22 Microscopic and mechanistic aspects of diffusion are treated in Chapters 7-10. An expression for the basic jump rate of an atom (or molecule) in a condensed system is obtained and various aspects of the displacements of migrating particles are described (Chapter 7). Discussions are then given of atomistic models for diffusivities and diffusion in bulk crystalline materials (Chapter 8), along line and planar imperfections in crystalline materials (Chapter 9), and in bulk noncrystalline materials (Chapter 10). CHAPTER 2 IRREVERSIBLE THERMODYNAMICS AND COUPLING BETWEEN FORCES AND FLUXES The foundation of irreversible thermodynamics is the concept of entropy produc- tion. The consequences of entropy production in a dynamic system lead to 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 existence of a conserved internal energy is a consequence of the first law of thermodynamics. Numerical values of a system’s energy are always specified with respect to a reference energy. 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 by devising a reversible path from a reference state to the system’s state and integrating dS = 6q,,,/T along that path. For a nonequilibrium system, a reversible path is generally unavailable. In statistical mechanics, entropy is related to the number of microscopic states available at a fixed energy. Thus, a state-counting device would be required to compute entropy for a particular system, but no such device is generally available for the irreversible case. To obtain a local quantification of entropy in a nonequilibrium material, con- sider a continuous system that has gradients in temperature, chemical potential, and other intensive thermodynamic quantities. Fluxes of heat, mass, and other ex- tensive quantities will develop as the system approaches equilibrium. Assume that Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 23 Copyright @ 2005 John Wiley & Sons, Inc. 24 CHAPTER 2: IRREVERSIBLE THERMODYNAMICS: COUPLED FORCES AND FLUXES the system can be divided into small contiguous cells at which the temperature, chemical potential, and other thermodynamic potentials can be approximated by their average values. The local equilibrium assumption is that the thermodynamic state of each cell is specified and in equilibrium with the local values of thermo- dynamic potentials. If local equilibrium is assumed for each microscopic cell even though the entire system is out of equilibrium, then Gibbs’s fundamental relation, obtained by combining the first and second laws of thermodynamics, can be used to calculate changes in the local equilibrium states as a result of evo- lution of the spatial distribution of thermodynamic potentials. U and S are the internal energy and entropy of a cell, dW is the work (other than chemical work) done by a cell, Ni is the number of particles of the ith component of the possible N, components, and pi is the chemical potential of the ith component. pi depends upon the energetics of the chemical interactions that occur when a particle of i is added to the system and can be expressed as a general function of the atomic fraction Xi: pi = pp + kT ln(yiXi) (2.2) The activity coefficient yi generally depends on X, but, according to Raoult’s law, is approximately unity for Xi x 1. Dividing dLI through by a constant reference cell volume, V,, where all extensive quantities are now on a per unit volume basis (i.e., densities).’ For example, v = V/V, is the cell volume relative to the reference volume, V,, and ci = Ni/Vo is the concentration of component i. The work density, dw, includes all types of (nonchemical) work possible for the system. For instance, the elastic work density introduced by small-strain deformation is dw = + xi x, aij dEij (where aij and ~ij are the stress and strain tensors), which can be further separated into hydrostatic and deviatoric terms as dw = Pdv - xi xj 6ij dzij (where 5 and t are the deviatoric stress and strain tensors, respectively). The elastic work density therefore includes a work of expansion Pdv. Other work terms can be included in Eq. 2.3, such as electrostatic potential work, dw = -4dq (where 4 is the electric potential and q is the charge density); interfacial work, dw = -ydA, in systems containing extensible interfaces (where y is the interfacial energy density and A is the interfacial area; magnetization work, dw = -d . d6 (where d is the magnetic field and b‘ is the total magnetic moment density, including the permeability of vacuum); and electric polarization work, dw = -E ’ dp’ (where l? is the electric field given by E’ = -V$ and p’ is the total polarization density, including the contribu- tion from the vacuum). If the system can perform other types of work, there must ‘Use of the reference cell volume, V,, is necessary because it establishes a thermodynamic reference state. 2.1. ENTROPY AND ENTROPY PRODUCTION 25 be terms in Eq. 2.3 to account for them. To generalize: where $j represents a jth generalized intensive quantity and <j represents its con- jugate extensive quantity densitye2 Therefore, C$j d<j = -Pdu + 4dq + 6ki dgkl + ydA + d6+ E. dp’ j (2.5) + pi dci + * . . + p~, dCN, + . . . The $j may be scalar, vector, or, generally, tensor quantities; however, each product in Eq. 2.5 must be a scalar. Equation 2.4 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 approx- imations to the increase of the quantities at a point. Such changes may reflect how a quantity changes in time, t, at a fixed point, r‘; or at a fixed time for a variable location in a point’s neighborhood. The change in the total entropy in the system, S, can be calculated by summing the entropies in each of the cells by integrating over the entire ~ystern.~ Equation 2.4, which is derived by combining the first and second laws, applies to reversible changes. However, because s, u, and the & are all state variables, the relation holds if all quantities refer to a cell under the local equilibrium assumption. Taking s as the dependent variable, Eq. 2.4 shows how s varies with changes in the independent variables, u and 0. In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium. Entropy increase plays a large role in ir- reversible thermodynamics. If each of the reference cells were an isolated system, the right-hand side of Eq. 2.4 could only increase in a kinetic process. However, 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. In a hypothetical system for modeling kinetics, the microscopic cells must be open systems. It is useful to consider entropy as a fluxlike quantity capable of flowing from one part of a system to another, just like energy, mass, and charge. Entropy flux, denoted by i, is related to the heat flux. An expression that relates to measurable fluxes is derived below. 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. The consequences of entropy production are developed below. 2.1.1 Entropy Production The local rate of entropy-density creation is denoted by Cr. The total rate of en- tropy creation in a volume V is Jv d. dV. For an isolated system, dS/dt = Jv Cr dV. 2The generalized intensive and extensive quantities may be regarded as generalized potentials and displacements, respectively. 3Note that S is the entropy of a cell, S is the entropy of the entire system, and s is the entropy per unit volume of the cell in its reference state. 26 CHAPTER 2: IRREVERSIBLE THERMODYNAMICS: COUPLED FORCES AND FLUXES However, for a more general system, the total entropy increase will depend upon how much entropy is produced within it and upon how much entropy flows through its boundaries. From Eq. 2.4, the time derivative of entropy density in a cell is C$j% ds 1 du 1 dt T dt T Using conservation principles such as Eqs. 1.18 and 1.19 in Eq. 2.6,4 _ _ - From the chain rule for a scalar field A and a vector g, Equation 2.7 can be written Comparison with terms in Eq. 1.20 identifies the entropy flux and entropy produc- tion: (2.10) (2.11) The terms in Eq. 2.10 for the entropy flux can be interpreted using Eq. 2.4. The entropy flux is related to the sum of all potentials multiplying their conjugate fluxes. Each extensive quantity in Eq. 2.4 is replaced by its flux in Eq. 2.10. Equation 2.11 can be developed further by introducing the flux of heat, JQ. Applying the first law of thermodynamics to the cell yields (2.12) where Q is the amount of heat transferred to the cell. By comparison with Eq. 2.4 and with the assumption of local equilibrium, dQ/Vo = Tds and therefore Tu = YQ -k c$j& i Substituting Eq. 2.13 into Eq. 2.11 then yields (2.13) (2.14) 4Here, all the extensive densities are treated as conserved quantities. This is not the general case. For example, polarization and magnetization density are not conserved. It can be shown that for nonconserved quantities, additional terms will appear on the right-hand side of Eq. 2.11. 2.1: ENTROPY AND ENTROPY PRODUCTION 27 2.1.2 Conjugate Forces and Fluxes Multiplying Eq. 2.14 by T gives (2.15) Every term on the right-hand side of Eq. 2.15 is the scalar product of a flux and a gradient. Furthermore, each term has the same units as energy dissipation density, J m-3 s-’, and is a flux multiplied by a thermodynamic potential gradient. Each term that multiplies a flux in Eq. 2.15 is therefore a force for that flux. The paired forces and fluxes in the entropy production rate can be identified in Eq. 2.15 and are termed conjugate forces and fluxes. These are listed in Table 2.1 for heat, component i, and electric charge. These forces and fluxes have been identified for unconstrained extensive quantities (i.e., the differential extensive quantities in Eq. 2.5 can vary independently). However, many systems have constraints relating changes in their extensive quantities, and these constrained cases are treated in Section 2.2.2. Throughout Chapters 1-3 we assume, for simplicity, that the material is isotropic and that forces and fluxes are parallel. This assumption is removed for anisotropic materials in Chapter 4. Table 2.1 presents corresponding well-known empirical force-flux laws that apply under certain conditions. These are Fourier’s law of heat flow, a modified version of Fick’s law for mass diffusion at constant temperature, and Ohm’s law for the electric current density at constant temperat~re.~ The mobility, Mi, is defined as the velocity of component i induced by a unit force. Table 2.1: Force-Flux Laws for Systems with Unconstrained Components, i. Selected Conjugate Forces, Fluxes, and Empirical Extensive Quantity Flux Conjugate Force Empirical Force-Flux Law* Heat J; -+VT Fourier’s J; = -KVT Component i x -Vp, = -Vat Modified Fick’s x = -Mzc, Vpz Charge J:, -v4 Ohm’s J’ 9- - -pv4 *K = thermal conductivity; Mi = mobility of i; p = electrical conductivity 2.1.3 The basic postulate of irreversible thermodynamics is that, near equilibrium, the local entropy production is nonnegative: Basic Postulate of Irreversible Thermodynamics (2.16) 5Under special circumstances, this form of Fick’s law reduces to the classical form & = -D, Vc,, where D, is the mass diffusivity (see Section 3.1 for further discussion). 28 CHAPTER 2: IRREVERSIBLE THERMODYNAMICS. COUPLED FORCES AND FLUXES Using the empirical laws displayed in Table 2.1, the entropy production can be identified for a few special cases. For instance, if only heat flow is occurring, then, using Eq. 2.15 and Fourier’s heat-flux law, & = -K VT (2.17) results in (2.18) which predicts (because of Eq. 2.16) that the thermal conductivity will always be positive. If diffusion is the only operating process, (2.19) i=l implying that each mobility is always positive. 2.2 LINEAR IRREVERSIBLE THERMODYNAMICS In many materials, a gradient in temperature will produce not only a flux of heat but also a gradient in electric potential. This coupled phenomenon is called the thermoelectric effect. Coupling from the thermoelectric effect works both ways: if heat can flow, the gradient in electrical potential will result in a heat flux. That a coupling between different kinds of forces and fluxes exists is not surprising; flows of mass (atoms), electricity (electrons) , and heat (phonons) all involve particles possessing momentum, and interactions may therefore be expected as momentum is transferred between them. A formulation of these coupling effects can be obtained by generalization of the previous empirical force-flux equations. 2.2.1 In general, the fluxes may be expected to be a function of all the driving forces acting in the system, Fi; for instance, the heat flux JQ could be a function of other forces in addition to its conjugate force FQ; that is, General Coupling between Forces and Fluxes 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 2.2: LINEAR IRREVERSIBLE THERMODYNAMICS 29 FQ = Fq = F1 = = FN, = 0. To first order: or in abbreviated form, where (2.21) (2.22) is evaluated at equilibrium (Fp = 0, for all P).6 In this approximation, the fluxes vary linearly with the forces. In Eqs. 2.20 and 2.22, the diagonal terms; L,,, are called direct coeficients; they couple each flux to its conjugate driving force. The off-diagonal terms are called coupling coeficients and are responsible for the coupling effects (also called cross efSects) identified above. Combining Eqs. 2.15 and 2.21 results in a relation for the entropy production that applies near equilibrium: TC~ = C L,~F,F~ (2.23) Pa The connection between the direct coefficients in Eq. 2.21 and the empirical force-flux laws discussed in Section 2.1.2 can be illustrated for heat flow. If a bar of pure material that is an electrical insulator has a constant thermal gradient imposed along it, and no other fields are present and no fluxes but heat exist, then according to Eq. 2.21 and Table 2.1, JG = LQQ (-TVT) 1 (2.24) Comparison with Eq. 2.17 shows that the thermal conductivity K is related to the direct coefficient LQQ by K=- LQQ (2.25) T 6Note that the fluxes and forces are written a: scalars,+cons@te@ with the assumption that the material is isotropic. Otherwise, terms like JQ = (~JQ/~FQ)FQ must be written as rank-two tensors multiplying vectors, and the equations that result can be written as linear relations (see Section 4.5 for further discussion). 30 CHAPTER 2: IRREVERSIBLE THERMODYNAMICS: COUPLED FORCES AND FLUXES If the material is also electronically conducting, the general force-flux relation- JQ = LQQFQ + LQqFq (2.26) Jq = LqQFQ + LwFq (2.27) 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 current must vanish. Hence Jq = 0 and a force ships are F LqQ FQ LW 4- (2.28) will arise. The existence of the force Fq indicates the presence of a gradient in the electrical potential, V4, along the bar. Therefore, using Eqs. 2.28 and 2.26, LQQ - -1 LQqLqQ pQ = - [% - -1 VT = -KVT (2.29) 4, TL4, In such a material under these conditions, Fourier's law again pertains, but the thermal conductivity K depends on the direct coefficient LQQ, as in Eq. 2.25, 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 In many cases, changes in one extensive quantity are coupled to changes in others. This occurs in the important case of substitutional components in a crystal devoid of sources or sinks for atoms, such as dislocations, as explained in Section 11.1. Here the components are constrained to lie on a fixed network of sites (i.e., the crystal structure), where each site is always occupied by one of the components of the system. Whenever one component leaves a site, it must be replaced. This is called a network constraint [l]. For example, in the case of substitutional diffusion by a vacancy-atom exchange mechanism (discussed in Section 8.1.2), the vacancies are one of the components of the system; every time a vacancy leaves a site, it is replaced by an atom. As a result of this replacement constraint, the fluxes of components are not independent of one another. This type of constraint will be absent in amorphous materials because any of the N, components can be added (or removed) anywhere in the material without exchanging with any other components. The dNi will also be independent for interstitial solutes in crystalline materials that lie in the interstices between larger substitutional atoms, as, for example, carbon atoms in body-centered cubic (b.c.c.) Fe, as illustrated in Fig. 8.8. In such a system, carbon atoms can be added or removed independently in a dilute solution. When a network constraint is present, NC YdN, = 0 u i=l (2.30) 2.2: LINEAR IRREVERSIBLE THERMODYNAMICS 31 Solving Eq. 2.30 for dNNC and putting the result into Eq. 2.3 yields N,-l Tds = du + dw - C (pi - PN,) dci i=l (2.31) Starting with Eq. 2.31 instead of Eq. 2.3 and repeating the procedure that led to Eq. 2.15, the conjugate force for the diffusion of component i in a network- constrained crystal takes the new form 4 Fi = -v (Pi - PN,) (2.32) The conjugate force for the diffusion of a network-constrained component i there- fore depends upon the gradient of the difference between the chemical potential of component i and N, rather than on the chemical potential gradient of compo- nent i alone. If in the case of substitutional diffusion by the vacancy exchange mechanism, the vacancies are taken as the component N,, the driving force for component i depends upon the gradient of the difference between the chemical po- tential of component i and that of the vacancies. The difference arises because, during migration, a site’s state changes from occupancy by an atom of type i to occupancy by a vacancy. This result has been derived and extended by Larch6 and Cahn, who investigated coherent thermomechanical equilibrium in multicomponent systems with elastic stress fields [l-41. In the development above, the choice of the N,th component in a system un- der network constraint system is arbitrary. However, the flux of each component in Eq. 2.21 must be independent of this choice [3, 41. This independence imposes conditions on the Lap coefficients. To demonstrate, consider a three-component system at constant temperature in the absence of an electric field, where compo- nents A, B, and C correspond to i = 1, 2, and 3, respectively. If component C is the N,th component, Eqs. 2.21 and 2.32 yield fA = -LAAv(PA - PC) - LABV(PB - PC) JB = -LBAv(PA - ~c) - LBBv(PB - ~c) fc = -LCAV(PA - ~c) - LCBV(PB - ~c) (2.33) On the other hand, if B is the N,th component, + JL + = -LAA~(PA - PB) - LAC~(PC - PB) JA = -LBA~(PA - PB) - LBC~(PC - PB) & = -LCAV(PA - PB) - ~ccv(pc - PB) Because $ must be the same as < and the gradient terms are not necessarily zero, Eqs. 2.33 and 2.34 imply that (2.34) LAA + LAB + LAC = 0 LBA + LBB + LBC = 0 LCA + LCB + LcC = 0 (2.35) or generally, NC c Lij = 0 (2.36) j=1 [...]... from Eqs 2. 21 and 2. 32: JF = L i i F i + LizF2 d = -L1 1- 8% -LIZ d @2 = -L11 d(P1 - Pv) - L 12 ( P 2 - PV) Jg + = -L2 1- 8% + dX dX dX dX =~ 5 2 1 ~ 1 L22F2 3 3 2 - L2 2- dX = -L21 d(P1 - P V ) dX dX - L 22 d(P2 (3.7) - Pv) dX and J$ = - ( J f J,C) by Eq 3.6 Assumption of local equilibrium permits the Gibbs-Duhem relation to be written (3.8) A net vacancy flux develops in a direction opposite that of the... ci (us- w,")= -Di- dCi (3.15) dX Two equations representing the contributions of components 1 and 2 to the volume flux are obtained by multiplying Eq 3.15 through by R1 and R2 The sum of these two equations, using Eqs A.8 and A.lO, is6 w," - [R1ClW,R+ R2C&] = dCl ( D l - D2) R 1- (3.16) dX Using Eqs 3.15 (for i = l),3.16, and A.8 yields c1 w1" - c1 [RlCl?J,R+ R2c2.,"] = - [c2R2D1 + dCl ClRlDZ] - dX (3.17)... dJf 0 -d X 1 dJ,v + R2-dJ,L = 0 1-+ R 2- + dX dX dJl dx dub dx (3 .29 ) However, with the use of Eq A.lO, (3.30) and therefore - = -t( R 1 - + R 2 adJ,v dV Jr) ax dX (3.31) dX which, integrated, gives ( dub = - R1/ Jt, x x=-L dJ,YiR2/x x=-L d~;) (3. 32) Because J:, and u; are zero at the specimen ends x = ikL, where L is large compared to the diffusion zone width, (3.33) Therefore, with Eq 3 .21 , u; = - (R,J,v... the use of Eq 3.19 t o derive the unique choice of the V-frame In Eq 3 .22 , the flux of 1 in the V-frame obeys Fick's law and can be written (3 .24 ) where the binary solution znterdiffusivity, designated by 5 is related to the intrinsic diffusivities of components 1 and 2 (measured in a local C-frame) by the relation - D =~lRlD2 c ~ R J D ~ + which is often approximated through (3 .25 ) R1 = Q2 = ( 0 )... volume through any plane is zero: hence, the term volume-fixed frame Then using JY = CJI," and Eqs 3.16, 3.17, and 3 .21 , (3 .22 ) and = 21 : dCl (Dl - D2) R 1- ax (3 .23 ) Equation 3 .23 for the velocity of a local C-frame with respect t o the V-frame is therefore the velocity of any inert marker with respect t o the V-frame The assumptions that R1 and Q2 are each constant throughout the material and thus that... quantity of heat, dQ, from the reservoir into the system in order to maintain constant temperature in the system; the total entropy change of the system plus reservoir, dS', will then be dQ T dS' = dS - - (2. 61) where dS and -dQ/T are the entropy changes of the system and surroundings, respectively For the system, du = dQ - PdV, and therefore dS' = 0 TdS - 0 - PdV 24 T (2. 62) Note that 6 = U + PV - T S.. . exist depending upon the types of components and fields present Table 2. 2: Conjugate Forces and Fluxes for Systems with Network-Constrained Components, i Quantity Heat Component i Charge 2. 2.3 Flux Conjugate Force J;k -$ VT -V ( p z - p N , ) = -vQ., -V# J:, Introduction of the Diffusion Potential Any potential that accounts for the storage of energy due to the addition of a component determines the... through (3 .25 ) R1 = Q2 = ( 0 ) as 5 = X1D .2 + X2Dl (3 .26 ) Using a similar procedure t o find the flux of component 2 in the V-frame yields J" 2 - -D- d C 2 ax (3 .27 ) The only remaining task is now to relate the V-frame to a laboratory frame suitable for experimental purposes This is provided by the laboratory frame (Lframe) illustrated in Fig 3.4 Here, the ends of the specimen are unaffected by the diffusion... York, 19 82 EXERCISES 2. 1 Using an argument based on entropy production, what can be concluded about the algebraic sign of the electrical conductivity? Solution If electronic conduction is the only operative process in a material at constant T, then Eq 2. 15 reduces t o TU =-& .Vc$ (2. 52) Using Ohm's law, Because U & = -pV4, T =p 1 c l U 0 $2 2 0 and lVc$12is positive, (2. 53) p must be positive 2. 2 An isolated... respect to the corresponding V-frame The D, are related to 5 as indicated - D Interdiffusivity - = $' = -DVci D clRlD2 + c2R2D1 V-frame V-frame 6 is the composition-dependent interdiffusivity in a chemically inhomogeneous system In a binary system, it relates the flux of either component 1 or 2 to its corresponding concentration gradient via Fick's law in a V-frame 3 .2 M A S S DIFFUSION IN A N ELECTRICAL . 2. 1 presents corresponding well-known empirical force-flux laws that apply under certain conditions. These are Fourier s law of heat flow, a modified version of Fick s law for mass diffusion. meaning is clear. 34 CHAPTER 2 IRREVERSIBLE THERMODYNAMICS COUPLED FORCES AND FLUXES The statistical-mechanics derivation of Onsager&apos ;s symmetry principle is based on microscopic reversibility. balance of forward and backward rates is characteristic of the equilibrium state, and detailed balance exists throughout the system. Microscopic reversibility therefore requires that the forward