H.2 Euler-Maclaur in Sum Formula
14.5 Equilibrium Shape from the ξ -Vector
Provided thatγ (nˆ)is differentiable, we can proceed to find an analytical formula for the equilibrium shape of a solid in contact with a fluid. Places where it is not differentiable can be handled as limiting cases as explained inSection 14.3.1. We proceed to minimize the grand potentialK for the entire solid, assumed to be constrained to have a fixed volume and maintained at fixed temperatureTand chemical potentialsμi. We write this potential in the form
K=
Vs
ωsvdV+
VF
ωvFdV+
Aγ (nˆ)dA, (14.79)
whereωsv is the grand potential per unit volume in the solid, which may be crystalline or amorphous, ωFv is the grand potential per unit volume in the fluid,Vs is the volume of the solid, VF is the volume of the fluid,A is the area of the interface that separates the solid from the fluid, andnˆ points from solid to fluid. For the moment, we assume that the areaAis bounded by some closed curveC. We presume that the interface can be represented in terms of parametersu,vby the equationr=r(u,v), as discussed in detail in Appendix C, Section C.2. We write the equilibrium shape in the formr=r0(u,v)and make an infinitesimal normal variation to a new position,
r=r0(u,v)+ ˆn0(u,v)η(u,v)≡r0(u,v)+δr(u,v), (14.80) where the infinitesimal quantityη(u,v)is arbitrary but differentiable. Then the variation of the total Kramers (grand) potential is
δK=
A(ωsv−ωFv)η(u,v)dA+δ
Aξã ˆndA, (14.81)
where we have replacedγ byξã ˆn. The second area integral can be written in the form δ
Aξã ˆndA=δ
u,vξãHdudv, (14.82)
whereH = ru×rv, withru = ∂r/∂uandrv = ∂r/∂v. Note thatnˆ = H/H. The second integral in Eq. (14.82) is over a fixed domain inu,v space. Thus we can takeδ inside the integral to obtain
δ
u,vξãHdudv =
u,vξãδHdudv, (14.83)
whereHãδξ = 0 because of Eq. (14.35). Then, as shown by Eq. (C.46) in Appendix C, Eq. (14.80) leads to
δH= ˆn0(u,v)H0(u,v)η(u,v)K0(u,v)−H0(u,v)∇sη(u,v), (14.84) whereK0(u,v)is the mean curvature of the surface in the unvaried state (see Eq. (C.28) for a general formula) and∇s is the surface gradient operator defined by Eq. (C.35). Thus Eq. (14.81) becomes
δK=
A(ωsv−ωFv)η(u,v)dA+
A
γKη(u,v)−ξã ∇sη(u,v)
dA, (14.85)
where we have dropped the zero subscripts. Next, we prepare to integrate by parts by writing
−ξã ∇sη(u,v)= −∇sã [ξη(u,v)] +η(u,v)∇sãξ, (14.86) where the terms on the right-hand side contain the surface divergence. According to the surface divergence theorem, Eq. (C.49), we have
A∇sã [ξη(u,v)]dA=
Cξtã ˆtη(u,v)d+
AγKη(u,v)dA, (14.87) wheretˆis a unit tangent vector pointing out of the area along the curveC, specifically ˆ
td=d× ˆn, where the direction ofˆtdis determined by the right-hand rule. Thus δK=
A
(ωsv−ωvF)+ ∇sãξ
η(u,v)dA−
Cξtã ˆtη(u,v)d. (14.88) To guarantee that no work is done along the curveC, we can takeη(u,v)=0 alongCand the equilibrium criterion becomes
0=δK=
A
(ωsv−ωFv)+ ∇sãξ
η(u,v)dA. (14.89)
Then sinceη(u,v)is arbitrary over the areaA, the integrand must vanish, and we obtain the equilibrium condition
ωFv−ωsv= ∇sãξ. (14.90)
If the solid is amorphous and therefore isotropic,ξ =γn,ˆ ∇sãξ =γKby Eq. (C.38), and ωFv−ωvs =ps−pF, so the Laplace equation (Eq. (13.71)) for fluids would apply.
Equation (14.90) is a nonlinear partial differential equation for the solid-fluid interface shape, so one would have to find a solution that attached to the bounding curveC, a difficult task. On the other hand, for aclosedsurface, the curveCcloses back on itself and the line integral in Eq. (14.88) vanishes without restriction onη(u,v). Then, since∇sãr=2 (see Eq. (C.41)) an obvious solution to Eq. (14.90) is
r= 2
(ωFv−ωsv)ξ, (14.91)
which is the equation for the equilibrium shape of a crystal.12 Note that a result of the same form would be obtained if one varied only the shape of the body while holding its volume constant. This could be done by using a Lagrange multiplier to put in the volume constraint. The present method identifies that Lagrange multiplier in terms of physical quantities so we obtain the size of the crystal in addition to its shape.
Let us return to Eq. (14.88) for a bounding curveCthat can move in a manner described by Eq. (14.80). Then the work done by an external force workfLper unit length is given by
12As explained in connection with the Wulff theorem, one must truncate the ears to get a convex body if there are missing orientations. Note in two dimensions that∇sãr=1, so the factor of 2 would be missing.
238 THERMAL PHYSICS
δW =
C
fLã ˆnη(u,v)d. (14.92)
The equilibrium criterion now becomesδK−δW =0. Equation (14.90) still holds over the area, but along the curveCwe would need
−
Cξtã ˆtη(u,v)d−
C
fLã ˆnη(u,v)d=0. (14.93) By usingtˆd=d× ˆnin the first integral, Eq. (14.93) becomes
C
ξt×(d/d)+fL
ã ˆnη(u,v)d=0. (14.94) Sinceη(u,v)is arbitrary alongC, we conclude that
fL+ξt×(d/d)
ã ˆn=0, (14.95)
which gives a normal force for curved surfaces that is the same as given by Eq. (14.58) for a planar surface. By considering a variation of the curveCin the tangential direction ˆ
tinstead of Eq. (14.80) one can obtain the tangential component of Eq. (14.58). It must be borne in mind, however, that this is only valid if the state of strain of the solid is not affected by the variation; otherwise, one would obtain the surface stress instead ofγ for a tangential force per unit length, as discussed inSection 14.1.2.
To evaluate the quantityωFv−ωvs for a single component, one usually uses
d(ωFv−ωsv)= −(sFnF−ssns)dT−(nF−ns)dμ, (14.96) which is only valid if the effect of shear strain in the solid can be ignored. Here, sF is the entropy per mole of the fluid,ss is the entropy per mole of the solid,nF is the molar density of the fluid,nsis the molar density of the solid,Tis temperature, andμis chemical potential. For the fluid, one also has
dμ= −sFdT+(1/nF)dpF, (14.97) wherepFis the pressure of the fluid. Then
d(ωFv−ωsv)= −ns(sF−ss)dT−(nF−ns)
nF dpF. (14.98)
We can examine two states, both at the same value ofpF. One such state corresponds to a planar interface for an infinite crystal, soωFv −ωvs = 0 andT becomes the nominal melting pointTMfor that chosen pressure, where we have chosen the fluid to be a liquid for the sake of illustration. The other state corresponds to the equilibrium state of a small crystal in equilibrium with its liquid melt at temperatureT. Then integration of Eq. (14.98) withLV:=ns(sF−ss)TM, the latent heat per unit volume of solid, assumed to be constant, gives
(ωFv−ωsv)=LV(TM−T)
TM . (14.99)
Thus Eq. (14.90) becomes
T=TM−(TM/LV)∇sãξ, (14.100) which is a form of theGibbs-Thomson equation for anisotropic γ. For isotropic γ it becomes
T =TM−(TMγ /LV)K, (14.101)
which is well known.
Another important option is to keepTfixed in Eq. (14.96) but allowμto vary from its valueμ∞for an infinite crystal withωFv −ωsv =0 to its valueμfor a finite crystal. Then by treating0=:(ns−nF)−1as a constant, Eq. (14.98) can be integrated to obtain
μ=μ∞+0∇sãξ. (14.102)
If∇s ãξ is evaluated at a point on the surface, Eq. (14.100) is equivalent to the Herring equation, which usually pertains to the case in which the fluid is a gas with negligible density, so that 0 ≈ (ns)−1. In the next section, we will develop that equation in detail.
The derivation of Eq. (14.90) was carried out in the context of global equilibrium between a crystal and a fluid, so that Eq. (14.91) is an equation for the equilibrium shape of the crystal. Under those conditions, the fluid is homogeneous, so its temperature and chemical potentials are uniform. On the other hand, Eqs. (14.100) and (14.102) are frequently regarded aslocal equilibriumconditions that apply at the surface of a crystal having any shape. In that case, for example, Eq. (14.102) would lead to a chemical potential that varied along the surface of the crystal. Such a nonuniform chemical potential would provide a driving force for diffusion processes that would lead to shape changes of the crystal and eventually to an equilibrium shape and a uniform chemical potential. For multicomponent systems, an equation similar in form to Eq. (14.100) can be obtained if the chemical potentialsμFi of the fluid can be maintained at fixed values. Then only the term in dT in Eq. (14.96) survives and Eq. (14.100) applies with LV replaced by LV = (sFnF −ssns)TM, where nowTM is understood to be the local bulk melting point of the multicomponent alloy. Note that it is not so easy to extend Eq. (14.102) to a multicomponent system becauseω = uV −TsV −
μini, so more than one chemical potential is involved. Such an extension is sometimes made, however, to the case of a Gibbs solid that has a fixed composition (Gibbs called this the substance of the solid) and does not contain other chemical components (if any) that are contained in the fluid.
For such a solid, the chemical potential μA of the substance of the solid (regarded as a supercomponentA that is made up of appropriate components of the fluid in fixed proportions) would obey Eq. (14.102) at its surface. The chemical potentials of any other components of the fluid would be unconstrained.
240 THERMAL PHYSICS