Appendix A Thermodynamic Functions and Legendre Transforms Background Chapter 2 prepared a brief thermodynamic basis for understanding transitions from crystalline solids to their conjugate melt phases, and vice versa. The internal energy, enthalpy, and Gibbs free energy played important roles in that development. Certain system parameters, such as the temperature, pressure, and component concentra- tions, were selected as independent intensive variables when discussing state func- tion changes under carefully prescribed conditions. The relationships among the many thermodynamic state functions and a system’s intensive and extensive param- eters can seem arbitrary, and occasionally confusing. Students often ask why and how the selection of state functions relates to these variables, and most of those who do not, have merely memorized the connections. This appendix provides a rigorous yet manageable alternative. Thermodynamic Functions and Variables Internal energy, U, is a fundamental thermodynamic state function providing the ten- dency of isolated, or ‘closed’ material systems to approach equilibrium. Moreover, U is a linear, homogeneous function, U(S, V, Ni ,...), of all the system’s so-called ‘natural’ extensive variables, including its entropy, S, volume, V , component molar masses, Ni , etc. Intensive parameters, by contrast, such as temperature, T , pressure, P, and chemical potential, μi , respectively, are also system variables that often are more conveniently measured under practical solidification conditions than would the ‘natural’ extensive quantities. With care, as we shall show, extensive thermodynamic variables may be easily substituted by alternative independent parameters of the system, each replacing their associated extensive variable. Substitution of intensive variables for extensive variables generates different thermodynamic functions, or potentials, that can equivalently determine the state of equilibrium for solidifying systems maintained under widely varying circumstances, such as melts communi- cating with a heat bath, a pressure source, or exchanging mass with a reservoir of chemical potential. M.E. Glicksman, Principles of Solidification, DOI 10.1007/978-1-4419-7344-3, 449 C Springer Science+Business Media, LLC 2011 450 A Thermodynamic Functions and Legendre Transforms Interestingly, all such ‘practical’ intensive variables, often imposed by the sys- tem’s environment, can be mathematically related to conjugate extensive thermody- namic variables. One, therefore, can reformulate the internal energy function, U, by replacing one or more of its extensive variables by intensive ones. A good example of needing such an alternative function is recognizing that a solidifying system’s temperature, T , is both easily measured using thermometry and controlled in an oven or furnace with a wide variety of standard laboratory instruments, such as ther- mocouples or pyrometers. By contrast, temperature’s conjugate extensive variable is the entropy, which, unfortunately, lacks practical devices for its direct measurement or control. How, therefore, might one transform the internal energy function, U, to provide an alternative thermodynamic function by replacing one, or more, of its extensive variables, viz., (S, V, Ni ,...), by its respective intensive variables, viz., (T, P,μi ,...)? Such a substitution may be accomplished using Legendre transforms—a standard mathematical procedure. Legendre transforms, among other things, lead directly to important thermodynamic ‘potentials’ that will be used throughout this book. These potentials provide alternatives for the internal energy. Some of these potentials might seem more familiar than others, as they remain in common use, but the formal procedure outlined next of how they are created remains identical, irrespective of the variables chosen. The Formal Math Problem Given a function Y defined as Y = Y(X0, X1, X2 ...Xn), (A.1) one desires to replace its independent variables, Xi , by one or more of their partial derivatives, Pi , where ∂Y P ≡ . i ∂ (A.2) Xi 1,2,...n =i The subscripts placed outside the parentheses in Eq. (A.2) denote the variables that are held fixed during partial differentiation. In fact, the Pi s represent the geometric ‘slopes’ of the function Y plotted with respect to the selected variable Xi , holding all other variables fixed. Consider the case of a function of just one independent variable, Y(X).This function represents a curve in X-Y space, composed of Cartesian coordinate pairs, ( , ) = dY ( ) X Y . The derivative, P dX, is the tangent to the curve Y X at any point. One may replace the independent variable, X,byP. Simply replacing the coordinate X by the function’s slope at that point to yield a new function, Y(P), however, doesn’t quite work, because, as will be shown, some information is lost by this direct substitution. Notice that for a single variable, Eq. (A.2) becomes an ordinary A Thermodynamic Functions and Legendre Transforms 451 differential equation, which, when integrated, yields the solution, Y(X). Indeed, this result occurs, but the solution so obtained is established only to within an as yet unknown constant of integration! See Fig. A.1. Thus, by using direct substitution some information is lost. Fig. A.1 Loss of information occurs by replacing the independent variable, X,inY(X) by the slope, P, to form a new transformed function Y(P). Clearly, knowing the slope at every point along the curve is not equivalent to knowing the coordinates, X, Y . The curves Y1, Y2,... all have identical P(X) values, but are distinctly different functions The original function Y(X) may, nonetheless, be represented without loss using the substitution of slopes via Plücker line geometry. Plücker line geometry shows that a ‘continuous’ envelope of tangents lines would suffice to define the curve, Y, as suggested in Fig. A.2. Individual tangents contribute an infinitesimal line segment that join together smoothly to form the curve Y(P) without any loss of information. This procedure is, therefore, equivalent to ordinary point geometry, where an infinite number of coordinate pairs connect to form a curve. The difference, however, is that each tangential segment contributing to the curve now fundamentally represents both a slope, or derivative, P, plus some positional information, or locus, obtained from the derived function, P(X). Full specification of all the tangent lines is gained by establishing the relationship between the Y -intercepts, Ψ , and their associated slopes, P. In short, these become the transformed new independent variables. Thus, Ψ(P) is mathematically equivalent to the original Y representation, namely, Y(X), with all information contained in that original function fully retained in the new Ψ representation. Thus, Ψ = (P). (A.3) 452 A Thermodynamic Functions and Legendre Transforms Fig. A.2 Shown here schematically is an equivalent representation of a function, Y(X), using the envelope of all its tangent lines,Pn(X). This type of Plücker line geometry requires the relation- ship, (P), which are the functions connecting the slopes of the tangent lines to their Y -intercepts, i.e., the Ψ s. No loss of information results from this representation Now the formal mathematics problem reduces to how one calculates (P) given Y(X). A well-known mathematical manipulation called the Legendre transforma- tion provides the appropriate procedure. Legendre Transformations One-Variable Transforms As indicated in Fig. A.2, the slope of the tangent line touching the curve Y(X) at the general coordinates X, Y ,is Y − Ψ P = . (A.4) X − 0 The linear form, Eq. (A.4), provides the definition of the new function, Ψ , which we term the Legendre transform of the function Y, with respect to X. Operationally, one writes Ψ = Y[X], which notes that Ψ is the function Y with its independent variable X replaced by the slope P = dY/dX,or Ψ = Y(X) − PX. (A.5) A Thermodynamic Functions and Legendre Transforms 453 The inverse transform, i.e., recovering Y(X) given the function (P),isthesym- metrical inversion, dΨ = dY(X) − PdX − XdP. (A.6) Recalling Eq. (A.2), which shows that dY = PdX,Eq.(A.6) reduces to dΨ =−XdP, or, equivalently, dΨ X =− . (A.7) dP Equations (A.5) and (A.7) together allow elimination of the variables Ψ and P from the transform function (P), thereby inverting the transform and returning the original function Y(X). The Legendre transform and its inverse are shown as a series of operational steps in the table below. Transform →←Inverse transform Y = Y(X)Ψ= (P) = dY − = dΨ P dX X dP Ψ =−PX + YY= XP+ Ψ Eliminate X and Y → Eliminate P and Ψ → Ψ = (P) Y = Y(X) Multivariate Transforms The generalization of the Legendre transform method to multivariate functions, such as U(S, V, Ni ,...), which are commonly encountered in thermodynamics, requires a straightforward extension. Starting with the fundamental equation Y = Y(X0, X1,...Xn) portrayed in an n + 2 dimensional space, and represented by the coordinates Y, X0, X1, X2,...Xn. The partial derivative, Pk, is defined in the usual way as ∂Y P ≡ , k ∂ (A.8) Xk 0,1,2,...,n =k and represents the partial slope of the hypersurface, Y. The total differential of the function Y is given as the usual sum, n dY = PkdXk. (A.9) k=0 In an analogous manner to the one-variable case, the hypersurface can be rep- resented either as the locus of all points satisfying the fundamental equation, 454 A Thermodynamic Functions and Legendre Transforms Y = Y(X0, X1,...Xn), or as the envelope of all its tangent hyperplanes. Each hyperplane may be characterized by its intercept, Ψ , which is a function of all the slopes P0, P1, P2 ...Pn and their associated independent variables, X0, X1, X2,...Xn, namely n Ψ = Y − Pk Xk. (A.10) k=0 Taking the differential of Eq. (A.10) yields n dΨ =− XkdPk, (A.11) k=0 and thus, ∂Ψ Xk =− . (A.12) ∂ Pk Thermodynamic Potentials The general function, Y(X0, X1,...), may now be chosen as the internal energy, U = U(S, V, Ni ,...).