Boltzmann superposition principle diffusion

Equations (10) and (11) can also be used to calculate several other situations which are of importance for DTA. For these calculations the Boltzmann superposition principle, already used in the writing of Eq. (8), will be employed. The actual changes that occur in the sample as a function of time are additive as a series of separate events, each of which is describable by Eqs. (10) and (11). Figure 4.15 shows the results for an analysis of heating through the glass transition. It is assumed that the thermal diffusivity jumps at time t from to kg . The thermal diffusivity k is assumed to be... 

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

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. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

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