Mathematical Foundation
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- : Oriented surface around a volume
General Set of Linear Equations
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Irreversible Thermodynamics
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Differential Change in Entropy
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- : Rate of entropy-density creation
- : Flux of heat
- : Conjugate force
- : Conjugate flux
Empirical Force-Flux Law
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Basic Postulate of Irreversible Thermodynamics
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The local generation of entropy, is nonnegative
Coupling Between Forces and Fluxes
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Abbreviated form:
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Force-Flux Relations with Constrained Extensive Quantities
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Onsager Symmetry Principle
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Driving Forces and Fluxes
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- : source or sink term
- : any flux in a V-frame
Linearization of Diffusion Equation
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- : enthalpy density
- : heat capacity
- : thermal diffusivity
Constant Diffusivity
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One-Dimensional Diffusion Along x from an Initial Step Function
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- Source strength,
Diffusivity as a Function of Concentration
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- Interdiffusivity:
Diffusivity as a Function of Time
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- Change of variable:
- Transformed equation:
- Solution:
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Diffusivity as a Function of Direction
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- The diagonal elements of are the eigenvalues of , and the coordinate system of defines the principal axes.
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- Relation of and :
Steady-State Solutions
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- Laplacian Operator:
- Integrate Twice and Apply the Boundary Conditions:
- Laplacian operator in spherical coordinates
Variable Diffusivity
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- Steady-state conditions
- varies with position
- Solution is obtained by integration:
Solutions with the Error Function
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- Uniform distribution of point, line, or plana source placed along
- Contribution at a general position from the source:
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- Integral over all sources:
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Method of Separation of Variables
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- System : Three Dimensions,
- Equation :
- Solution :
- Laplace transform of a function
Atomic Models of Diffusion
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Model of One-Particle with Step Potential-Energy Wells
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Model of One-Particle with Step Potential-Energy Wells
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Diffusion as Series of Discrete Jumps
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Diffusivity and Mean-Square Particle Displacement
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Relation of Macroscopic Diffusivity and Microscopic Jump Parameters
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- Macroscopic Diffusivity and Microscopic Parameters:
Atomic Models of Diffusivity
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