the crystal
face is independent of position, and the number of ions
deposited on the circular area πr2 must be balanced by
the diffusional flux through the cylindrical boundary of
area 2πr. Weyl further assumed the equilibrium concentration to be linear function of surface stress, cσ =
c0 + bσ (which is a good approximation for moderate supersaturations, see Equation (2)) and crystal growth kinetics to be sufficiently fast for the concentration at any
position to be considered equal to the stress dependent
equilibrium concentration, c(r) = cσ (r). Under these
assumptions, it was found that if the load F is such that
F /π R2 < σc /2, where R is the radius of the circular crystal face, growth should take place on a rim surrounding
a hollow core. If the film thickness h is independent of
normal stress, the ratio of rim width δ to crystal radius is
a function of the applied mean stress σ = F /π R2 only:
δ
= 1−
R
r
1−2
σ
.
σc
(3)
If the film thickness decreases exponentially with stress,
an explicit formulation for δ / R can not be found, but
the functional form is similar to Equation (3) (see [20,
fig.19]). The growth rims reported by ref. [10, 11] were
cited by Weyl as a qualitative confirmation of this theory.
a)
d)
b)
δ
I
II
2X2
c)
δh
δh
2X1
FIG. 2: a) The crystal is resting on a microscope cover glass,
placed on an o-ring on the bottom of a plastic beaker containing the supersaturated solution. The beaker is immersed in a
thermostat controlled bath. b) Crystals are unloaded, loaded
with calibration weights or clamped. c) Cross-section of crystal showing rim and staircase morphologies. d) View of the
loaded crystal face showing I rim (width δ) and II central face.
In this paper, we present systematic measurements
of the width of the growth rims on crystals subject to
a range of normal loads in a controlled supersaturated
solution. It turns out that the thin film model fails to
describe our results. We propose instead that the rim
development is analogous to the formation of "hopper"
crystal faces as a consequence of the concentration gradient parallel to the confined crystal face. In our experiments, there is no indication of the loaded surface progressing towards a smooth state as predicted by the continuum model.
II.
EXPERIMENT
Supersaturated solutions are unstable by definition,
and performing controlled experiments in a supersaturated solution is notoriously difficult. If new crystals
spontaneously form in the solution, it has the effect of
lowering the solution concentration to some unknown
value. The tendency for nucleation to occur increases
with the degree of supersaturation and with the duration of the experiment (unless a freshly prepared solution is continuously supplied to the system). Experiments at high supersaturations therefore need to be of
short duration in order for the supersaturation to remain constant; longer experiments can be conducted at
lower supersaturation, but with correspondingly slower
growth rates. Temperature control also becomes more
critical when the solution concentration is close to equilibrium. In situ measurements of crystal growth is also
difficult because any measuring equipment which is introduced into the system will act as thermal bridges
to the more thermally unstable surroundings, and very
slight thermal gradients can be enough to facilitate nucleation.
For the present study, we chose to simplify the system as much as possible. A small volume of solution was subcooled in order to generate supersaturation. When thermal equilibrium was reached, a small
3
Sample
100511Aa
100511Ba
100511Cab
100512Ha
100515A1c
100515A2c
100515B
100518A
100518B
100518C
100518D
100519A
100519B
100519C
100519D
100519E
100601A
100601B
100603A2d
100603D
100607Ae
100607Ate
100922A
101020A
101021Af
101021Bf
101025A
101025C
101026A
101026C
Ω
∆t
w
σ /σc
1.024
1.024
1.024
1.034
1.027
1.027
1.027
1.020
1.020
1.020
1.020
1.021
1.021
1.021
1.021
1.021
1.020
1.020
1.020
1.020
1.020
1.020
1.019
1.021
1.028
1.028
1.020
1.020
1.020
1.020
1440
1440
1440
1440
15
27
15
10
10
10
10
15
15
15
15
15
20
20
40
40
130
130
150
20
20
20
20
20
20
20
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
100
20
900
900
900
900
0
10
1
10
20
0
0
200
8.3 × 10−6
1.1 × 10−5
1.1 × 10−5
9.1 × 10−6
5.1 × 10−6
5.1 × 10−6
6.1 × 10−6
9.9 × 10−6
8.4 × 10−6
9.7 × 10−6
7.6 × 10−6
3.4 × 10−6
8.1 × 10−6
5.2 × 10−6
1.9 × 10−5
1.6 × 10−5
2.2 × 10−2
3.1 × 10−3
2.5 × 10−1
4.0 × 10−1
3.6 × 10−1
3.6 × 10−1
1.5 × 10−5
4.2 × 10−3
6.4 × 10−4
4.4 × 10−3
7.0 × 10−3
1.8 × 10−5
3.9 × 10−6
2.8 × 10−1
2X0,1 2X0,2
2364
2913
1305
1319
3086
3086
3132
3321
3165
3435
3212
2609
4297
2354
4967
4808
4188
5002
3864
3187
3247
3247
5078
3595
2459
2797
3620
7426
782
1814
2128
2423
1165
1258
3036
3036
3119
3052
3143
3417
3210
2212
3407
2158
4432
3285
4105
4825
3779
2984
2801
2801
3001
2655
2068
2550
3303
6650
698
1730
y0
1305
1588
630
754
990
990
1194
1506
1284
1497
1161
509
1289
815
3045
2659
1627
2177
1452
1217
1065
1065
2160
1780
758
943
1170
2750
500
965
a due to the large extent of crystal growth in these long experiments,
the final stress was substantially larger than the initial (reported)
stress.
b in this experiment, a second crystal formed next to the original crystal. This resulted in very asymmetric growth of the measured crystal.
c the topography of this crystal was measured after 15 minute exposure; the crystal then was re-immersed for another 12 minutes.
d this crystal was examined after 15 minutes, and no visible growth
rim had developed. It was then re-immersed for another 40 minutes.
e both of the confined faces on this spring-loaded crystal were measured.
f the initial crystal dimensions were not measured, those reported are
final dimensions.
crystal (loaded or unloaded, as explained below) was
immersed in the solution for a set amount of time, and
then removed and quickly dried off. The dimensions of
the crystals and the topography of the loaded faces were
measured before and after the experiment.
Cubic crystals of NaClO3 (v0 = 7.07 × 10−29 m3 [21],
temperature dependent solubility shown in Figure 3)
140
xsalt (g / 100 g water)
TABLE I: Overview of experimental conditions. Ω is supersaturation, ∆t is the experiment duration in minutes, w is the
applied load in g, σ is the mean stress on the crystal surface,
σc is the theoretical equilibrium stress, and 2X0,1 , 2X0,2 and y0
are the initial crystal dimensions in µm.
130
120
110
xsalt = 0.91*T+78
100
90
80
10
20
30
40
50
60
T (oC)
FIG. 3: Solubility of NaClO3 as a function of temperature [21],
given as xsalt = mass of salt (g) per 100 g of water. The fitted
line was used to calculate the solubility at different temperatures for the preparation of solutions.
were grown from solution in the lab. Crystals grown
from solution in a glass dish are never perfect on the
bottom side, but the upper faces are reasonably smooth.
For the crystals that were loaded only on one side, we
therefore always used what had been the top face.
The duration ∆t of the experiments ranged from 10
minutes to 24 hours (Table I).
The solution was prepared by precisely weighing salt
and water to obtain the desired concentration, heating and stirring to dissolve the salt, and adding 30 ml
of filtered solution to a plastic beaker which was immersed in a thermostat controlled bath and left until
thermal equilibration was reached at the desired temperature. The supersaturation Ω varied between 1.019
and 1.034 ± 0.002 and the temperature T of the solution was 20.8 ± 0.1◦ C for all experiments except those
where ∆t = 24 hours (Table I). In the latter, the solution beakers were placed in a controlled air enclosure
with slightly poorer temperature control. Because of the
large extent of crystal growth, the solution concentration
at the end of the long experiments was probably lower
than the initial value. This was not measured.
The corresponding values for σc (Equation (2)) ranged
from 2.2 to 3.8 ± 0.2 MPa (larger variation for the 24
hour experiments). The initial crystals varied in size
from 700 to 7430 µm in the horizontal dimension and
from 500 to 3050 µm in height (±2 µm, Table I). Before
and after each experiment, the dimensions of the crystal was measured with a micrometer, and the topography of the loaded crystal faces were measured optically
using a white light interferometer (Veeco from WYKO),
with horizontal and vertical resolutions of 2 µm and 20
nm, respectively.
A number of experiments were made with no applied
load on the crystals. The load on the bottom face of
the crystal, which was resting on a glass surface, was
merely the weight of the crystal itself. In other experiments calibration weights of mass between one and 200
g were glued to the top of the crystal. The highest loads
4
TABLE II: Experimental results: Measured changes in side
lengths ∆X1 , ∆X2 and ∆y (where y is the crystal height), mean
and standard deviation of measured rim widths δ, and mean
and standard deviation of measured height δh of rim above
the crystal interior or neighbouring step. All dimensions are
in µm.
Sample
∆y δmean std(δ) δhmean std(δh)
∆X1 ∆X2
100511A
335 274 199
100511B
750 694 494
100511C
636 612 1376
100512H
1655 1181
73
100515A1a
100515A2
45
49
73
100515B
50
43
89
100518A
16
6
19
100518B
8
8
4
100518C
5
9
-3
100518D
9
4
8
100519A
44
38
51
100519B
40
68
42
100519C
14
20
38
100519D
25
16
9
100519E
24
22
33
100601Ab
121 148
100601Bb
167 181
100603A2
33
47
2
100603D
32
25
26
100607A
123 150
-2
100607At
123 150
-2
100922Aa
101020Ab
64
32
101021Ac
101021Bc
101025A
30
30
14
101025C
39
34
30
101026A
140 108 100
101026Cd
-7
-1
35
80
133
36
46
66
55
90
51
119
97
77
50
71
58
110
74
96
140
12
7
76
67
136
40
46
43
75
49
37
38
20
59
17
19
13
19
34
29
55
33
66
15
22
24
45
24
55
116
25
13
30
46
47
37
23
38
42
27
19
42
45
39
5
23
8
14
1
2
2
1
8
19
9
4
6
6
50
0.2
0.5
1
1
22
1
4
11
10
3
24
4
6.4
6.2
2.7
7.6
2.8
4.6
0.8
0.7
0.6
1.0
3.1
5.7
4.6
1.7
4.2
5.4
28
0.4
0.9
0.8
1.6
6.7
1.1
1.6
8.6
4.1
2.0
3.0
3.4
a final
crystal dimensions were not measured.
crystal height was not measured beacuse the top face of the
crystal was glued to a calibration weight.
c initial crystal dimensions were not measured.
d the edges of the crystal were damaged during loading.
b final
were obtained by squeezing the crystals between two
small pieces of microscope objective glass using a spring
loaded clamp, calibrated to be (9 ± 1) N (Figure 2, Table
I).
III.
A.
RESULTS
Growth rim observations
A distinct growth rim was found on the loaded face
of all samples in the final topography measurements
(Figures 4 and 5), while the topography of the central face was only slightly different from the initial topography. Some of the crystals, and particularly those
which had been exposed for long times, developed a
succession of rims giving rise to a staircase morphology
(100511A-100512H). Crystals which had been loaded off
axis due to a slightly tilted glass surface or non-centered
applied load developed more clearly asymmetric rims
(100519C, 100601B, 101021B, 101026C). On the crystals
which carried a high load, the rim was sometimes
completely absent on one or more sides (100603A2,
100607At, 101020A,101025C). Where the rim was missing along parts of the edge, it seemed to form smoothly
from the initial surface (Figure 6a).
The rim surfaces displayed a great deal of topography. Sometimes discontinuous steps with a height difference of several µm had formed (100515A1,100519B,
100922A), and most rim surfaces showed some
smoother undulations of several µm height (Figure 6b).
Small scale topographical features were also present on
apparently flat portions of the rim surfaces (Figure 6a).
The rim width δ (Figure 2d) and height above the central surface or neighbouring step, δh (Figure 2c), were
measured at 16 positions along the rims of each crystal. Where a succession of rims had formed, only the
outermost rim was measured. There were considerable
variations in both rim width and rim height for each individual sample. Mean rim widths ranged from 7 ± 12
to 140 ± 116µm, and the mean rim height ranged from
0.2 ± 0.4 to 50 ± 28µm. The changes in crystal dimensions and measured widths and heights of the rims are
summarized in Table II.
B.
Correlation with experimental parameters
According to the theoretical predictions, the ratio of
rim width and crystal width δ / X should be given by the
normalised mean stress σ /σc . There should also be a
clear correlation between the rim width δ and the crystal
width X. No such correlations were observed (Figure 7).
There were also no correlations found between the
rim width δ and the duration of the experiment ∆t or
the extent of growth of the free surfaces, ∆X = X1 − X0
(Figure 8).
C.
Time evolution
The time evolution of rims on crystal faces subject
to low stress (σ /σc < 10−4 ) is represented by samples
100518A-C (∆t = 10 min), 100519A-E (∆t = 15 min),
100922A (∆t = 150 min) and 100511A-C (∆t = 1440
min) (see Figures 4 and 5).
After 10 minutes, a rim is starting to form in a rather
patchy manner around the perifery of the crystal face.
The initial topography is still clearly visible, and the
height of the rim is in many places lower than interior
parts of the crystal surface. After 15 minutes, the rim is
distinct and cleary higher than the middle portion of the
surface. The rim is substantially higher but qualitatively
5
100511A, 3030, 84.3, 1440
100515A1, 3180, 20.1, 15
100518B, 3180, 6.5, 15
100519B, 3540, 39, 15
100511B, 4410, 91.9, 1440
100511C, 2580, 132, 1440
100512H, 4630, 136, 1440
100515A2, 3180, 56.6, 27
100515B, 3230, 33.5, 15
100518A, 3350, 12.4, 10
100518C, 3450, 5.8, 10
100518D, 3230, 9.2, 10
100519A, 2700, 19, 15
100519C, 2380, 17.8, 15
100519D, 5020, 17.5, 15
100519E, 3330, 24, 15
FIG. 4: Measured topographies of the loaded faces in the first 16 experiments. The color denotes height, with blue as the lowest
and red as the highest points on the surface. Captions of each subimage shows sample name, horizontal size of image in µm,
range of color scale in µm and experiment duration in minutes. The sample name is shown in red for the samples in which the
entire load is carried by the rim. On 100511C, there was more of a smooth tilt towards the initial crystal surface than a staircase
shape. The angle of the tilted surface was too large for the WLI topography measurements, which is why only the rim surface is
shown.
6
100601A, 4430, 19, 20
100607A, 3100, 6.0, 130
101020A, 3720, 14.7, 20
101025A, 3680, 26.1, 20
100601B, 5340, 114, 20
100607At*, 3100, 31.2, 130
100603A2, 3930, 15, 40
100603D*, 3250, 23.7, 40
100922A, 5080, 53.4, 150
101021A, 3680, 9.7, 20
101025C, 7500, 13.2, 20
101021B, 2550, 42.6, 20
101026A, 1060, 29.6, 20
101026C, 1800, 15.3, 20
FIG. 5: Measured topographies of the loaded faces in the last 14 experiments.The color denotes height, with blue as the lowest
and red as the highest points on the surface. Captions of each subimage shows sample name, horizontal size of image in µm,
range of color scale in µm and experiment duration in minutes. The sample name is shown in red for the samples in which the
entire load is carried by the rim. * The wide rim-like features on these crystals were parts of the initial surface topography. The
newly formed rim can be seen in the upper left-hand corner of 100603D and along the left and bottom sides of 100607At.
similar after 150 minutes. However, the left hand side
of the face on sample 100922A, a second rim has formed
with its edge closer to the crystal perifery than the initial
rim, forming a two step staircase. After very long exposures, the crystal faces have developed a succession of
steps resembling a staircase morphology.
D.
Effect of stress
On crystal faces subject to a high mean stress, the development of rims progressed substantially slower than
for faces at low stress. One example is 100607A, which
after 130 minutes of exposure resemble the developing
rims seen after 10 minutes on the unloaded crystals. The
7
a)
0.20
a)
0.15
FIG. 6: a) 3D view of rim on top left corner of 100603D (marked
with rectangle on Figure 5), where the rim had only formet on
part of the perifery, and the transition from rim to flat surface
is shown. Measured area is 1200 × 300 µm, and the maximum height difference is 11.6 µm. b) 3D view of rim along
the bottom side of 100519B (marked with rectangle on Figure
4). Measured area is 3400 × 110 µm, and the maximum height
difference is 26 µm.
δ/X
b)
0.10
0.05
0
-6
400
-5
-4
-3
log(σ/σc)
-2
-1
0
b)
350
width of the rim is not correlated with stress (Figure 7a).
δ (µm)
300
E. Surface roughness
250
200
150
The roughness of line profiles parallel to the edge
of initial crystal surfaces, rim surfaces and central
faces (Figure 2d) of three samples (100515A, 100519B,
101025A) were analyzed by calculating the average
roughness amplitude w(l ) as a function of line segment
length l using the relation [22]
w(l ) = h[h( x) − hl ( x)]2 i x ,
(4)
where h(l ) and hl ( x) are the local and average surface
heights (Figure 9c). The Hurst exponents, which describes the scaling of the roughness curves (w L (l ) ∝ l H
[22]) was calculated by fitting a straight line to the entire w(l ) curve for the rim surfaces and to the portion of
the curve before the crossover for the other. Measured
values of H ranged from 0.46 to 0.68 for initial surfaces,
0.76 to 0.82 for rims and 0.51 to 0.68 for middle surfaces.
100
50
0
0
500
1000
1500
2000
X (µm)
2500
3000
3500
4000
FIG. 7: a) Ratios of measured rim widths δ and mean crystal
half-widths X plotted against mean stress σ normalized by the
theoretical maximum stress σc given by Equation (2). The results for each crystal is shown as a column spanning from δmin
to δmax with the mean δ marked a circle. The result of Weyl’s
model (Equation 3) is shown as a solid line for a cylinder with
radius X and rim width δ. The stress axis is logarithmic in
order to better resolve the results at low stresses. b) Measured
rim withs δ versus crystal half-width X measured at the end of
the experiment. Rectangles span from [ X1 , δmin ] to [ X2 , δmax ]
for each crystal while mean values are marked with circles.
In both panels, results for samples in which the rim is high
enough to conclude that no part of the central face is carrying
load are shown in red.
F. Mean growth velocities
IV.
Mean growth velocities v in all directions were found
from the ratio of increase in side length to elapsed time.
The growth velocities of the unloaded sides (∆X /∆t) is
[6.0 ± 7.0] × 10−8 m/s, while in the loaded direction
(v y = 12 ∆y/∆t; for the unloaded crystals this is the
mean velocity of the free upper surface and the confined
lower surface) it is [1.1 ± 1.4] × 10−8 m/s. These results
are comparble with the growth velocity of 2.7×10−8 m/s
which was found for unstressed faces of NaClO3 at
Ω = 1.02 by Ristic et al. [23].
A.
DISCUSSION
Surface stress distribution
For the solution concentrations used in these experiments, the thickness of liquid films which are capable
of sustaining normal stresses of some magnitude is less
than 10 nm. Since the roughness of the crystal surfaces
is much larger than this, the stress cannot be smoothly
distributed on the loaded surface. Instead, the stress
is transmitted in localized contact points, while the re-
8
150
150
a)
100
δ (µm)
δ (µm)
100
50
0
b)
50
103
104
∆t (s)
105
0
0
500
1000
X-X0
1500
FIG. 8: Mean rim width δ plotted against a) exposure time ∆t
and b) extent of growth of the free faces, X − X0 , where X0 is
the mean of initial crystal widths. The insert in b) is a close-up
of the region 0 < X − X0 < 200µm. Results for samples in
which the rim is high enough to conclude that no part of the
central face is carrying load are shown in red.
a)
w (µm)
100 c)
10-1
h (µm)
10-2
0
2 b)
4
6
8
0
2
4
6
X (mm)
101
8
102
l (µm)
10
103
12
FIG. 9: a) Measured topography of the loaded face of one crystal (σ /σc = 5.1 × 10−6 , ∆t = 15 min). b) 2D cross-sectional
profile along the rim. The colors correspond with the arrows
in a). In this profile, the surface has been tilted manually to obtain the maximum apparent contact area. The h axis shows the
distance from the imagined glass plate. c) Average roughness
amplitude w(l ) of 2D cross-sectional profiles of the rim profile (blue), central face (red, profile along white dashed line
in a), and initial surface (green) for three different samples
(100515A, 100519B, 101025A).
mainder of the surface is not subject to normal stress
[18, 19].
As the rims are starting to form, the load on the crystal is probably distributed on points on the rim as well
as points on the middle part of the surface. With continued growth of the rims, the middle part of the surface is
moved away from the confining surface, so that the entire load is carried by contact points which are situated
on the rim.
For the crystal to move the initial surface away from
the confining surface, part of the growth must have
taken place on the actual contact points; if the crystal was only growing on the unloaded portions of the
surface it would have become smooth, and no lifting
could have taken place. The highest normalized load
on a crystal where lifting had definitely taken place was
σ /σc = 7.0 × 10−3 (101025A). For the stress at the contact points to be below the theoretical equilibrium stress
σc , this means that more than 0.7% of the crystal surface
must be in contact. The rim area is on this sample found
to be about 8% of the total area of the loaded face. This
means that more than about 9% of the rim surface must
be in contact in order for lifting to take place without
violating the thermodynamic limit.
B.
Effect of mean stress
According to the thin film theory of Weyl [20], the rim
width should be a strong function of applied stress. This
is evidently not the case in these experiments, where
the mean rim widths seem to be independent of applied
stress (Figure 7a). In addition, there is no clear correlation between crystal width X and rim width δ (Figure
7b), which is also in disagreement with the model.
Considering the assumptions in Weyl’s model, this
disagreement is not surprising. The substantial roughness of the rim surfaces (Figures 6 and 9) is a clear violation of the model assumption of perfectly smooth
surfaces. While the normal stress in Weyl’s model is
a smooth function of position, it is in our experiments
transmitted at localized contact points, leaving the major part of the rim surface unstressed. This in turn means
that diffusion from the bulk solution can not be described as being due to the stress gradient alone.
C.
Correlation with free face growth
Taber [11] proposed that is that growth on the lower
side takes place only "along the advancing outer edge".
If this implies that no growth takes place on the initial crystal surface, then the width of the rim should be
given by the extent of growth on the unloaded faces of
the crystal, δ = X − X0 . No such correlation was found
in our data (Figure 8b). For about half of the samples,
δ > X − X0 , suggesting that part of the rim has formed
on the original crystal.
D.
Proposed mechanism for rim development
The staircase morphology which is found to develop
on crystals after long exposures (Figure 2c), and which
was reported by references [10, 11], resemble the morphology of so-called hopper crystal faces [24, 25]. For a
polyhedral crystal in a diffusional field, the surface concentration at the edges and corners is higher than in the
center of a flat face [25, 26]. Growth kinetics at low bulk
supersaturation is predominantly controlled by screw
9
roots of this equation are ±1/ L, where
c
c
L=
FIG. 10: Step bunching instability due to step propagation in
the direction of lower concentration. Steps nucleate near the
edge, where the concentration c is highest, and move toward
the center of the crystal (a 2D cross-section of a portion of the
crystal near the edge is shown). The step propagation velocity,
shown by the length of the arrows, decreases with decreasing
supersaturation. As steps catch up with the step in front, a
macrostep is formed, giving rise to a hopper morphology (on
free crystal faces above the 2D nucleation threshold) or growth
rim (on a confined crystal surface).
dislocations, and since these are more frequent in central portions of crystal faces, steps will predominantly
nucleate here and propagate towards the edges, accelerating due to the positive concentration gradient. This
leads to a stable flat surface. Hopper crystals develop
when the supersaturation along the edges of the crystal
exceeds the critical threshold for 2D nucleation. In this
situation, steps will have a tendency to nucleate in the
edge regions and propagate towards the center of the
face. Due to the negative concentration gradient steps
will decelerate and bunch up, resulting in the formation of macrosteps parallel to the edges and little or no
growth in the center of the face (Figure 10).
E.
Characteristic length scale for rim formation
The growth kinetics of NaClO3 is dislocation controlled at least up to Ω = 1.04 [23]. However, diffusion
of ions from the bulk solution leads to a strong imposed
concentration gradient on the confined crystal face. This
gives a much stronger driving force for growth near the
edges than in the middle portion of the confined face,
and steps which nucleate near the edges may slow down
and bunch up as they propagate towards the center of
the face. In the fluid film between the crystal and glass
plate, diffusion is essentially two dimensional, and if local growth can be approximated as a linear function of
the excess concentration then mass balance requires
Dh
d2 c
− K (c − c0 ) = 0,
dx2
(5)
where D is the diffusion constant of ions in the solution,
h is the film thickness, K is the kinetic constant for surface growth and c( x) is the local ion concentration. The
r
Dh
K
(6)
gives a characteristic length scale for the concentration
gradient. For a film thickness of h = 1 µm, which is
within the typical roughness of the rim surfaces, then
with D = 1.5 × 10−9 (average diffusion constant for
−5
Na+ and ClO−
3 ions [21]) and K = 4 × 10 m/s (estimated from [23] for the free face growth of NaClO3 ) the
characteristic length becomes L = 6µm, which is a bit
smaller than the typical rim width, but well within the
observed range.
F.
Staircase formation
After some time, secondary rims are found to form
closer to the crystal perifery than the initial rim, until a
staircase morphology is formed for the longest experiments. The development of stairs is probably due to the
growth on the unloaded faces. As the unloaded faces
grow, the distance from the edge of the face to the rim
edge becomes longer. At some point, it becomes so long
that the instability responsible for rim formation causes
a second rim to form closer to the perifery of the face.
G.
Effect of crystal size
The roughness measurements showed that the height
difference along the rim increases with system size. This
means that we can expect larger film thicknesses to occur on larger
√ crystals. Since the characteristic length L
scales with h, we would expect that the maximum rim
width δmax for a given crystal would scale as δmax ∝
X H /2 , with H ≈ 0.75. We do observe a tendency for the
maximum rim width to increase with crystal size (Figure
8), which is in qualitative agreement with this interpretation. Due to the low exponent, a larger range of X is
required in order to quantitatively validate this power
law dependence.
H.
Growth dynamics of loaded crystal surface
The observed difference in roughness scaling between
free surfaces and the rims (Figure 9) indicates a difference in growth mechanism. On the free faces, large scale
roughness is suppressed. The crossover length scale
around 100 µm may be related to the thickness of the
diffusional boundary layer, which is of the order 100 µm
for the growth of ionic crystals from an aqueous solution
[25]. On the rim surface, where the distance to the glass
is much less than the boundary layer thickness, this suppression is not present. Furthermore, the large Hurst
10
exponent of the rim surface is indicative of a mechanism which amplifies roughness on large scales. Large
Hurst exponents (0.7 < H < 1) seem to be characteristic of many surfaces grown by molecular beam epitaxy,
and it may in part be explained by asymmetric surface
currents resulting from Schwoebel barriers [27]. In our
case the boundary conditions are different, and we may
interpret the large Hurst exponent as being consistent
with preferential growth on high portions of the surface, which are also regions where the distance to the
glass plate is smaller. This is somewhat counter intuitive and does not agree with continuum considerations
as explained in the Introduction. Another possible explanation for the large Hurst exponent is that it is related
to the dimensionality of the process. If the growth on the
rim is substantially different in the direction parallel to
and perpendicular to the edge of the crystal, this may
result in a different scaling when comparing to growth
processes which are isotropic in two dimensions.
If the scaling is due to preferential growth on confined portions of the surface, we propose three processes which may be responsible for this phenomenon.
1: Preferential growth in regions of higher concentration, which in a diffusion field would coincide with the
corners. Corners are often higher than other areas of
the rims, but not always. This may be due to the effect of free convection currents on the local concentration field. 2: Preferential growth at or near contacts due
to higher dislocation densities caused by large normal
stresses. This follows the idea of Wilcox [24] that edge
instabilities arise more frequent in stirred solutions due
to dislocations produced by collisions between crystals.
3: Preferential growth at the contacts due to some electrochemical effects in the thin liquid film, or as a consequence of diffusion of ions towards higher concentrations due to the large pressure gradients near the contacts [20, 28]. In situ observations of the surface evolution would be necessary to discriminate between these
mechanisms. We hope to be able to do this at some later
stage.
The only experimental support for Weyl’s model [20]
on the force of crystallization has been the observed
growth rims in the experiments of Becker and Dey [10]
and Taber [11]. Although we know nothing about the
exact morphology of the rims they observed, it seems
likely that they may have formed due to the mechanism
we have proposed. We suggest on the basis of our experiments that Weyl’s thin film model is not generally
applicable for macroscopic crystals growing to excert a
force of crystallization.
In the context of pressure solution, which may be
viewed as the inverse process of the force of crystallization [20], the structure and transport properties of
the interface has been investigated in some detail. Since
observed rates of solution compaction are much larger
than what can be explained by diffusion through a
smooth thin film [29], various models for rough interfaces have been proposed, where diffusion through regions of thicker films allows for increased transport.
Some degree of interfacial roughness has been confirmed by experimental studies [30–32]. Our results suggest that these considerations may also be valid in the
context of the force of crystallization.
On the local scale, the crystal growth rate can not be
taken to be linearly dependent on the excess chemical
potential. In order to fully understand the growth dynamics of a crystal face subject to normal stress, the effect of normal stress on the nucleation and propagation
of steps as well as the transport of ions in a confined
film with varying thickness and in the presence of large
pressure gradients needs to be considered. There rate
of growth of a crystal performing mechanical work on
its surroundings is important for processes such as replacement and vein formation [6], but there is at present
no model which can be used to predict such rates with
any degree of accuracy. Both theoretical and in situ experimental studies are required in order to advance this
subject further.
[1] N. L. Watts, Geology 6, 699 (1978).
[2] G. W. Scherer, Cement and Concrete Research 34, 1613
(2004).
[3] C. Rodriguez-Navarro and E. Doehne, Earth Surface Processes and Landforms 24, 191 (1999).
[4] R. J. Flatt and G. W. Scherer, Cement and Concrete Research 38, 325 (2008).
[5] R. G. Maliva and R. Siever, Geology 16, 688 (1988).
[6] R. C. Fletcher and E. Merino, Geochimica et Cosmochimica Acta 65, 3733 (2001).
V.
CONCLUSION
Growth rims observed on crystal faces subject to a
normal load has been taken as support for a model
which assumes a perfectly smooth crystal and steady
state growth. Our experimental results show that these
assumptions are violated and that the growth rims are
formed by a different mechanism, most likely as a step
bunching mechanism due to the concentration gradient
which develops parallel to the confined crystal surface.
Observations of rim surface roughness show no indication of the system progressing towards a smooth state.
The development of the observed roughness can not be
explained in the framework of the present theories describing growth of crystal faces subject to normal stress.
Further theoretical and experimental studies are needed
in order to understand the dynamics of growth for crystals which perform mechanical work on their surroundings.
11
[7] D. V. Wiltschko and J. W. Morse, Geology 29, 79 (2001).
[8] B. C. Regan, S. Aloni, K. Jensen, R. O. Ritchie, and A. Zettl,
Nano Letters 5, 1730 (2005).
[9] J. Lavalle, Comptes Rendus de l’Academie des Sciences
36, 49 (1853).
[10] G. F. Becker and A. L. Day, Proceedings of the Washington
Academy of Sciences 8, 283 (1905).
[11] S. Taber, American Journal of Science 41, 532 (1916).
[12] C. W. Correns, Discussions of the Faraday Society 5, 267
(1949).
[13] J. W. Gibbs, in The Scientific Papers of J. W. Gibbs (Longmans, Toronto, Ont., 1906), vol. 1, pp. 55–349.
[14] M. S. Paterson, Reviews of Geophysics 11, 355 (1973).
[15] M. Steiger, Journal of Crystal Growth 282, 455 (2005).
[16] R. M. Espinosa-Marzal and G. W. Scherer, Accounts of
chemical research 43, 897 (2010).
[17] A. Anzalone, J. Boles, G. Greene, K. Young, J. Israelachvili,
and N. Alcantar, Chemical Geology 230, 220 (2006).
[18] J. Greenwood and J. Williamson, Proceedings of the Royal
Society of London Series a-Mathematical and Physical
Sciences 295, 300 (1966).
[19] B. N. J. Persson, Surface Science Reports 61, 201 (2006).
[20] P. K. Weyl, Journal of Geophysical Research 64, 2001
(1959).
[21] D. R. Lide, CRC handbook of chemistry and physics: a ready-
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reference book of chemical and physical data (CRC Press, Boca
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A.-L. Barabási and H. Stanley, Fractal concepts in surface
growth (Cambridge University Press, Cambridge, 1995).
R. Ristic, J. N. Sherwood, and K. Wojciechowski, The Journal of Physical Chemistry 97, 10774 (1993).
W. R. Wilcox, Journal of Crystal Growth 38, 73 (1977).
I. Sunagawa, Crystals - growth, morphology and perfection
(Cambridge University Press, 2005).
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P. Meakin, Fractals, scaling and growth far from equilibrium
(Cambridge University Press, 1998).
W. K. Heidug, Journal of Geophysical Research-Solid
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N. Alcantar, J. Israelachvili, and J. Boles, Geochimica et
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J. P. Gratier, R. Guiguet, F. Renard, L. Jenatton, and
D. Bernard, Journal of Geophysical Research-Solid Earth
114 (2009).
Paper III
Experimental investigation of surface energy and subcritical crack growth in
calcite
Anja Røyne, Jan Bisschop and Dag Kristian Dysthe (2011)
Journal of Geophysical Research - Solid Earth, 116, B04204
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, B04204, doi:10.1029/2010JB008033, 2011
Experimental investigation of surface energy and subcritical crack
growth in calcite
Anja Røyne,1 Jan Bisschop,2 and Dag Kristian Dysthe1
Received 1 October 2010; revised 15 December 2010; accepted 10 January 2011; published 9 April 2011.
[1] Subcritical cracking behavior and surface energies are important factors in geological
processes, as they control time‐dependent brittle processes and the long‐term stability of
rocks. In this paper, we present experimental data on subcritical cracking in single calcite
crystals exposed to glycol‐water mixtures with varying water content. We find upper
bounds for the surface energy of calcite that decrease with increasing water concentration
and that are systematically lower than values obtained from molecular dynamics
simulations. The relation of surface energy to water concentration can explain water
weakening in chalks. The rate of subcritical crack growth in calcite is well described by a
reaction rate model. The effect of increasing water on crack velocity is to lower the
threshold energy release rate required for crack propagation. The slope of the crack
velocity curve remains unaffected, something which strongly suggests that the mechanism
for subcritical cracking in calcite does not depend on the water concentration.
Citation: Røyne, A., J. Bisschop, and D. K. Dysthe (2011), Experimental investigation of surface energy and subcritical crack
growth in calcite, J. Geophys. Res., 116, B04204, doi:10.1029/2010JB008033.
1. Introduction
[2] Subcritical crack growth, refers to the phenomenon of
slow crack propagation in brittle materials at stresses below
the nominal failure stress [Lawn, 1993; Olagnon et al.,
2006]. This phenomenon in of importance in the Earth’s
crust, where water and other active species are present and
displacements and stresses are small. Subcritical cracking is
argued to be the main mechanism of brittle creep of rocks
[Heap et al., 2009; Scholz, 2002], and to control the time to
failure at constant stress. Subcritical crack growth has thus
been used as the underlying mechanism in models of slow
earthquakes and aftershocks of major earthquakes [Helmstetter
and Shaw, 2009].
[3] Calcitic rocks are abundant in many hydrocarbon
bearing environments and other settings where reactive fluid
transport is important. Faulting and fracturing of carbonate
reservoirs have major implications for hydrocarbon fluid
pathways [Agosta et al., 2007] and carbonates host many
active seismic fault zones [Miller et al., 2004]. Experimental
studies show that subcritical cracking plays an important
role during the compaction of carbonate sediments [Croizé
et al., 2010]. The physical processes controlling subcritical
cracking in calcite are still, however, poorly known.
Experiments on single calcite crystals [Dunning et al., 1994]
have shown a nontrivial dependency on pH and ionic concentrations, while studies on calcitic rocks [Henry et al.,
1977; Atkinson, 1984] have indicated that these rocks dis-
play a complex behavior, different to quartz bearing rocks
and glasses. Various mechanisms including dissolution and
microplasticity have been proposed to explain subcritical
crack growth in calcite [Atkinson, 1984]. Unlike many other
rocks, calcitic rocks display both plastic and brittle behavior
at low stresses and temperature [Turner et al., 1954; Fredrich
and Evans, 1989; Schubnel et al., 2006], and we therefore
expect the fracture behavior of calcite to be more complex
than the more commonly studied minerals such as mica and
quartz.
[4] Chalk, a highly porous rock composed almost entirely
of calcite, has been extensively studied due to its importance
as an oil and gas producing reservoir rock. The mechanical
strength of water saturated chalk is significantly lower than
that of dry or oil saturated chalk. This so‐called water
weakening effect has been the subject of extensive study,
but the underlying mechanism is still not fully understood.
Risnes et al. [2005] showed that the strength of chalk
decreases systematically with water content when saturated
with water‐glycol mixtures. Glycol is fully miscible with
water, and its effect on chalk strength is similar to that of oil.
Mixtures of water and glycol therefore allow for a systematic study of the effect of water concentration. Risnes et al.
[2005] concluded that the water weakening effect is caused
by the adhesion properties of water on the calcite surfaces.
[5] Brittle fracture propagation is controlled by the Griffith
equilibrium condition, which can be stated as
G0 2se ¼ 0;
1
Physics of Geological Processes, University of Oslo, Oslo, Norway.
Institute for Building Materials, ETH Hönggerberg, Zurich,
Switzerland.
2
Copyright 2011 by the American Geophysical Union.
0148‐0227/11/2010JB008033
where G is the mechanical energy release rate with G0 corresponding particularly to the lower limit of crack propagation and g es is the surface energy of the solid in contact with a
given chemical environment. The presence of a chemically
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Figure 1. Schematic diagram of subcritical crack propagation [after Maugis, 1985]. The curve to the right corresponds
to propagation in vacuum. The physical interpretation of
regions I, II, and III is given in section 1.
active environment lowers the value of G0 and as a result the
value of v increases for a given value of G (Figure 1).
Measured crack velocity curves v(G) can generally be
described by three regions of behavior. In region I, at low G,
crack propagation is controlled by reaction kinetics. When
transport of active species to the crack tip becomes rate
limiting, the system enters region II. The slope of v(G) is
typically much lower than in region I. Region III behavior
takes place close to the critical energy release rate Gc, where v
is a very strong function of G.
[6] Subcritical crack propagation in region I (Figure 1)
can be understood as a thermally activated process characterized by forward and backward energy barriers which are
functions of G − 2g es [Vanel et al., 2009], and reaction rate
theory gives the following relation for crack velocity v [Wan
et al., 1990a]:
G 2se
DF
sinh
v ¼ 20 a0 exp ;
kT
kT
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Progress has been made in direct measurements of surface
energies by advanced use of the surface forces apparatus
[Alcantar et al., 2003; Anzalone et al., 2006] and recently
also with atomic force microscopy [Hamilton et al., 2010],
but most results from these methods are obtained on mica
because of its atomically smooth surface. Molecular simulation studies have been used to determine the surface
energies of calcite [de Leeuw and Parker, 1997; de Leeuw
et al., 1998; de Leeuw and Cooper, 2004; Wright et al.,
2001; Kerisit et al., 2003; Kvamme et al., 2009] with
greatly varying results. Experimental measurements of the
surface energy of calcite have been performed using fast
fracture experiments with poor control on the environmental
conditions [Gilman, 1960; Santhanam and Gupta, 1968];
surface energies have also been calculated from precipitation studies [Donnet et al., 2005, 2009]. As both of these
methods are associated with large uncertainties, better
experimental measurements of the surface energies of calcite
are needed in order to test the numerical predictions. Subcritical fracture studies represent a complementary method
for direct measurements of the surface energies of brittle
materials in a range of chemical environments.
[8] In this paper, we present experimental results on the
subcritical growth curves and surface energies of calcite in
water‐glycol mixtures with a range of water concentrations.
The results are used to find upper bounds for the surface
energies of calcite in these liquids, and to study the effect of
water on the crack propagation velocities.
2. Experimental Setup
2.1. Double Torsion Testing
[9] We used the double torsion method [Evans, 1972] in
our experiments (Figure 2). In this method, a flat sample
with an initial notch or starting crack is loaded as shown in
Figure 3. Bending of the sample results in propagation of a
ð1Þ
where n 0 = kT/h is a fundamental lattice vibration frequency,
k is the Boltzmann constant, T is temperature, a0 is some
characteristic atomic spacing, DF is the quiescent value of
the energy barrier, and a is an activation area. Experimental
data from mica, sapphire and soda‐lime silicate glass have
been successfully fitted to this model [Wan et al., 1990a].
The phenomenological reaction rate theory does not specify
the nature of the process involved.
[7] Surface energies of mineral‐fluid interfaces control the
presence of fluids at grain boundaries [de Gennes, 2003],
create an upper bound for the stresses that can be generated
by crystal growth in pores [Espinosa‐Marzal and Scherer,
2010], and control mineral surface morphologies during
growth and dissolution [de Leeuw and Parker, 1997].
2 of 10
Figure 2. Schematic of the double torsion rig.
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RØYNE ET AL.: SURFACE ENERGY AND SCC IN CALCITE
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only one crack length measurement is needed. This is particularly useful for nontransparent materials where identification of the crack tip is difficult. In measurements on
transparent materials, calculation of v from equation (5) is
complementary to direct measurement of crack velocity
from the recorded crack length a as a function of time.
Figure 3. Loading of sample and sample dimensions in the
double torsion method.
mode I crack with the highest tensile stress at the bottom
side of the sample. This method is well suited for stable
crack propagation studies. The energy release rate can be
considered to be independent of crack length for the middle
portion of the sample, where it can be calculated as [Shyam
and Lara‐Curzio, 2006]
G¼
3P2 Sm2
;
2St 4 G
ð2Þ
where P is the applied load, Sm is half the distance between
the supports, S and t are sample width and thickness, y = 1 −
0.6302t + 1.20t exp(−p/t) is a geometric correction factor
with t = 2t/S, and G is the shear modulus for calcite (32.8 GPa
[Chen et al., 2001]).
[10] Equation (2) is derived from the analytical expression
for the sample compliance C [Shyam and Lara‐Curzio,
2006],
C¼
D
3S 2
3 m a;
P St G
ð3Þ
where D is the displacement of the loading point. Experimentally, the compliance of the specimen is found to follow
the relationship
C¼
D
¼ Ba þ D;
P
ð4Þ
where B and D are scaling constants. The linear dependence
of compliance on crack length is only observed in the
middle part of the sample, where edge effects are negligible.
Reliable measurements can therefore be made for the region
of crack lengths where C(a) is linear. The parameter D reflects
the compliance of the loading system, while B should be
equal to the prefactor in equation (3).
[11] The load relaxation method, where displacement of
the loading point D is increased quickly and then left constant while the decay of the load P is recorded, is perhaps
the most commonly used method in double torsion tests.
When the D is constant, then the crack velocity v = da/dt
can be calculated from the relation
v¼
Pi
D dP
;
ai þ
B dt
P
ð5Þ
where Pi and ai are instantaneous measurements of load and
crack length. Ideally, D/B a and can be ignored so that
2.2. Sample Preparation and Loading
[12] Calcite samples measuring 30 × 10 × 1 mm were cut
with the 10 × 30 mm face parallel to the {1014} cleavage
plane. All of the other faces were cut normal to the {1014}
plane and thus miscut with respect to the calcite rhomb
(Figure 4). The samples were prepared by Photox Optical
Systems Ltd. from mined calcite crystals. The 10 × 30 mm
surfaces were optically polished. No heterogeneities were
visible in the samples when examined between crossed
polarizers.
[13] A starting crack was made by scratching near the
edge of the sample to produce a slightly rough cleavage
crack, 4–5 mm in length. The sample was then loaded in the
rig and the crack was forced to propagate to the middle
portion of the sample before measurements commenced. In
other materials, a guide groove is often necessary to avoid
significant crack deflection, and the results can sometimes
be affected by the shape of the groove [Shyam and Lara‐
Curzio, 2006], but this was not necessary in our samples
due to the strong cleavage of calcite. Crack propagation in
other crystallographic directions was not attempted as it is
extremely difficult to fracture a single calcite crystal at an
angle to a cleavage plane.
[14] All of the experiments were performed at room
temperature, which varied with ±0.5°C around 22°C. The
water concentration was varied using a mixture of distilled
water in ethylene glycol, following the idea from Risnes
et al. [2005]. The solubility of calcite in pure glycol is
30% lower than in pure water [Sandengen, 2006] and both
fluid–air surface tension and dielectric constant of glycol are
half those of water [Lide, 2008]. Several experiments were
performed at each mole fraction of 0, 0.1, 0.3, 0.5, 0.7 and 1
of water in glycol. Distilled water was used in all mixtures.
The pure glycol may contain as much as 3% water as
reported by the manufacturer. Glycol from the same bottle
was used for all experiments.
2.3. Experimental Procedure
[15] Due to the small size of the samples (which was
limited by the availability of high‐quality crystals), the
stiffness of the rig D was high compared with the sample
compliance B (theoretical value 1.67 × 10−4 N−1, measured
values B = 2.5 ± 0.4 × 10−4 N−1 and D = 1.3 ± 0.3 ×
10−5 N−1 m). When the sample stiffness is comparable to the
stiffness of the loading rig, only a narrow range of crack
velocities can be measured in a single load relaxation
experiment because P decays less with a than in a stiffer rig.
In order to obtain data for as wide a range of stresses as
possible, we used a combination of load relaxation measurements (Figure 5) and measurements where the displacement of the loading point was continuously increased.
The rate of loading point displacement was increased from
10−9 m/s at small crack velocities to 10−6 m/s at high
velocities. The crack velocities were calculated from the
local derivatives of crack length and/or load relaxation
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Figure 4. Crystallographic orientation of sample. (a) Sample
orientation in relation to the calcite rhomb. (b) Top view of
sample, showing the two possible crack directions. The
deviation from the center line is greatly exaggerated for
clarity. The direction of crack 1 is expected to be more
energetically favorable than crack 2. (c) Side view showing
the two possible sample orientations. The angle between
the cleavage plane and the crack front is larger in A than
in B, which may facilitate crack propagation in samples
with the B orientation.
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intensity along a profile parallel to the crack and recording
where a threshold light intensity was exceeded (Figure 6).
Since calcite is a transparent material, crack length measurements are much less problematic than in nontransparent
and disordered materials. The linear relationship between
compliance and crack length confirms that the crack length
detection algorithm is consistent for the entire sample
length. Pictures were obtained with a resolution of about
7.5 mm/pixel at a maximum rate of 3 Hz.
[17] After each experiment, the crack surface was examined using a white light interferometer (Wyko NT1100 from
Veeco). Scans over the entire crack surface were made with
a vertical resolution of about 20 nm. More detailed scans
were also made of areas of size 120 × 90 mm with a vertical
resolution of about 2 nm. Smaller features, approaching the
unit step height of calcite (5 Å) were also visible in these
scans, but not quantifiable.
[18] The crack velocities were found by measuring the
derivative of crack length with respect to time, and also
using equation (5) during load relaxation measurements.
The derivatives were estimated by fitting a straight line to a
suitable number of data points. This method resulted in an
uncertainty in the measured velocity of up to a factor two,
depending on the number and quality of pictures or load
measurements in the relevant range. The uncertainty in G
calculated from equation (2) is due mainly to the uncertainty
in sample thickness t (3%) and shear modulus G (5–10%)
and absolute position of the supports Sm (5%) and is estimated to be about 15%. The discrepancy between measured
and theoretical sample compliance B may reflect a systematic error leading to a systematic overestimation of G.
[19] The rig was tested for relaxation several times before
and between measurements, by leaving it without a sample
or with a dummy sample made of stainless steel, at loads
higher than those used in the experiments. The relaxation
measured in the rig was negligible compared to that measured due to crack growth. In order to test repeatability, we
performed some experiments on 76 × 26 × 1 mm glass
measurements (equation (5)) and the energy release rate G
was found from instantaneous load measurements using
equation (2).
2.4. Data Acquisition, Processing, and Uncertainty
Analysis
[16] The position of the loading point was controlled by a
linear actuator (PI M‐227.50). The load P was recorded by
an Omega load cell with a maximum range of 40 N, and
read with a Keithley 2002 multimeter, giving a sensitivity in
the load readings of about ±0.001 N. The crack was imaged
by adjusting the light source to get maximum reflection
from the crack surface, and the crack tip was identified
automatically using Matlab software by plotting the light
Figure 5. Load P (Newtons) and crack length a (millimeters) measured during one load relaxation experiment.
The velocities measured from this plot are the upper portion
of measurements shown in blue at cw = 0.7 in Figure 7.
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Figure 6. (a) Instantaneous image of the crack tip. The image was rotated to have the crack propagate
along the x axis of the picture. (b) Plot along the dotted line in Figure 6a, together with a threshold function (dashed line) which reflects the background intensity. The crack position was taken to be the point
where the intensity crossed the threshold curve. When the crack tip was near locations in the cell with
high reflections (seen at 10–11 mm in the plot), this had to be corrected manually, and some images were
discarded. (c) Compliance curve for this sample. The dashed line shows a linear fit, C = 2.26 × 10−4a +
1.07 × 10−5 (with a in millimeters).
microscope slides with a very sharp starting crack and
without a guide groove, loaded in the same way as the
calcite samples. The measured v(G) curves for the glass
samples showed a high degree of reproducibility, much
more than what we experienced for the calcite samples (see
section 4).
3. Results
3.1. Crack Velocities
[20] The measured crack velocity v as a function of energy
release rate G from all our experiments are shown in Figure 7.
Measured crack velocities range from 10−8 to 10−2 m/s. It can
be seen that the crack velocity at a given G is increased by
several orders of magnitude when the water concentration is
raised.
[21] We find a significant variation in the shapes of the
v(G) curves that corresponds to variations in velocity from a
factor 2 (pure water) to almost 3 orders of magnitude (pure
glycol). This variability is found to be independent of
loading method, sample alignment, crack deflection and
crack roughness (see Figure 9) and strongly contrast the
reproducibility of the tests we made on glass samples. The
variation is of similar magnitude for experiments performed
both on a single sample and for results for different samples.
For any material, a higher energy release rate should always
correspond to a higher crack velocity. However, we observe
situations in a number of our experiments where v decreases
with G (see section 4).
[22] In the purely brittle regime all of the mechanical
energy is converted to crack extension. It follows that the
maximum velocity observed at any given G is the velocity
closest to the true brittle v(G) curve, while lower velocities
are caused by additional dissipative processes. For each
water fraction we have picked a number of maximum
velocities (open black symbols in Figure 7) which we take
to represent the “brittle v(G) curve” for the respective
conditions.
3.2. Surface Energies
[23] Conservative estimates of g es = G0/2 were made by
recording the lowest measured G at which crack growth was
observed, which gives upper bounds for the true values of
G0. With increasing water concentration, we obtained surface energies of 0.32, 0.30, 0.23, 0.12, 0.14 and 0.15 J/m2
(Table 1 and Figure 8).
3.3. Crack Surface Topography
[24] Figure 9 shows the crack surface topography for one
of the experiments (cw = 0.5). The crack surface morphologies are typical for cleavage surfaces, with steps repeating
down to the nanometer scale. All of the crack surfaces show
steps which are orthogonal to the crack front. This type of
step is expected to form on cleavage surfaces due to crack
interaction with screw dislocations [Gilman, 1959]. The
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RØYNE ET AL.: SURFACE ENERGY AND SCC IN CALCITE
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Figure 7. Results from the crack experiments. Each panel corresponds to a different water concentration, cw. Data points from all experiments are included and are shown in different colors. G0 is shown
as a vertical dotted line. Open black symbols denote high‐velocity points that we have extracted and
shown in the collapse in Figure 10. Note that the velocity scale is the same for all frames, while
the G scale varies.
ratio of vertical to horizontal displacement is 10−4 which
means that the direct contribution of fracture steps normal to
the fracture direction is negligible. One may also observe in
Figure 9 that the surface step direction and step density does
not seem to correlate with the crack velocity.
4. Discussion
4.1. Surface Energy of Calcite
[25] The measured surface energy of calcite (Figure 8)
decreases with water concentration, and the value of g es is
more than doubled when the water concentration goes from
1 to close to 0. This is consistent with the ratio between the
dielectric constants and air/fluid surface tensions of glycol
and water [Lide, 2008], which are 0.5.
[26] In Figure 8 we have compared our surface energy
measurements with the molecular dynamics simulation data
of de Leeuw and Parker [1997] for surface energies of the
calcite cleavage plane at varying degrees of water coverage,
from vacuum to one full monolayer at zero Kelvin. Our data
and the data of de Leeuw and Parker [1997] show similar
trends, suggesting that water causes an almost linear
decrease in surface energy. Our data extend the already
noticed discrepancy between results from experiments and
modeling. For wet calcite surfaces, we would have expected
experimental results to be higher than the numerical values,
because the experimental calcite surfaces always contain
steps. The lower experimental values for wet surfaces
therefore strongly suggest that there are relaxation effects at
water covered calcite surfaces that are not taken into account
by current atomistic models. For the dry surfaces, we can
explain the lower experimental values by considering that
there is always some residual water adsorbed on the calcite
surfaces, and in our case “dry” surfaces are covered with
glycol, which also lowers the surface energy from the
vacuum state.
[27] The trend in our data is consistent with the decrease
in hydrostatic yield stress with increasing water activity that
was found by Risnes et al. [2005] for chalk saturated with
water‐glycol mixtures. Our findings therefore support
Risnes’ hypothesis that the principal mechanism causing the
water weakening effect is related to decreased cohesion of
the chalk grains.
4.2. Crack Velocity Model
[28] Subcritical crack growth in region I (see Figure 1) can
be understood as a thermally activated process as described
by equation (1). Figure 10 shows a fit of this model to our
data. Assuming a characteristic spacing of a0 = 5 Å [Stipp
and Hochella, 1991], we get good agreement with our
data with the parameters for the activation area a = 2.9 ×
10−20 m2 and the quiescent energy barrier DF = 7.6 × 10−20 J.
6 of 10
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Table 1. Comparison of Surface Energies g es of Dry and Fully
Hydrated Calcite Surfaces Found Experimentally and Numerically
for the {1014} Surface as Reported in Literature
g es , Dry (J/m2)
g es , Wet (J/m2)
Experimental Resultsa
This study
0.32
Donnet et al. [2005]
Donnet et al. [2005]
Donnet et al. [2009]
Gilman [1960]
0.23
Santhanam and Gupta [1968]
0.347 ± 0.045
0.15
0.046 ± 0.007
0.135 ± 0.029
0.039–0.164
Reference
Simulation Results
de Leeuw and Parker [1997]
0.60
de Leeuw et al. [1998]
0.59
de Leeuw and Cooper [2004]
0.59
Wright et al. [2001]
0.32
Kerisit et al. [2003]
0.59
Kvamme et al. [2009]
0.86
0.30
0.17
0.33
0.23
0.21
0.29
a
The measurements of Donnet et al. [2005] are from precipitation
studies, while those of Gilman [1960] and Santhanam and Gupta [1968]
are from fracture experiments.
This can be compared with the results of Wan et al. [1990a]
for mica, which are a = 1.5 × 10−19 m2 and DF = 7.1 ×
10−20 J with a0 = 4.6 Å. There is no indication of a systematic change of fitting parameters with water concentration. This indicates that a single mechanism, independent of
water concentration, controls subcritical crack growth in
calcite at the velocities we have studied. The exact physical
B04204
mechanism controlling subcritical crack growth in calcite
cannot be indentified from these measurements. The activation areas correspond to length scales of 1.4 Å for calcite
and 3.9 Å for mica, which are both on the order of the
atomic spacing. This supports the notion of atomically sharp
cleavage cracks in brittle crystals.
[ 29 ] Our findings and those of Dunning et al. [1994]
indicate that the low‐velocity plateau described by Henry
et al. [1977] and Atkinson [1984] does not exist in single
calcite crystals. If it is a real effect, it must be caused by
some property of polycrystalline rocks rather than processes
in calcite cleavage. This could be related to processes on the
grain boundaries, or to effects related to plasticity in grains
which are oriented favorably for twinning.
4.3. Origin of Variable Crack Velocity Data
[30] Our measured v(G) curves (Figure 7) are highly variable, in particular at low cw and high G. A number of other
studies have found subcritical crack growth to be more
erratic with decreasing water concentration. For metals, this
has been explained by plasticity in the form of local stick‐
slip [Briggs et al., 1981] while in the case of mica, it has
been attributed to surface charge effects [Deryagin and
Metsik, 1960; Wan et al., 1990b]. In calcite, it has been
shown experimentally that twins readily develop at the crack
tip at room temperature. This process is reversible up to a
certain stress threshold, but it is still a dissipative, nonelastic
process [Bowden and Cooper, 1962]. It is therefore likely
that there are stress‐dependent, intermittent, dissipative
Figure 8. Surface energies g es of calcite as a function of water concentration (mole fraction or surface
cover fraction). Filled circles are estimates from experiments on calcite (this study). The measured g es
are plotted with asymmetric error bars: the upper shows the experimental uncertainty, while the lower
is undefined, indicating that our values represent upper bounds. Open squares are surface energies for
calcite surfaces with partial coverage of water, calculated by atomistic simulations by de Leeuw and
Parker [1997]. Ellipses show the range of numerical results from literature for dry and wet calcite surfaces, while rectangles show the range of experimental results from literature (see Table 1).
7 of 10
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Figure 9. Crack surface topography of one of the 50% water experiments (plotted in green in Figure 7)
together with the surface height along the middle of the sample (solid line) and crack velocity (dashed
line) as a function of crack length. No correlation was found between the variations in velocity and crack
surface topography. Some 120 × 90 mm high‐resolution surface images are also included, showing typical
cleavage surface steps down to the crystal unit step height.
processes occurring at the crack tip in calcite which do not
leave a visible trace on the crack surfaces.
[31] Variations may also be due to effects of crystallographic orientation. On the small scale, some left‐right
wandering around the cleavage plane is observed (Figure 9),
which would result in variation of the ratio of type 1 to
type 2 cleavage cracks (see Figure 4). Since type 1 cracks
are expected to more energetically favorable, this may have
an effect on the observed crack velocity. Slight sample
misalignment may cause these effects to become more
Figure 10. High‐velocity points shown in Figure 7, plotted as a function of G − G0. The line shows a fit
to the model of Wan et al. [1990a] (equation (1)) for region I subcritical crack propagation, with fitting
parameters a = 2.9 × 10−20 m2 and DF = 7.6 × 10−20 J.
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RØYNE ET AL.: SURFACE ENERGY AND SCC IN CALCITE
dominant. More research is needed in order to fully understand the origin of the crack velocity variations.
5. Conclusion
[32] Double torsion experiments on subcritical cracking in
calcite represent an alternative method for measuring the
effect of chemical environment on the surface energy of
calcite. Our results for different water concentrations are
consistent with previous experimental results, and are consistently lower than results from numerical simulations. The
discrepancy points to relaxation effects that are not properly
accounted for in the simulations.
[33] The observed large variations in crack velocity at a
given energy release rate may be caused by intermittent
plastic processes in calcite, or by some other unidentified
process. The highest measured velocities at any given G,
which are taken to represent the truly brittle behavior, can be
well described using a reaction rate model with parameters
similar to those for mica. These findings support the notion
of an atomically sharp cleavage crack. There is no indication
that the mechanism of subcritical crack growth in calcite is
dependent on the water concentration.
[34] Subcritical crack measurements such as those presented here represent an opportunity for measuring the
surface energies of different minerals in a wide range of
chemical environments. Improved experimental control and
higher optical resolution could yield better constrained estimates of G0. These experimental results would be highly
valuable for the validation of numerical simulations. More
knowledge on the surface energies of different minerals could
also provide us with new insight on how fluids move and
reactions take place in a heterogeneous rock.
[35] Acknowledgments. We thank Paul Meakin and Francois Renard
for helpful comments to the manuscript. Comments made by the two
reviewers significantly improved the paper. This study was supported by
a Center of Excellence grant from the Norwegian Research Council to
the Physics of Geological Processes (PGP) Center.
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Paper IV
Subcritical crack propagation driven by crystal growth
Anja Røyne, Paul Meakin, Anders Malthe-Sørenssen, Bjørn Jamtveit and
Dag Kristian Dysthe
submitted to PRB
Subcritical crack propagation driven by crystal growth
Anja Røyne,∗ Anders Malthe-Sørenssen, Bjørn Jamtveit, and Dag Kristian Dysthe
Physics of Geological Processes (PGP), University of Oslo, P.O. Box 1048, 0316 Oslo, Norway
Paul Meakin
Physics of Geological Processes (PGP), University of Oslo, P.O. Box 1048, 0316 Oslo, Norway
Carbon Resource Management Department, Idaho National Laboratory, USA and
Institute for Energy Technology, Kjeller, Norway
(Dated: April 12, 2011)
Crystals that grow in confinement may exert a force on their surroundings and thereby drive crack
propagation in rocks and other materials. We describe a model of crystal growth in an idealized
crack geometry in which the crystal growth and crack propagation are coupled through the stress
in the surrounding bulk solid. Two non-dimensional parameters, which relate the kinetics of crack
propagation and crystal growth to the supersaturation of the fluid and the elastic properties of the
surrounding material, determine whether dynamic or subcritical crack propagation will take place.
I.
INTRODUCTION
Crystallization in pores can generate stresses sufficient to cause damage in a wide range of porous materials, including rocks, masonry, and concrete. It is also
believed to be the principal mechanism driving weathering related to freeze-thaw cycles [1, 2] or infiltration of
salt-containing fluids [3, 4], and it may also cause vein
formation [5, 6] and grain fragmentation in rocks [7].
While a large body of work has focused on damage caused by crystal growth in pores [8], and an equilibrium thermodynamic framework for the bulk deformation that this process may lead to has been established [9], it is still not fully understood how crystal
growth controls fracture propagation in the surrounding medium. Subcritical crack growth plays a key role
in fracture development when stresses are generated
slowly [10, 11], and subcritical crack growth coupled
to water transport through a partially frozen rock has
proved to be a successful model for fracture propagation during freezing [1].
In this paper, we use an idealized fracture model to
analyze the coupling between stress generation by crystal growth from a supersaturated solution and subcritical crack propagation. This provides a basis for understanding the formation of crack networks driven by
crystal growth and the associated force of crystallization.
II.
THEORETICAL BACKGROUND
A.
Stress due to crystal growth
The thermodynamic equilibrium of a nonhydrostatically stressed solid in contact with its
∗ anja.royne@fys.uio.no
solution has been thoroughly established [12–14],
although it has not, to date, been firmly demonstrated
experimentally [15]. In the absence of stress, the excess
chemical potential ∆µ driving crystal growth is given
by ∆µ = kT ln β, where k is the Boltzmann constant,
T is the temperature and β is the supersaturation
defined as the ratio between the ion activity product in
solution and the solubility product of the solid. It can be
shown that a surface normal stress σ reduces the excess
chemical potential to [13]
∆µ = kT ln β − σ v0 ,
(1)
where v0 is the specific volume (per particle) of the mineral corresponding to the precipitation reaction used to
calculate the supersaturation. Consequently, the equilibrium normal stress σc , corresponding to zero crystal
growth, is given by [14]
σc =
kT ln β
.
v0
(2)
This approximation is valid for crystal sizes above ≈
1µm, below which σc is lowered by the effect of curvature [16].
In order for a growing crystal to perform mechanical work by displacing a confining wall in the direction
of normal load, a liquid film must be present between
the crystal face and the confining surface through which
ions are supplied to the growing crystal surface [17–
19]. The normal stress between the crystal face and the
confining wall is transmitted by the disjoining pressure
in the liquid film, which increases with decreasing film
thickness [20] and becomes significant when the thickness of the film is less than about 10 nm [21]. Disjoining pressures are well described by colloidal theory and
have been measured for a range of materials and fluid
environments [22].
The growth velocity of a confined crystal face subject
to a normal load has not been measured experimentally.
However, growth kinetics are known to be rate limiting for the inverse process, pressure solution, for quartz
2
[23] and if growth kinetics are rate limiting, the supersaturation β will be constant across the confined crystal
surface. A good first order approximation is then (using standard crystal kinetic theory [24]) that the rate of
crystal growth is proportional to the excess chemical potential. This gives
dw
= Kv0 (σc − σ ),
dt
(3)
where K is a kinetic constant and dw
dt is the growth velocity of one face of a crystal of thickness 2w when that face
is subject to a normal stress σ. It has also been shown
[18] that, close to equilibrium, the surface velocity varies
linearly with stress for diffusion limited growth. We will
therefore assume that the rate of crystal growth is given
by equation (3) in our model.
If the stress on the crystal surface is not uniform,
then according to equation (3), portions of the surface
at which the stress is lower than the mean surface stress
will experience a higher growth velocity, while areas
where the stress is higher will grow more slowly. As
a consequence, the distance to the confining wall and
subsequently the surface normal stress will tend to be
uniform across the confined crystal surface.
Studies of the influence of mechanical stress on the
growth and dissolution rates of crystals have also shown
that variation in the growth/dissolution rate is due primarily to the change in free energy of the stressed surface [25]. It follows that, apart from differences due to
crystal anisotropy, the kinetic constant K is the same
for both the stressed and unstressed crystal faces. The
growth rate of the stress free face for a crystal of radius
a is then
da
= Kv0σc .
dt
B.
where ν0 = kT /h is a characteristic lattice vibration frequency, a0 is a characteristic atomic spacing, ∆F is the
quiescent value of the energy barrier, and α is an activation area [27]. As G increases, a second regime with a
close to constant crack growth velocity is often encountered. Here, the rate of crack growth is limited by the
rate of transport of active species to the crack tip. In
the third regime, the crack velocity increases sharply as
G → Gc . Here we consider only the first regime of crack
growth, described by equation (5).
Experimental data on subcritical cracking in rocks is
most commonly fitted to the empirical Charles’ law [10],
which expresses the crack velocity as a power of the
stress intensity factor. Using this model instead of equation (5) does not qualitatively change our results.
III.
The conceptual model used in this work consists of
a crystal growing from a supersaturated solution in the
aperture of a fluid-filled penny-shaped crack (Figure 1)
of radius c and maximum opening 2w. The crack is connected to a larger fluid reservoir, so that the supersaturation in the fluid is constant during crystal growth.
The crystal is assumed to nucleate in the center of the
crack, and to grow in such a manner that the side faces
are stress free while there is a uniform normal stress σ on
the crystal faces that are separated from the crack walls
by a thin fluid film.
When a penny-shaped crack is subject to a radially
symmetric stress distribution σr (r), where r is the radial
coordinate, then the stress intensity factor at the crack
tip is given by [26]
(4)
Subcritical crack growth
The stress field at the tip of a crack may be characterized by the energy release rate G , which is a function
of the applied stress, crack length and crack geometry
[26]. When G = G0 , the elastic energy released during incremental crack growth exactly matches the incremental surface energy of the material under the prevailing environmental conditions [11], and the fracture is
in thermodynamic equilibrium. If a stress sufficient to
make G exceed a critical value, Gc , is applied sufficiently
rapidly to the fracture, the fracture will propagate at
a velocity which approaches the Rayleigh wave velocity in the material. If, on the other hand, G is smaller
than Gc but larger than G0 , much slower subcritical crack
growth will take place. Using the theory of thermally activated processes, the crack velocity can be described by
the equation
dc
∆F
α (G − G0 )
= 2ν0 a0 exp −
sinh
, (5)
dt
kT
kT
THE MODEL
2
KI = √
πc
Z c
0
rσ (r)
√
dr.
c2 − r2
In the model, the stress distribution is given by
σ
0≥r≥a
σr (r ) =
0
r>a
(6)
(7)
which gives
2γ √
K I = √ σ c,
π
where
γ = 1−
r
1−
a 2
c
(8)
.
(9)
The wall displacement wr (r) of a penny-shaped crack
subject to the stress distribution described by equation
(7) is given by [28]
r
r 2
wr (r )
4 =
σ γ 1−
+
0
c
πE
c
a
[E (r/ a) − E (arcsin (r/c) , a/c)] , (10)
c
3
where E0 = E/(1 − ν 2 ) under plane strain conditions,
E (φ, k) is the second elliptical integral and E (k) is the
complete second elliptical integral. Since the rate of
r
opening dw
dt during crystal growth and crack extension is smallest in the center of the crack, equation (10)
can be solved for the maximum crack opening w where
r = 0. In this case, k = 0 which gives E(0) = π2 and
E(φ, 0) = arcsin(φ). The displacement in the crack center is then given by
w
4
=
σ (γ + ψ)
c
π E0
(11)
with ψ = ( a/c) [π /2 − arcsin( a/c)].
Using the relationship between the energy release rate
G and the stress intensity factor K I [26],
G=
K I2
,
E0
(12)
σ
2w
c
a
FIG. 1: Conceptual model. A crystal (grey) is growing from
a supersaturated solution (hatched) in a penny-shaped crack
with radius c and maximum opening 2w. The crystal is located
in the center of the crack and has radius a and thickness 2w,
which matches the aperture of the crack (the fluid film thickness is assumed to be negligible). The growing crystal exerts a
normal stress σ on the crack walls, which is transmitted by the
disjoining pressure in the confined liquid film.
the relationship
π E0 w2
G=
4 c
γ
γ +ψ
2
(13)
can be derived using equations (8) and (11).
For a crack subject to an internal fluid pressure p, the
energy release rate is given by
G=
4 2
p c.
π E0
Therefore an equivalent pressure
w
γ
p=
,
c γ +ψ
(14)
(15)
which corresponds to the fluid pressure which would be
required to achieve the same value of G as that observed
in the system, can be assigned. Note that σ = p when
a = c.
which can be solved to yield (with σ (0) = 0)
σ = σc 1 − e−t/t0 ,
0
. This implies that as long as the crack
where t0 = π E4c0 Kv
0
is stationary, the normal stress on the crack walls will
approach σc ; this will happen more slowly if the crystal
radius a takes a long time to reach c0 . Often the "crystallization pressure" is simply assumed to be a function
of supersaturation. However, equation (16) indicates
that this pressure only asymptotically approaches σc , at
a rate which is determined by t0 .
Crack propagation is initiated when
G , given by equaq
π E G0
tion (13), reaches G0 . If σc <
4c0 , this limit will
never be reached and the crack will not grow. The critical crack radius, cc , is thus given by
0
cc =
IV.
ANALYSIS
As the crystal nucleates and begins to grow, the energy release rate is initially not large enough for the fracture to propagate. The evolution of stress σ and equivalent pressure p are determined by the crystal growth
kinetics and fracture compliance. If the crystal rapidly
grows to fill the crack, the increase in σ = p if a = c = c0
(where c0 is the initial crack radius) is given by equation
(11) so that
dσ
π E0 dw
=
.
dt
4c0 dt
Substituting the expression for
this equation gives
dw
dt
from equation (3) into
dσ
π E0 dw
π E0 Kv0
=
=
(σc − σ ),
dt
4c0 dt
4c0
(16)
π E0 G0
.
4σc2
(17)
c
The ratio tc = π E4c0 Kv
provides a natural time scale
0
for the system, and the dimensionless variables t∗ =
t/tc , c∗ = c/cc , w∗ = w/cc , a∗ = a/cc , σ ∗ = σ /σc and
p∗ = p/σc completely characterize the model. Introducing the dimensionless groups
8ν0 a0
∆F
α G0
4σc
A=
exp −
,B =
,C =
,
0
π E Kv0
kT
kT
π E0
the system is described in dimensionless form by the
equations
σ∗ =
1 w∗ 1
,
C c∗ γ + ψ
(18)
p∗ =
1 w∗ γ
,
C c∗ γ + ψ
(19)
4
0
Asinh B(c∗ p∗2 − 1)
if c∗ p∗2 ≤ 1
if c∗ p∗2 > 1
dw∗
= C (1 − σ ∗ ),
dt∗
da∗
=
dt∗
C
if a∗ < c∗
if a∗ = c∗
(20)
1
0.8
II
(21)
σ*, p*
dc∗
=
dt∗
0.6
(22)
0.4
Crack propagation due to a filled pore (a∗ = c∗ ) is
completely described by the parameters A and B, which
determine how the crack growth and crystal growth kinetics are related to the elastic moduli of the material.
The dimensionless parameter C, which is a function of
the supersaturation, only controls crystal growth in the
unloaded direction, and therefore determines the maximum crack velocity at which the crystal continues to
fill the crack. It also determines how long it takes to
build up the initial equivalent pressure p∗ in the crack
and subsequently the time at which the fracture propagation is initiated.
In a filled pore, the time required for the crack propagation to initiate, t∗ = t∗1 , can be found by solving the dimensionless form of equation (16) for c∗σ ∗2 = 1, which
gives
s !
1
t∗1 = −c∗0 ln 1 −
.
(23)
c∗0
0.2
dc∗
dt∗
This function has a minimum when c∗0 ≈ 2. Furthermore, as c∗0 → 1, t∗0 → ∞, because small cracks require
an effective pressure very close to σc in order for p∗σ ∗2 to
reach the value of 1, which is required for growth (equation 20). The effect of smaller C is to increase t∗1 . The
effect is greater for larger c∗0 , so that the minimum of t∗1
is shifted to lower c∗0 .
V.
NUMERICAL RESULTS
The model was investigated by solving the set of
equations (18)-(22) numerically for a range of parameters A, B, C and c0 . Figure 2 shows the resulting time
evolution of the stress and equivalent pressure in a single crack. Initially, the stress σ ∗ is higher than the effective pressure p∗ due to the small area over which the
on the crystallization pressure is applied. As the crystal
grows unrestricted in the radial direction, the effective
pressure increases more rapidly than σ ∗ because stresses
that are applied closer to the crack tip contribute more
to the stress intensity factor than stresses applied in the
center of the crack (see equation (8)). At point I, the crystal has grown to fill the fracture (a∗ = c∗ ), and subsequent crystal growth results in an increase in p∗ = σ ∗
as described by equation (16). Crack propagation is initiated at point II, after which p∗ and σ ∗ fall off from
0
III
I
0
5
10
15
20
t*
FIG. 2: Time evolution of the crystal stress σ ∗ (dashed blue
line) and effective pressure p∗ (solid line) for A = B = C = 1
and c∗0 = 2. The red dashed line shows the asymptotic increase
in stress which is predicted for a stationary, filled crack according to equation (16). The circles show: I. how σ ∗ = p∗ increases according to equation (16) when the crystal has grown
to fill the crack, so that a∗ = c∗0 ; II. the initiation of crack prop∗
dc∗
agation; and III. the point where da
dt∗ < dt∗ so that the crystal
∗
∗
no longer fills the crack, and σ > p .
the exponential curve and decrease as the effect of increased fracture compliance becomes dominant. After
point III has been reached, the velocity of the crack tip is
faster than the growth rate of the free crystal face, so that
a∗ < c∗ . This increases σ ∗ and consequently decreases
the rate of crystal growth, which makes the rate of crack
propagation slower than it would have been if the crack
had remained filled.
The time at which fracture propagation is initiated,
and whether the crystal grows to fill the crack before it
starts to propagate, is determined by the initial crack radius c∗0 , as shown in the upper panel of figure 3.
d2 p∗
After the point at which dc∗2 > 0, the dimensionless
equivalent pressure p∗ and the normal stress σ ∗ do not
depend on the initial dimensionless crack radius c∗0 , but
are uniquely defined by c∗ . This is evident in the lower
panel of figure 3. The implication is that after an initial
transient, memory of the initial conditions is lost, so that
d2 p∗
a crack for which dc∗2 > 0 could have started out at any
c∗0 < c∗ .
It is reasonable to assume that the subcritical crack
growth model (equation (5)) is valid for c∗ p∗2 ≤ 2. If
c∗ p∗2 exceeds a value of 2 within a short time or before the crack has grown appreciably, the system may
enter the dynamic crack propagation regime, where inertial effects and dissipative mechanisms not considered
in this model may become important. At c∗ p∗2 = 2,
dc∗
B
dt∗ ≈ Ae . This means that the combination of paramB
eters Ae and C should determine whether the crack re-
5
≥2
2.0
1
1.9
p*,σ*
0.8
1.5
0.6
0.4
0
5
10
15
20
0.5
1.5
1.4
0
1
0.8
p*,σ*
log(C)
1.6
t*
1.3
1.2
-0.5
0.6
1.1
0.4
-1.0
-2.0
0.2
0
1.7
1.0
0.2
0
1.8
-1.5
-1.0
-0.5
0
0.5
1.0
log(AeB)
1
2
3
4
5
6
7
8
c*
FIG. 3: Evolution of crystal stress σ ∗ (dashed line) and effective
pressure p∗ (solid line) as a function of time t∗ (upper panel)
and crack radius c∗ (lower panel) for cracks of initial radius 1.2
(blue), 2 (green) and 5 (red) cc . In this simulation, A = 10, B =
2 (so that Ae B = 73.9) and C = 10−0.5 .
FIG. 4: The value of c∗ p∗2 reached at c∗ = 1000 as a function
of AeB and C for c∗0 = 2. For c∗ p∗2 > 2 the model is out of
its range of validity, and the system may enter the dynamic
fracture growth regime. The range of parameters are chosen
to show the crossover from a power law dependence on C at
low Ae B and C to a regime which is insensitive to C (because
C is large enough to maintain a = c at any crack velocity of interest), and the crossover from dynamic to subcritical cracking
as Ae B increases.
mains subcritical.
Figure 4 shows a contour plot of the value of c∗ p∗2
that is reached when c∗ = 1000 for a range of values of
AeB and C. The simulations were stopped if c∗ p∗2 > 2.
The cutoff at c∗ = 1000 was chosen because very large
cracks are expected to interact with other cracks in realistic systems, and we do not include the effects of
crack-crack interactions. The interpretation of figure 4
is that blue colors in the plot correspond to more "subcritical" behaviour. If AeB is high, crack propagation is
sufficiently fast compared with the rate of stress generation due to crystal growth that even if the crystal fills
the crack at all times, the system can approach a steady
state in which stress generation by crystal growth is almost balanced by crack propagation. For low AeB , fast
crystal growth and slow crack propagation allows c∗ p∗2
to reach a large value before the stress is relieved by
crack propagation, and the system may enter the dynamic fracture regime. The effect of lowering C is to
increase the range of AeB at which crack propagation
is subcritical. This happens because the ratio a∗ /c∗ becomes smaller as the crack is propagating, so that the
effective pressure p∗ decreases faster than it would for a
filled crack. This has a stabilizing effect on the value of
c∗ p∗2 as the crack grows, which allows longer periods
of subcritical crack growth (because the fracture acceleration never goes completely to zero, all cracks in this
model will become unstable at long enough times).
VI.
APPLICATIONS
In nature, there are huge variations in both crystal
growth and fracture propagation kinetics. Data available in the scientific literature [6, 27, 29] yield values for
AeB which span almost 20 orders of magnitude, ranging
from 4.2 × 10−5 for gypsum growing in glass, through
3.3 for calcite in glass to 1.1 × 1015 for quartz in mica.
Based on Figure 4, we would expect quartz to produce
subcritical fractures in mica under all conditions (the
model for mica from [27] is only valid for c∗ p∗2 < 1.2,
but this is irrelevant at such a high value of AeB because
c∗ p∗2 approaches steady state at a value very close 1).
Calcite growing in glass would cause it to break dynamically after some time provided that the supersaturation
is high enough. In order for gypsum to produce subcritical fractures in glass, the supersaturation would need
to be very low.
It has been shown that statistical properties of fracture networks may be strongly influenced by the degree
of subcritical crack propagation [30, 31]. In light of the
current results, this would imply that the effect of fracture propagation in rocks due to crystal growth may be
highly variable depending on the mineralogy and chemistry of the host rock and growing minerals.
6
VII. CONCLUSION
To summarize, we have shown that the coupling between crystal growth in a crack and subcritical crack
growth has implications for the evolution of fractures
when fracture propagation is driven by growing crystals. The dimensionless parameters AeB (which relates
the crack growth kinetics and the crystal growth kinetics) and C (which characterizes the degree of supersaturation) determine whether cracks will propagate dynamically or subcritically. For a wide range of parameters, subcritical crack growth cannot be ignored, and
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