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Irreversible
Thermodynamics
Thermodynamics and Kinetics
� Thermodynamics is precise about what cannot
happen.
� How can thermodynamics be applied to systems
that are away from equilibrium?
� How are concepts from thermodynamics useful
for building kinetic models?
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A few math concepts
� Fields, scalar
� Fields, vector
� Fields, tensor
� Variations of scalar fields
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� Fields, scalar
Values of something are specified as a function of position,
� r = xi + yj + zk , and (in kinetics) time.
� r ,tExample: composition field c( ) from a point source
diffusing into a body
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� Fields, vector
Values of a vector quantity are specified as a function of position
and (in kinetics) time
Example: velocity is a vector field, e.g. in a flow field � �
r ,tv( )
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� Fields, tensor
A second-rank tensor called the stress tensor
�� r� ij ( ) relates force F at a point to an oriented area A
at that point
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� Variations of scalar fields
� • Stationary field, moving point r (a bug on a surface…)
� c� ��v dt
�)v dt
� r + = c(
�)r +c(
dc =
dt � v
� c� �
�• Evolving field, moving point r (a bug on a surface in
an earthquake…) dc
= � v +
� c� �
�c dt �t
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� Continuum limits
Is it possible to define a local value for concentration
in the limit �V � 0 ?
See Figures 1.5 and 1.6 in Balluffi, R. W., S. M . Allen & W. C. Carter. Kinetics of Materials.
New York: Wiley & Sons, forthcoming.
It is, with suitable care.
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� Fluxes�
The flux vector Ji represents mass flow ofcomponent i at a point per unit oriented area at thatpoint
�Mi (�A � )
= Ji � nlim �A�0 �A
Example: swimming fish
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� Accumulation�
Flow field, J (r) ˙Rate of production in dV, �i
� Accumulation of species i
�ci = �� � Ji + �i �t
��u For a conserved quantity like internal energy = �� � Ju
�t and for a non-conserved quantity like entropy
��S= �� � JS + �i
�t
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System at Equilibrium
� Characterized by uniform values of various “potentials” throughout the system e.g. T, P, µi
� Densities of conjugate extensive quantities e.g. S,can be inhomogeneous at equilibriumV N, i
q � S is maximized in a system at constant U
� For an isolated system dS � T
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System away from Equilibrium
� Kinetics concerns the path, mechanisms, andrates of spontaneous and driven processes.
� Irreversible thermodynamics attempts to apply thermodynamics principles to systems that are not in equilibrium and to suggest principles by which they relax toward equilibrium or steady state.
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Postulate 1
� It is generally possible to use principles of the continuum limit to define meaningful, useful, local values of various thermodynamics quantities, e.g., chemical potential µi.
� Useful kinetic theories can be developed by assuming a functional relationship between the rate of a process and the local departure from equilibrium (“driving force”).
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Postulate 2
� For any irreversible process the rate of entropy production is everywhere � 0.
��S � = + �� JS � 0 �t
� More restrictive than dStot � 0 for isolated system.
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Postulate 3
� Assume linear coupling between fluxes J and forces X
J = L X �,� = Q,q, I ,..., Nc� �� �
J = �� L X X� �� � �� �
� Electrical + heat conduction
J = L Xq + LqQXQ Direct coefficients, Lqq LQQq qq
JQ = LQqXq + LQQXQ Coupling coefficients, LqQ LQq
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Postulate 4
� Onsager symmetry postulate
� L = L�� (microscopic reversibility). ��
�Jq �JQL = L � =�� �� �XQ �Xq
Similar to Maxwell’s relations in thermodynamics
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Force/Flux Pairs
Flux Conjugate Force Empirical law
Heat
Charge
Mass
� JQ � Jq� Ji
1 �T�T
� JQ=�K�T Fourier
��� � J Ohm� �= � �q
��µi
� Ji = �Mici�µi Fick
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Multiple forces and fluxes
Consider simultaneous flow of heat and charge� JQ = �LQQXQ � LQqXq� Jq = �LQqXQ � LqqXq
Compare heat flow in an electrical insulator with heat flow in an electrical conductor
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