Difference between revisions of "Alkanethiol Self-Assembled Monolayers"

Latest revision as of 00:48, 6 December 2016

_AUTOLINKER{1|equilibrium|stable} Interface free energy is the contribution to the free energy of a system due to the presence of an interface separating two coexisting phases at equilibrium. It is also called surface tension. The content of the article is the definition and main properties of the interface free energy from first principles of statistical mechanics.

1 Interface free energy in statistical mechanics

1.1 Definition of the interface free energy

Consider a physical system at equilibrium in a vessel [math]V[/math] at a first order phase transition point with bulk phases [math]A[/math] and [math]B\ .[/math] The interface is the common boundary of the two phases when they coexist in [math]V\ .[/math] At the macroscopic scale, when the length of the vessel [math]V[/math] is the reference length, a flat interface perpendicular to a unit vector [math]{\mathbf n}[/math] is described mathematically by a plane perpendicular to [math]{\mathbf n}\ ;[/math] above this plane the state of the system is specified by the value of the order-parameter of one of the phases, and below by that of the other phase. The interface free energy [math]\tau({\mathbf n})[/math] is the free energy of that interface (per unit area). The way of defining [math]\tau({\mathbf n})[/math] is quite general and can be applied in principle to most systems; its origin can be traced back to the monumental work of J.W. Gibbs, On the Equilibrium of Heterogeneous Substances (1875-1878). The basic postulate is that the various contributions to the overall free energy [math]F(V)[/math] (taking into account the interactions of the system with the walls) can be separated into the bulk free energy, which is proportional to the volume of [math]V\ ,[/math] and a term proportional to the surface of [math]V[/math] (up to a negligible correction term). Thus, at a point of first order phase transition, when only phase [math]A[/math] is present,

[math] F_A(V)=-\frac{1}{\beta}\ln Z_A(V)=f_{{\rm bulk} }(A)|V| + f_{{\rm wall} }(A)|\partial V|+ o(|\partial V|) [/math]

where [math]Z_A(V)[/math] denotes the partition function of the system for phase [math]A\ ,[/math] [math]\beta[/math] the inverse temperature, [math]|V|[/math] the volume of [math]V[/math] and [math]|\partial V|[/math] the area of the boundary [math]\partial V[/math] of the vessel. A similar expression holds when phase [math]B[/math] is present. Under specific conditions it is possible to obtain macroscopic inhomogeneous states with a planar interface separating the two coexisting bulk phases. In such cases there is an additional contribution to the free energy and the postulate is that the free energy can be written as

[math] F_{AB}(V)=-\frac{1}{\beta}\ln Z_{AB}(V)=f_{{\rm bulk} }(AB)|V| + f_{{\rm wall} }(AB)|\partial V|+ \tau({\mathbf n})|I({\mathbf n})|+ o(|\partial V|) [/math]

with

[math] f_{{\rm wall} }(AB)=\alpha f_{{\rm wall} }(A)+ (1-\alpha)f_{{\rm wall} }(B). [/math]

The term [math]|I({\mathbf n})|=O(|\partial V|)[/math] is the area of the interface in [math]V[/math] perpendicular to the unit vector [math]{\mathbf n}[/math] and [math]\alpha[/math] is the proportion of the walls of [math]V[/math] in contact with phase [math]A\ .[/math] At a first order phase transition point [math]f_{{\rm bulk} }(AB)=f_{{\rm bulk} }(A)=f_{{\rm bulk} }(B)\ ,[/math] and if the postulate is correct, one eliminates the terms [math]f_{{\rm wall} }(AB)\ ,[/math] [math]f_{{\rm wall} }(A)[/math] and [math]f_{{\rm wall} }(B)[/math] by considering the ratio of partition functions

[math]\label{1.3} -\frac{1}{\beta}\ln\frac{Z_{AB}(V)}{Z_A(V)^{\alpha}Z_B(V)^{1-\alpha}}=\tau({\mathbf n})|I({\mathbf n})|+ o(|\partial V|). [/math]

An obvious difficulty is that [math]\tau({\mathbf n})[/math] is defined only when there is phase coexistence. This is why in many situations one proceeds differently in Physics. One models directly the interface in order to bypass these problems and then the interface free energy is simply identified with the free energy of the model for which one has standard methods for evaluating it. This is often an adequate way to proceed, but it cannot be applied always, for example when one is studying how the coexisting phases are spatially distributed inside the vessel [math]V\ .[/math]

1.2 Macroscopic states and interface free energy in Ising model

The ideas of the preceding section are implemented for the Ising model for which the mathematical results are the most complete. We expose the main results for three-dimensional Ising model. The two-dimensional case is also of interest. The model is defined on

[math] \Lambda_{LM}:=\{t=(t_1,t_2,t_3)\in{\mathbf Z}^3\,{:}\; \max(|t_1|,|t_2|)\leq L\,,\;|t_3|\leq M \}. [/math]

The energy of the system is equal to

[math] H_{LM}(\underline{\sigma})=-\frac{1}{2} \sum_{t\in\Lambda_{LM}}\sum_{t^\prime\in\Lambda_{LM}}J(t,t^\prime)\,\sigma(t)\sigma(t^\prime) -\sum_{t\in\Lambda_{LM}}h\,\sigma(t) [/math]

with coupling constants [math]J(t,t^\prime)=0\ ,[/math] except if [math]t,t^\prime[/math] are nearest neighbors, in which case [math]J(t,t^\prime)=J\gt 0\ .[/math] An inhomogeneous magnetic field [math]J^\prime \eta(t)\ ,[/math] which acts only on the spins located at the boundary of the box [math]\Lambda_{L,M}\ ,[/math] models the interaction of the system with the walls, which is defined by

[math] W_{LM}^\eta(\underline{\sigma}):=-\sum_{t\in\partial\Lambda_{LM}}J^\prime\eta(t)\sigma(t). [/math]

Here [math]\partial\Lambda_{LM}:=\{t\in\Lambda_{LM}\,:\,|t_3|=M\; \text{or}\; \max(|t_1|,|t_2|)=L \}[/math] and [math]J^\prime\gt 0\ ;[/math] the value of [math]\eta(t)[/math] is fixed, either [math]\eta(t)=1[/math] or [math]\eta(t)=-1\ .[/math] Different kinds of walls are modeled by choosing different values for [math]\eta(t)[/math] and for the coupling constant [math]J^\prime\ .[/math] The overall energy of the system is [math]H_{LM}+W_{LM}^\eta\ .[/math] According to statistical mechanics the free energy of the system is the logarithm of the partition function [math]Z_{LM}^\eta\ ,[/math]

[math] F_{LM}^\eta(\beta,h,J^\prime):=-\beta^{-1}\ln Z_{LM}^\eta\quad\text{with}\quad Z_{LM}^\eta=\sum_{\underline{\sigma}^\prime}{\rm e}^{-\beta(H_{LM}(\underline{\sigma}^\prime)+W_{LM}^\eta(\underline{\sigma}^\prime))}\,. [/math]

At the thermodynamical limit the bulk free energy per spin

[math] f_{{\rm bulk} }(\beta,h)=\lim_{L\rightarrow\infty}\frac{1}{(2L+1)^d}F_{LL}^\eta(\beta,h,J^\prime) [/math]

is independent on the choice of [math]J^\prime\gt 0[/math] and [math]\eta\ .[/math] It is well-known that the model exhibits a first order phase transition at [math]h=0[/math] and [math]\beta\gt \beta_c(3)[/math] ([math]\beta_c(d)[/math] is the inverse critical temperature of the [math]d[/math]-dimensional Ising model, [math]d\geq 2[/math]). At that transition the bulk free energy [math]f_{{\rm bulk} }(\beta,h)[/math] is not differentiable at [math]h=0\ ,[/math] the spin-flip symmetry of [math]H_{LM}[/math] is broken and there is a positive spontaneous magnetization [math]m^*(\beta)\ ,[/math]

[math] 0\lt m^*(\beta)=\frac{d}{dh}f_{{\rm bulk} }(\beta,h)|_{h=0^+}=-\frac{d}{dh}f_{{\rm bulk} }(\beta,h)|_{h=0^-}. [/math]

From now on the external magnetic field [math]h=0[/math] and [math]\beta\gt \beta_c(3)\ .[/math] The coarse-grained description of the model at the macroscopic scale is obtained by taking the macroscopic limit. In this limit the state of the system is given by a magnetization profile. Let [math]0\lt a\lt 1[/math] and for simplicity set [math]L=M\ ;[/math] the set [math]\Lambda_{LL}[/math] is partitioned into cubic cells [math]C_i[/math] of linear size [math]L^a[/math] and the averaged magnetization over [math]C_i[/math] is

[math] m_{C_i}(\underline{\sigma}):=|C_i|^{-1}\sum_{t\in C_i}\sigma(t). [/math]

All lengths are scaled by [math]L^{-1}\ ,[/math] so that the distance between neighboring spins becomes [math]L^{-1}\ .[/math] For each point [math]x[/math] of the macroscopic box [math]V=\{(x_1,x_2,x_3)\in{\mathbf R}^d\,{:}\; |x_i|\leq 1\}[/math] the magnetization profile is defined by

[math] \rho_L(x|\underline{\sigma}):=m_C(\underline{\sigma})\quad\text{if}\; (Lx_1,Lx_2,Lx_3)\in C_i\,. [/math]

The probability of the profile [math]\rho_L(x|\underline{\sigma})[/math] is the joint probability of the block-spins [math]m_{C_i}(\underline{\sigma})[/math] induced by the usual Gibbs measure. The macroscopic limit is obtained by taking the limit [math]L^{-1}\rightarrow 0\ .[/math] (In probability theory this corresponds to the regime of the law of large numbers.) For pure boundary conditions, that is [math]\eta(t)\equiv+1\ ,[/math] respectively [math]\eta(t)\equiv -1\ ,[/math]

the interactions with the walls favor the bulk phase with positive spontaneous magnetization [math]m^*(\beta)\ ,[/math] respectively negative magnetization [math]-m^*(\beta)\ .[/math] In the macroscopic limit the probability measure on the density profiles becomes concentrated on the unique magnetization profile [math]\rho(x)\equiv m^*(\beta)\ ,[/math] respectively [math]\rho(x)\equiv -m^*(\beta)\ ,[/math] for any value of [math]J^\prime\gt 0\ ;[/math] this constant profile describes the macroscopic state of the [math]+[/math]-phase, respectively [math]-[/math]-phase, of the model. A mixed boundary condition is related to the emergence of a planar interface perpendicular to [math]{\mathbf n}=(n_1,n_2,n_3)\ ,[/math]

[math] \eta^{\mathbf n}(t):=+1\quad \text{if}\; t_1n_1+t_2n_2+t_3n_3\geq 0\quad\text{and}\quad \eta^{\mathbf n}(t):=-1\quad \text{if}\; t_1n_1+t_2n_2+t_3n_3\lt 0. [/math]

Thus [math]\eta^{{\mathbf n} }(t)=1[/math] if and only if [math]t[/math] is above or in the plane [math]\pi({\mathbf n})[/math] perpendicular to [math]{\mathbf n}[/math] and passing through the origin, otherwise [math]\eta^{{\mathbf n} }(t)=-1\ .[/math] Let [math]Z_{LM}^{{\mathbf n} }:=Z_{LM}^{\eta^{\mathbf n} }\ ;[/math] using the symmetry [math]Z_{LL}^+=Z_{LL}^-[/math] the interface free energy [math]\tau({\mathbf n})[/math] is defined by \eqref{1.3} and is given by

[math]\label{tau} \tau({\mathbf n})=-\frac{1}{\beta|I({\mathbf n})|}\lim_{L\rightarrow\infty}\frac{1}{L^{d-1}}\ln\frac{Z_{LL}^{{\mathbf n} } }{Z_{LL}^+}. [/math]

One can prove:

1. the limit \eqref{tau} is independent on [math]J^\prime\geq J\ ;[/math]
2. for [math]\beta\gt \beta_c(3)[/math] the function [math]\tau({\mathbf n})[/math] verifies the basic properties 1), 2) and 3) of an interface free energy (see below, next section);
3. in the macroscopic limit the measure on the density profiles is concentrated on the unique magnetization profile  :[math] \rho_{{\mathbf n} }(x):=+ m^*(\beta)\;\text{if}\; x\; \text{is above}\; \pi({\mathbf n}) \quad\text{and}\quad \rho_{{\mathbf n} }(x):=-m^*(\beta)\;\text{if}\; x\; \text{is above}\; \pi({\mathbf n})\,. [/math]

The condition [math]J^\prime\geq J[/math] is important, because for some values of [math]J^\prime\lt J[/math] and [math]\beta[/math] the physics near the walls of the system is different: a surface phase transition may take place and portions of the interface may be pinned to the walls. As a consequence of this phenomenon, in the macroscopic limit the interaction of the system with the walls given by [math]\eta^{\mathbf n}[/math] may not induce an interface perpendicular to [math]{\mathbf n}[/math]. For example, in the two-dimensional case, the macroscopic state may have an interface making an angle with the vertical walls of the vessel, whose value is given by the Young-Herring equation, so that \eqref{tau} may not be equal to [math]\tau({\mathbf n})\ ,[/math] or, if [math]J^\prime[/math] is small enough and the macroscopic box is a square, then the whole interface may even be pinned to the walls so that there is no interface through the macroscopic box and the magnetization profile is constant, either equal to [math]m^*(\beta)[/math] or to [math]-m^*(\beta)\ .[/math] In such cases the limit \eqref{tau} depends on [math]J^\prime\ .[/math] The condition [math]J^\prime\geq J[/math] has a simple physical interpretation; it ensures that the walls of the box [math]V[/math] are in the complete wetting regime, so that the interface cannot be pinned to the walls. In the literature the standard choice for ferromagnetic models is [math]J^\prime=J\ ,[/math] so that \eqref{tau} gives the correct definition of [math]\tau({\mathbf n})\ .[/math] These results illustrate the fact that one must avoid the possibility of pinning the interface to the walls when using definition \eqref{1.3}. On the other hand any wall interactions, which induce a macroscopic state with an interface perpendicular to [math]{\mathbf n}[/math] and such that otherwise \eqref{1.3} is independent of the chosen interactions, are admissible for defining the interface free energy.

Several other definitions for [math]\tau({\mathbf n})[/math] have been proposed for the Ising or similar models. Most of them involve a ratio of partition functions and are based on the same pattern leading to \eqref{tau} (see references below). A possibility of avoiding the above problem with the walls is to suppress (partially) the walls of the system by taking (partial) periodic boundary conditions. Then one imposes a condition implying the existence of a single planar interface perpendicular to [math]{\mathbf n}\ .[/math] There are also variants of \eqref{tau} where one considers a box [math]\Lambda_{LM}[/math] instead of [math]\Lambda_{LL}[/math] and take first the limit [math]M\rightarrow\infty[/math] before taking [math]L\rightarrow\infty\ .[/math] When [math]J^\prime\lt J[/math] this limit may give a different answer as the limit \eqref{tau}. On the other hand, if [math]J^\prime\geq J\ ,[/math] then one can take the limits in any order, first [math]L\rightarrow\infty[/math] and then [math]M\rightarrow\infty[/math] or vice-versa, or simultaneously [math]L\rightarrow\infty[/math] and [math]M\rightarrow\infty\ .[/math] The reason is that the walls are in the complete wetting regime and the interface is not pinned to the walls.

The surface tension for the two-dimensional Ising model can be computed exactly. Onsager computed it for [math]{\mathbf n}=(0,1)\ ,[/math]

[math] \beta\tau((0,1))=2(K-K^*)\,,\;\beta\gt \beta_c(2)\quad\text{and}\quad \tau((0,1))=0\,,\;\text{otherwise,} [/math]

where [math]K^*[/math] is defined by [math]\exp(-2K^*)=\tanh K[/math] and [math]K=\beta J\ .[/math] Onsager did not use the definition \eqref{tau}; the computation of [math]\tau((0,1))[/math] defined by \eqref{tau} is due to Abraham and Martin-Löf. The full interface free energy has been computed by McCoy and Wu. In general it is not easy to show that reasonable definitions give the same value for [math]\tau({\mathbf n})\ .[/math]

2 Basic properties of the interface free energy

2.1 Convexity of the interface free energy

Assume that [math]{\tau}({\mathbf n})\gt 0[/math] for each unit vector [math]{\mathbf n}[/math] is given. By convention [math]\tau({\mathbf n})\ ,[/math] with [math]\|{\mathbf n}\|=1\ ,[/math] is the physical value of the interface free energy of an interface perpendicular to [math]{\mathbf n}\ .[/math] It is convenient to extend the definition of [math]\tau[/math] to any [math]{\mathbf x}\ ,[/math] as a positively homogeneous function, by setting

[math] \tau({\mathbf x}):=\|{\mathbf x}\|\tau({\mathbf x}/\|{\mathbf x}\|)\,. [/math]

Let [math]\langle\,{\mathbf x}|{\mathbf y}\,\rangle:=x_1y_1+x_2y_2+x_3y_3[/math] be Euclidean scalar product. The convex set [math]W_\tau\ ,[/math] which is the intersection of the half-spaces [math]H({\mathbf n})=\{{\mathbf x}\,:\,\langle\,{\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\}\ ,[/math]

[math] W_\tau=\{{\mathbf x}\,{:}\; \langle\, {\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\, {\mathbf n}\}, [/math]

is called the equilibrium shape because it gives the solution of the following isoperimetric problem. Let [math]K[/math] be a subset of [math]\mathbf R^3[/math] with [math]{\rm vol}(K)={\rm vol}(W_\tau)\ .[/math] If inside [math]K[/math] there is phase [math]A[/math] and outside [math]K[/math] phase [math]B\ ,[/math] then the (surface) free energy associated with the boundary of [math]K[/math] is given by the surface integral

[math] {\mathcal F}(\partial K)=\int_{\partial K}\tau(n)\,dS. [/math]

Among all sets [math]K[/math] with [math]{\rm vol}(K)={\rm vol}(W_\tau)[/math] the minimum of the surface integral is attained for, and only for, [math]K=W_\tau[/math] or a translate of [math]W_\tau\ .[/math] It is Wulff (1901) who gave the geometrical construction of the solution of this isoperimetric problem. This is why the set [math]W_\tau[/math] is also called Wulff crystal.

The main property of an interface free energy is its convexity: for two distinct phases [math]A[/math] and [math]B[/math] at equilibrium, the interface free energy is a continuous convex function, which is positive and sublinear, that is

1. [math]\tau({\mathbf x})\gt 0\quad {\mathbf x}\not=0\ ,[/math]
2. [math]\tau(t{\mathbf x})=t\, \tau({\mathbf x})\quad\forall \,{\mathbf x}[/math] and all [math]t\geq 0\ ,[/math]
3. [math]\tau({\mathbf x}+{\mathbf y})\leq\tau({\mathbf x})+\tau({\mathbf y})\quad \forall \,{\mathbf x},{\mathbf y}\ .[/math]

By a classical result of Minkowski the interface free energy [math]\tau[/math] is the support function of the convex set [math]W_\tau\ ,[/math] that is

[math] \tau({\mathbf x})=\sup\{\langle\,{\mathbf x}|{\mathbf y}\,\rangle\,{:}\; {\mathbf y}\in W_\tau\}\,. [/math]

The next simple thermodynamical argument shows the convexity of [math]\tau\ .[/math] Let [math]{\mathcal P}[/math] be a right prism whose base is a triangle with vertices [math]a,b,c[/math] and whose length [math]L[/math] is very large. Let [math]\ell_0\ ,[/math] respectively [math]\ell_1\ ,[/math] [math]\ell_2\ ,[/math] be the side of the triangle opposite to the vertex [math]c\ ,[/math] respectively [math]b\ ,[/math] [math]a\ .[/math]

The length of the side [math]\ell_i[/math] is [math]|\ell_i|[/math] and [math]{\mathbf n}_i[/math] is the outward unit normal to [math]\ell_i[/math] (in the plane of the triangle), so that

[math] |\ell_0|{\mathbf n}_0+|\ell_1|{\mathbf n}_1+|\ell_2|{\mathbf n}_2=0. [/math]

We set [math]{\mathbf n}:=-{\mathbf n}_0=|\ell_1|/|\ell_0|{\mathbf n}_1+ |\ell_2|/|\ell_0|{\mathbf n}_2\ .[/math] In the plane spanned by [math]{\mathbf n}_1[/math] and [math]{\mathbf n}_2[/math] let [math]{\mathbf m_1}[/math] and [math]{\mathbf m_2}[/math] be reciprocal vectors to [math]{\mathbf n}_1[/math] and [math]{\mathbf n}_2\ ,[/math] [math]\langle\,{\mathbf m_i}|{\mathbf n_j}\rangle=\delta_{ij}\ .[/math] Then

[math] \sum_{i=1}^2\frac{|\ell_i|}{|\ell_0|}\tau({\mathbf n}_i)= \langle\,\sum_{i=1}^2\tau({\mathbf n}_i){\mathbf m}_i|{\mathbf n}\,\rangle\equiv\langle\,{\mathbf z}|{\mathbf n}\rangle. [/math]

The vector [math]{\mathbf z}=\sum_{i=1}^2\tau({\mathbf n}_i){\mathbf m}_i[/math] belongs to the intersection of the boundaries of the half-spaces [math]H({\mathbf n}_1)[/math] and [math]H({\mathbf n}_2)[/math] since [math]\langle\,{\mathbf z}|{\mathbf n}_i\,\rangle=\tau({\mathbf n}_i)\ .[/math] Suppose that [math]\langle\,{\mathbf z}|{\mathbf n}\rangle\lt \tau({\mathbf n})\ ;[/math] then

[math] L\ell_0\tau({\mathbf n})\gt L\ell_1\tau({\mathbf n}_1)+L\ell_2\tau({\mathbf n}_2)\,, [/math]

and an interface perpendicular to [math]{\mathbf n}[/math] can be deformed using the right prism [math]{\mathcal P}\ ,[/math] with long enough length [math]L\ ,[/math] so that the deformed interface has a lower free energy. At equilibrium such a planar interface cannot exist since its free energy must be minimal. Notice also that the plane [math]\{{\mathbf x}\,:\,\langle\,{\mathbf x}|{\mathbf n}\,\rangle=\tau({\mathbf n})\}[/math] cannot intersect [math]W_{\tau}\ .[/math] Therefore at equilibrium,

[math]\label{2.5} |\ell_0|\tau({\mathbf n})\leq|\ell_1|\tau({\mathbf n}_1)+|\ell_2|\tau({\mathbf n}_2). [/math]

Since [math]\tau[/math] has been defined as a positively homogeneous function, it is immediate to see that for all choices of [math]{\mathbf n}_1\ ,[/math] [math]{\mathbf n}_2\ ,[/math] [math]\ell_1[/math] and [math]\ell_2[/math] \eqref{2.5} is equivalent to

[math] \tau({\mathbf x}+{\mathbf y})\leq\tau({\mathbf x})+\tau({\mathbf y})\quad \forall \,{\mathbf x},{\mathbf y}. [/math]

By definition an interface perpendicular to [math]{\mathbf n}[/math] is thermodynamically stable if

[math] \tau({\mathbf x}+{\mathbf y})\lt \tau({\mathbf x})+\tau({\mathbf y})\quad \forall \,{\mathbf x},{\mathbf y}\; \text{linearly independent, such that}\; {\mathbf x}+{\mathbf y}={\mathbf n}\,. [/math]

In general the choice of the normal to the interface does not matter, so that [math]\tau({\mathbf n})=\tau(-{\mathbf n})\ .[/math]

2.2 Stable interfaces and polar set of the equilibrium shape

Assume that [math]\tau[/math] is given, verifying properties 1), 2) and 3) above (but not necessarily that [math]\tau({\mathbf n})=\tau(-{\mathbf n})[/math]). Under these assumptions [math]W_\tau[/math] is a bounded closed convex set with [math]0[/math] as an interior point. In convex analysis there is another natural set associated with [math]W_\tau\ ,[/math] which is the polar set [math]W^*_\tau\ .[/math] It is defined by the dual relationship between non-zero vectors [math]{\mathbf v}[/math] and closed half-spaces [math]{\mathbf v}^*[/math] containing the origin, [math] {\mathbf v}^*:=\{{\mathbf x}\,{:}\; \langle\,{\mathbf v}|{\mathbf x}\,\rangle\leq 1\}. [/math] The polar dual or polar set [math]W^*_\tau[/math] of [math]W_\tau[/math] is

[math] W_\tau^*:=\bigcap\{{\mathbf x}^*\,{:}\; {\mathbf x}\in W_\tau\}= \{{\mathbf u}\,{:}\; \langle\,{\mathbf x}|{\mathbf u}\,\rangle\leq 1\quad\forall\,{\mathbf x}\in W_\tau\}. [/math]

It is also a bounded closed convex set with [math]0[/math] as an interior point and [math]W_\tau=W_\tau^{**}\ .[/math] It is not difficult to show that

[math] W^*_\tau=\{{\mathbf u}\,{:}\; \tau({\mathbf u})\leq 1\}\quad\text{and}\quad \tau({\mathbf x})=\min\{t\geq 0\,{:}\; {\mathbf x}/t\in W^*_\tau\}\,. [/math]

These statements mean that [math]\tau[/math] is the gauge function of [math]W^*_\tau\ .[/math] Hence the interface free energy can be interpreted either as the support function of [math]W_\tau\ ,[/math] or as the gauge function of [math]W^*_\tau\ .[/math] The boundary [math]\partial W^*_\tau[/math] of the polar set is simply the level-[math]1[/math] surface of [math]\tau\ .[/math] Since [math](\partial W^*_\tau)^*= W_\tau^{**}[/math] and [math]{\mathbf n}^*=H({\mathbf n})[/math] for any [math]{\mathbf n}\in \partial W^*_\tau\ ,[/math] the boundary points of [math]W^*_\tau[/math] give a natural labeling of the support planes of [math]W_\tau\ .[/math] Moreover, the extremal points of [math]W_\tau^*[/math] label precisely the support planes of [math]W_\tau[/math] which represent stable interfaces. Therefore the equilibrium shape can be written as

[math] W_\tau=\{{\mathbf x}\,{:}\; \langle\,{\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\,{\mathbf n}\in{\rm ext}W_\tau^*\}\,. [/math]

One can measure experimentally [math]\tau({\mathbf n})[/math] only for [math]{\mathbf n}\in {\rm ext}W_\tau^*\ .[/math] All tangent planes of [math]W_\tau[/math] represent stable interfaces, but there are support planes of [math]W_\tau[/math] which are not tangent planes when [math]W_\tau[/math] has an edge or a corner and which represent also stable interfaces.

2.3 Summary

Provided that one can construct a macroscopic state with a planar interface perpendicular to [math]{\mathbf n}\ ,[/math] one can use formula \eqref{1.3} to define [math]\tau({\mathbf n})\ .[/math] The fundamental property of the interface free energy is that it is a convex function. The interface free energy can be measured experimentally at equilibrium only for the interfaces which are thermodynamically stable. By convention the physical value of the interface free energy [math]\tau({\mathbf n})[/math] is given for a unit vector [math]{\mathbf n}\ .[/math] But, using the extension of [math]\tau[/math] as an homogeneous function, this function can be interpreted either as the support function of the equilibrium shape [math]W_\tau=\{{\mathbf x}\,{:}\; \langle\,{\mathbf x}|{\mathbf n}\,\rangle\leq \tau({\mathbf n})\,,\;\forall\, {\mathbf n}\}\ ,[/math] or as the gauge function of [math]W^*_\tau=\{{\mathbf x}\,{:}\; \tau({\mathbf x})\leq 1\}\ .[/math] Stable interfaces are labeled by the extremal points of [math]W^*_\tau\ .[/math]

3 Bibliographical notes

(Herring 1953) and (Rotman, Wortis 1984) are reviews of physics on interfaces and equilibrium shapes of crystals. The review (Abraham 1986) is a review about exact results. Comparisons of several definitions of the interface free energy are carefully discussed and references can be found there. The results of the computation of the interface free energy of the two-dimensional Ising model can be found in (Rotman, Wortis 1981). The macroscopic limit for the two-dimensional Ising model and the role of the wetting transition is discussed in (Pfister, Velenik 1999). Mathematical results on wetting phenomenon for Ising systems are in (Fröhlich, Pfister 1987). The up-to-date reference concerning proofs of existence and convexity of surface tension for ferromagnetic models is (Messager et al. 1992). The basic reference for the thermodynamical properties of [math]\tau[/math] is (Herring 1951). The argument proving the convexity of [math]\tau[/math] is adapted from (Herring 1951). Instead of the polar set Herring uses for studying [math]\tau[/math] the surface tension plot, which is the set of points [math]\{{\mathbf x}\,{:}\; {\mathbf x}=\tau({\mathbf n})\,{\mathbf n}\,,\;\|{\mathbf n}\|=1\}\ .[/math] This is the standard way of presenting [math]\tau[/math] in physics. One gets the surface tension plot from [math]\partial W_\tau^*[/math] by an inversion on the unit sphere (or the unit circle in dimension 2). Affine parts of [math]\partial W_\tau^*[/math] become spherical parts, or circular parts, of the surface tension plot. The theory of convex sets used for studying the interface free energy and its equilibrium shape is classical and due essentially to Minkowski. A good recent reference is chapters 1 and 2 of (Schneider 1993). An extended version of this article with further references can be found in (Pfister 2009).

4 References

Abraham D.B. (1986): Surface Structures and Phase Transitions–Exact Results, pp. 1–74 in Phase Transitions and Critical Phenomena vol 10, eds Domb C., Lebowitz J.L., Academic Press, London.

Fröhlich J., Pfister C.-E. (1987): The wetting and layering transitions in the half–infinite Ising model, Europhys. Lett. 3, 845–852.

Herring C. (1951): Some Theorems on the Free Energies of Crystal Surfaces, Phys. Rev. 82, 87–93.

Herring C. (1953): The Use of Classical Macroscopic Concepts in Surface-Energy Problems, pp.5–81 in Structure and Properties of Solid Surfaces, eds. Gomer R., Smith C.S., The University of Chicago Press, Chicago.

Messager A., Miracle-Sole S., Ruiz J. (1992): Convexity Properties of the Surface Tension and Equilibrium Crystals, J. Stat. Phys. 67, 449–470.

Pfister C.-E. (2009): Interface free energy or surface tension: definition and basic properties, arXiv:0911.5232 (2009).

Pfister C.-E., Velenik Y. (1999): Interface, Surface Tension and Reentrant Pinning Transition in the 2D Ising Model, Commum. Math. Phys. 204, 269–312.

Rotman C., Wortis M. (1981): Exact equilibrium crystal shapes at nonzero temperature in two dimensions, Phys. Rev. B 11, 6274–6277.

Rotman C., Wortis M. (1984): Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions, Phys. Rep. 103, 59–79.

Schneider R. (1993): Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 44 (chapters 1 and 2), Cambridge University Press, Cambridge.

5 See Also

Ising model: exact results, Wulff shape of crystals