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Materials Science and Engineering/Equations/Kinetics

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Mathematical Foundation

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Time-Dependent Field

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: Velocity
: Time-Dependent Field

Accumulation

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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

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: Oriented surface around a volume

General Set of Linear Equations

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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

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

Transformation of Rank-Two Tensor

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Irreversible Thermodynamics

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Differential Change in Entropy

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Entropy Production

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

Empirical Force-Flux Law

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Fourier's

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Modified Fick's

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Ohm's

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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:

 

Force-Flux Relations with Constrained Extensive Quantities

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Diffusion Potential

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Onsager Symmetry Principle

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Driving Forces and Fluxes

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Diffusion in Absence of Chemical Effects

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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
 

Diffusion in a Volume-Fixed (V-Frame)

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  • 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,
 

Diffusion of Charged Ions in Ionic Conductors

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

Electromigration in Metals

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  • 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

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

Mass Diffusion Driven by Capillarity

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  • 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

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Diffusion Equation in the General Form

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

Fick's Second Law

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Linearization of Diffusion Equation

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Heat 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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Localized Source

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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:

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.
  • Relation of and :
 

Steady-State Solutions

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Harmonic Functions

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One Dimension

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Cylindrical Shell

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

Spherical Shell

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  • Laplacian operator in spherical coordinates

Variable Diffusivity

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

Infinite Media with Instantaneous Localized Source

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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:
  • Integral over all sources:
 

Method of Separation of Variables

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

Method of Laplace Transforms

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  • 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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Many-Body Model

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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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Diffusion and Correlated Jumps

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  • Correlation factor:
 
  • Macroscopic Diffusivity and Microscopic Parameters:
 
 
 

Atomic Models of Diffusivity

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Metals

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Correlation Factor

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Isotope Effect

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