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