Spinodal Decomposition - Gradient Energy

Gradient Energy

Gradient energies associated with even the smallest of compositional fluctuations can be evaluated using an approximation introduced by Ginzburg and Landau in order to describe magnetic field gradients in superconductors. This approach allows one to approximate the energy associated with a concentration gradient C. Thus, as a result of series expansions with respect to ( c – c_o ), this energy can be expressed in the form κ(C)2

• Note: In a three-dimensional Cartesian coordinate system R3 with coordinates ( x, y, z ), del is defined in terms of partial derivative operators as

are the unit vectors in the respective coordinate directions.

The vector derivative of a scalar field f is called the gradient, and it can be represented as:

Cahn & Hilliard used such an approximation to evaluate the free energy of a small volume of non-uniform isotropic solid solution as follows:

or:

where:

= particle density (#/vol)

is the free energy of the homogeneous solution.

The κ(C)2 term, is a measure of the free energy of a composition gradient and is strongly dependent on local composition. (The constant κ is related to derivatives of the free energy with respect to composition.) The interfacial energy associated with this compositional gradient therefore increases with the square of C.

Since we shall be concerned with testing the stability of an initially homogeneous solution to infinitesimal composition (or density) fluctuations, the gradients will also be infinitesimal and the second term will be completely sufficient to describe the contribution from the incipient 'surfaces" (between regions differing in composition). Higher order gradient energy terms will be negligible, except at very large gradients. We may also expand f (c) about the average composition c_o as follows:

The difference in free energy per unit volume (or free energy density) between the initial homogeneous solution and one with a composition given by:

is given by:

Note that both terms are quadratic in the amplitude, so the stability criterion is initially independent of amplitude.

Thus, ΔF is positive if the second derivative of the free energy with respect to composition (hereafter referred to as f'' ) is positive, because the contribution of the surface energy in the second term is always positive. In this case, the system is stable against all infinitesimal fluctuations in composition since the formation of such fluctuations would result in an increase in the free energy of the system.

In contrast, if f'' is negative, then ΔF is negative when:

The formation of fluctuations can therefore be accompanied by a decrease in the free energy of the system within this region provided the scale or wavelength of the fluctuation is large enough. Within this context, such gradual changes in composition maintain small values for the gradient term C.