Irreversible Thermodynamics and Coupling between Forces and Fluxes
The foundation of irreversible thermodynamics is the concept of entropy production
There is a natural and general coupling of the driving forces and corresponding fluxes that are present in a nonequilibrium system.
2.1 Entropy and Entropy Production
The conservation of internal energy is a consequence of the first law of thermodynamics.
The existence of the entropy state function is a consequence of the second law of thermodynamics.
In classical thermodynamics, the value of a system's entropy is not directly measurable but can be calculated.
In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium.
Consider a continuous system that demonstrates gradients in temperature, chemical potential, and other intensive thermodynamic quantities.
Fluxes of heat, mass, and other extensive quantities develop as the system approaches equilibrium.
Divide the system into small contiguous cells
The local equilibrium assumption is that the thermodynamic state of each cell is specified and in equilibrium with the local values of thermodynamic potentials.
Gibbs' fundamental relation can be used to calculate changes in the local equilibrium states as a result of evolution of the spatial distribution of thermodynamic potentials.
Divide dU through by a constant reference cell volume in order that all extensive quantities are on a per unit volume basis.
Equation 2.4 (derived by combining the first and second law) can be used to define the continuum limit for the change in entropy in terms of measurable quantities.
The differential terms are the first-order approximations to the increase of quantities at a point.
Because energy, heat, and mass may flow between cells during kinetic processes, they cannot be treated as isolated systems, and application of the second law must be generalized to the system of interacting cells.
It is useful to consider entropy as a fluxlike quantity capable of flowing from one part of a system to another, like energy, mass, and charge.
Entropy flux, denoted by Js, is related to heat flux.
Mass, charge, and energy are conserved quantities and additional restrictions on the flux of conserved quantities apply.
However, entropy is not conserved--it can be created or destroyed locally.
2.1.1 Entropy Production
The local rate of entropy-density creation is denoted by sigma dot.
Integrate to find the total rate of entropy creation in a volume.
In a general system, the total entropy increase depends upon how much entropy is produced within it and how much entropy flows through its boundaries.
The entropy flux is related to the sum of all potentials multiplying their conjugate fluxes.
Introduce the flux of heat.
2.1.2 Conjugate Forces and Fluxes
Regarding Equation 2.15, every term on the right-hand side is the scalar product of a flux and a gradient with the same units as energy dissipation density.
Each term that multiplies a flux is a force of that flux. The paired forces and fluxes in the entropy production rate are termed conjugate forces and fluxes.
There can be constraints in systems relating changes in extensive quantities.
Assume in Chapters 1 - 3 that the material is isotropic and that forces and fluxes are parallel.
Assumption is removed in anisotropic materials in Chapter 4.
There are well-known empirical force-flux laws that apply under certain conditions.
Extensive Quantity | Flux | Conjugate Force | Empirical Force-Flux Law | Equation |
|---|---|---|---|---|
Heat |
|
| Fourier's |
|
Component i |
|
| Modified Fick's |
|
Charge |
|
| Ohm's |
|
2.1.3 Basic Postulate of Irreversible Thermodynamics
The basic postulate of irreversible thermodynamics is that, near equilibrium, the local entropy production is nonnegative
Using empirical laws displayed in the table above, the entropy production can be identified in a few special cases.
Use Equation 2.15 and Fourier's heat-flux law to predict that thermal conductivity is always positive.
If diffusion is the only operating process, an equation implies that each mobility is always positive.
2.2 Linear Irreversible Thermodynamics
In many materials, a gradient in temperature produces not only flux of heat but also a gradient in electric potential.
If heat can flow, the gradient in electrical potential will result in a heat flux.
This coupled phenomenon is called the thermoelectric effect.
Flows of mass, electricity, and heat all involve particles possessing momentum, and interactions may be expected as momentum is transferred between them.
A formulation of the coupling effects can be obtained by generalization of the previous empirical force-flux equations
2.2.1 General Coupling between Forces and Fluxes
The fluxes are a function of the driving force acting on the system.
Assuming that the system is near equilibrium and the driving forces are small, each of the fluxes can be expanded in a Taylor series near the equilibrium point
In the approximation in the book, the fluxes vary linearly with the forces.
Direct coefficients (diagonal terms): coupling of each flux to its conjugate driving force.
Coupling coefficients (off-diagonal terms): coupling effects or cross effects
Illustrate connection between direct coefficients and the empirical force-flux laws
Consider a pure material with a constant thermal gradient imposed along it and that is electronically conducting
If a constant thermal gradient is imposed and no electrically conductive contacts are made at the ends of the specimen, the heat flow is in a steady state and the charge-density vanishes.
Fourier's law pertains, but the thermal conductivity K depends on the direct coefficient as well as on the direct and coupling coefficients associated with electrical charge flow.
In general, the empirical conductivity associated with a particular flux depends on the constraints applied to other possible fluxes.
2.2.2 Force-Flux Relations when Extensive Quantities are Constrained
Changes in one extensive quantity are in many cases coupled to changes in others.
Network constraint: whenever one component leaves a site, it must be replaced.
As a result of the replacement constraint, the fluxes of components are not independent of each other.
This type of constraint is absent in amorphous materials and with interstitial solutes in crystalline materials that lie in the interstices between larger substitutional atoms.
The conjugate force of the diffusion of a network-constrained component i depends upon the gradient of the difference between the chemical potential of component i and Nc rather than on the chemical potential gradient of i alone.
The difference arises because during migration a site's state changes from occupancy by an atom of type i to occupancy by a vacancy.
In the development above, the choice of the Ncth component in a system under network constraint system is arbitrary. The flux of each component in Equation 2.21 must be independent of the choice, and this independence imposes conditions on the L alpha beta coefficients.
The sum of the entries in any row or column of the matrix Lij is zero.
Below is a table of conjugate forces and fluxes that are obtained when the only constraint is a network constraint.
Quantity | Flux | Conjugate Force |
|---|---|---|
Heat |
|
|
Component i |
|
|
Charge |
|
|
2.2.3 Introduction of the Diffusion Potential
Any potential that accounts for the storage of energy due to the addition of a component determines the driving force for the diffusion of that component.
Diffusion potential: The total conjugate force of a diffusing component. The sum of all supplemental potentials, including the chemical potential.
The conjugate force of the flux of a component is always of the form F = - grad Phi
2.2.4 Onsager's Symmetry Principle
Three postulates were utilized to derive relations between forces and fluxes. They do not follow from the first and second laws of thermodynamics
- The rate of entropy change and the local rate of entropy production can be inferred by invoking equilibrium thermodynamic variations and the assumption of local equilibrium.
- The entropy production is nonnegative
- Each flux depends linearly on all the driving forces
Onsager's principle supplements these postulates and follows from the statistical theory of reversible fluctuations.
The principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are symmetric, which simplifies the coupled force-flux equations and has led to experimentally verifiable predictions.
All eigenvalues of Equation 2.21 will be real numbers.
The kinetic matrix will be positive definite. All the eigenvalues are nonnegative.
The equation shows that the change in flux of some quantity caused by changing the direct driving force of another is equal to the change in flux of the second quantity caused by changing the driving force for the first.
These equations resemble the Maxwell relations from thermodynamics.
The statistical-mechanics derivation of Onsager's symmetry principle is based on microscopic reversibility of systems near equilibrium. The time average of a correlation between a driving force of type alpha and the fluctuations of quantity beta is identical with respect to switching of alpha and beta.
A demonstration of the role of microscopic reversibility in the symmetry of the coupling coefficients can be obtained for a system consisting of three isomers, A, B, and C.
When microscopic reversibility is present in a complex system composed of many particles, every elementary process in a forward direction is balanced by one in the reverse direction.
The balance of forward and backward rates is characteristic of the equilibrium state, and detailed balance exists throughout the system.
The forces and reaction rates are conjugate
Regarding Equation 2.15, every term on the right-hand side is the scalar product of a flux and a gradient with the same units as energy dissipation density.
Each term that multiplies a flux is a force of that flux. The paired forces and fluxes in the entropy production rate are termed conjugate forces and fluxes.
There can be constraints in systems relating changes in extensive quantities.
Assume in Chapters 1 - 3 that the material is isotropic and that forces and fluxes are parallel.
Assumption is removed in anisotropic materials in Chapter 4.
There are well-known empirical force-flux laws that apply under certain conditions.
Extensive Quantity | Flux | Conjugate Force | Empirical Force-Flux Law | Equation |
|---|---|---|---|---|
Heat |
|
| Fourier's |
|
Component i |
|
| Modified Fick's |
|
Charge |
|
| Ohm's |
|
2.1.3 Basic Postulate of Irreversible Thermodynamics
The basic postulate of irreversible thermodynamics is that, near equilibrium, the local entropy production is nonnegative
Using empirical laws displayed in the table above, the entropy production can be identified in a few special cases.
Use Equation 2.15 and Fourier's heat-flux law to predict that thermal conductivity is always positive.
If diffusion is the only operating process, an equation implies that each mobility is always positive.
2.2 Linear Irreversible Thermodynamics
In many materials, a gradient in temperature produces not only flux of heat but also a gradient in electric potential.
If heat can flow, the gradient in electrical potential will result in a heat flux.
This coupled phenomenon is called the thermoelectric effect.
Flows of mass, electricity, and heat all involve particles possessing momentum, and interactions may be expected as momentum is transferred between them.
A formulation of the coupling effects can be obtained by generalization of the previous empirical force-flux equations
2.2.1 General Coupling between Forces and Fluxes
The fluxes are a function of the driving force acting on the system.
Assuming that the system is near equilibrium and the driving forces are small, each of the fluxes can be expanded in a Taylor series near the equilibrium point
In the approximation in the book, the fluxes vary linearly with the forces.
Direct coefficients (diagonal terms): coupling of each flux to its conjugate driving force.
Coupling coefficients (off-diagonal terms): coupling effects or cross effects
Illustrate connection between direct coefficients and the empirical force-flux laws
Consider a pure material with a constant thermal gradient imposed along it and that is electronically conducting
If a constant thermal gradient is imposed and no electrically conductive contacts are made at the ends of the specimen, the heat flow is in a steady state and the charge-density vanishes.
Fourier's law pertains, but the thermal conductivity K depends on the direct coefficient as well as on the direct and coupling coefficients associated with electrical charge flow.
In general, the empirical conductivity associated with a particular flux depends on the constraints applied to other possible fluxes.
2.2.2 Force-Flux Relations when Extensive Quantities are Constrained
Changes in one extensive quantity are in many cases coupled to changes in others.
Network constraint: whenever one component leaves a site, it must be replaced.
As a result of the replacement constraint, the fluxes of components are not independent of each other.
This type of constraint is absent in amorphous materials and with interstitial solutes in crystalline materials that lie in the interstices between larger substitutional atoms.
The conjugate force of the diffusion of a network-constrained component i depends upon the gradient of the difference between the chemical potential of component i and Nc rather than on the chemical potential gradient of i alone.
The difference arises because during migration a site's state changes from occupancy by an atom of type i to occupancy by a vacancy.
In the development above, the choice of the Ncth component in a system under network constraint system is arbitrary. The flux of each component in Equation 2.21 must be independent of the choice, and this independence imposes conditions on the L alpha beta coefficients.
The sum of the entries in any row or column of the matrix Lij is zero.
Below is a table of conjugate forces and fluxes that are obtained when the only constraint is a network constraint.
Quantity | Flux | Conjugate Force |
|---|---|---|
Heat |
|
|
Component i |
|
|
Charge |
|
|
2.2.3 Introduction of the Diffusion Potential
Any potential that accounts for the storage of energy due to the addition of a component determines the driving force for the diffusion of that component.
Diffusion potential: The total conjugate force of a diffusing component. The sum of all supplemental potentials, including the chemical potential.
The conjugate force of the flux of a component is always of the form F = - grad Phi
2.2.4 Onsager's Symmetry Principle
Three postulates were utilized to derive relations between forces and fluxes. They do not follow from the first and second laws of thermodynamics
- The rate of entropy change and the local rate of entropy production can be inferred by invoking equilibrium thermodynamic variations and the assumption of local equilibrium.
- The entropy production is nonnegative
- Each flux depends linearly on all the driving forces
Onsager's principle supplements these postulates and follows from the statistical theory of reversible fluctuations.
The principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are symmetric, which simplifies the coupled force-flux equations and has led to experimentally verifiable predictions.
All eigenvalues of Equation 2.21 will be real numbers.
The kinetic matrix will be positive definite. All the eigenvalues are nonnegative.
The equation shows that the change in flux of some quantity caused by changing the direct driving force of another is equal to the change in flux of the second quantity caused by changing the driving force for the first.
These equations resemble the Maxwell relations from thermodynamics.
The statistical-mechanics derivation of Onsager's symmetry principle is based on microscopic reversibility of systems near equilibrium. The time average of a correlation between a driving force of type alpha and the fluctuations of quantity beta is identical with respect to switching of alpha and beta.
A demonstration of the role of microscopic reversibility in the symmetry of the coupling coefficients can be obtained for a system consisting of three isomers, A, B, and C.
When microscopic reversibility is present in a complex system composed of many particles, every elementary process in a forward direction is balanced by one in the reverse direction.
The balance of forward and backward rates is characteristic of the equilibrium state, and detailed balance exists throughout the system.
The forces and reaction rates are conjugate