Materials Science and Engineering/Equations/Kinetics

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
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  •  
  •  
  •  

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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