Abstract
Elastically mediated interactions between surface domains are classically described in terms of point forces. Such point forces lead to local strain divergences that are usually avoided by introducing a poorly defined cutoff length. In this work, we develop a selfconsistent approach in which the strain field induced by the surface domains is expressed as the solution of an integral equation that contains surface elastic constants, S_{ij}. For surfaces with positive S_{ij} the new approach avoids the introduction of a cutoff length. The classical and the new approaches are compared in case of 1D periodic ribbons.
Introduction
The classical approach used to calculate the strain field that surface domains induce in their underlying substrate consists of modeling the surface by a distribution of point forces concentrated at the domain boundaries [13], the force amplitude being proportional to the difference of surface stress between the surface domains [36]. However, point forces induce local strain divergences, which are avoided by the introduction of an atomic cutoff length. Hu [7,8] stated that the concept of concentrated forces is only an approximation valid for infinite stiff substrates. Indeed if the substrate becomes deformed by the point forces acting at its surface, the substrate in turn deforms the surface and then leads to a new distribution of surface forces so that the surface forces have to be determined by a selfconsistent analysis. In this paper, we show that when elastic surface properties are properly considered, the strain field induced by the surface domains may be expressed as the solution of a selfconsistent integrodifferential equation.
Results and Discussion
Let us consider (see Figure 1a) a semiinfinite body whose surface contains two domains (two infinite ribbons) A and B characterized by their own surface stress s^{A} and s^{B}. The 1D domain boundary is located at x_{o} = 0. Note that for the sake of simplicity only the surface stress components are taken to be different from zero (see Appendix I for the Voigt notation of tensors).
In the classical approach [68] the strain field generated in the substrate is assumed to be generated by a line of point forces (with δ(x) being the Dirac function) and is given by:
where D_{xx}(x/x',z) is the xx component of the Green tensor and where the component f_{x}(x) = Δs_{1} originates from the surfacestress difference at the boundary between the two surface domains. The Green tensor valid for a semiinfinite isotropic substrate can be found in many text books [1,2,9] so that the deformation at the surface ε_{1}(x,z = 0) finally reads:
where with E_{subs} and ν_{subs} being the Young modulus and the Poisson coefficient of the substrate (supposed to be cubic). The strain at the surface (Equation 2) exhibits a local divergence at the boundary x = x_{0} = 0. The elastic energy can thus be calculated after introduction of an atomic cutoff length to avoid this local divergence [6,10].
However, the concept of point forces is only an approximation. If the substrate is deformed by point forces acting at its surface, the substrate in turn deforms the surface and then leads to a new distribution of surface forces. In the following, we consider that, due to the elastic relaxation, the surface stress at equilibrium exhibits a Hooke’slawlike behavior along the surface [9,11,12]:
with i = A, B according to whether x lies in region A or B. In Equation 3, is the surface stress far from the domain boundary (or in other words the surface stress before elastic relaxation) and the surface elastic constants properly defined in terms of excess quantities (see Appendix). The surface force distribution due to the surface stress variation (see Figure 1b) is obtained from force balance and reads f_{x}(x,z = 0) = ds_{1}/dx.
By using the Green formalism again, we obtain at the surface, z = 0:
where ε_{1,x} = dε_{1}/dx.
This equation replaces the classical result of Equation 2. Equation 4 is an integrodifferential equation that has to be solved numerically. At mechanical equilibrium the absence of surface stress discontinuity at the domain boundary, combined to the constitutive Equation 3 leads to the following boundary condition
When the elastic constants of the surface are positive, Equation 4 can be easily numerically integrated. Figure 2a shows (black dots) the result obtained by integration of Equation 4 with the boundary condition
that means for . We also plot in Figure 2a the classical result calculated from Equation 2 (continuous red curve). It is clearly seen that the new expression avoids the local strain divergence that is now replaced by a local strain jump Δs_{1}/S_{11} at x_{0} = 0.
Since the solutions of Equation 4 depend on the values of hS_{11} and Δs_{1} we report in Figure 2b the results obtained for different typical values of hS_{11} and Δs_{1} data obtained from [11]. More precisely, since the classical expression scales as 1/x, we plot ln ε versus x. As can be seen, in the limit of large x all solutions tends towards the classical one (common red asymptote in Figure 2b). Moreover we can clearly see that the classical approach is recovered in the limit S_{11}→ 0.
The elemental solution of Equation 4 enables to describe more complex experimental configurations as the one that corresponds to the spontaneous formation of 1D periodic stripes by a foreign gas adsorbed on a surface (as for instance O/Cu(110) [13]). In the classical model each stripe (width 2d) is modeled by two lines of point forces one located at d and the other at −d with the opposite sign f_{x}(x) = Δs_{1}(δ(x − d) − δ(x + d))) so that for a set of periodic ribbons of the period L the elastic field is obtained by a simple superposition of the elemental solutions given in Equation 2. In the classical case it reads
whereas within the new approach the elastic field is solution of the integral equation:
The results are shown in Figure 3 in which two cases are reported. In the first case d/L = 1/2, whereas in the second case d/L = 3/10. Again both solutions (classical and new approach) are quite similar since the only difference lies in the local divergences of the classical model (red curves in Figure 3) that are now replaced by local strain jumps.
For surfaces with negative surface elastic constants Equation 4 does not present stable solutions. It is quite normal since in this case, the surface is no more stable by itself but is only stabilized by its underlying layers (see Appendix I). From a physical point of view it means that, for mechanical reasons, we have to consider a “thick surface” or, in other terms, that the surface has to be modeled as a thin film the thickness a of which corresponds to the smaller substrate thickness necessary to stabilize the body (bulk + surface). It can be shown that this is equivalent to modify the integrodifferential equation for S_{11} < 0, by changing the kernel:
In Figure 4 we show the result obtained from numerical integration of Equation 8 for the test value hS_{11} = −0.01. In this case a = 2hS_{11} is the minimum value necessary to stabilize Equation 8. Since s_{1} is positive but S_{11} is negative, there is a sign inversion of ε close to the boundary. For vanishing a this local oscillation propagates on the surface and is at the origin of the instabilities that do not allow to find stable solutions to Equation 4. However we cannot exclude that the total energy of materials with s_{1}S_{11} < 0 could be reduced by some local morphological modifications of their surface. In such a case, the Green tensor used for this calculation should be inadequate.
In conclusion, the selfconsistent approach expressed in terms of surface elastic constants is more satisfactory than the classical approach, particularly in the case of stable surfaces (characterized by positive surface elastic constants) for which there is no need to introduce a cutoff length. In case of unstable surfaces (negative surface elastic constants) a cutoff length is still necessary, its value is connected to the minimum substrate thickness necessary to stabilize the body (surface + underlying bulk). Even if the model only deals with 1D structures it can be generalized to other structures such as 2D circular domains. The soobtained equations are less tractable but the main result remains the same (see Appendix II).
Appendix I: Surface elasticity
From a thermodynamic point of view all extensive quantities may present an excess at the interface between two media (for a review see [9]). For a system formed by a body facing vacuum the following excess quantities can be defined [9]:
where
is the second order strain development of the energy of a body of volume V_{0} limited by a surface of area A_{0} and
is the second order development of a piece of body of same volume V_{0} but without any surface. In these expressions are the bulk stress components and C_{ijkl} the bulk elastic constant.
The sodefined surface quantities depend on a typical length scale at which surface effects are disentangled from bulk effects. Actually, in surface energy calculations, this length is unambiguoulsy determined by a Gibbs dividing surface construction [14]. Surface stress and surface elastic constants values can thus be calculated from strain derivatives of the welldefined surface energy quantity [11].
In contrast to surface energy density and bulk elastic constants, surface stress components and surface elastic constants do not need to be positive. [9,11]. This does not violate the thermodynamical stability condition since actually a surface can only exist when it is supported by a bulk material. Hence the stability of the solid is ensured only by the total energy (surface + volume).
Finally, in the body of the paper we use the Voigt notation so that the surface stress can be written as the components of a 3D vector s = (s_{xx},s_{yy},s_{xy}) = (s_{1},s_{2},s_{6}), while surface and bulk elastic constants are written as the components of 3D matrices S_{ij} and C_{ij}, respectively.
Appendix II: 2D circular domains
In case of a circular domain of radius R, the classical approach considers a force distribution f_{r}(r) = Δs_{0}δ(r−R) that generates a displacement field expressed in terms of complete elliptic integrals K(x) and E(x) as:
In the distributed force model, we use the stressstrain relations valid at the surface expressed in polar coordinates:
again with the Voigt notation in polar coordinates A_{rr}≡A_{r}, A_{θθ}≡A_{θ}.
By using the classical mechanical equilibrium equation and strain–displacement relations expressed in polar coordinates we obtain the following force distribution
The displacement can thus be obtained from the selfconsistent equation (which replaces Equation 11)
The necessary boundary conditions, analog to Equation 5, must now be written for normal and tangential strains
The integral equation for the displacement field, Equation 15, only needs the surface elastic constant S_{11}, but the edge condition introduces the need of the other surface elastic constant S_{12}. Qualitatively the result is similar to the one shown in Figure 2.
Acknowledgments
We thank A. Saul for fruitful discussions. This work has been done thanks to PICS grant No. 4843 and ANR 13 BS000402 LOTUS Grant.
References

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1.  Mindlin, R. D. J. Appl. Phys. 1936, 7, 195–202. doi:10.1063/1.1745385 
2.  Landau, L.; Lifshitz, E. Theory of elasticity; Pergamon Press: Oxford, United Kingdom, 1970. 
3.  Maradudin, A. A.; Wallis, R. F. Surf. Sci. 1980, 91, 423. doi:10.1016/00396028(80)903428 
1.  Mindlin, R. D. J. Appl. Phys. 1936, 7, 195–202. doi:10.1063/1.1745385 
2.  Landau, L.; Lifshitz, E. Theory of elasticity; Pergamon Press: Oxford, United Kingdom, 1970. 
9.  Müller, P.; Saúl, A. Surf. Sci. Rep. 2004, 54, 157. doi:10.1016/j.surfrep.2004.05.001 
6.  Alerhand, O. L.; Vanderbilt, D.; Meade, R. D.; Joannopoulos, J. D. Phys. Rev. Lett. 1988, 61, 1973–1976. doi:10.1103/PhysRevLett.61.1973 
7.  Hu, S. M. J. Appl. Phys. 1979, 50, 4661–4666. doi:10.1063/1.326575 
8.  Hu, S. M. Appl. Phys. Lett. 1978, 32, 5–7. doi:10.1063/1.89840 
7.  Hu, S. M. J. Appl. Phys. 1979, 50, 4661–4666. doi:10.1063/1.326575 
8.  Hu, S. M. Appl. Phys. Lett. 1978, 32, 5–7. doi:10.1063/1.89840 
3.  Maradudin, A. A.; Wallis, R. F. Surf. Sci. 1980, 91, 423. doi:10.1016/00396028(80)903428 
4.  Marchenko, V. I. J. Exp. Theor. Phys. 1981, 54, 605–607. 
5.  Marchenko, V. I. J. Exp. Theor. Phys. 1981, 33, 381–383. 
6.  Alerhand, O. L.; Vanderbilt, D.; Meade, R. D.; Joannopoulos, J. D. Phys. Rev. Lett. 1988, 61, 1973–1976. doi:10.1103/PhysRevLett.61.1973 
9.  Müller, P.; Saúl, A. Surf. Sci. Rep. 2004, 54, 157. doi:10.1016/j.surfrep.2004.05.001 
11.  Shenoy, V. B. Phys. Rev. B 2005, 71, 094104. doi:10.1103/PhysRevB.71.094104 
13.  Kern, K.; Niehus, H.; Schatz, A.; Zeppenfeld, P.; Goerge, J.; Comsa, G. Phys. Rev. Lett. 1991, 67, 855–858. doi:10.1103/PhysRevLett.67.855 
9.  Müller, P.; Saúl, A. Surf. Sci. Rep. 2004, 54, 157. doi:10.1016/j.surfrep.2004.05.001 
14.  Nozières, P.; Wolf, D. E. Z. Phys. B: Condens. Matter 1988, 70, 399–407. doi:10.1007/BF01317248 
9.  Müller, P.; Saúl, A. Surf. Sci. Rep. 2004, 54, 157. doi:10.1016/j.surfrep.2004.05.001 
11.  Shenoy, V. B. Phys. Rev. B 2005, 71, 094104. doi:10.1103/PhysRevB.71.094104 
12.  Müller, P. Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity. In Mechanical Stress on the Nanoscale; Hanbrücken, M.; Müller, P.; Wehrspohn, R. B., Eds.; WileyVCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2011; pp 27–59. 
6.  Alerhand, O. L.; Vanderbilt, D.; Meade, R. D.; Joannopoulos, J. D. Phys. Rev. Lett. 1988, 61, 1973–1976. doi:10.1103/PhysRevLett.61.1973 
10.  Kern, R.; Müller, P. Surf. Sci. 1997, 392, 103–133. doi:10.1016/S00396028(97)005360 
9.  Müller, P.; Saúl, A. Surf. Sci. Rep. 2004, 54, 157. doi:10.1016/j.surfrep.2004.05.001 
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