Elasticité de surface P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004).
The concept I Physical origin Definition Applications Surface stress and crystallographic parameter of small crystals Surface stress plays a role each time a surface is deformed Calculation and measurements Calculations: main results Method of measurements
Surface stress: a simple approach Increase area upon creation (cleavage) Increase area upon stretching (deformation) W 2A A da W Surface energy Work per surface area : g [J/m^2] = W/2A Scalar, anisotropic Origin : missing neighbors Surface stress Work per surface area s ij [mj/m^2] = W/dA Tensor, anisotropic Origin : modification of the bond strength P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004).
g connected to broken bonds is the excess of grand potential s connected to surface forces is the excess of bulk stress g W 2A 2 1 A n It is a scalar 11 n2 2 Plays each time a surface is created... It is a 2D tensor ga s 1 A s s 0 xx yx s s xy yy 0 0 0 0 Plays each time a surface is deformed
A toy model surface bulk M-2 surface Slab Equivalent bulk Contribution of the surface
First neighbours interaction at the surface [ev] Justification of ks and as Dimers first neighbours distance [A] The bond strength increases when the coordination number decreases The effective pair interaction V eff increases V eff (surf) 2.5 2.0 1.5 1.0 0.5 Cd Zn Re V Cr Fe Zr Ru Ir Hf Rh Pt Ti CoNi La Y Pd Sc Cu Au Ag Mo W Ta Nb 0.0 0.0 0.5 1.0 1.5 2.0 2.5 First neighbours interaction in the bulk [ev] V eff (bulk) The inter-atomic distance d 0 3.5 3.0 d 0 (dimer) 2.5 2.0 decreases Rh Ru Mo Fe Co Pd Ag Ti Sc 2.0 2.5 3.0 3.5 Bulk first neighbours distance [A] d 0 (bulk) Y P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004).
Surface stress as an excess quantity Gibbs s excess of extensive quantities Exemples of extensive quantities: masse (adsorption) energy (surface energy) entropy The interfacial elastic energy is an extensive quantity
No relative gliding at the interface B Mechanical equilibrium at the interface A Surface stress Surface strain
For a free surface: dw surf el s d Surface stress thus is the isothermal work need to deform a free surface at constant number of surface atoms. Surface stress as an excess s s 0 xx yx s s xy yy 0 0 0 0 Surface strain as an excess 0 0 e zx e 0 0 yz e e e xz yz zz
Des questions de vocabulaire et de référentiel E s g E E g 0 L s g L 0 Formulae written for isotropic case R.Schuttleworth, Proc. Roy. Soc. London 163 (1950) 644 Excès de contrainte Excès d énergie Solides Contrainte s = g dg/d Energie g Liquides Tension s = g Tension g
Theoretical II Physical origin Discussion about its definition Applications Surface stress and crystallographic parameter of small crystals Surface stress plays a role each times a surface is deformed Calculation and measurements Calculations: principle and examples Method of measurements Towards surface elasticity constants
Mesures de contraintes de surface : déformation spontanée d un cristal Cristal soumis à son propre surface stress s A À l équilibre 2 W f ( C) g( s) * 1 xx 4 2 E Contraction du cristal soumis à ses propres contraintes de surface A A s A Cas 2D: W / V W Ah 1 F.Streitz et al. PRB 49, 1994, 10699 1 W / A he 2 2s 2 3 E W / V 2 ' 2 * 2s Eh E(1 Bs/ h)
Theoretical III Physical origin Discussion about its definition Applications Surface stress and crystallographic parameter of small crystals Example of a thermodynamic process in which s plays a role Calculation and measurements Calculations: principle and examples Method of measurements Towards surface elasticity constants
Calcul et mesure des contraintes de surface Une grandeur physique délicate à calculer Pour un matériau pur (Ag) Energie d un atome isolé : 144000 ev Energie de cohésion : 3 ev Energie de surface : 0.7 ev Energie d une déformation de =0.01 : 0.015 ev Une grandeur physique difficile à mesurer : pas de valeurs expérimentales absolues
Differences between surface energy and surface stress plots Surface energy one branch (scalar) minima at low index Surface stress two branches (parallel) red (perpendicular to the steps) most disperse than the green (parallel) diagonal at high symmetry points maxima at low index orientations SMA potential P. Muller and A. Saul Surf. Sci Rep. 54, 157 (2004).
Surface stress and adsorption Laser beam S1 interferogram Si(001) CCD S2 separator x mirror z Silver cell G.Degand, P.Müller, R.Kern, Surf. Rev. Lett. 4 (1997) 1047 Ni/W110) DS(J/m2) z( mm) 0 0-0.1-1 -0.2-0.3-2 -3-4 -0.4-5 -0.5-0.6-0.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 q(mc) -6-7 -8 D.Sander, PRB 57, 1998, 1406
1. Definition Conclusions From the excess of elastic energy one can define surface stress and surface strain in the Gibb s sense. Surface stress plays a role each time a surface is deformed Surface stress is a size effect 2. Surface stress calculation Larger anisotropy than for surface energy Larger sensitivity to surface realxation Maxima of s correspond to minima of g 3. Experimental determination of anisotropy of Si From small crystal deformation From curvature measurements From facetting