PRL 96, 146103 (2006)
PHYSICAL REVIEW LETTERS
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Instabilities and Coarsening of Stressed Crystal Surfaces in Aqueous Solution
Jan Bisschop* and Dag Kristian Dysthe†
Physics of Geological Processes, Postbox 1048 Blindern, N-0316 Oslo, Norway
(Received 19 December 2005; published 11 April 2006)
Strong pattern formation occurs on polished miscut surfaces of sodium chlorate (NaClO3 ) single
crystals that are uniaxially stressed perpendicular to the step edge direction and placed in a saturated
aqueous solution. The wavelength of the stress-induced surface instability increased continuously in
experiments up to 9 days after placed in the solution. There were three successive regimes of coarsening:
(i) one-dimensional step bunching with t1=4 until an undulation transition was reached, (ii) a twodimensional coarsening mechanism with t1=2 , and a gradual transition to (iii) Ostwald ripening-like
coarsening with t1=3 . The coarsening of the surface patterns towards a stable, flat surface implies the
spontaneous formation of a stress-free skin on the surface of the stressed solid.
DOI: 10.1103/PhysRevLett.96.146103
PACS numbers: 68.08.De, 68.37.d
Introduction.—It is theoretically well known that the
application of stress to a solid in diffuse contact with like
molecules (melt, solution, or surface diffusion in vacuum)
may lead to spontaneous roughening of the surface through
a number of different mechanisms [1–6]. These surface
instabilities are important in heteroepitaxial thin film
growth, a technique widely applied in the manufacture of
semiconductor devices [4,5,7,8]. In addition to the technological interest, there is a basic academic interest in nonequilibrium physics of describing surface evolution during
crystal growth from the knowledge of microscopic processes [6]. Consequently, a great number of theoretical and
numerical studies on growth of stressed solids and many
experimental studies on instabilities during heteroepitaxial
film growth have been published the past 20 years [1,9,10].
Recently, the importance of stress-induced roughening
of rock forming mineral surfaces in geophysics has also
been suggested [11–15]. The Earth’s crust consists of
polycrystalline solids with an abundance of aqueous solutions or melts at grain boundaries and in pores between the
minerals. In the Earth’s crust, the rocks are, more often
than not, nonhydrostatically stressed, and the rock forming
minerals respond by intragranular plastic flow or by deformation through dissolution and reprecipitation mediated
by the interstitial fluid. It has recently been shown that,
upon the application of stress, ionic solids in contact with
an aqueous solution develop grooves, ripples, or islands
with an initial characteristic spacing [11] or that are continuously coarsening [12,13]. These are, to our knowledge,
the first clear experimental examples of stress driven surface instabilities in solid-aqueous systems. The understanding of these systems is important for our ability to
understand and model the continuous dissolution and
growth of minerals under elastic stress in the Earth’s crust.
Two types of models for instabilities have been applied
to stressed solid-vapor systems: purely stress-induced instabilities (thermodynamic models) [2,3,7,8] and kinetic
models related to directional anisotropies of kinetics or of
diffusion [9]. In solid-aqueous systems, diffusion, dissolution (desorption), and growth (adsorption) are mediated by
0031-9007=06=96(14)=146103(4)$23.00
an aqueous solution and not by adatom motion on the solid
surface (in vacuum). It is therefore doubtful (but it remains
to be shown) that the kinetic models of stress-induced
instabilities are applicable to solid-aqueous systems. It is
necessary to perform careful experiments on solid-aqueous
(and solid-melt) systems in order to test the applicability of
the existing theories. In this Letter, we present new in situ
observations of stress-induced instabilities on miscut surfaces of NaClO3 single crystals. We will show that stress
induces macrosteps on the surface, the macrosteps coarsen
in time by different mechanisms, and the surface evolves
towards an end state with an unstressed skin in equilibrium
with the solution.
Experiments. —We developed a new technique to apply
a high compressive surface stress to brittle ionic crystals
such as sodium chlorate (NaClO3 ). First, thin sections of
NaClO3 single crystals with a constant thickness of 90 m
were prepared on a microscope cover glass with a thickness
of 200 m. This double layer sample was glued into a
Perspex holder with a uniform concave curvature radius of
20 cm. The plane of zero strain of the elastically bent
double layer lies within the glass piece, and as a result
the whole NaClO3 slice was under compression and did not
break upon bending. The surface stress was calculated
using the solution for bending of elastically isotropic plates
[16] assuming perfect stick between the NaClO3 slice and
cover glass. Young’s moduli and Poisson’s ratios used in
the calculations were, respectively, 49 GPa and 0.25 for
NaClO3 [17] and 72.9 GPa and 0.208 for the cover glass
(manufacturer’s specification). NaClO3 does not deform
plastically at room temperature up to stresses of at least
20 MPa [11].
Experiments reported in this Letter were carried out on
sample surfaces polished with a 5 miscut to f100g. Crystals were first embedded in epoxy resin, carefully ground
with a 5 angle to f100g, and firmly glued on the cover
glass. The principal stress direction by bending was parallel to the h100i crystallographic direction. Up to 230 C,
NaClO3 has a cubic crystal symmetry; it grows as cubic
crystals from aqueous solution at least up to 40 C [18].
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© 2006 The American Physical Society
PRL 96, 146103 (2006)
PHYSICAL REVIEW LETTERS
In situ experiments were performed in a Peltier controlled temperature box under a Leica stereo microscope
(with a Leica DFC320 digital camera) at 24, 25, or 27 0:1 C. All experiments were carried out in distilled water
saturated with NaClO3 . The fluid chamber was created by
placing a cover glass over the curved samples. In this way,
the fluid chamber height was 100 m and possible
convection effects minimized. Immediately after a drop
of saturated solution was sucked into the fluid chamber
(by capillary forces), the chamber was sealed off with wax
and placed in the temperature box under the microscope.
Experiments were ended at different times, the fluid cell
opened, and the surface dried quickly before the surface
topography was measured by white light interferometry.
Results.—We have performed four experiments on
stress-free and five experiments on highly stressed
NaClO3 single crystals with a 5 miscut to f100g. All
miscut samples showed some net dissolution of the crystal
surfaces as a new single crystal rim formed at the four
edges of the samples. The rim growth was probably driven
mainly by the surface energy of the miscut surface. The
stress-free samples showed no roughening of the crystal
surfaces in saturated solutions in the four experiments with
durations of up to 96 hours. We also performed two experiments on stressed NaClO3 with surfaces polished parallel
to f100g crystallographic planes. These 70 hour long experiments showed no significant stress-induced pattern.
We present in detail results of five experiments, S1–S5
on crystals with miscut surfaces and uniaxial compressive
stress of 24 –29 MPa and temperature 24 –27 C. In all
these experiments, a strong surface pattern with dark lines
perpendicular to the stress developed. The average spacing
of the lines was initially (5 minutes after adding the saturated solution) 15 m in all five experiments and, subsequently, increased continuously during the experiments
(Fig. 1). The pattern formation was highly uniform across
the sample surfaces away from sample edges.
Figure 1 shows gray regions of the crystal surface that
are parallel to the f100g plane. The dark lines in the images
are macrosteps accommodating the miscut of the surface,
and we will term them ‘‘positive steps.’’ Bright lines in
Figs. 1(b) and 1(c) correspond to macrosteps in the other
direction and will be referred to as ‘‘negative steps.’’ The
mean spacing of positive steps, also termed the wavelength
, of the surface corrugation was measured for all images
along three 1.5 mm long transects parallel to the principal
stress direction ( k ).
~ The ex situ interferograms confirmed that the mean amplitude of steps was proportional
to the wavelength. Figure 2 shows wavelength versus time
for all five experiments. Over 3 orders of magnitude in
time, the overall evolution of the wavelength in the five
experiments was close to t1=3 . However, the deviations
from the overall t1=3 coarsening rate were significant
and related to three different dominant coarsening mechanisms: (a) 1D step bunching, (b) 2D step ‘‘zipping,’’ and
(c) reducing curvature.
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FIG. 1. Detail in situ optical photographs of stress-induced
step coarsening in sample S1. Images (a), (b), and (c) are at
the same sample location. (a) Pattern 15 minutes after adding
saturated solution ( 17 m): long straight steps coarsening
by 1D step bunching. (b) After 5 hours ( 48 m): many
undulating step edges and step coarsening by 2D ‘‘step zipping.’’
Black lines are ‘‘positive’’ steps; bright lines are ‘‘negative’’
steps formed by grooves in terraces as illustrated by the inset
surface profile. (c) After 31 hours ( 111 m): smoothening
and rounding of steps by Ostwald ripening-like coarsening.
(a) In experiments S1 (images in Fig. 1 and circles in
Figs. 2 and 3) and S2, the first image showed straight
positive steps ? ~ with a mean wavelength of 15 m.
During the first 70 –80 minutes, the steps remained relatively straight [see Fig. 1(a)] and smaller steps moved
‘‘uphill’’ k ~ by dissolution until they joined larger steps
that moved ‘‘downhill’’ by precipitation. Since steps were
dominantly straight and ? ~ and motion was perpendicular to the steps, the surface evolution is fully described by a
1D transect along the surface k .
~ The evolution of wavelength versus time for the first 100 minutes was close to
t1=4 (see Fig. 3).
(b) After about 70 –80 minutes, the steps started to
undulate in the crystal surface plane, and the amplitude
of the undulations grew until neighboring steps touched
each other and created a cellular pattern [see Fig. 1(b)]. At
the same time that the undulations joining steps became
dominant, the first negative steps appeared at an angle of
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PHYSICAL REVIEW LETTERS
PRL 96, 146103 (2006)
2.2
log10(λ)
2
1.8
1.6
1.4
1.2
1
2
3
4
log (t)
10
FIG. 2 (color online). Log distance m between macrosteps on a miscut surface of sodium chlorate crystals under uniaxial strain versus log time t min. The five individual experiments (S1–S5) had surface stresses s 28 (circles), 29 (plusses), 28 (squares), 29 (solid squares), and 24 MPa (diamonds),
respectively. The dashed-dotted line indicates (it is not a fit) the
global cubic root of time dependence t1=3 .
40 –70 to .
~ Negative steps appear to be caused by
stress-induced dissolution of the terraces. The coarsening
process proceeded by joints propagating ? ~ like a zipper
and by fading away due to overgrowth of smaller steps.
The transition between the two modes of coarsening was
gradual, but we have determined an undulation threshold
experimentally from the observed change in pattern
(Fig. 1) and from the break in slope of vs t (Fig. 3): u 27 1 m and tu 75 5 min . At this transition, the
surface pattern and the coarsening mechanism became
two-dimensional and the coarsening rate changed slope
to close to t1=2 .
(c) The last stage (30 –190 hours) was characterized by
smooth, round, cellular structures [see Fig. 1(c)] that increased in size and decreased in curvature. In this stage, the
1/3
λ∼t
0.1
10
u
u
log ((λ/λ )(t / t)1/3)
0.15
1/2
λ∼t1/4
λ∼t
0.05
0
−1
0
1
log10(t / tu)
2
FIG. 3 (color online). Wavelength as a function of time relative
to the undulation transition (u ; tu ) and with the cubic root time
dependence divided out. The undulation transition points are
S1 (circles): (26 m, 70 min); S2 (plusses): (29 m, 80 min);
S3 (squares): (52 m, 360 min); S4 (solid squares): (52 m,
480 min); S5 (diamonds): (52 m, 380 min). The dashed-dotted
lines are straight lines to guide the eye (they are not fitted) with
time exponents of (from left to right) 1=4, 1=2, and 1=3.
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stress-induced grooves gradually disappeared and the
coarsening rate was t1=3 .
Samples S3, S4, and S5 showed a similar pattern evolution as samples S1 and S2, except that during the onedimensional regime there was a fast growth in the number
of negative steps. The negative steps in these experiments
differed from those in the first experiments, in that they
were ? ~ and they all nucleated in front of and close to
positive steps, creating a kind of ‘‘step pairing.’’ The
difference between S1 and S2 and S3–S5 may have been
due to slight differences in miscut ? ~ or due to slightly
different initial fluid conditions.
(a) The pattern on S3, S4, and S5 was soon dominated by
positive and negative steps ? ,
~ with very little undulation, and the time evolution during this 1D step bunching
was close to t1=4 . The undulation threshold from 1D
step bunching to 2D step coarsening in experiments S3–S5
occurred at almost twice the wavelength u 52 0:4 m of S1 and S2 and at tu 360–480 min (see
Fig. 3). (b) Following the undulation transition, the coarsening mechanism as described for experiments S1 and S2
took over and the coarsening rate increased, approaching
t1=2 until the experiments were stopped.
Discussion.—One of the best known stress-induced instabilities is the so-called Asaro-Tiller-Grinfeld (ATG) instability [2,3] that eventually leads to cracking of the solid.
The ATG linear stability analysis for solids with smooth
and unfaceted surfaces predicts an exponential growth of
surface corrugations with maximum unstable wavelength
max 4–6 m and growth time scale of 0.2 – 0.5 hours
for the conditions reported here [17]. We observe surface
corrugations after 5 minutes that are 3– 4 times max where
the wavelength grows, but the amplitude does not grow
exponentially but is simply linear with wavelength. It is
also important to note that, contrary to the ATG analysis,
the experiments on stressed crystals with surfaces parallel
to f100g showed no groove formation. Our observations are
consistent with the theoretical prediction of Tersoff [19]
that close to equilibrium (small supersaturation) the maximum unstable wavelength diverges for faceting surfaces.
This result has the important implication that the traditional ATG instability analysis is not applicable to slowly
growing, faceting minerals in aqueous solution under compressive stress. This probably applies to mineral-water
interfaces in most of the Earth’s crust. The observed surface patterns are (as in ATG) caused by elastic stress but in
a different manner, which we will attempt to explain.
The initial evolution of the stressed, miscut surfaces
was dominated by motion of crystallographically oriented
macrosteps. The coarsening rate t1=4 for this onedimensional step bunching seems to agree with predictions
for elastically driven step bunching in the no growth regime [7] with adatom diffusion and no surface tension.
Leonard and Tersoff [8] have also explored the limits of
stability of the one-dimensional step bunching mechanism.
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PHYSICAL REVIEW LETTERS
They predict a transition to undulation of step bunch edges
for elastically interacting steps with a step line tension on a
stressed solid [8]: Neighboring step edges are attracted,
and, as the distance between the steps () increases, the
line tension of step edges can no longer stop the undulation
instability. We do not have the material parameters necessary to estimate the transition wavelength as given in their
analysis but note that the observed ‘‘step pairing’’ in experiments S3, S4, and S5 may have increased the effective
line tension of the steps that stabilized the 1D structure,
thereby doubling the crossover wavelength u .
As the pattern became two-dimensional, the mechanism
of coarsening changed. The dissolving grooves are sites of
maximum stress .
~ The amplitude now became so large
that we propose that the newly grown ‘‘hill tops’’ of the
surface were unstressed. This created a constant chemical potential difference between stressed and unstressed
material. The pattern developed by local dissolution of
stressed material and precipitation of unstressed material
and acted like a zipper joining adjacent steps and by covering smaller steps by unstressed material. The rate limiting
process was diffusional mass transport between stressed
and unstressed sites (constant chemical potential difference) which has a time scale tD / 2 , thus explaining the
coarsening behavior approaching t1=2 in this stage.
The transition to the third stage is very gradual, and the
duration of the second stage is not equal in experiments S1
and S2. This may be related to the lack of a specific
transition length scale since, unlike the first transition,
there is no change in dimensionality.
In the final stage of coarsening, we observe that most of
the original, stressed surface was covered by newly grown
material that we claim was unstressed. The reason for
formation of rounded concave step corners on the crystal
facet (also observed in the rim around the crystal and in
fluid inclusions) remains to be explained. The evolution of
the pattern resembles surface tension driven coarsening or
so-called Ostwald ripening [1], and the coarsening rate
t1=3 is characteristic of this process. This time dependence of interface roughness coarsening was also found to
control the creep rate of crystals deforming by dissolution
precipitation creep [12].
The existence of a newly grown unstressed crystalline
skin has yet to be proven experimentally, but there are good
arguments in favor of this proposition. Dislocations nucleating in heteroepitaxial growth resulting in relaxation of
stress in the growing crystal is well known [20 –22]. If all
the material at the surface were still under elastic stress in the last stage of the experiment, there is no reason why
the step bunching mechanism should not be operative. This
is coherent with the argument by Duport et al. [20] that, if
dislocations relax the strain during growth, the original
instability should be forbidden.
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Conclusions.—The main result of this study is that miscut sodium chlorate surfaces respond to applied stress by
roughening, coarsening, and growth of a smooth, unstressed skin. The coarsening happens in successive regimes with different time dependence: 1D step bunching
(t1=4 ), 2D zipping (t1=2 ), and finally by Ostwald ripeninglike coarsening (t1=3 ). The study also indicates that it is
unlikely that the ATG instability leading to cracking applies to slowly growing, faceting minerals under compressive stress in the Earth’s crust.
*Electronic address: jan.bisschop@fys.uio.no
†
Electronic address: d.k.dysthe@fys.uio.no
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