Eur. Phys. J. Special Topics 178, 123–132 (2009)
c EDP Sciences, Springer-Verlag 2010
DOI: 10.1140/epjst/e2010-01185-3
THE EUROPEAN
PHYSICAL JOURNAL
SPECIAL TOPICS
Regular Article
Thermodynamics of stressed solids: Slow
deformation and roughening of material interfaces
L. Angheluta1 and J. Mathiesen1,2,a
1
2
Physics of Geological Processes, University of Oslo, Oslo, Norway
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen O, Denmark
Abstract. At every turn in nature we are confronted with complex patterns.
Patterns often formed in multiphase systems by an intricate dynamics of mass
transport, e.g. diffusion and/or advection, and mass exchange between individual
phases. Here we consider instabilities of phase boundaries in idealized stressed
multiphase systems. Specifically, we study the growth of small perturbations of
surfaces by considering mass transport from regions, where the stress and chemical
potential is high, to surrounding regions where the stress and chemical potential
is low. We present a linear stability analysis for various stress configurations and
their corresponding stability diagrams.
1 Introduction
Pattern formation in multiphase systems is a central subject in the research on nonlinear dynamics. The interest has been sparked by the ubiquitous appearance of spectacular patterns on
all scales in nature. A common goal for research on pattern formation has been to illuminate
the underlying fundamental mechanisms and the coupling between them. Here we shall demonstrate in few model systems how basic physical principles can explain complex morphologies
formed in deformable and reactive materials. The evolution of solid surfaces is usually governed
by stress or thermally activated physical processes. It involves mechanical deformation often
coupled with chemical alteration e.g. surface growth by dissolution and precipitation.
A classical example of surface instability is thermal grooving triggered by evaporation and
condensation [11]. At surfaces of stressed solids, mass is typically transported by surface diffusion from regions of relatively high stress (high chemical potential) to regions of low stress (low
chemical potential). This process corrodes the surface and gives rise to the Asaro-Tiller-Grinfeld
instability [4,6]. Often the mass transport is mediated by an interstitial fluid via dissolution in
stressed regions and the subsequent precipitation at free sites, a process also known as “pressure solution” [17]. Sutured grain boundaries in sandstone [14] is one example where undulating
surfaces are believed to be generated by pressure solution.
From a modeling perspective, we consider interfaces as transition regions over which material
properties (densities, rheological properties, stress, velocity) undergo steep gradients. In the
limit where the thickness of these regions is much smaller than any other relevant length scale
of the system, one may consider the internal structure of a material to be represented by
continuous regions connected at discontinuity surfaces or sharp interfaces. The term interface
is used throughout this text to denote borders that separate different phases or individual but
homogeneous regions. Below, we shall present in details the stability analysis of various sharp
interfaces in two-phase systems. We show that the interfacial stability depends on the stresses
in the system, its rheology (viscous, elastic) and other discontinuities in material properties.
a
e-mail: mathies@nbi.dk
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The change in field variables, as small undulations of size h(x) develop along an otherwise
flat interface, is conveniently calculated using linear perturbation theory. When the amplitude
is small enough, the new fields may be written perturbatively as an expansion around the
values of the flat interface. Formally, this is written as
U (x, y) = U (0) (x, y) + U (1) (x, y) + O(2 ),
(1)
where U (0) (x, y) is the solution to a planar interface and U (1) (x, y) is a first order correction
accounting for small undulations. Evaluated at a point on the interface y = h(x), the expansion
becomes
U (x, h(x)) = U (0) (x, 0) + h(x)∂y U (1) (x, y)|y=0
+ U (1) (x, 0) + O(2 ).
(2)
In Section 2 we summarize essential parts of interfacial perturbation schemes using as an example the classical problem of viscous fingering. This instability has important similarities to
instabilities of liquid-solid (Section 3) and solid-solid stressed interfaces presented in Section 4.
2 Viscous fingering
Viscous fingering is realized when a less viscous fluid displaces a more viscous fluid contained
in the narrow gap between two glass plates (a Hele-Shaw cell). When the less viscous fluid is
inserted through a gap at one side of the glass plates, a finger-like pattern is formed at the
interface separating the two immiscible fluids. The fluid velocity v of the displaced fluid in a
Hele-Shaw cell satisfies Darcy’c law, where p is the pressure.
Namely, a less viscous fluid (e.g. water) migrates into a more viscous fluid (e.g. oil) by
developing fingers which move ahead of the interface at various speeds. Contrarily, the interface
remains planar when the water is displaced by the oil. This instability is known as SaffmanTaylor instability [15] or viscous fingering.
The governing equations for an incompressible viscous flow are given by Stokes equations
combined with the continuity equation,
µ∇2 v − ∇(p + ρgz) = 0
∇2 p = 0,
(3)
(4)
where v = (vx , vy , vz ) is the velocity vector field which is a solenoidal field when the fluid density
ρ is constant. µ is the kinematic fluid viscosity. The gravitational field g is pointing downwards
opposite with respect to the vertical z-axis. The pressure field is denoted by p(x, y, z, t). When
the flow is confined between two parallel plates in the (x, z)-plane, the velocity in the y-direction
vanishes (see Fig. 1). For a single phase flow, we can assume a homogeneous flow in the xdirection and therefore obtain essentially an uniaxial flow in the z-direction with vz (y, t).
For small perturbations, it is enough to study the stability of an arbitrary mode
hk (t) exp(ikx), where hk (t) ∼ exp(ωt), i.e. a mode with a growth rate ω. If ω > 0 the mode
will be unstable and have an initial exponential growth rate. Inserting this perturbation in the
governing equation we obtain the usual dispersion relation for Saffman-Taylor fingering
gH 2 ρ1 − ρ2
µ1 − µ2
ω
=
+ V (0)
.
k
12 µ1 + µ2
µ1 + µ2
(5)
From the above relation, we see that the stability of the interface depends on the direction
of the flow (by the sign of V (0) ) and the relative density ρ1 − ρ2 and viscosity µ1 − µ2 . The
interface becomes unstable when the flow is under gravity with the denser fluid on the top of a
lighter fluid or when the flow is upwards with a positive V (0) and the less viscous fluid at the
bottom migrating into the more viscous fluid above it. The growth rate is linearly proportional
to the wavenumber, and therefore there is no mode selection, e.g. the interface is either stable
Order, Robustness and Instabilities in Complex Systems
125
Fluid 1
g
H
Fluid 2
z
x
y
Fig. 1. Basic setup used to study the Saffman-Taylor instability.
or unstable at all length scales. In the presence of surface tension, there will be an additional
term in the growth rate related to the curvature and surface tension and one can show that this
term√is stabilizing the interface perturbations with wavelengths smaller than a critical value
2πH σ[12V (0) (µ1 − µ2 ) + (ρ1 − ρ2 )gH 2 ]−1/2 , with σ being the surface tension [15].
Some of the features of the Saffman-Taylor instability are recovered when two stressed linear
elastic solids are in contact and when the solids are allowed to exchange mass along the contact
interface. That is, the solids cannot flow and deform like the viscous fluids, however if mass
can be transferred across contacts, the contact interface evolves and in certain cases becomes
unstable.
3 Morphological evolution of a liquid-solid interface
The dissolution and precipitation of a solid surface in contact with a surrounding fluid at
rest may induce a net transport of mass along the surface and in certain cases even lead to
morphological instabilities. Diffusion-controlled growth by material deposition or heat flow give
rise to the Mullins-Sekerka instability [12] which has been studied e.g. in systems of growing
ice crystals in contact with undercooled water [8]. In the nonlinear regime, the same instability
may give rise to side-branching and tip-splittings [9]. Like the thermal fluxes, stress can bring
the solid out of chemical equilibrium with the surrounding fluid and thus induce local changes
in the interface morphology. Surface corrosion controlled by stress variations at a liquid-solid
interface has been studied in [4]. The linear instability caused by stress corrosion was discovered
independently by Asaro and Tiller in [4], Grinfeld in [6] and is known as the ATG-instability.
Nonlinear stability analysis combined with numerical simulations reveal that the interfacial
shape evolves into cusp-like singularities when higher order terms in the amplitude expansion
are retained [18,19]. The ATG instability has been studied extensively using diffuse interface
models, e.g. [7,20]. Hereby, we shall present its equivalent formalism for sharp interfaces.
When a stressed solid is in contact with a saturated fluid, the chemical potential at the solid
surface becomes a function of stress on the form [16]
µ(s) = F(s)V − σnn (s)V,
(6)
where s is a parameterization of the surface, V is the molar volume of the solid component, F is
the Helmholtz free energy density, σnn is the normal component of the stress vector. For a free
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The European Physical Journal Special Topics
Fig. 2. Basic setup of two stressed linear elastic solids in contact. The solids are allowed to exchange
mass across the interface such that the interface evolves in time. We show that depending on the elastic
parameters of the solids the interface may be either morphologically stable or unstable.
surface σnn = 0, while if the surface is in contact with the fluid then σnn = −p with p being the
hydrostatic pressure in the fluid. Unless the normal stress is small or vanishes, the Helmholtz
free energy is in general small compared to the last term and therefore, to the leading order
approximation, can be neglected.
The surface gradients in the chemical potential produce a drift of surface atoms with a flux
given by [11]
Ds a ∂µ(s)
,
(7)
J =−
kT ∂s
where a is the surface density of atoms, Ds is the surface diffusion coefficient and kT is the
Boltzmann’s constant times temperature. Note that depending on the system, Ds could represent diffusion along the solid surface or along a thin fluid film in contact with the interface.
If we take the divergence of the mass flux, we achieve an expression for the local change in
number of atoms per unit area per unit time which is directly related to the normal velocity of
the interface via
Ds aV ∂ 2 µ(s)
∂J
=
V = −V
.
(8)
∂s
kT
∂s2
For small undulations on a planar interface, the above equation can be approximated by
∂h(x, t)
∂2
= M 2 µ(h(x, t)),
∂t
∂x
(9)
s aV
where M = DkT
is a positive-defined mobility coefficient. The above equation describes the
morphological evolution due to mass transport by diffusion. This is an alternative mass transport mechanism to the Mullins-Sekerka instability where the mass exchange takes place across
the interface and thus is driven by the jumps in the chemical potential,
∂h(x, t)
= M µ(h(x, t)) .
∂t
(10)
Previous studies of the ATG instability treat the limit where the fluid is at rest and thus
possible shear stresses at the interface are neglected. However, many natural interfaces are
able to sustain shear stresses. In a recent work, shear stresses induced by a flowing fluid are
shown to have a stabilizing effect [3] on the surface growth. In the next section, we discuss the
morphological evolution of stressed solid-solid interface.
Order, Robustness and Instabilities in Complex Systems
127
4 Morphological stability of two contacting solids
4.1 Phase transformation kinetics in one dimension
It is illustrative to consider the dynamics in a one-dimensional system of two linear elastic solids
separated by a single interface. The solids are allowed to exchange mass at a rate determined
by their chemical potential. Assume that a force σ is applied at one boundary of the system and
that the other boundary is kept fixed. Each solid phase, i, is characterized by Young’s modulus
Ei (i = 1, 2), undeformed density ρ0i and length L0i . When the external force is applied, the
system deforms to a length Li = L0i (1+σ/Ei ). Similarly, the density is changed to ρi = ρ0i L0i /Li .
We shall now follow the analysis presented in [2]. For the solid i, the total specific free energy
is given by
1
σ2
.
(11)
fi =
2 ρi (Ei + σ)
In this simple setup we do not allow new phases to nucleate within the solids and we only
consider the propagation of a single interface separating the two solids. Moreover, we assume
that the system is isothermal and that there is no bulk diffusion of mass. The interface moves
as one phase, slowly transforms into the other and an amount ρ1 dL1 , of solid 1 is replaced by
an amount ρ2 dL2 of solid 2 such that the total mass is conserved. The phase transformation is
assumed to be irreversible and to occur on time scales that are much larger than the time it
takes for the system to relax mechanically under the deformational stresses.
The local mass exchange rate Q is proportional to the jump in the Gibbs potential across
the interface, i.e. Q ∼ f − σ/ρ, where 1/ρ is the mass specific volume and the jump condition,
a = a1 − a2 , is defined as the difference in the quantity a when approaching the interface from
each material phase. In that way, we can write the change in mass of e.g. solid 1 as
2
2
σ
σ
σ
σ
−
+ 0 ,
(12)
ṁ1 = −K
=K
2ρ0 E
ρ
2ρ0 E
ρ
with K > 0 being some dimensional constant of proportionality. In most cases, the contribution
from the jump in the elastic energy density will be small compared to the contribution from
the work term σ/ρ (because σ/E 1, within the linear elasticity regime). The change in the
total length will in general follow the sign of the stress
1
ρ1
L̇ = L̇1 1 −
= ṁ1
ρ2
ρ
2
1
σ
σ
σ
+
+
.
=K
2Eρ0
ρ0
ρ0
Eρ0
If the densities in the undeformed states are identical, ρ01 = ρ02 , the change in the total length
is given by
2
σ3 1
L̇ = K 0
,
(13)
2ρ E
whereas a jump in the referential densities (ρ01 = ρ02 ) will result in a work term given by
L̇ ≈ Kσ
1
ρ0
2
.
(14)
In summary, a compressional load will favor growth of the dense phase at the expense of the
less dense phase (if the two phases have the same Young’s modulus). If the two phases have
the same density, the soft phase grows at the expense of the hard phase, such that overall the
system responds to the external force by shrinking. The one-dimensional model cannot predict
the morphological stability of the propagating phase boundary in two dimensions.
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The European Physical Journal Special Topics
4.2 Stability analysis in 2D
Under the assumption that the system is instantaneously relaxing to its equilibrium configuration, we consider the steady state of the momentum equations for both elastic solids. The
stress of an elasto-static two-dimensional configuration is conveniently calculated using the Airy
stress function, U (x, y) [13], which satisfies the biharmonic equation ∆2 U = 0. Here, we have
∂2
∂2
introduced the Laplace operator ∆ = ∂x
2 + ∂y 2 . Once the stress function has been found, the
stress tensor components readily follow from the relations
σxx =
∂2U
,
∂y 2
σyy =
∂2U
,
∂x2
σxy = −
∂2U
.
∂x∂y
(15)
Here we solve the elasto-static equations with the boundary conditions of a normal load applied
in the y direction at infinity, i.e. σyy → −|σ∞ | < 0 and σxy = 0 for y → ±∞. The continuity
of the stress vector across the interface follows from force balance. In addition we require that
the displacement in the x-direction vanish for ux (±∞, y) = 0.
For a flat interface, the stress field is homogeneous in space. This implies that the Airy stress
function is quadratic in x and y, with coefficients determined by the boundary conditions. With
the boundary conditions specified above, the stress function for the i-th phase can be written
in the form
|σ∞ | 2
(x + νi y 2 ),
Ui (x, y) =
(16)
2
where νi is the Poisson’s ratio of phase i.
From this stress function we can calculate the Gibbs potential which in the case of dissimilar
phases is discontinuous across the interface. The velocity of the phase transformation readily
follows from the potential
1
1
1 − 3ν2
|σ∞ |2 1 − 3ν1
(0)
(0)
0
(0)
= M |σ∞ |
V
= M F /ρ + W
− 0 −
− 0
.
ρ01
ρ2
4
ρ01 G1
ρ2 G2
(17)
The superscript of the free energy density and the work term refers to an unperturbed interface.
From the above equation, we see that the direction of propagation depends on the jump in the
material properties in a similar way to Saffman-Taylor instability. When the referential densities
are different, the above expression is dominated by the first term and predicts that the phase
transformation is directed from the denser phase into the lighter phase. In the case where the
referential densities are the same, the second term becomes the leading order and, for ν < 1/3,
gives a reverse propagation from the softer phase (higher shear modulus) into the harder phase
(lower shear modulus).
In the case of an arbitrarily shaped interface separating the two phases, the analytical
solution to the stress field is in general far from trivial. In-plane problems can in some cases be
solved using conformal mappings or perturbation schemes [5,10,13]. Here, we solve the stress
field around a small undulation of flat interface employing the linear perturbation scheme
introduced above. Using the linear stability analysis, we now study the growth of an arbitrary
harmonic perturbation with wavelength k, i.e. h(x, t) = Aeωt cos(kx) with A 1. The Airy
stress function can be written as a superposition of the solution to the flat interface and a small
correction due to undulation, U (x, y) = U (0) (x, y) + U (1) (x, y), where U (1) (x, y) is determined
from the interfacial constraints of continuous stress vector and displacement field. When the
wave number k is much smaller than the cutoff introduced by the surface tension, we obtain
the following expressions for the Airy stress functions
(1)
U1 (x, y) =
(1)
U2 (x, y) =
−|σ∞ |h(x) exp(−ky)(α1 y + β)
k(G2 κ1 + G1 )(G1 κ2 + G2 )
|σ∞ |h(x) exp(ky)(α2 y − β)
k(G2 κ1 + G1 )(G1 κ2 + G2 )
(18)
Order, Robustness and Instabilities in Complex Systems
129
i
where κi = 3−ν
1+νi , Gi is the shear modulus of phase i and we have introduced the material
specific constants,
α1 = −k(1 − ν1 )(G2 − G1 )(G1 κ2 + G2 )
α2 = k(1 − ν2 )(G1 − G2 )(G2 κ1 + G1 )
and
1 − ν2
1 − ν1
ν1 − ν2
.
− 2G22
+ 4G1 G2
1 + ν2
1 + ν1
(1 + ν2 )(1 + ν1 )
In order to evaluate the jump in Gibbs energy density, i.e. F/ρ0 + W , we need to determine
the stress field around the interface by solving the elastostatic equations. We have that under
plane stress conditions, the local strain energy density can be written on the form
1
ν
2
2
2
(σxx + σyy )2 + 2σxy
F=
+ σyy
−
(19)
σxx
4G
1+ν
β = 2G21
and the work term is defined as
= −σnn ρ−1
W = −σnn ρ−1
i
i,0 (1 + Tr()).
(20)
The trace of strain is given in terms of stress by
Tr() =
1 − 2ν
(σxx + σyy ).
2G(1 + ν)
(21)
Note that we could as well have formulated the problem under plane strain conditions; however,
the generic behavior in both plane stress and strain is the same although the detailed dependence
on the material parameters is altered.
From the Airy stress functions, we then calculate the stress components using Eq. (15) and
find the jumps in the Gibbs energy density from Eqs. (19) and (20). The evolution of the shape
perturbation relative to a uniform translation of the flat interface is a dispersion relation given
as
M F + W − V (0)
ω=
.
(22)
h
Below follows an evaluation of the growth rate for a small harmonic perturbation to a flat
interface. For this perturbation, the general expression for the growth rate follows directly upon
insertion of the Airy functions in Eq. (18) and then in Eq. (22), however, the growth rate is not
easily expressed in a short and readable form and we have therefore limited our presentation
to a few special cases. The growth rate is a function of the six material parameters (νi , µi , ρi )
and the external stress. Naturally, the stability of the growing interface is invariant under the
interchange of the solid phases and correspondingly the region of the stability diagram that we
have to study is reduced.
4.3 First and second order phase transitions
Whenever the system is stressed, only one of the two phases will be stable, i.e. the two phase
system will evolve to a global equilibrium state consisting of a single phase. In the absence of
stress it is possible for two phases to coexist without any phase transformation taking place at
their interface. In the aforementioned 1D model system, the specific Gibbs energy is given by
the stress applied to the system σ,
g(σ) =
σ
σ2
− .
2Eρ0
ρ
(23)
0
−0.
03
−0.01
ρ2, (ρ1 = 1)
0.5
ρ2, (ρ1 = 1)
0.5
2
0.10
1.5
ρ2, (ρ1 = 1)
0.15
µ2,
1
(µ1 = 1)
05
−0
.
7
1.5
0
1
µ2,
(µ1 = 1)
1.5
2
−0.10
0 10
−0.15
−0.20
0.15
0.10
0.05
0
0.0
−0.05
−0.10
2
0
0.
6
1
0.5
0.5
0.00
−0.02
.06
−0
.1
−0
0.02
0.5
4
0.5
1
−0.05
6
−0.0
1
−0.
.14
−0
−0
.04
0
−0.02
.02
−0
0.
0
0.00
−0.02
0.
04
0.00
0
0.08
0.1
2
0.0
2
−0.0
2
ρ2, (ρ1 = 1)
ρ2, (ρ1 = 1)
ρ2, (ρ1 = 1)
ρ2, (ρ1 = 1)
ρ2, (ρ1 = 1)
−0
.06
−0.0
4
.0
4
−0
0.01
5
−0.0
3
−0.01
08
−0.
−0.0
−0
0.10
10
2
1
−0.15
2
1
0.05
0.0
(µ1 = 1)
1.5
0.15
6
0.0
−0.
11
1.5
0.01
µ2,
1
4
0.0
0.5
0.02
0.09
0.5
2
4
0.0
.14
1
−0.10
0.0
6
1.5
0
05
0.05
−0.05
−0.15
−0.2
0.10
0.02
(µ1 = 1)
0.04
2
µ2,
1.5
−0.05
1
(µ1 = 1)
08
0.5
0.05
1
2
.06
−0
.1
−0
0.02
0.03
−0
0.0
4
2
0.10
µ2,
0.10
−0.01
03
−0.
0.01
−0.10
0.00
0.5
06
0.
1
−0.05
0.5
1.5
2
14
−0.1
.2
−0
0.5
0.
1.5
−0.
−0.20
0.00
3
1
8
−0.0
0 05
0.05
0
(µ1 = 1)
−0.15
0.0
−0.04
1.5
07
0.
µ2,
−0.10
1.5
0.0
3
0.0
0.5
0.01
0.5
0.0
5
1.5
0.1
0
1
4
0.0
1
1
(µ1 = 1)
−0.05
0.02
0.00
µ2,
.16
5
0.0
2
0.05
1
.0
−0
ρ2, (ρ1 = 1)
0.5
3
0.0
0
2
1.5
−0
0.5
0.01
−0.02
−0.15
0. 0 4
0.02
−0.
1
−0.
08
−0.
06
−0.0
4
1
−0
.08
1.5
06
−0.
.1
−0
−0
.05
1
(µ1 = 1)
µ2,
−0.02
0.08
.
−0
−0.10
0.0
2
6
0.0
3
0.0
0.5
0 .0 1
0. 0
5
0.5
−0.05
0.00
1
0.0
6
15
1.5
4
0.0
1
−0.0
.03
−0
6
.0
−0
2
0 05
0.05
04 .06
0.
0
1
0.0
4
2
0.10
15
1.5
0.00
2
0
−0.01
0.05
15
1.5
0.
04
2
0.0
−0.0
2
3
0.0
0.01
2
3
The European Physical Journal Special Topics
−0.
05
130
µ2,
1
(µ1 = 1)
1.5
−0.05
−0.10
2
Fig. 3. The rows from top to bottom are stability diagrams with Poisson’s ratio ν2 = 0.25,0.33,0.40,
for solid 2. Note that µ in the figure represents the shear modulus G. From the left to the right, the
columns show stability diagrams computed using for solid 1 Poisson’s ratio ν1 = 0.25,0.33,0.40. Note
that the symmetry is broken since ρ1 = 1 and G1 = 1.
We define a first order phase transformation process when the first derivative of the specific
Gibbs energy with respect to σ is discontinuous at the critical point σ = 0. From the above
relation, we see that this happens when the two phases have different referential mass densities.
By a second order phase transition, we mean that there is a finite jump in the second order
derivative which is related to the discontinuity of the Young modulus, E. We adopt the same
terminology for the interfacial phase transformation in 2D.
In the two dimensional system, we have that for the second order phase transition where
both solids have the same referential densities ρ01 = ρ02 = ρ0 and when the Poisson’s ratios
ν1 = ν2 = ν are identical, the dispersion relation assumes a simple form given by
ω
(3ν − 1)(1 − ν)(G1 + G2 )(G2 − G1 )2
= 0
k
ρ G1 G2 (G1 + G2 κ)(G2 + G1 κ)(1 + ν)
(24)
where κ is the fraction introduced right below Eq. (18) and k the wave number of the perturbation. The expression reveals an interesting behavior where the interface is stable for Poisson’s
ratio less than 1/3 and is unstable for Poisson’s ratio larger than 1/3. Fig. 3 shows stability
diagrams for the specific case where G1 = 1 and ρ01 = 1 (in arbitrary units). The diagonal panels
are calculated for two solids that have the same Poisson’s ratio, i.e. values ν = 0.25, 0.33, 0.40.
The second order phase transition occurs along the horizontal cut ρ02 = 1 and is marked by a
dashed gray line. We observe that ω/k is negative along this line and the interface is therefore
stable. For ν larger than 1/3 (not shown in the figure) the horizontal zero level curve will flip
around and the gray dashed line will then be covered with unstable regions. In order to see this
flip, we expand Eq. (22) around the point (1,1), i.e. in terms of ρ02 − 1 and G2 − 1, and achieve
the following expression for the zero curve
ρ02 ≈ 1 +
(1 − 2ν − 3ν 2 )(G2 − 1)
.
ν(7 + ν)
(25)
Order, Robustness and Instabilities in Complex Systems
0.75
131
(A)
0.5
0.25
0.0
0.75
(B)
0.5
0.25
0.0
-1.0
-0.75
-0.5
-0.25
0.0
0.25
0.5
0.75
1.0
Fig. 4. Simulations taken from [1] of the temporal evolution of solid-solid interfaces for first order
transitions. Panel (A) shows a simulation using ρ1 = 1.0, G1 = 1.0 and ρ2 = 1.05, G2 = 2.0. Both
phases have identical Poisson’s ratio, ν1 = ν2 = 0.45. Panel (B) is a simulation run with densities and
shear modules similar to panel (A) but with a different Poisson’s ratios, ν1 = ν2 = 0.25.
Note that the right hand side is in units of ρ1 . We directly observe that the horizontal zero curve
flips around at the critical point ν = 1/3, which is also observed by following the diagonal in
Fig. 3. In the case when the two solids are identical, i.e. at the point (1,1) in the stability diagram,
all modes will as expected remain unchanged and the interface therefore remains unaltered. The
other parts of the zero levels lead to marginal stability but will in general induce a motion of
the interface with a constant velocity.
We now consider a cut in the stability diagram where the two solids have the same shear
modules, G1 = G2 = G, but different densities (first order phase transition) and Poisson’s
ratios. For different Poisson’s ratios the dispersion relation Eq. (22) becomes
(ν2 − ν1 )(ν1 ρ02 − ν2 ρ01 + 2(ρ02 − ρ01 )G)
ω
=
.
k
4ρ01 ρ02 G
(26)
From this expression we see that the vertical zero line observed in Eq. (24) and in Fig. 3
only exists for identical Poisson’s ratios. When the solids have different Poisson’s ratios, the
separatrix or intersection of the two zero curves located at (1,1) in the diagonal panels of Fig. 3
will split into two non-intersecting zero curves. The off-diagonal panels show stability diagrams
for solids with different Poisson’s ratios.
In general the stability diagram is characterized by four quadrants, two stable and two
unstable, delimited by neutral zero curves. The physical regions would typically correspond
to the quadrants I and III under the assumption that higher density implies higher shear
modulus. In these quadrants the growth rate is typically positive (i.e. the interface is unstable)
except for a thin region at the borderline between a first and second order phase transition, i.e.
when ρ2 ρ1 .
5 Concluding remarks
The linear stability analysis of an interface separating two reactive solids reveals an intricate
stability diagram where the stability strongly depends on the material properties and densities
of the two solids. In Fig 4, we show a figure where we have explored this stability beyond the
linear regime using numerical methods. The numerics is based on solving the bulk elastostatic
equations using the Galerkin finite element method. The discontinuous jumps appearing in
the normal interfacial velocity are computed at the outer border of the interface. In addition,
we have imposed periodic boundary conditions to minimize the possible influence of the finite
system size in the x-direction (parallel to the interface).
132
The European Physical Journal Special Topics
In Fig. 4 we present numerical simulations of the phase transformation kinetics using parameter regions where the interface is either stable or unstable. The simulations represent interface
snap shots of a first order phase transition dynamics. In panel (A), the values of the parameters
were chosen in a region of the stability diagram where the interface is predicted to be unstable
and in panel (B) we have used parameters corresponding to a stable evolution of the interface.
Note that the interface in both cases is moving from the denser phase into the lighter phase
independent of its stability. For Poisson’s ratio smaller than 1/3, the kinetics is stable and the
phase of small shear modulus grows into the phase of higher shear modulus while for higher
values of Poisson’s ratio the behavior is reversed and the interface roughens with time.
In general, it turns out that contrasts in the referential densities of the two solids often
lead to the formation of finger-like structures aligned with the principal direction of the far
field stress. In cases where the referential densities are identical the stability depends on the
“compressibility” of the material and it turns out that Poisson’s ratio plays a central role in the
stability of the interface.
References
1. L. Angheluta, E. Jettestuen, J. Mathiesen, Phys. Rev. E 79, 031601 (2009)
2. L. Angheluta, E. Jettestuen, J. Mathiesen, F. Renard, B. Jamtveit, Phys. Rev. Lett. 100, 096105
(2008)
3. L. Angheluta, J. Mathiesen, C. Misbah, F. Renard, Morphological instabilities of stressed and
reactive geological interfaces (2009) (preprint)
4. R. Asaro, W. Tiller, Met. Trans. 3, 1789 (1972)
5. H. Gao, Int. J. Solids Struct. 28, 703 (1991)
6. M.A. Grinfeld, Dokl. Akad. Nauk SSSR 290, 1358 (1986)
7. K. Kassner, C. Misbah, J. Muller, J. Kappey, P. Kohlert, Phys. Rev. E 63 (2001)
8. J.S. Langer, Rev. Mod. Phys. 52, 1 (1980)
9. O. Martin, N. Goldenfeld, Phys. Rev. A 35, 3 (1987)
10. J. Mathiesen, I. Procaccia, I. Regev, Phys. Rev. E 77 (2008)
11. W.W. Mullins, J. Appl. Phys. 28, 3 (1957)
12. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35, 2 (1964)
13. N.I. Muskhelishvili, Some basics preblems of the mathematical theory of elasticity (P. Noordhoff
Ltd., Groningen-Holland, 1953)
14. F. Renard, P. Ortoleva, J.-P. Gratier, Techtoniphysics 280, 257 (1995)
15. P.G. Saffman, G. Taylor, Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. 245, 312 (1958)
16. R.F. Sekerka, J.W. Cahn, Acta Mater. 52, 1663 (2004)
17. P.K. Weyl, J. Geophys. Res. 64, 11 (1959)
18. Y. Xiang and W.E., J. Appl. Phys. 91, 11 (2002)
19. W.H. Yang, D.J. Srolovitz, Phys. Rev. Lett. 71, 10 (1993)
20. D.-H. Yeon, P.-R. Cha, M. Grant, Acta Mater. 54 (2006)