- : Velocity
- : Time-Dependent Field
Rate of accumulation is the negative of the divergence of the flux of the quantity plus the rate of production
- : Rate of production of the density of in
- : The divergence of
- : Rate at which flows through area
- : Oriented surface around a volume
The vector equation is equivalent to a matrix equation of the form
where M is an m×n matrix, x is a column vector with n entries, and y is a column vector with m entries.
- : square matrix or tensor
- : eigenvector (special vector)
- : eigenvalue (special scalar multiplier)
- : Rate of entropy-density creation
- : Flux of heat
- : Conjugate force
- : Conjugate flux
The local generation of entropy, is nonnegative
Abbreviated form:
Force-Flux Relations with Constrained Extensive Quantities
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- Components diffuse in chemically homogeneous material
- Diffusion measured with radioactive tracer
- Fick's law flux equation derived when self-diffusion occurs by the vacancy-exchange mechanism.
- The crystal is network-constrained
- There are three components:
- Inert atoms
- Radioactive atoms
- Vacancies
- C-frame: single reference frame
- Vacancies assumed to be in equilibrium throughout
- Raoultian behavior
Diffusion of i in Chemically Homogeneous Binary Solution
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Diffusion of Substitutional Particles in Concentration Gradient
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- Constraint associated with vacancy mechanism:
- Difference in fluxes of the two substitutional species requires net flux of vacancies.
- Gibbs-Duhem relation:
- Chemical potential gradients related to concentration gradients:
Flux is proportional to the concentration gradient
Assumptions that simplify
- Concentration-independent average site volume
- The coupling (off-diagonal) terms, and , are small compared with the direct term
- Velocity of a local C-frame with respect to the V-frame: velocity of any inert marker with respect to the V-frame
- Flux of 1 in the V-frame:
- The interdiffusivity, , can be simplified through
- The L-frame and the V-frame are the same
Diffusion of Interstitial Particles in Concentration Gradient
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- Evaluate by substitution of interstitial mobility,
- : Electric field
- Absence of concentration gradient:
- Electrical conductivity:
- Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal
- : Flux of conjuction electrons
- : Flux of interstitials
- Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs:
- The system consists of two network-constrained components:
- No mass flow within the crystal (the crystal C-frame is also the V-frame)
- Constant temperature and no electric field
- : source or sink term
- : any flux in a V-frame
- : enthalpy density
- : heat capacity
- : thermal diffusivity
One-Dimensional Diffusion Along x from an Initial Step Function
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- Source strength,
- Interdiffusivity:
- Change of variable:
- Transformed equation:
- Solution:
- The diagonal elements of are the eigenvalues of , and the coordinate system of defines the principal axes.
- Relation of and :
- Laplacian Operator:
- Integrate Twice and Apply the Boundary Conditions:
- Laplacian operator in spherical coordinates
- Steady-state conditions
- varies with position
- Solution is obtained by integration:
- Uniform distribution of point, line, or plana source placed along
- Contribution at a general position from the source:
- Integral over all sources:
- System : Three Dimensions,
- Equation :
- Solution :
- Laplace transform of a function
Relation of Macroscopic Diffusivity and Microscopic Jump Parameters
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- Macroscopic Diffusivity and Microscopic Parameters: