Eigenvalues diffusivity matrix

Similar to binary diffusivities, each element in the diffusivity matrix is expected to depend on composition, sometimes strongly, especially for highly nonideal systems. If the nonideality is strong enough to cause a miscibility gap, the eigenvalues would vary from positive to zero and to negative. If there is no miscibility gap, the eigenvalues are positive but can still vary with composition. 

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

The zeroth eigenvalues of matrix A and G are zero, so that without any approximation, one can write an equation for diffusion mode... 

Let us note, that the matrixes A and G are approximations of the real situation though, in any case, the zeroth eigenvalues of the matrixes must be zero and equation (4.1) for diffusive mode is valid, the other eigenvalues of matrix G depends, generally speaking, on the mode label. In fact, the written equations for the relaxation modes are implementation of the statements that the motion of a single macromolecule can be separated from others, and the motion of a single macromolecule can be expanded into an independent motion of modes. 

Matrix of Fick diffusion coefficients relative to a reference diffusivity [ — ] Matrix of turbulent diffusion coefficients [m /s] zth eigenvalue of [D] [m7s]... 

Figure 3.5 Plots of the eigenvector components for the eigenvalues of largest and smallest magnitudes for a 1-D diffusion matrix.

Near computational linear dependence in the basis set was monitored in all calculations reported in this paper by diagonalizing the overlap matrix. A 30s basis subset was centred on each of the points defining a particular distributed basis set. Diffuse basis fimctions were deleted from off-atom basis sets until the smallest eigenvalue of the overlap matrix, e, satisfied the condition e < 10 . So, for example, the basis set designated 30s ac 28s oa ac) [nj = 5] which arises in... 

To solve a diffusion equation, one needs to diagonalize the D matrix. This is best done with a computer program. For a ternary system, one can find the two eigenvalues by solving the quadratic Equation 3-lOOe. The two vectors of matrix T can then be found by solving... 

The eigenvalues of this matrix may be computed from the general Eqs. 8.3.48. However, for Stefan diffusion in a ternary mixture, it is possible to derive simple analytical expressions for the eigenvalues as... 

In their original development of the linearized theory Toor (1964) and Stewart and Prober (1964) proposed that correlations of the type given by Eqs. 8.8.5 and 8.8.7 could be generalized by replacing the Fick diffusivity D by the charactersitic diffusion coefficients of the multicomponent system that is, by the eigenvalues of the Fick matrix [ >]. The mass transfer coefficient calculated from such a substitution would be a characteristic mass transfer coefficient an eigenvalue of [/c]. For example, the Gilliland-Sherwood correlation (Eq. 8.8.5) would be modified as follows ... 

The sensitive state of the system corresponds to the eigenvalues A meeting the requirement Re(212) = 0. We shall investigate the appearance of sensitive states depending on the diffusion coefficients, Dx, Dy, and on the control parameters on which the elements of the stability matrix au rely. 

Scriven [334] showed that the stability of spatially discrete homogeneous reaction-diffusion systems can be analyzed in terms of the structural modes of the network, i.e., the eigenvectors of the Laplacian matrix L. We have extended that approach [305], and the eigenvalues and eigenvectors of the matrix... 

The investigation of relaxation times and diffusion coefficients requires the determination of the eigenvalues of the matrix corresponding to the system of algebraic equations obtained from Eqs. (18) after Fourier-Laplace transformation (s, Laplace transform of time q, Fourier transform of the space coordinate). The roots of the secular equation are... 

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