Materials Science and Engineering/Equations/Kinetics

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Mathematical Foundation[edit | edit source]

Time-Dependent Field[edit | edit source]

: Velocity
: Time-Dependent Field

Accumulation[edit | edit source]

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

Divergence Theorem[edit | edit source]

: Oriented surface around a volume

General Set of Linear Equations[edit | edit source]


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.

Eigenvalue Equation[edit | edit source]

: square matrix or tensor
: eigenvector (special vector)
: eigenvalue (special scalar multiplier)

Transformation of Rank-Two Tensor[edit | edit source]


Irreversible Thermodynamics[edit | edit source]

Differential Change in Entropy[edit | edit source]


Entropy Production[edit | edit source]

: Rate of entropy-density creation
: Flux of heat
: Conjugate force
: Conjugate flux

Empirical Force-Flux Law[edit | edit source]

Fourier's[edit | edit source]


Modified Fick's[edit | edit source]


Ohm's[edit | edit source]


Basic Postulate of Irreversible Thermodynamics[edit | edit source]

The local generation of entropy, is nonnegative


Coupling Between Forces and Fluxes[edit | edit source]


Abbreviated form:


Force-Flux Relations with Constrained Extensive Quantities[edit | edit source]


Diffusion Potential[edit | edit source]


Onsager Symmetry Principle[edit | edit source]


Driving Forces and Fluxes[edit | edit source]

Diffusion in Absence of Chemical Effects[edit | edit source]

  • 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[edit | edit source]


Diffusion of Substitutional Particles in Concentration Gradient[edit | edit source]

  • 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

Diffusion in a Volume-Fixed (V-Frame)[edit | edit source]

  • 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[edit | edit source]

  • Evaluate by substitution of interstitial mobility,

Diffusion of Charged Ions in Ionic Conductors[edit | edit source]

  • : Electric field
  • Absence of concentration gradient:
  • Electrical conductivity:

Electromigration in Metals[edit | edit source]

  • Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal
    • : Flux of conjuction electrons
    • : Flux of interstitials

Mass Diffusion in Thermal Gradient[edit | edit source]

  • Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs:

Mass Diffusion Driven by Capillarity[edit | edit source]

  • The system consists of two network-constrained components:
    • Host atoms
    • Vacancies
  • No mass flow within the crystal (the crystal C-frame is also the V-frame)
  • Constant temperature and no electric field

Fick's Second Law[edit | edit source]

Diffusion Equation in the General Form[edit | edit source]

: source or sink term
: any flux in a V-frame

Fick's Second Law[edit | edit source]


Linearization of Diffusion Equation[edit | edit source]


Heat Equation[edit | edit source]

: enthalpy density
: heat capacity
: thermal diffusivity

Constant Diffusivity[edit | edit source]

One-Dimensional Diffusion Along x from an Initial Step Function[edit | edit source]

Localized Source[edit | edit source]

  • Source strength,

Diffusivity as a Function of Concentration[edit | edit source]

  • Interdiffusivity:

Diffusivity as a Function of Time[edit | edit source]

  • Change of variable:
  • Transformed equation:
  • Solution:

Diffusivity as a Function of Direction[edit | edit source]

  • The diagonal elements of are the eigenvalues of , and the coordinate system of defines the principal axes.
  • Relation of and :

Steady-State Solutions[edit | edit source]

Harmonic Functions[edit | edit source]


One Dimension[edit | edit source]


Cylindrical Shell[edit | edit source]

  • Laplacian Operator:
  • Integrate Twice and Apply the Boundary Conditions:

Spherical Shell[edit | edit source]

  • Laplacian operator in spherical coordinates

Variable Diffusivity[edit | edit source]

  • Steady-state conditions
  • varies with position
  • Solution is obtained by integration:

Infinite Media with Instantaneous Localized Source[edit | edit source]


Solutions with the Error Function[edit | edit source]

  • 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 | edit source]

  • System : Three Dimensions,
  • Equation :
  • Solution :

Method of Laplace Transforms[edit | edit source]

  • Laplace transform of a function

Atomic Models of Diffusion[edit | edit source]

Model of One-Particle with Step Potential-Energy Wells[edit | edit source]


Model of One-Particle with Step Potential-Energy Wells[edit | edit source]


Many-Body Model[edit | edit source]


Diffusion as Series of Discrete Jumps[edit | edit source]


Diffusivity and Mean-Square Particle Displacement[edit | edit source]


Relation of Macroscopic Diffusivity and Microscopic Jump Parameters[edit | edit source]


Diffusion and Correlated Jumps[edit | edit source]

  • Correlation factor:
  • Macroscopic Diffusivity and Microscopic Parameters:

Atomic Models of Diffusivity[edit | edit source]

Metals[edit | edit source]

Correlation Factor[edit | edit source]


Isotope Effect[edit | edit source]