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Nonlinear contributions to the relation between surface stress and surface energy
Surface Science 0 North-Holland
57 (1976) 446-448 Publishing Company
NONLINEAR
CONTRIBUTIONS
TO THE RELATION
BETWEEN SURFACE STRESS AND SURFACE ENERGY Received
12 November
1975; manuscript
received
in final form
12 February
1976
Shuttleworth [I], Herring [2] and ~ull~ns [3] have shown that the surface stress tensor is related to the surface energy for a planar surface lying normal to the x3 direction by the relation
where i, j, k, 1, m, n = I, 2 and 6ij is the Kronecker delta function. The surface energy and surface stress as usual [2] are referred to the reference state of a Gibbs dividing surface. The above expression is a slight, but obvious, modification of the earlier treatments [l-3], in that the initial state for the infinitesimal reversible change of the system, by an imposed 6eij, is a state of strain E,, . Eq. (I) has been verified to high precision in an atomic computer simulation study [4,5] of bee iron surfaces, using the empirical potential of Johnson [6]. The details of the calculations are presented in refs. [4] and [S]: briefly, a simulated crystal was strained macroscopically with and without Poisson contraction, i.e. constraint in the x,~ direction, allowed to relax to satisfy free surface boundary conditions, strained a small increment 6eii under conditions where other strain increments were held constant and where relaxation both was and was not subsequently pe~itted in the x3 direction, and y computed. The Gibbs dividing surface was selected such that there was no excess mass. Surface stresses at emn were determined both from eq. (1) and directly by the technique [2,7] of making a virtual cut normal to the surface and containing the line aj on which ~j acts, summing virtual forces fin) acting on the cut and referring these to a reference force fiacting on the Gibbs dividing surface. For the strained case the equivalent forces FF(‘) acting in the bulk must be subtracted, so f.. = _ c (@r)-@(n9 V n ai
(2)
While the potential used is artificial, it did permit calculations to six-figure accuracy and with initial strains extending into the nonlinear elastic regime. Some results for the case of uniaxial strain with 92 = 0 are presented in table 1. The surface stresses also were nonlinear in response to the strain range shown. Since the surfave relaxations of the layers nearest the free surface were large, corresponding to a tensile strain of seven percent, well into highly nonlinear behavior, we tested the
J.P. Hirth, C. W. Price/Relaxation
between surface stress and surface energy
447
Table 1
Surface energies 7 and surface stresses for an (001) iron surface, relaxed over 20 atom layers, as a function of strain ~11 El1
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
-Y
fll from
fil from
fll from
(J/m*)
eq. (1) (N/m)
eq. (2) (N/m)
eq. (4) (N/m)
1.272613 1.273063 1.273505 1.273940 1.274368 1.274788 1.275201 1.275606 1.276005 1.276396 1.276780
1.72573 1.71890 1.71206 1.70520 1.69832 1.69144 1.68454 1.67762 1.67069 1.66375 1.65679
1.72573 1.71890 1.71206 1.70520 1.69833 1.69144 1.68454 1.67762 1.67069 1.66375 1.65680
1.12573 1.71885 1.71196 1.70506 1.69815 1.69124 1.68432 1.67739 1.67046 1.66351 1.65656
surface stress data to determine how well it could be fitted by perturbation theory, i.e. by introducing the equivalent of third-order elastic constants. This could provide some guideline for the extrapolation of eqs. (1) and (2) for other highly strained situations such as that near the tip of a sharp crack. To second order, the expansion of eq. (1) about the zero-strain value is
(3) it being understood that T and other strain components are held constant partial differentiation. For the data of table 1, eq. (3) reduces to
in a given
=~~+(~)o+[(-$)o+(~)J ‘11
l-11
+;[($)o+(f$J41 +....
(4)
Values for the derivatives in eq. (4) were obtained in a straightforward manner by the finite difference method, giving (&y/&ll)o = 0.453 J/m2, (?12r/&:,), = -7.333 J/m2, (a3~/&~,), = 0. Results forfll from eq. (4) using these numbers are given in table 1. As can be seen the agreement is very good. Even at one percent elastic strain, which is a reasonable upper limit for experiment, corresponding to about the elastic limit for erfect metal w_hiskers, the deviation is less than 0.015%. At ell = 0.01, the term (a ! r/&e~,), contributes 35% of the variation in fll . The
448
J.P. Hirth, C. W. PricefRelaxaiion
between surface stress and surface energy
0.015% error arises because of the roundoff error in the computations which, of course, have the largest effect in the (a2r/&f1)u term. The fit of eqs. (3) with (1) and (2) would be even better if (a2r/ae;t)u were adjusted to -7.311 J/m2 which is within the accuracy in the second derivative arising from eight-place accuracy in the atomic calculations (such improvement in fits of truncated Taylor series by small adjustments of coefficients is well known mathematically). Thus, even though the surface layer is strained by seven percent, the surface stress, which to some extent reflects such surface relaxations, can be accurately modelled by secondorder terms in an expansion about zero strain, equivalent to the inclusion of thirdorder surface elastic constants. This research was supported by the National Science Foundation under Grant Gh-34568X. Comments by a referee on the properties of general Taylor series were helpful. J.P. IIIRTII
Department of Metallurgical Engineering, The Ohio State University, Columbus, Ohio 43210, USA and C.W. PRICE
Metal Sciences Group, Battelle Columbus Laboratories, Columbus, Ohio 43201, USA
References [l] R. Shuttleworth, Proc. Phys. Sot. (London) A63 (1950) 444. [2] C. Herring, in: Structure and Properties of Solid Surfaces (Univ. Chicago Press, Chicago, 1952) p. 5. [3] W.W. Mullins, in: Metal Surfaces (Am. Sot. Metals, Cleveland, 1963) p. 17. [4] C.W. Price, Ph.D. Thesis, Ohio State Univ., Columbus (1975). [5] C.W. Price and J.P. Hirth, Surface Sci. 57 (1976), to be published. [6] R.A. Johnson, Phys. Rev. 134A (1964) 1329. [7] J.P. Hirth, in: Structure and Properties of Metal Surfaces (Maruzen, Tokyo, 1973) p. 10.
57 (1976) 446-448 Publishing Company
NONLINEAR
CONTRIBUTIONS
TO THE RELATION
BETWEEN SURFACE STRESS AND SURFACE ENERGY Received
12 November
1975; manuscript
received
in final form
12 February
1976
Shuttleworth [I], Herring [2] and ~ull~ns [3] have shown that the surface stress tensor is related to the surface energy for a planar surface lying normal to the x3 direction by the relation
where i, j, k, 1, m, n = I, 2 and 6ij is the Kronecker delta function. The surface energy and surface stress as usual [2] are referred to the reference state of a Gibbs dividing surface. The above expression is a slight, but obvious, modification of the earlier treatments [l-3], in that the initial state for the infinitesimal reversible change of the system, by an imposed 6eij, is a state of strain E,, . Eq. (I) has been verified to high precision in an atomic computer simulation study [4,5] of bee iron surfaces, using the empirical potential of Johnson [6]. The details of the calculations are presented in refs. [4] and [S]: briefly, a simulated crystal was strained macroscopically with and without Poisson contraction, i.e. constraint in the x,~ direction, allowed to relax to satisfy free surface boundary conditions, strained a small increment 6eii under conditions where other strain increments were held constant and where relaxation both was and was not subsequently pe~itted in the x3 direction, and y computed. The Gibbs dividing surface was selected such that there was no excess mass. Surface stresses at emn were determined both from eq. (1) and directly by the technique [2,7] of making a virtual cut normal to the surface and containing the line aj on which ~j acts, summing virtual forces fin) acting on the cut and referring these to a reference force fiacting on the Gibbs dividing surface. For the strained case the equivalent forces FF(‘) acting in the bulk must be subtracted, so f.. = _ c (@r)-@(n9 V n ai
(2)
While the potential used is artificial, it did permit calculations to six-figure accuracy and with initial strains extending into the nonlinear elastic regime. Some results for the case of uniaxial strain with 92 = 0 are presented in table 1. The surface stresses also were nonlinear in response to the strain range shown. Since the surfave relaxations of the layers nearest the free surface were large, corresponding to a tensile strain of seven percent, well into highly nonlinear behavior, we tested the
J.P. Hirth, C. W. Price/Relaxation
between surface stress and surface energy
447
Table 1
Surface energies 7 and surface stresses for an (001) iron surface, relaxed over 20 atom layers, as a function of strain ~11 El1
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
-Y
fll from
fil from
fll from
(J/m*)
eq. (1) (N/m)
eq. (2) (N/m)
eq. (4) (N/m)
1.272613 1.273063 1.273505 1.273940 1.274368 1.274788 1.275201 1.275606 1.276005 1.276396 1.276780
1.72573 1.71890 1.71206 1.70520 1.69832 1.69144 1.68454 1.67762 1.67069 1.66375 1.65679
1.72573 1.71890 1.71206 1.70520 1.69833 1.69144 1.68454 1.67762 1.67069 1.66375 1.65680
1.12573 1.71885 1.71196 1.70506 1.69815 1.69124 1.68432 1.67739 1.67046 1.66351 1.65656
surface stress data to determine how well it could be fitted by perturbation theory, i.e. by introducing the equivalent of third-order elastic constants. This could provide some guideline for the extrapolation of eqs. (1) and (2) for other highly strained situations such as that near the tip of a sharp crack. To second order, the expansion of eq. (1) about the zero-strain value is
(3) it being understood that T and other strain components are held constant partial differentiation. For the data of table 1, eq. (3) reduces to
in a given
=~~+(~)o+[(-$)o+(~)J ‘11
l-11
+;[($)o+(f$J41 +....
(4)
Values for the derivatives in eq. (4) were obtained in a straightforward manner by the finite difference method, giving (&y/&ll)o = 0.453 J/m2, (?12r/&:,), = -7.333 J/m2, (a3~/&~,), = 0. Results forfll from eq. (4) using these numbers are given in table 1. As can be seen the agreement is very good. Even at one percent elastic strain, which is a reasonable upper limit for experiment, corresponding to about the elastic limit for erfect metal w_hiskers, the deviation is less than 0.015%. At ell = 0.01, the term (a ! r/&e~,), contributes 35% of the variation in fll . The
448
J.P. Hirth, C. W. PricefRelaxaiion
between surface stress and surface energy
0.015% error arises because of the roundoff error in the computations which, of course, have the largest effect in the (a2r/&f1)u term. The fit of eqs. (3) with (1) and (2) would be even better if (a2r/ae;t)u were adjusted to -7.311 J/m2 which is within the accuracy in the second derivative arising from eight-place accuracy in the atomic calculations (such improvement in fits of truncated Taylor series by small adjustments of coefficients is well known mathematically). Thus, even though the surface layer is strained by seven percent, the surface stress, which to some extent reflects such surface relaxations, can be accurately modelled by secondorder terms in an expansion about zero strain, equivalent to the inclusion of thirdorder surface elastic constants. This research was supported by the National Science Foundation under Grant Gh-34568X. Comments by a referee on the properties of general Taylor series were helpful. J.P. IIIRTII
Department of Metallurgical Engineering, The Ohio State University, Columbus, Ohio 43210, USA and C.W. PRICE
Metal Sciences Group, Battelle Columbus Laboratories, Columbus, Ohio 43201, USA
References [l] R. Shuttleworth, Proc. Phys. Sot. (London) A63 (1950) 444. [2] C. Herring, in: Structure and Properties of Solid Surfaces (Univ. Chicago Press, Chicago, 1952) p. 5. [3] W.W. Mullins, in: Metal Surfaces (Am. Sot. Metals, Cleveland, 1963) p. 17. [4] C.W. Price, Ph.D. Thesis, Ohio State Univ., Columbus (1975). [5] C.W. Price and J.P. Hirth, Surface Sci. 57 (1976), to be published. [6] R.A. Johnson, Phys. Rev. 134A (1964) 1329. [7] J.P. Hirth, in: Structure and Properties of Metal Surfaces (Maruzen, Tokyo, 1973) p. 10.