Pattern Selection in Biaxially Stressed Solids

P H Y S I C A L R E V I E W L E T T E R S week ending2 MAY 2003VOLUME 90, NUMBER 17

Pattern Selection in Biaxially Stressed Solids

Patrick Berger,1 Peter Kohlert,2 Klaus Kassner,2 and Chaouqi Misbah1

1Laboratoire de Spectrometrie Physique GREPHE, Universite Joseph Fourier (CNRS), Grenoble I, BP 87,38402 Saint Martin d’Heres, France

2Institut fur Theorstische Physik, Otto-von-Guericke-Universitat, Magdeburg Postfach 4120, D-39016 Magdeburg, Germany(Received 9 October 2002; published 2 May 2003)

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We analyze the morphological instability of the surface of a solid which is subject to a biaxial stress.The stability calculation reveals a new favored pattern: a diamond morphology. This occurs if the stressis tensile in one direction and compressive in the orthogonal one and the ratio exceeds a certain value. Anonlinear analysis shows that the bifurcation is subcritical and hints to a nontrivial competitionbetween tilted stripes and diamonds. This study opens a new line of inquiries in the field of stress-induced pattern selection.

DOI: 10.1103/PhysRevLett.90.176103 PACS numbers: 68.35.Md, 05.70.Np, 62.20.–x, 81.15.Aa

We find a generic breaking of rotational symmetrywhen a tensile stress coexists with a compressive one.

can go back and forth). One must, however, write it bytaking the liquid pressure as a reference (as expressed by

Although elasticity is synonymous of small displace-ments of atoms around their equilibrium positions, itseffect can drastically affect the macroscopic and micro-scopic behaviors of stressed solids. For example, an elas-tic mismatch of a few percent between a growing filmand a substrate is known to lead in the purely elasticregime (dislocation-free films) to the breakup of thefilm into islands[1] (e.g., quantum dots). This is the resultof a morphological instability driven by stress, whichplays an essential role for self-organization of nanostruc-tures [2].

Asaro and Tiller and Grinfeld (ATG) [3,4] have shownthat in a uniaxially stressed solid the initial planar inter-face undergoes inevitably a morphological instability: amodulated surface has a lower elastic energy. Besides itstechnological importance, this instability raises funda-mental questions, such as those pertaining to patternselection or the development of finite time singularities[5,6] . . . , and is now a topic of much current interest invarious communities ranging from physics [7] to geo-physics [8].

The ATG instability has been in most analytical worksstudied for one dimensional deformations only (uniaxialstresses), which is the exception rather than the rule. Theeffect of a biaxial stress is not only of obvious practicalinterest but also raises important questions: (i) What kindof surface pattern can be expected? (ii) What happenswhen the stress is tensile in one direction and compres-sive in the other direction? Would the stress favor, amonga large manifold of possibilities, a special pattern? Theaim of this Letter is to address these questions. Works inthe presence of a biaxial stress were addressed previouslyin [9] regarding the stress concentration and in [10,11]concerning a linear stability analysis. A study of thephase diagram of the possible solutions, together withthe bifurcation structure and the nonlinear competition ofthe patterns, is lacking.

0031-9007=03=90(17)=176103(4)$20.00

The emerging pattern is of diamond type. This occursabove a threshold value for �0X=�0Y , where �0X and �0Yare the applied stresses in the X and Y directions. Anonlinear analysis shows that the bifurcation is subcrit-ical. The calculation reveals that a tilted stripe patternmay override the diamond, but in view of the develop-ment of finite time singularities along the two diamonddirections (e.g., their amplitudes grow unstably withoutbound), the ultimate stage is nontrivial. This findingshould open new lines of inquiries in the field of patternselection (e.g., in quantum dots) via a stress monitoring.

The solid is in contact with its melt, its vapor, or withvacuum.We consider the first two cases; the latter one canbe dealt with without additional complication. The basicsituation is that of a semi-infinite solid with a planarsurface located at a height z � 0. Under dynamics, thefront evolves by a melting-crystallization process (or viamass diffusion). The instantaneous position of the inter-face obeys F�x; y; z; t� � z� h�x; y; t� � 0.

For simplicity we restrict ourselves to isotropic andlinear elasticity. The stress tensor �ij obeys the followingrelations (in which repeated subscripts are to be summedover) in the bulk and interface, respectively:

@j�ij � 0; �ijjhnnj � �pLnni; (1)

where pL is the pressure of the liquid (or vapor) phase,and nn � rF=krFk is the normal vector to the surface.

A basic quantity of interest is the difference, due tostress, of the chemical potential between the solid and theliquid phases at the interface. The expression used byNozieres in 2D [12] reads in 3D

��el �1

2E��1 ��~��ijjh ~��ijjh � ��~��kkjh�2: (2)

The tensor ~��ij � �ij � �nn�ij. Note that Eq. (2) is thedensity energy written in terms of the stress (usually it isgiven in terms of the strain, and by using Hooke’s law one

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the presence of the tilted variable). From now on we workwith ~��ij only, and we drop the tilde.

The stress contribution to the chemical potential mustbe supplemented with those due to capillarity and gravityand are given by

��cap � ��; and ��grav � ��gh: (3)

E is the Young modulus, � is the Poisson ratio of thesolid, � represents the interfacial free energy, � is itsmean curvature, and �� L � �S the difference ofthe solid and liquid densities.

In the basic state, the stress in the solid is uniform andpurely tangential to the interface (�zz � 0). The appliedstresses along X and Y are denoted by �0X and �0Y (seeFig. 1).

It is convenient to introduce compression � � ��0X �0Y�=2 and a shear stresses �� 0X � �0Y�=2. For aplanar interface, (2) implies a recession down to a newposition z � �hmelt where the gravity effect counterbal-ances the destabilizing elastic part. One can easily checkthat

hmelt ��1� ��2 �1 ��2

E��g:

From now on, we redefine the basic planar front positionto be z � 0 by subtracting hmelt.

We first perform a linear stability analysis of the planarfront as in [10]. Because of the 3D character of thedisplacement, the use of the Airy function [13], as in[12], is not helpful. We find it convenient to rewrite theLame equation in terms of the stress tensor [13]:

�1 ��@k@k�ij @i@j�kk � 0: (4)

This equation is known under the name of Beltrami-Michell. From these relations and (1), it follows that

r2�kk � 0; and r2r2�ij � 0: (5)

Like the Airy function the stress tensor is a biharmonicfunction. Two remarks are in order: (i) the stress is fullythree dimensional and no special assumption (e.g., planestress) is made; (ii) we consider the steady version of

z0

hmelt

σ0X σ0X

σ0Y

σ0Y

X

Y

Z

0

FIG. 1. Stressed solid in contact with its melt.

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elasticity since mass transport is slow in comparison tosound propagation.

Any fluctuation of the planar interface can be decom-posed in its Fourier’s modes hq�x; y; t� � H�t�ei�qXxqYy�c:c:, characterized by the wave vector q � �qX; qY� �q�cos!; sin!�. We find

�ij � �Aij�t�z Bij�t�eqzei�qXxqYy� c:c: (6)

Introducing the notation Q � �iqX; iqY; q�, one can re-write (6) as �ij�q� � �Aijz Bij� exp�Qx� c:c: Thenr2�ij�q� � 2qAij exp�Qx�, so that (5) gives Akk � 0and Eq. (4) yields

Aij � �QiQj

2q�1 ��Bkk: (7)

It follows from Eq. (1) in the bulk that

QjBij � �AiZ: (8)

The A’s and B’s are obtained from (1). Using the generalsolution for �ij�q� [given by Eq. (6)] we obtain BXZ �iqXH�0X, BYZ � iqYH�0Y , and BZZ � 0. Using (8) fori � Z one finds that AZZ � q2H�k, where we have intro-duced the diagonal stress in the direction parallel to q:

�k �q2X�0X q

2Y�0Y

q2� �0Xcos

2! �0Ysin2!:

It can be checked from (7) that Aij can be written in acompact form Aij � QiQjH�k. Once the elastic field iscompletely determined (and so are all stress componentsin the solid), we are in a position to evaluate (2). The finalexpression can be put in a simple form

��el�q� � �2qh�x; y�E

f�1� �2�� cos2!�2

�1 �� sin2!�2g: (9)

This expression agrees with the Grinfeld result [10].��el is negative (i.e., the solid chemical potential is lowerthan the liquid one) for h > 0 (in the mound) and positivefor h < 0 (in the groove). The biaxial stress leads toseveral new features discussed below. The solid melts inthe grooves and recrystallizes at the top of the interface.Note that in the case of solid/vacuum the instability takesplace via surface mass diffusion: mass diffuses along thesurface from the grooves towards the top of the interface.Before analyzing the biaxial result, we write down thecapillary and gravitational contributions to the chemicalpotential which are given in the linear regime by

��cap�q� � �q2h�x; y�; ��grav�q� � ��gh�x; y�:

(10)

These terms stabilize the planar interface at short andlong wavelengths, respectively. Thus, there must be athreshold value for the applied stress beyond which aplanar interface is unstable.

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The first novelty in the biaxial case is that the chemicalpotential does not only depend on jqj, but also on theorientation of the wave vector as shown by the presenceof ! in expression (9). Let us first consider some limitingcases. (i) For ! � 0 one recovers the results of a uniaxiallystressed solid [3,4,12] with the external stress given by�0X. (ii) For �� 0 (which corresponds to the situationof an isotropic biaxial stress, �0X � �0Y), one obtains thesame result provided that one substitutes �0X by �.

The main question is whether a solution with ! � 0becomes more favorable than that with ! � 0. Since !enters the chemical potential via the elastic contributiononly, it suffices to focus on the behavior of (9).

A differentiation with respect to ! reveals two extrema.The first extremum corresponds to the trivial solutionsin�2!� � 0, implying ! � 0 or (=2. This means thatthe 2D emerging surface pattern should correspond to‘‘stripes’’ along X (! � 0) or along Y (! � (=2). Themost unstable situation corresponds to ! � 0 (respec-tively, ! � (=2) if j�0Xj > j�0Yj—sector IX in Fig. 2—(respectively, if j�0Xj< j�0Y j—sector IY in Fig. 2). Thenew interesting feature corresponds to the second ex-tremum (actually a pair of absolute minima correspond-ing to �!�; note that ! � 0 becomes a maximum of thechemical potential; see Fig. 3) which exists under thefollowing condition:

cos2!� � �1� ��=��: (11)

In the vicinity of the critical point given by the con-dition �� c � �1� ��=��, ! is small, and ex-pansion of (9) up to fourth order in ! yields

I

I

I

I

II

II

Y

X

X

Y

σ

σ0Y

0X

Stable

FIG. 2. Phase diagram of the biaxially stressed solid (seetext).

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��el�q; !�=�2qh=E� � � �1� �2��20X

��1 ��c��c ��!2

� ��1 ��2c!

4=4: (12)

This is a Landaulike expansion for a second order phasetransition with ! being the order parameter. The angle!� �

�� c

p. This means that the solution with

! � 0 bifurcates from the solution ! � 0 (or ! � (=2)which loses its stability against a tilted pattern. This is apitchfork (or supercritical) bifurcation from a straightinto a tilted pattern.

Let us discuss the implication of this result. First of all,due to thermodynamical stability one must have �1<�< 1=2. For most substances � > 0 condition (11) isfulfilled if the absolute value of the right-hand side issmaller than unity. Substituting � and �� by their ex-pressions in terms of �0X and �0Y , we obtain from thecondition j cos�2!�j< 1 two sectors, denoted II in Fig. 2in the plane �0X-�0Y , satisfying

�0Y <��1� 2��0X; �0X <��1� 2��0Y: (13)

These sectors correspond to two tilted patterns in thedirections �!�. Since gravity and capillarity are stabiliz-ing (10), the instability is suppressed if the stress is toosmall, and this defines the threshold for the planar frontinstability. More precisely in sectors II the elastic con-tribution may be written as

��el ��2qh�x; y�

E

��1 ��2

�1� ��

�2

�: (14)

Since the gravity and the capillary effects are indepen-dent of the applied stress [Eq. (10)], comparison of thesum of these two contributions to (14) defines an ellipse in

FIG. 3. Elastic part of the chemical potential as a function of! for different values of �0Y at constant �0X. Dashed lines anddotted lines correspond to a shear stress, respectively, belowand above the transition.

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the plane �0X-�0Y inside which the planar front is stable(Fig. 2). If condition (13) is not fulfilled (outside sectors IIin Fig. 2) only a straight pattern is possible (! � 0 or ! �(=2). These correspond to sectors IX and IY in Fig. 2.Since in sector IX (! � 0) the chemical potential (9) isindependent of �0Y the boundary of the planar frontinstability is not anymore an ellipse but a straight lineparallel to the �0Y axis. A similar scenario holds for IY(! � (=2) with Y playing the role of X. In summary theboundary inside which a planar front is stable is com-posed of portions of an ellipse connected with horizon-tal and vertical lines, as shown by the thick solid line inFig. 2.

The new solution (sectors II in Fig. 2) consists of asuperposition of the two orthogonal modes and the modu-lation amplitude takes the form h�x; y� � H+e

iq�x H,e

iq�x c:c: [the two unstable modes have wave num-bers q+ � �1; 1�q=

��2

pand q, � �1;�1�q=

��2

p]. This cor-

responds to a diamond surface pattern. This is the linearlyfastest mode. In the early stage of the instability, thispattern prevails. If the ultimate stage of dynamics (if anywithin elasticity theory) is to be determined, or if thelong time behavior of the instability is to be ascertained,then a nonlinear analysis is necessary. Note that while thefront can undergo a large enough deformation (this cor-responds to mass rearrangements) the elastic deformationremains small.

We have performed a weakly nonlinear analysis as anexpansion of the field equations and the boundary con-ditions in terms of the amplitudes H+ and H,. Thevicinity of the instability threshold is described by asmall linear growth rate - proportional to ��c.To third order in the expansion a double solvability con-dition must be imposed, yielding

_HH + � -H+ �1 ��2

EH+�GjH+j

2 gjH,j2�;

(15)

_HH , � -H, �1 ��2

EH,�GjH,j2 gjH+j2�:

(16)

The form of the equations could also be inferred fromtranslational and rotational symmetries. The expressionsof the third order coefficients are

G �11

2and g � �

�22��2

p� 27� 14�

��2

p

�3� 2��2

p� 2�

��2

p : (17)

SinceG > 0 and g < 0 one deduces that the bifurcation issubcritical and that in addition the solution satisfyingH+=H, � R � 1 is unstable. Indeed manipulation of(15) and (16) allows one to write _RR=R / �G� g��R2 �1�. Writing R � 1 �R and linearizing, one obtains_��R / 2�G� g��R. Since the prefactor is positive, R � 1

is an unstable solution. This means that one of the two

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amplitudes will take over the other in the course of time.The diamond pattern (corresponding initially to H+ �H,) may be overridden in the course of time by a puretilted solution along the !� or �!� solution. However,on the basis of the full numerical solutions[7] in 1D bothamplitudes should ‘‘diverge’’ by developing a finite timesingularity. This means that once the initial weakly non-linear diamond pattern appears, the amplitude increasesdramatically in time. An important task for future inves-tigation is thus to elucidate the question of whether the‘‘divergence’’ of the amplitudes in the diamond morphol-ogy is, or not, a fast process in comparison to the dynam-ics of the R variable. In the affirmative diamonds win, andin the negative diamonds would be overridden by tiltedstripes. For thin epitaxially grown films, the diamondmorphology may be engaged until the breakup of the filminto mounds. A highly nonlinear numerical analysisseems necessary before drawing a conclusive answer.

In summary, we have shown that a biaxially stressedsolid offers a rich variety of patterns and raises newquestions for future highly nonlinear studies [14]. Thisrichness occurs when the applied stress is tensile in onedirection and compressive in the orthogonal one. A non-trivial ultimate dynamics is expected from the competi-tion between tilted stripe and diamond patterns. Thiswork has focused on isotropic materials. It is an impor-tant task to elucidate in the future the effect of anisotropy.This enters in the surface properties (surface energy. . .) aswell as in the elastic bulk problem. While the first ques-tion can be dealt with in some cases analytically, thesecond one requires a full numerical analysis.

This work is supported by a German-French coopera-tion programme, PROCOPE. Financial support isacknowledged.

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Pattern Selection in Biaxially Stressed Solids