H.2 Euler-Maclaur in Sum Formula
14.1 Planar Solid-Fluid Interfaces
We now treat planar interfaces, such as depicted in Figure 13–1, except that one phase is a solid (superscript s) and the other is a fluid (superscriptF). The bulk solid is assumed to be homogeneous; in particular, it is in a state of homogeneous stress and strain. If the solid is a crystal, we treat a constrained equilibrium such that the planar interface has a definite direction with respect to the crystallographic axes. Such an interface might not be stable with respect to breakup into a hill and valley structure (made up of facets) but we will examine this possibility later inSection 14.4. For amorphous solids, stability of a planar interface with respect to faceting is not an issue.
When a bulk solid is in equilibrium at temperature T with a bulk fluid across a hypothetical surface element with normal nˆ pointing from solid to fluid, the following boundary conditions are valid at that surface element [31–33]:
usv−Tssv−
i
μFiρsi = −pF; (14.1)
α
nβσαβ= −pFnβ, (14.2)
whereusv is the internal energy density of the solid, withαandβ representing Cartesian coordinates, ssv is the entropy density of the solid,ρis is the partial density of chemical component i in the solid, σαβ is the symmetric Cauchy stress tensor in the solid, pF is the pressure of the fluid, and μFi is the chemical potential of component i in the fluid. Equation (14.2) is just a balance of forces at the surface element. If we take the surface element to be perpendicular to thez-axis, it becomesσzz= −pF,σxz =σyz=0, consistent with the fact that a fluid in equilibrium cannot support shear. Equation (14.1) is a thermodynamic condition. If the mobility of the chemical components of the solid was unrestricted and the solid was in chemical equilibrium,1its chemical potentialsμsi =μFi. If the solid behaved like a fluid, it would be in a state of hydrostatic stress, soσαβ= −psδαβ
and the left-hand side of Eq. (14.1),viaits Euler equation, would be the negative of its
1Generally speaking, solids are quite rigid and mobility of chemical components within them is quite slow, although not zero. On practical time scales, mobility of such components can sometimes be ignored. This leads to the concept of a Gibbs solid in which the “substance of the solid” is fixed and immobile. Alternatively, movement of solid components can be allowed to occur but restricted to obey certain rules. For example, in a Larché-Cahn (LC) solid [31, 32], components that can only reside on a lattice are allowed to move only by virtue of exchange with point defects, namely lattice vacancies. For the LC solid, and with vacancies assumed to be a conserved species within a single crystal, LC define diffusion potentialsMithat are formally equal to the differences of chemical potentials of chemical components and chemical potentials of vacancies, calculated in that extended description. So for an LC solid, theirMiwould be equal to ourμsi.
pressureps. In that case, Eqs. (14.1) and (14.2) would coalesce and become simplyps =pF. But a solid in a general state of stress has no such simple Euler equation.
For a homogeneous solid, however, the left-hand side of Eq. (14.1) is uniform (indepen- dent of position) so one can multiply by the volume of the solidVsto obtain
Us−TSs−μFiNis= −pFVs, pseudo-Euler, homogeneous cylindrical solid, (14.3) which resembles an Euler equation except thatμFi andpF pertain to the fluid. If such a homogeneous solid were surrounded by a fluid, Eq. (14.2) would compel the solid to be in a state of hydrostatic stress. On the other hand, for aconstrained equilibriumin which a homogeneous cylindrical solid that is only in contact with the fluid across a planar interface perpendicular to the generators of its cylindrical surface, Eq. (14.3) applies and the solid can be in a state of nonhydrostatic stress. The fact that Eq. (14.3) also applies to a homogeneous solid and a homogeneous liquid separated by an actual planar region of discontinuity can be seen by considering a layer bounded by imaginary planes located in homogeneous phases on opposite sides of the region of discontinuity, just as was done for fluid-fluid interfaces. Then one can study variations in which the layer is unchanged but translates intact in either direction perpendicular to the planes that bound it. For such variations, changes of the homogeneous phases are the same as if the layer did not exist and were replaced by a mathematical plane.
Armed with the pseudo-Euler Eq. (14.3), we can define an excess pseudo-Kramers potential for a system having a homogeneous solid and a homogeneous liquid and a planar solid-fluid planar Gibbs dividing surface by means of the equation
Kxs=U−TS−
i
μFiNi−(Us−TSs−μFiNis)−(UF−TSF−μFiNiF)
=Uxs−TSxs−
i
μFiNixs=U−TS−
i
μFiNi+pFV, (14.4) where the last expression is clearly independent of the location of the dividing surface that separates the homogeneous solid from the homogeneous fluid. Here,Uxs=U−Us−UF, Sxs = S−Ss−SF, andNixs = Ni−Nis−NiFbutVxs = V −Vs−VF = 0, since the bulk phases meet at the dividing surface. Then we can define an excess potential per unit area by dividing by a suitable area. Following Cahn [28] or [29, pp. 379-399], we distinguish two cases,γ obtained by dividing by the areaAof the actual strained state and γ0 obtained by dividing by the areaA0of a homogeneous reference state of the solid, by definition the state of zero strain. Specifically,
γA=γ0A0=U−TS−
i
μFiNi+pFV=Uxs−TSxs−
i
μFiNixs. (14.5)
We could also use these same definitions ofγ andγ0for alayer model, similar to that for the fluid-fluid case, Eqs. (13.33)–(13.36), which gives
γA=γ0A0=U−TS−
i
μFiNi+pFV=UL−TSL−
i
μFiNiL+pFVL, (14.6)
218 THERMAL PHYSICS
where the bulk phases only extend to the imaginary planes that bound the layer, soVL = V −Vs−VF =0. As in the fluid case, most of these excess quantities and layer quantities have no physical meaning because they depend on the location of the dividing surface or the bounding planes, but their combinations used to formγ orγ0are independent of these locations and do have physical meaning.
We treat the more general layer model first and then indicate the slight modification needed to treat the dividing surface model. To do this, we adopt an equation for dUL of the form
dUL=TdSL−pFdVL+
i
μFidNiL+A
αβ
fαβLdεαβ, (14.7)
which is similar to Eq. (13.41) except for the last term. This last term accounts forstretching of the interface that accompanies straining of the bulk solid and replaces γdA for a fluid-fluid interface. Here, the Cartesian indices α and β take on the values x and y for an interface perpendicular to the z-direction, as above. The 2 ×2 tensor εαβ is a symmetric strain tensor (see Eq. (14.17))measured in the bulk homogeneous solidandfαβL is a symmetric stress tensor. This stress tensor must be consistent with the symmetry of the underlying solid, anisotropic if crystalline and isotropic if amorphous.
14.1.1 Adsorption Equation in the Reference State
By combining Eq. (14.7) with the differential of Eq. (14.6), and recognizing that dA0 = 0, we obtain
dγ0= −SL A0
dT+VL A0
dpF−
i
NiL A0
dμFi + A A0
α,β
fαβL dεαβ, (14.8) which is the counterpart to the Gibbs adsorption equation for fluids, Eq. (13.44). Similar to the fluid case, the variables T, μFi, pF, and εαβ are not independent because of the equations of bulk equilibrium of the solid and fluid phases. Two of these can be chosen as dependent variables and their differentials expressed in terms of the differentials of the others, most elegantly by using the determinant formalism discussed in terms of Cahn’s layer model of fluids in Section 13.1.3. To do this, we need a Gibbs-Duhem equation for the fluid, which is just
SFdT−VFdpF+
i
NiFdμFi =0, (14.9)
but also an equivalent Gibbs-Duhem equation for a cylinder of homogeneous bulk solid in equilibrium with that fluid across a plane perpendicular to thez-axis. This equation can be written in the form
SsdT−VsdpF+
i
NisdμFi −Vs
αβ
σαβlatdεαβ=0, (14.10) where
σα,βlat ≡
κ,λ
∂xα
∂xκ(σκλ+pFδκλ)∂xβ
∂xλ. (14.11)
The last term in Eq. (14.10), in which all sums are only overxandycoordinates, is present because for a cylindrical solid in a nonhydrostatic homogeneous state of stress, the forces needed to stretch it laterally are different from those needed to stretch it longitudinally.
Here,σκλis the Cauchy stress tensor of the homogeneous solid, the coordinatesxare those of a hydrostatic reference state, and the coordinatesxare those of the actual state. If the solid were in a state of hydrostatic stress in its actual state,σκλ= −pFδκλand the last term would vanish. We can therefore write (with a notation similar to Eq. (13.46))
dγ0= −[(SL/A0)/XY]dT+ [(VL/A0)/XY]dpF− κ
i=1
[(NiL/A0)/XY]dμFi +
α,β
fαβC dεαβ, (14.12)
whereXandYare the extensive conjugates to two distinct intensive variables of the setT, μFi,pF that are chosen to be dependent variables.2As with fluids, the two coefficients of the dependent intensive variables will vanish. Here,
fαβC ≡ 1 A0
AfαβL XL YL Vsσαβlat Xs Ys 0 XF YF Xs Ys
XF YF
(14.13)
and is independent of the choice of the planes that bound the layer, although it does depend on the choice of independent variables. Consequently,
∂γ0
∂εαβ
A0andκindependent intensive variables=fαβC. (14.14) The 2×2 tensorfαβC is the surface stress defined by Cahn [28] or [29, pp. 379-399]. As he points out, the application of tractions to the bulk solid usually produces only a small shift in the other intensive variables and can frequently be ignored. If the actual state of the solid is hydrostatic,σαβlat = 0 and we have simply fαβC = (A/A0)fαβL. If the actual state is taken to be a state of zero strain (coincident with a hydrostatic reference state), then fαβC =fαβL.
Note that Eq. (14.12) also holds formally for the Gibbs excess quantities with the understanding thatVL=0, which does not require the coefficient of dpFto be zero unless eitherXorY is chosen to beV.
2We retainεαβas independent variables because the application of tractions to the solid makes only a small second-order change to the relationships among the setT,μFi,pF. An estimate by Sekerka and Cahn [34] for a single component Gibbs solid shows that the equilibrium temperature would be lowered by about 10−3K for a shear stress of the order of 10 MPa.
220 THERMAL PHYSICS
14.1.2 Adsorption Equation in the Actual State
We now examine the parallel development whenγ is defined with reference to the area A of the actual state. For the layer model, we combine the differential of Eq. (14.6) with Eq. (14.7) to obtain
dγ = −SL
A dT+VL
A dpF−
i
NiL
A dμFi +
α,β
fαβLdεαβ−γdA
A . (14.15)
From a well-known relation [35, p. 16] in elasticity theory with coordinatesxin the actual state andxin the reference state, one has (in a 2×2 space)
dA
A =d lnA=d ln det ∂xα
∂xβ
=
κλ
∂xλ
∂xκd∂xκ
∂xλ =
κλν
∂xλ
∂xκ
∂xν
∂xκdελν, (14.16) where
ελν=1 2
∂xρ
∂xλ
∂xρ
∂xν −δλν =1 2
∂uν
∂xλ +∂uλ
∂xν +
ρ
∂uρ
∂xλ
∂uρ
∂xν
(14.17) is the full nonlinear strain tensor andu = x−xis the displacement. Thus, the last two terms in Eq. (14.15) can be combined to yield
α,β
fαβLdεαβ−γdA A =
α,β
fαβL −γ
κ
∂xα
∂xκ
∂xβ
∂xκ
dεαβ. (14.18)
Then the counterpart to Eq. (14.12) is
dγ = −[(SL/A)/XY]dT+ [(VL/A)/XY]dpF− κ i=1
[(NiL/A)/XY]dμFi +
α,β
fαβA dεαβ, (14.19)
where
fαβA ≡ 1 A
A[fαβL −γ
κ(∂xα/∂xκ)(∂xβ/∂xκ)] XL YL Vsσαβlat Xs Ys
0 XF YF
Xs Ys
XF YF
. (14.20)
Consequently
∂γ
∂εαβ
A0andκindependent intensive variables=fαβA. (14.21) If the actual state of the solid is chosen to be hydrostatic, we have noted previously that σαβlat =0, in which casefαβA =fαβL −γ
κ(∂xα/∂xκ)(∂xβ/∂xκ). If the actual state of the bulk solid is coincident with a hydrostatic reference state, then simplyfαβA =fαβL −γ δαβ.
Returning to the general case, we can expand the determinant in the numerator of Eq. (14.20) to obtain
AfαβA =
AfαβL XL YL Vsσαβlat Xs Ys 0 XF YF Xs Ys
XF YF
−Aγ
κ
∂xα
∂xκ
∂xβ
∂xκ. (14.22)
Then comparison with Eq. (14.13) shows that fαβC = A
A0
γ
κ
∂xα
∂xκ
∂xβ
∂xκ +fαβA
, (14.23)
which can also be written
∂γ0
∂εαβ = A A0
γ
κ
∂xα
∂xκ
∂xβ
∂xκ + ∂γ
∂εαβ
. (14.24)
In the case that the actual state of the bulk solid is coincident with a hydrostatic reference state, Eq. (14.24) becomes
∂γ0
∂εαβ =γ δαβ+ ∂γ
∂εαβ. (14.25)
If the solid behaved like a fluid, then∂γ /∂εαβ=0 and the surface stress would be isotropic and equal toγ, as we found previously for a fluid-fluid interface.
As Cahn points out, the relationship Eq. (14.25) can be based on the fact thatγ0A0=γA and the geometrical relationship ofAtoA0because of strain. Then by usingA0dγ0=γdA+ dγand Eq. (14.16), one would obtain the full nonlinear result Eq. (14.24). For small strain, one has simplyA/A0=1+
νεννso
∂γ0
∂εαβ ≈
1+
ν ενν γ (δαβ−2εαβ)+ ∂γ
∂εαβ ≈γ δαβ+ ∂γ
∂εαβ (14.26)
to lowest order. This is a linearized version of Eq. (14.24) and happens to agree with the exact Eq. (14.25) for the special states chosen in that case. So Eq. (14.23) and equivalently Eq. (14.24) are always true for geometrical reasons, even in the nonlinear case. Note, however, that these derivatives ofγ0and ofγ are only simply related tofαβL whenσαβlat is zero unless the actual bulk solid is hydrostatic or the small effect of shear stress on the bulk equilibrium (embodied byVsσαβlat) is negligible.