Boltzmann transformation

Several general methods are available for solving the diffusion equation, including Boltzmann transformation, principle of superposition, separation of... 

This section introduces the method of Boltzmann transformation to solve onedimensional diffusion equation in infinite or semi-infinite medium with constant diffusivity. For such media, if some conditions are satisfied, Boltzmann transformation converts the two-variable diffusion equation (partial differential equation) into a one-variable ordinary differential equation. 

In experimental studies of diffusion, the diffusion-couple technique is often used. A diffusion couple consists of two halves of material each is initially uniform, but the two have different compositions. They are joined together and heated up. Diffusive flux across the interface tries to homogenize the couple. If the duration is not long, the concentrations at both ends would still be the same as the initial concentrations. Under such conditions, the diffusion medium may be treated as infinite and the diffusion problem can be solved using Boltzmann transformation. If the diffusion duration is long (this will be quantified later), the concentrations at the ends would be affected, and the diffusion medium must be treated as finite. Diffusion in such a finite medium cannot be solved by the Boltzmann method, but can be solved using methods such as separation of variables (Section 3.2.7) if the conditions at the two boundaries are known. Below, the concentrations at the two ends are assumed to be unaffected by diffusion. 

The above transformation from Equation 3-10 to 3-34 is called the Boltzmann transformation. The two variables x and t are replaced by a single variable p, and the partial differential equation becomes an ordinary differential equation. The transformation works only if the initial and boundary conditions can also be written in terms of the single variable p. To solve the transformed ordinary differential equation, define w = dC/dp. Hence, Equation 3-34 becomes... 

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

Comparing Equation 3-37a with Equations 3-37b and 3-37c shows that Equation 3-37a is identical to Equations 3-37b and 3-37c. That is, both the initial condition and the boundary conditions are transformed to the same two equations using the variable Only when this happens can the problem be solved using Boltzmann transformation. If the diffusion duration is long, and C at x = oo changes with time, it would be impossible to write boundary condition 2 using a single... 

Using Boltzmann transformation, the initial and boundary conditions become... 

Each side satisfies separately the conditions for applying Boltzmann transformation (r[=x/V4Dt) hence, the solution is... 

The mathematical translation of the plane-source problem is as follows. Initially, there is a finite amount of mass M but very high concentration at a = 0, i.e., the density or concentration at a = 0 is defined to be infinite (which is unrealistic but merely an abstraction for the case in which initially the mass is concentrated in a very small region around a = 0). The initial condition is not consistent with that required for Boltzmann transformation. Hence, other methods must be used to solve the case of plane-source diffusion. Because this is the classical random walk problem, the solution can be found by statistical treatment as the following Gaussian distribution ... 

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

When D depends on space coordinates, the diffusion equation is in general difficult to solve because D cannot be taken out of the differential in Equation 3-9, and Boltzmann transformation cannot be applied. The solution given in Equation 3-44 may be viewed as a special case of D depending on x D takes the value of Dl at X < 0 and Dr at > 0. 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

Applying Boltzmann transformation to Equation 3-114a with u = a(D/f) (similar to Section 3.2.4), after some steps (see Section 4.2.2.1), the solution is... 

Use the Boltzmann transformation to solve the following diffusion equation ... 

Using Boltzmann transformation and following similar steps as in the case of diffusion in the melt, the solution is... 

The problem may be solved using Boltzmann transformation, and the solution for W is... 

If diffusion distance is small compared to the size of either phase, because each side satisfies separately the conditions for applying the Boltzmann transformation (r =x/s/4Dt), the solution for species 1 in each phase is... 

Recently Ruoff (Rll) has rederived the Stefan and Neumann solutions using the Boltzmann transformation. 

The short time solution is obtained using die Boltzmann transformation on the indqrendent variables... 

For short times, where diffusion effects have not reached the end of the cell, the Boltzmann transformation is used to yield an expression for the diffiisivity at concentration Ca ... 

The transient concentration in the semi-infinite medium can be obtained by solving the Pick parabolic mass diffusion equations using the Boltzmann transformation, q = xl->j4Dt, as follows. The governing equation for molecular diffusion in one dimension using Pick s second law can be written as... 

Thus, a PDE of the second order in space and time can be transformed into an ordi-nary differential equation (ODE) in one variable. The transformation q = xlyf4D is called the Boltzmann transformation. The solution to the ODE in the transformed variable, q, can be written as... 

The semi-infinite medium is employed to study the spatiotemporal patterns that the solution of the non-Fick damped wave diffusion and relaxation equation exhibits. This medium has been used in the study of Pick mass diffusion. The boundary conditions can be different kinds, such as constant wall concentration, constant wall flux (CWF), pulse injection, and convective, impervious, and exponential decay. The similarity or Boltzmann transformation worked out well in the case of the parabolic PDF, where an error function solution can be obtained in the transformed variable. The conditions at infinite width and zero time are the same. The conditions at zero distance from the surface and at infinite time are the same. 

Transient diffnsion in a semi-infinite medium was studied under a constant wall concentration bonndary condition using Pick s second law of diffusion and the damped wave diffnsion and relaxation equation. The latter can acconnt for finite speed of propagation of mass. A new procedure called the method of relativistic transformation was developed to obtain bounded and physically realistic solntions. These solntions were compared with the solution from Pick s second law of diffusion obtained nsing the Boltzmann transformation and the solution presented in the literatnre by Baumesiter and Hamill [32]. Four different regimes of the solution... 

The equation can be solved by the method of Boltzmann transformation, as in Understanding Voltammetry, or alternatively by a mathematical method known as Laplace transformation, in which an integral transform is used to convert a partial differential equation into an ordinary differential equation. The transformation is ... 

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