Spinodal Decomposition - Diffusion Equation

Diffusion Equation

The mathematical theory of spinodal decomposition is based largely on the development of a generalized diffusion equation. A diffusion equation relates a spontaneous flux of material to a gradient in composition. Fundamental thermodynamic principles dictate that in order for the flux to be spontaneous, it must be associated with a net decrease in the free energy of the system. Consider the following diffusion equation relating the flux of two species ( JA and JB ) to the gradient of the chemical potential difference:

As pointed out by Cahn, this equation can be considered as a phenomenological definition of the mobility M, which must by definition be positive. It consists of the ratio of the flux to the local gradient in chemical potential.

The quantity ( μA - μB ) is the change in free energy when we reversibly add a unit amount of A atoms ( ΔF = + μA ) and simultaneously remove an equal number of B atoms ( ΔF = - μB ). This term may include factors such as composition, compositional gradients, stresses, and magnetic fields. For a homogeneous system:

The quantity f is the free energy of that number of lattice points in the crystal which initially occupied a unit volume. Substituting,

and defining the interdiffusion coefficient D by:

We can then define the interdiffusion coefficient D as follows:

Note that since M must always be positive, D takes its sign from the sign of f", which is negative within the spinodal. This has often been referred to as "uphill diffusion".

The above derivation of the diffusion coefficient is valid for concentration gradients that are so small that, for all practical purposes, each atom finds itself in surroundings which are similar to that which it would have in a homogeneous material of identical composition. If, however, concentration gradients are so large that within the range of interaction of an atom the average concentration has changed appreciably, then the atom will be aware of its inhomogeneous environment. This leads to a change in its chemical potential, and for fluids:

Substitution yields:

By taking the divergence, we obtain the new diffusion equation:

Alternatively, since:

the flux equation can be written as:

For a system in equilibrium, the chemical potentials, and hence their difference, are constant throughout the system. Thus this equation for the flux satisfies the physical requirement that the net flux should go to zero as equilibrium is approached. For the time dependence of the composition we obtain on differentiation:

Comparing this equation with the usual statement of Fick's second law

it is seen that the mobility is related to the interdiffusion coefficient by the following:

It then follows from the solution to be described next that a particular solution to this new diffusion equation is given by:

in which co is the average composition and A(β,t) is the amplitude of the Fourier component of wavenumber β at time t. In terms of the initial amplitude at time zero:

where R(β) is an amplification factor given by:

Other articles related to "diffusion equation, diffusion, equations, equation":

Radiative Transfer Equation And Diffusion Theory For Photon Transport In Biological Tissue - Solutions To The Diffusion Equation - Point Sources in Infinite Homogeneous Media
... A solution to the diffusion equation for the simple case of a short-pulsed point source in an infinite homogeneous medium is presented in this section ... The source term in the diffusion equation becomes, where is the position at which fluence rate is measured and is the position of the source ... The diffusion equation is solved for fluence rate to yield The term represents the exponential decay in fluence rate due to absorption in accordance with Beer's law ...
Convection–diffusion Equation
... The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical ... Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, Smoluchowski equation (after Marian Smoluchowski), or (generic) scalar ...
Diffusion Equation - Discretization (Image)
... The product rule is used to rewrite the anisotropic tensor diffusion equation, in standard discretization schemes ... Because direct discretization of the diffusion equation with only first order spatial central differences leads to checkerboard artifacts ... The rewritten diffusion equation used in image filtering where "tr" denotes the trace of the 2nd rank tensor, and superscript "T" denotes transpose, in which in image ...
Eddy Current - Explanation - Diffusion Equation
... The derivation of a useful equation for modelling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's Law, providing an expression for the magnetizing field ... The diffusion equation therefore is ...
Radiative Transfer Equation And Diffusion Theory For Photon Transport In Biological Tissue - Diffusion Theory - The Diffusion Equation
... Using the second assumption of diffusion theory, we note that the fractional change in current density over one transport mean free path is negligible ... The vector representation of the diffusion theory RTE reduces to Fick's law, which defines current density in terms of the gradient of fluence rate ... Substituting Fick's law into the scalar representation of the RTE gives the diffusion equation is the diffusion coefficient and μ'sμs is the reduced scattering coefficient ...

Famous quotes containing the word equation:

Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.
Anna Quindlen (b. 1952)