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Linear Irreversible Thermodynamics
Diffusion Potential
Network Constraint
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Self-diffusion
Interdiffusion
Diffusion reference frames
Linear Irreversible Thermodynamics
Expand y(xo + dx) = y(xo) + dy / dx |xo dx
Expand JQ(FQ, Fq, Fi,...) + ...
Expand the heat flux when there are operative any number of independent variables, such as heat flow, charge flow, etc.) about a point where the driving forces are small.
The point where the driving force goes to zero is where the flux is zero
JQ = dJQ / dFQ FQ + dFQ / dFq Fq + ...
Jq = dJq / dFQ FQ + dJQ / dFq Fq + ...
The direct terms are dJQ / dFQ FQ and dJQ / dFq Fq
The smaller terms are dFQ / dFq Fq and dJq / dFQ FQ
Heat flux
Heat flux arises where there are many driving forces present.
There could be a gradient in electrical potential or in any component that can carry heat
Flux couples with the driving force
Generalize a heat flux to an expression
JQ = Lalphabeta F beta, where Lalphabeta = dJ / dF
This represents the entire set of equations. The repeated subscript implies summation.
Entropy production
T sigma dot = sum beta sum alpha Lalphabeta Falpha Fbeta
Entropy production is the product of flux and a corresponding driving force.
Multiply flux times a driving force.
Lalphabeta are related to macro transport coefficients as in heat flow alone. (only force is from a temperature gradient)
Fourier's Law
JQ = LQQ FQ
JQ = LQQ (-1/T grad T)
JQ = -LQQ / T grad T
Identify with macroscopic quantity
K = -LQQ / T
The thermal conductivity is K
Heat flow in conductor
Put material in a thermal gradient
Consider heat flow in a conductor in an open circuit condition
Allow the case that there is a flux of heat with no electrical connection
JQ = LQQ ( - 1/T grad T ) + LQq q grad phi
Jq = LqQ ( - 1/T grad T ) + Lqq q grad phi
If there is no current flow, Jq = 0
Eliminate grad phi in the terms
JQ = - (LQQ / T - LqQ LQq / (T Lqq)) grad T
The thermal gradient causes a potential gradient in the material
More charge carriers are generated with increased temperature
There is charge transport
The thermal conductivity is the coefficient in front of grad T, and there is more than one contribution.
Onsager Symmetry Postulate
Lalphabeta = Lbetaalpha
The rate of change of the flux of quantity alpha with respect to force beta
d Jalpha/ d Fbeta = d Jbeta / d Falpha
Consider a model of order/disorder transition
A key point in the theory involves entropy production, which is the product of forces and fluxes.
The product is in units of energy dissipation.
There is no unique way to pick conjugate forces and fluxes
Consider a simple case with temperature gradient, etc. There are other combinations that result in units of energy dissipation.
A warning is that the theory requires careful choices of force, flux pairs, which when multiplied give energy dissipation or production.
Network Constraint
Constraint on independent variables in system of interest, such as components in a system, atoms and vacancies
A most relevant example is diffusion in a crystal by a vacancy mechanism
The only way that the atom can move is if it exchanges with a vacancy, and this is discussed in Chapters 7 and 8
The vacancy approximately executes a random walk
Simulate the random walk of a vacancy.
The effect is an exchange with a distinguished atom.
Consider a crystal in which there are a fixed number of vacancies and the vacancy concentration is constant
When the there are no sources or sinks of vacancies, the total number is fixed
In a crystalline system, it is the network of sites that remains fixed
Vacancies are in thermal equilibrium, and the number of A atoms plus the number of B atoms plus the number of vacancies is fixed.
There is a network of sites that can be occupied that remains fixed (constant)
The occupancy of sites is changing, but the network is static. There is not adding or removing of sites.
Interdiffusion
Vacancy sources and sinks are like atom sources and sinks.
Inject atoms in one part of a crystal and remove in another
Nothing happens far away
The crystal expands in one place and contracts in another
The middle part is being carried along with respect to the ends of the sample
The A atoms move at a different rate than the B atoms, and there is a drift velocity.
The flux looks different when considering a moving plane
There may not be three independent variables even with two types of atoms and vacancies due to constraints
Vacancies are considered as a special component
Reference frames should make good sense
Diffusion Potential
The chemical potential is defined as the change of energy that results from adding a component
Fick's law relates a gradient in chemical potential with diffusion
Consider a concentration gradient and the carry of charge
There may be an aliovalent impurity in an ionic material
Excess positive charge may be associated with an impurity
There is diffusion of the species associated with a gradient
Adding an electric field results in a gradient of electrical charge
What gives rise to a force?
The flux is proportional to force, and a contribution of force is from the rate of change of concentration and charge with distance
The diffusion potential contains mu and q phi
An example is the diffusion of a charged species within a concentration gradient and electric potential gradient
Ji = grad mu i + q grad phi
The electrochemical potential is used to describe kinetic parameters
Be careful about the two parts
The term mui is due to chemical effects and qi phi is due to charged species
sometimes qi phi is folded into mu i
A definition is that the diffusion potential is any potential that accounts for change of energy...
Add charge with electric field and add energy
Understand how to set up
Where does Fick's First Law come from?
It comes from an analysis of experiments that was consistent with a theory that helps inform where equations come from.
There are detailed derivations of flux equations of various types of diffusion using Laphabeta coefficients
There may be a need to invoke a network constraint with the diffusion in crystalline materials
Dispense with a constraint when considering diffusion in non-crystalline materials
Diffusion
Components move due to gradients of diffusion potential
Consider a simplest case wherein the force on a component is equal to the gradient mu i
Fi = - grad mu i
There are more concentration gradients invoked in Chapter 3, such as the electrochemical potential, the effect of stress on diffusion, and capilary diffusion
The simplest case leads to Fick's 1st law
Ji = -Di grad c
The term Di is the diffusivity
Diffusion in three increasingly complicated systems all in crystals
Self-diffusion in a pure material
Gradients can't be measured in a one component system
Diffusion in crystals involve a vacancy mechanism
What are the mobilities at a given temperature?
There is a clever experiment used to measure or describe diffusion of a radio tracer species in an element.
The experiment utilizes stable isotopes versus radioactive isotopes with a long half-life
Prepare a system with *1 and a stable isotope called 1
Form a diffusion couple where there is a small amount of C*1
Note that there is the same chemical composition. There is the same number of atoms on both sides, and there is no reason there should be non-ideality. The only difference is that there is a detectable mass difference. With the same chemical composition and bonding characteristics, there is ideal solution behavior.
Equations are written below that describe the system. The term "c" refers to a "crystal frame" or "c-frame". The flux is attached to a given plane.
J1C = L11 grad mu i + L1*1 grad mu*1 + vacancy term
J*1C = L1*1 grad mu i + L*1*1 grad mu*1 + vacancy term
Jv
Look at the entire system. Invoke a network constraint and assume that vacancies are in local equilibrium. Terms can be regrouped. Through the network constraint, redundant variables are removed.
Write a 2 x 2 system of equations with components of choosing. The terms grad (mu1 - muv) and grad (mu*1 - muv) are associated with J1C.
Reference frames
Describe diffusion with respect to a network. Use a given plane in a crystal as a reference coordinate system. The network is assumed to be constrained and to remain in the same place.
The system is constrained and the sum of fluxes is equal to zero.
J1C + J*1C + JvC = 0
Assume that vacancies are in thermal equilibrium
dq / dC1 = 0
Consider how to define a chemical potential of vacancies and Rault's law. Define a partial of chemical potential with respect to x.
d mui / dx = kT d lnC / dx
Define a final relationship
J1C = - kT [L11 / C1 - L1*1 / C*1 ] dC1 / dx
J*1C = - kT [L*11 / C1 - L1*1 / C*1 ] dC*1 / dx
Derive a final expression
JC*1 = -kT [ L11 / C1 - L1*1 / C*1] dC1 / dx
JC*1 = - *D dC*1 / dX
The theory of linear irreversible thermodynamics was used to write an expression of coupled forces and fluxes.
Extract expression of flux of radio tracer. Linear proportionality is expected. This is consistent with a multi-interacting coupled system.