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Equations
Flow field
Rate of production
Accumulation
Conserved quantity, internal energy
Nonconserved quantity, entropy
A flow field is a vector field
Discussion
Accumulation was demonstrated graphically in Mathematica animation in the previous lecture
Accumulation is the negative of the divergence of the flux plus the creation or destruction of material
Consider the accumulation of internal energy, a conserved quantity. It is equal to the negative of the divergence of internal energy.
Entropy change is due to net flux, and there is a production term.
Preview
The book Kinetics of Materials is divided into five parts
Half of this class is devoted to diffusion
Study how fast composition readjusts itself
Continuity equation and Fick's second law
Most things interested in are not in equilibrium
Predict how properties and performance change over time
Use concepts from thermodynamics to ascribe values, thermodynamic potentials to system
From distribution of thermodynamic potential determine rate to equilibrium
Diffusive flux is equal to a constant times the gradient of the chemical potential
The chemical potential is a function of local composition
Solve Fick's second law
Irreversible thermodynamics is about ascribing thermodynamic values to nonuniform systems.
Fundamental basis provided today
The field of irreversible thermodynamics is not on as rigorous footing as thermodynamics
Chapter 2 is highlighted today
The first part of the book provides an idea of where diffusion equations comes from
The remaining part of the class is about how microstructure evolves in the absense of phase transformation
Diffusion
Diffusion is the motion of species, components, or matter. Fluid transportation involves the diffusion of momentum
Differential equations are used in a macroscopic description
Mechanisms are described at a microscopic level
The details of atomistic mechanisms is used to understand the macroscopic details
Mechanisms influence the proportionality constant
Fick's 1st and 2nd Laws
The first law involves diffusive flux, which is proportional to gradient of concentration
The details of this description came originally from empirical observation. Imagine setting up gradients, measuring, and making plots. The plot is essentially linear.
How could that relationship come about
Entropy and Entropy Production
Entropy production is key in irreversible thermodynamics
Divide a system into small volume elements
Imagine that we can monitor local values of thermodynamic quantities
There need not be the same values of local quantities
There can be flux between volume elements
A basic postulate of irreversible thermodynamics is that entropy production is always positive at each point in a system
An isolated system at fixed energy evolves to highest entropy
In every small volume element, entropy is increasing
(note is equations that uppercase letters indicate total amount)
U: internal energy
u: energy per unit volume
TdS = du + dw + sum(mu dc)
There are many contributions of work to dw, such as pdV work, interfacial energy, stress fields
- Psi d zeta = dw + sum(mu dc)
Pressure, p, is a thermodynamic potential and dV is related to an extensive quantity
Consider sigma d epsilon. A terms consist of something acting as a potential and a differential of an extensive quantity