h2. Last time
h3. Equations
Flow field
Rate of production
Accumulation
Conserved quantity, internal energy
Nonconserved quantity, entropy
A flow field is a vector field
h3. 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.
h2. 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
h2. 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
h2. 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
h2. 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
Consider Equation 2.6 in Kinetics of Materials.
Use the continuity equations and the first law to derive an expression that relates entropy production to local fluxes at a point in a system.
Consider Equation 2.15 in Kinetics of Materials
Entropy production rate is related to the flux of heat.
Units are in terms of energy density per time
Determine how fast the energy density is changing.
Energy density change is termed dissipation
Units of energy density dissipation is J m \-3 s-1
There is an unstated assumption that temperature does not vary with time
Nonuniform may refer to variations in space while not constant refers to changes in time
Imagine a system with spatial variations of temperature
Consider heat transfer and maintaining temperature. Find temperature at any point. The system is not in equilibrium but is changing with time. There is heat constantly added to one end and extracted to maintain a temperature gradient.
Terms on the right involve fluxes dotted with a gradient
T sigma dot = - JQ / T grad T - sum J grad Psi
Flux of chemical component, heat, etc. involves a pairing of fluxes and forces. There are gradients of some potential
There is a postulate that entropy production is positive
h3. Entropy Production in Heat Flow
h4. Fourier's law relates heat flux
JQ = - K grad T
K is thermal conductivity
If heat is the only thing that is flowing, the entropy production is minus the heat flux dotted with grad T / T
sigma dot = - JQ grad T / T
sigma dot = K (grad T) \^2 / T
The entropy production is postulated to be greater than zero, which means that the thermal conductivity is greater than zero
K > 0
A positive thermal conductivity means that heat flows from a positive body to a negative body.
h4. Fick's Law
J = M c grad mu
when sigma dot is greater than zero, the mobility is always positive
The diffusion coefficient in traditional form of Fick's law
D is either positive or negative
There is a relationship between flux and chemical potential
Must apply to sum of components or each individual component
h2. Linear Irreversible Thermodynamics
Systems create entropy locally due to fluxes of many things
Predict what fluxes are and define kinetic models
There is a relationship between fluxes and driving forces especially with multiple driving forces (applied simultaneously), most generally flux of heat (q), charge (q), components (i)
JQ = JQ (FQ, Fq, Fi)
Look at systems not far from equilibrium
Fi: components carry energy
Fq: electronic contribution to heat flow{latex}
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