\
Pergamon Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/97 $17.00 + 0.00 PII: S1359-6454(96)00411-9THE EFFECT OF SURFACE STRESS ON THE STABILITY
OF SURFACES OF STRESSED SOLIDS
C. H. WU1, J. HSU2and C.-H. CHEN2
1Department of Civil and Materials Engineering, University of Illinois at Chicago, 842 West Taylor Street,
Chicago, IL 60607-7023, U.S.A. and2Institute of Applied Mechanics, National Taiwan University, Taipei,
Taiwan, R.O.C.
(Received23August1996;accepted26November1996)
Abstract—The chemical potential associated with surfaces of stressed solids is found to consist of four terms, instead of the widely accepted two. The boundary conditions are also affected by the presence of a surface deformation-dependent surface stress. The stability of the surfaces of stressed solids is reinvestigated, using the revised equations. It is found that the stability condition is sensitive to the sign of the applied stress.71998Acta Metallurgica Inc.
1. INTRODUCTION
The first theoretical investigation of the stress-driven morphological instability in solids appears to have been given by Asaro and Tiller [1]. They examined the linear stability of a planar surface separating a stressed, two-dimensional semi-infinite solid from a fluid and found that the planar surface was unstable to small disturbances with wavelengths greater than a characteristic length. The same result was later rediscovered by Grinfeld [2] and Srolovitz [3]. The starting point of these theoretical investigations is the chemical potentialC given by
C = (W − G0K)V (1)
where W is the strain energy density of the solid (evaluated on the surface in question),V the atomic volume, G0 the assumed constant surface energy
density, and K the mean curvature, all expressed in terms of a chosen referential configuration. The critical wavelength bcgoverning the instability of the
surface Z2= 0, bounding the elastic half-space
Z2Q 0, is
bc= pEG 0
(1 −n2)T2 (2)
where E and n are, respectively, Young’s modulus and Poisson’s ratio and T is the value of the horizontal stress in the solid. The elasticity solution associated with the above result satisfies the traction-free conditions along the perturbed planar boundary, i.e.
tNN= 0, tSN= 0, (3)
where S and N signify, respectively, the tangential and normal directions to the boundary. It is noted that bc is not affected by the sign of T.
Stressing the importance of an assumed constant elastic surface stress (surface tension) S0, Grilhe [4]
replaced the first of equations (3) with, in terms of our notation,
tNN=S0K (4)
and obtained a new elasticity solution for the evaluation of W. Grilhe’s modified critical wave-length bcg is given by
bcg= bc(1 −S0/Tbc+ · · ·) (5)
which also makes use of equation (1).
In the context of finite elasticity and with the assumption that surface energy is a function of surface deformation, the completeness of the chemical potential, together with the associated governing equations, was re-examined by Wu [5]. It was found that, even after a reduction to linear elasticity, the expression (1) is incomplete. Moreover, even the modified condition (4) is still incomplete. The correct equations and conditions are presented in Section 2.1.
It has been commonly accepted that for solids surface tension is not the same as surface energy [6]. On the other hand it has also been not uncommon to find the two terms used in synonymous ways when applied to solids. One of the reasons for this ambiguity is perhaps due to the lack of known surface deformation-dependent surface energy expressions. The determination of surface energy is already a feat in itself [7]. We are not aware of any experimental or theoretical work that deals with the surface strain dependence of surface energy. Using an experimen-tally justified universal binding-energy–distance curve [8–11], a plausible surface energy expression is obtained in this paper. This is given in Section 2.2.
The stability of the surfaces of stressed solids is used as an example to illustrate the effect of the corrected equations. It is shown that the corrected critical wavelength bcwis bcw= bc
0
1 − 4 1 −n eT− 1 − 2n (1 −n)2e 2 T+ · · ·1
(6)whereoT= (1 −n2)T/E. This result is to be compared
with equations (1) and (5). The derivation for the above is given in Section 3.
2. GOVERNING EQUATIONS
2.1. Finite plane elastostatics with surface strain dependent surface energy
As in Wu [5], we consider a two-dimensional material body with (two-dimensional) reference configuration B and (two-dimensional) deformed configuration b and designate by Z = ZIeI and
z = ziei, respectively, the referential and spatial
positions of a typical particle relative to a common origin 0 and common base4e1, e2, e35. The referential
and spatial positions of a typical particle are linked by
z =F(Z), zi=Fi(ZI) on B, (7)
and the associated deformation gradient tensor F is just
F = GradF, FiI=Fi,I=1Fi/1ZI. (8)
It is assumed that the material body is composed of a homogeneous elastic material with strain-energy function W(F) measured per unit volume in the referential configuration.
Let the bounding surface1B of B be parametrically represented by
1B:Z = X(L) (9)
where L measures the arc length along the boundary. The unit tangent S is
S = dX/dL (10)
and the sense of L is fixed in such a way that, together with the outward normal N and e3, (S, N, e3) is
right-handed. It is convenient to express the unit tangent in terms of a single angle U(L) so that
S = cosUe1+ sinUe2 (11) and dS dL= KN, dN dL= −KS, (12)
where K = dU/dL is the curvature.
For an arbitrary point Z sufficiently close to a point X(L) on 1B, it is convenient to introduce the shell-coordinates representation:
Z = X(L) + YN(L) (13)
where Y measures the distance along N. We have dZ = [(1 − YK) dL]S + dYN (14) which defines the metric associated with the curvilinear coordinates (L, Y). Subjecting equation (14) to the deformation (7), we obtain
F(X + YN) dZ =L(S)(L, Y)[(1 − YK) dL]s
+L(N)(L, Y) dYnd (15)
where s and ndare unit vectors, and, to the first order
in Y, L(S)(L, Y) =L +FS L·[N·Grad F]SY, (16) L(N)(L, Y) =LN+ FN LN ·[N·Grad F]NY, (17) in which L = L(S)(L, 0), LN=L(N)(L,0). (18)
It can be easily checked that
FS =Ls, FN = LNndon1B (19)
and s, ndare, in general, not orthogonal. It is now
assumed that the material surface1B is characterized by an isotropic surface energy function G(L), measured per unit surface area in the referential configuration. We may now write the total free energy of the system as
P =
g
B
W dA +
g
1B
G dL, (20)
and all the governing conditions can be deduced from a variational consideration of the above.
In terms of the curvilinear coordinates (L, Y) of equation (13) and for linear elasticity, the strains and stresses may be denoted by (oSS,oNN,oSN) and
(tSS,tNN,tSN), respectively. They are the 2 × 2
portions of o and t, which are related by
tij= 2moij+ldijokk (21)
where l and m are Lame constants. For the correct and complete chemical potential, we have
C =
$
W −GK + S 1oSS 1Y−1L1 (2SoSN)%
V (22) where W =1 2tijoij, (23) G = G(o), S = 1G1o, o = oSS(L, 0). (24)The boundary conditions for the elasticity problem to be solved are tNN=Sk, tSN= (1 −o)dS dL (25) where k = (1 −oNN)K −1o SS 1Y+ 21o1LSN (26) in which K is the curvature associated with 1B. A detailed account of the above results may be found in Wu [5] where the development was based on the premise that the surface energyG is a function of the surface deformation, i.e. L of equation (18) or o of equation (24). We are not aware of any known theoretical or experimental results dealing with the dependence of G on o. However, it is commonly accepted that G and S are of comparable orders of magnitude [6]. A simple and plausible relation is established in the rest of the section based on certain known universal features of the equation of state of metals [8–11].
2.2. Effect of surface strain on surface energy It has been shown in a series of papers by Rose et al. [8–11] that the metallic binding-energy–distance curves can be approximately scaled into a single universal relation in each of the following cases: (1) chemisorption on a metal surface; (2) metallic and bimetallic adhesion; and (3) the cohesion of bulk metals. In each case, the energy relation, expressed in terms of our notation, can be written as
ET(a) =DETE*(a*) (27)
where ET(a) is the total energy as a function of
the interatomic separation distance a, DET is the
equilibrium binding energy, and E*(a*) is an approximately universal function which describes the shape of the binding-energy curve. The dimensionless separation a* is a scaled distance defined by
a* = (a − ae)/l (28)
where ae is the equilibrium separation and l is a
scaling length characterizing the material(s) involved. A suitable choice for E* is
E*(a*) = (1 + a*) e−a* (29)
and the scaling length l may be determined from
l =
$
DET>
d2E T(a) da2b
a = ae%
1/2 . (30)This result has been used by Gupta in an evaluation of the interface tensile strength–roughness relation-ship. Our purpose here is to deduce a surface strain-dependent toughness from the universal features of the above energy relation.
Let a1, a2 and a3 be the lattice constants of an
orthorhombic crystal. If a is the interatomic separation distance in the a3direction, then ET(a) of
equation (27) is the total energy between a couple of atoms divided by a1a2. Thus ET(a) is the total energy
per unit area in the referential (a1, a2)-plane. In
applying equation (27), however, aeof equation (28)
should be replaced by a3. Differentiating equation
(27), we obtain s(a) = 1ET 1a =DE T l a* e −a*, (31) 1s 1a=1 2E T 1a2= DET l2 (1 − a*) e −a*, (32)
where s is the stress defined per unit area in the (a1, a2)-plane. In terms of a strain variableo, we have
1s 1a=a13 1s 1o, o = a − a3 a3 . (33)
Equating the last two expressions for 1s/1a and settingo = a* = 0, we obtain
DET= l2E/a3 (34)
where E is the modulus in the a3 direction. The
intrinsic toughness G0 and the surface energy G0,
defined per unit area in the (a1, a2)-plane are given by
G00 2G0=
g
a a3s(a) da
= ET(a) − ET(a3) =DET. (35)
In particular, for a cubic lattice a1= a2= a3= a0
G0= l2E/2a0 (36)
where E is Young’s modulus.
Consider now a cubic lattice with lattice constant a0. The crystal is deformed into an ‘‘orthorhombic
system’’ with
a1=L1a0, a2=L2a0, a3=L3a0, (37)
whereL1,L2andL3are the principal stretch ratios
associated with the deformation of the cubic lattice. Let W(L1,L2,L3) denote the strain energy density of
the crystal. The stretch ratio L3 of equation (37) is
actually a function ofL1andL2determined from
1W 1L3
b
L3=L0 3(L, L2)
= 0. (38)
It is in this sense that a3of equation (37) is referred
to as an equilibrium separation. We may now apply equations (27)–(35) to the orthorhombic system (37) to conclude that the surface energy G*0 defined per
unit area in the (a1, a2)-plane of equation (37) is
It follows that the surface energyG defined per unit area in the original (a0, a0)-plane is L1L2G*0, i.e.
G = G(L1,L2) = L 1L2
L0
3(L1,L2)G
0, (40)
where G0 is given by equation (36). For small to
moderately large strains, one may use equations (23) and (38) to obtain
L0 3= 1 −
n
1 −n(L1+L2− 2). (41) The functionG(o) of equation (24) may be deduced from the last two expressions. It is
G(o) = 1 +o 1 − n
1 −n o
G0. (42)
The elastic surface stress (or surface tension) is just
S = 1G1o=S0
0
1 +2n
1 −n o+ · · ·
1
(43) where S0=G0(1 −n) is the constant portion of thesurface stress.
3. STABILITY OF SURFACES OF STRESSED SOLIDS
Consider the lower half-space defined by Z2Q h(Z1) = a coskZ1= a cos
2pZ1
b , (44) where a is the amplitude and b = 2p/k is the wavelength of the bounding surface. The sinusoidal fluctuation of the surface is assumed to be small in the sense that ak1. The half-space is uniformly stressed at infinity by t11= T and the boundary
condition along Z2= h(Z1) is of the type specified by
equation (25).
The following stress function [3] is introduced: F = f +T
2Z
2
2, f = (A + BkZ2) ekZ2coskZ1, (45)
where A and B are constants to be determined. The associated equilibrium stresses are:
t11= T +f,22, t22=f,11, t12= −f,12 (46)
where commas denote partial differentiation and f(Z1, Z2) is biharmonic.
The kinematics associated with the boundary Z2= h(Z1) are defined by equations (10)–(12), and
sinU = −ak sin kZ1+ . . . , cosU = 1 + · · · , (47)
K = −ak2coskZ
1+ · · · . (48)
It follows from equation (47) that the curvilinear components (tSS,tNN,tSN) of the stress tensor are
tSS= T +f,22+ · · · , tNN=f,11+ · · · ,
tSN= Tak sin kZ1−f,12+ · · · . (49)
The curvilinear components (oSS,oNN,oSN) of the
strain tensor, for plane strain, are just oSS= 1 E[(1 −n 2)T +(1 −n2)f ,22−n(1 + n)f,11+ · · ·], oNN=1 E[−n(1 + n)T + (1 − n 2)f ,11 −n(1 + n)f,22+ · · ·], oSN= 1 +n E [Tak sin kZ1−f,12+ · · ·]. (50)
The curvature k of the deformed boundary may be calculated from equation (26)
k = −
6$
1 −(1 +n)(2 − n)T E%
ak 2 −(1 +n)k 3 E [A − (1 − 2n)B]7
coskZ1. (51)Along the boundary Z2= h(Z1), the boundary
conditions (25) must be satisfied. Making all the appropriate substitutions and retaining terms accu-rate to the order of ak, we obtain
$
1 +(1 +En)kS0%0
kAaT1
−(1 +n)(1 − 2n)kSE 00
kBaT1
=$
1 −(1 +n)(2 − n)T E%
kS0 T , (52)0
kA aT1
+0
kB aT1
+ 1 = −$
1 − (1 −n2)T E%
×2n(1 + n)kS0 (1 −n)E$0
kA aT1
+ 2(1 −n)0
kB aT1%
. (53) We remark again that the critical length bc, (2), is theresult of the approximation kA aT= 0,
0
kA aT1
+0
kB aT1
+ 1 = 0.Grilhe’s result, (5), is a consequence of replacing the first of the above with
0
kA aT1
=kS0
T .
In plane strain, the state of stresses (t11,t22,t33) = (T, 0,nT) is paired with (o11,o22,o33)=
[oT, −(n/1 − n)o, 0] where
In terms ofoTandS0of equation (43), the expression
bcand the associated wave numberkcmay be written
bc= 2p kc = pEG0 (1 −n2)T2= pG0 ToT =p(1 − n)S0 ToT . (55) It is now convenient to scale the wave numberk by kc, so that k = bkc, kS 0 E = 2bo2 G (1 −n)2(1 +n). (56)
The boundary conditions (52) and (53) may now be solved in terms of a power series of the small parameteroT. We have kA aT= 2b 1 −n oT− 6b 1 −n o 2 T+ · · · , (57) kB aT= −1 − 2b 1 −n oT+ 2b(3 + n) (1 −n)2 o 2 T+ · · · . (58)
The chemical potential C may now be computed from equation (22). Retaining terms linear in the amplitude a, we have C V=12ToT−
0
1 2ToT1
4ak6
−kB aT− 1 2(1 −n) kA aT − EkS0 2(1 +n)T2− kS0 2(1−n)T$
kA aT+ (3 − 2n) k B aT%
+ kS0 (1 −n)T$
1 + kA aT+ kB aT%7
coskZ1. (59)Making use of equations (57), (58) and retaining terms accurate to the order of oT, we finally obtain
C V=12ToT
$
1 − 4ak6
1 −b + 4b 1 −n oT +o2 T[4b2− (3 + 2n)b] 1 (1 −n)27
coskZ1%
. (60)If we treat the amplitude of equation (44) as a function of time, a(t), then the surface evolution governed by surface diffusion satisfies
1h 1t= D 1 2 1Z2 1 (C/V), (61)
where D is a positive diffusion constant. It follows that at= 4k3
0
1 2ToT1
D6
1 −b + 4b 1 −n oT +o2 T[4b2− (3 + 2n)b] 1 (1 −n2)7
a, (62)and the flat surface becomes unstable if b Q bc= 1 + 4 1 −n oT+ 17 − 2n (1 −n)2o 2 T (63) or, equivalently, bq bcw= bc
$
1 − 4 1 −n oT− 1 − 2n (1 −n)2o 2 T%
, (64)where bc is given by equation (2). It is seen that a
tensile applied stress enhances the instability whereas a compressive applied stress inhibits the instability region. It is important to note that the change is brought about by the correct equations (22)–(26), not merely the inclusion of the surface stress in the form of equation (14). Finally, in the context of epitaxially-strained solid films [12–14] it is not uncommon to findoT as large as24%.
4. SUMMARY AND DISCUSSION
Flat surfaces of stressed solids are unstable against diffusional perturbations with wavelengths larger than the critical wavelength bcgiven by equation (2).
In fact the maximally unstable mode has the wavelength b = bm= (4/3)bc. This result, however, is
independent of the sign of the underlying stress. Using a refined chemical potential and an explicit surface deformation-dependent surface energy, we have readdressed the instability phenomenon and obtained a new critical wavelength that is sensitive to the sign of the underlying stress. It is found that tensile stresses (positive oT) enhance the instability
while compressive stresses (negative oT) inhibit the
instability. The full implication of the full chemical potential on the nonlinear evolution of the surface remains to be calculated. It would be premature to predict at this point the exact nature of the equilibrium surface morphology resulting from the refined potential. It suffices to say that the outcome would be quite different from that of Spencer and Meiron [15].
It is well known that bulk strain plays an important role in determining surface morphology [16, 17]. The continuum approach presented in this paper ignores, by necessity, the presence of different surface steps. Based on molecular dynamics simulations, the effect of the sign of the underlying bulk strain on semiconductor surface roughness was investigated by Xie et al. [18]. It was found that the surface is flat (smooth) under tension but rough under compression.
The study employed Ge0.5Si0.5 films grown on
GexSi1 − x cap layers which, in turn, were grown
on compositionally graded buffer layers. The cap layers are therefore treated as the substrate in the following discussion. Using aSi= 0.5433 nm and
aGe= 0.5660 nm, we compute the effective lattice
spacing for the substrate as= xaGe+ (1 − x)aSi=
0.5433 + 0.0227x. The effective lattice spacing for the film is just af= 0.5546 nm. The misfit strain in the
film, considered as a function of x and denoted by oT(x), is just
oT(x) = (as− af)/af= −0.02(1 − 2x),
ranging from 2% tensile on a 100% Ge substrate to 2% compressive on a 100% Si substrate. Substantial surface roughness was found for the film under 2% compression, while no detectable roughness was observed for films under tensile strains of up to 2%. Based on the atomic force micrographs given in the paper, the most pronounced wavelength for the roughness is in the range of 0.05–0.1mm. Inserting G0= 1 J/m2, E = 160 GPa, n = 1/4 and oT=20.02
into equation (55) yields bc= 0.046mm. The refined
critical wavelength turns out to be bcw= (13 0.107)
0.046mm, where the signs apply to oT2 0.02. The
maximally unstable mode has the wavelength bm2 4/3 bcw= (13 0.107) 0.0612 mm. For a 2%
com-pressive strain, bm2 0.0677 mm. It is hoped that a
nonlinear evolution calculation based on our refined theory would lead to the desired steady film profile.
REFERENCES
1. Asaro, R. J. and Tiller, W. A., Metall. Trans., 1972, 3, 1789.
2. Grinfeld, M. A., Sov. Phys. Dokl., 1986, 31,
831.
3. Srolovitz, D., Acta metall., 1989, 37, 621. 4. Grilhe, J., Acta metall. mater., 1993, 41, 909. 5. Wu, C. H., J. Mech. Phys. Solids, in press.
6. Murr, L. E., Interfacial Phenomena in Metals and
Alloys. Addison-Wesley Publishing, Co., New York,
1975.
7. Gilman, J. J., J. appl. Phys., 1960, 31, 2208. 8. Rose, J. H., Ferrante, J. and Smith, J. R., Phys. Rev.
Lett., 1981, 47, 675.
9. Ferrante, J., Smith, J. R. and Rose, J. H., Phys. Rev.
Lett., 1983, 50, 1385.
10. Rose, J. H., Smith, J. R. and Ferrant, J., Phys. Rev. B., 1983, 28, 1835.
11. Rose, J. H., Smith, J. R., Guina, F. and Ferrante, J.,
Phys. Rev. B., 1984, 29, 2963.
12. Freund, L. B. and Jonsdottir, F., J. Mech. Phys. Solids, 1993, 41, 1245.
13. Gao, H., J. Mech. Phys. Solids, 1994, 42, 741. 14. Nix, W. D., Metall. Trans., 1989, 20A, 2217. 15. Spencer, B. J. and Meiron, D. I., Acta metall. mater.,
1994, 42, 3629.
16. Eaglesham, D. J. and Cerullo, M., Phys. Rev. Lett., 1990, 64, 1943.
17. van der Merwe, J. H., J. Electronic Mater., 1991, 20, 793.
18. Xie, Y. H., Gilmer, G. H., Roland, C., Silverman, P. J., Buratto, S. K., Cheng, J. Y., Fitzgerald, E. A., Kortan, A. R., Schuppler, S., Marcus, M. A. and Citrin, P. H.,
