16 CHAPTER 1: INTRODUCTION which is a requirement that the homogeneous equation, Eq. 1.34, has a nontrivial solution. After the eigenvalues have been determined, the directions of the eigen- vectors Z can be determined by solving Eq. 1.34. A rank-two property tensor is diagonal in the coordinate system defined by its eigenvectors. Rank-two tensors transform like 3 x 3 square matrices. The general rule for transformation of a square matrix into its diagonal form is -1 eigenvector eigenvect or matrix matrix ] = [ column ] [ 'quare ] [ column ] (1.36) matrix matrix where the ith member of the diagonal matrix is the eigenvalue corresponding to the eigenvector used for the ith column vector of the transformation matrix. Nearly all rank-two property tensors can be represented by 3 x 3 symmetric matrices and necessarily have real eigenvalues. Bibliography 1. 2. 3. 4. 5. 6. S.M. Allen and E.L. Thomas. The Structure of Materials. John Wiley & Sons, New York, 1999. R. Clausius. The Mechanical Theory of Heat: With Its Applications to the Steam- Engine and to the Physical Properties of Bodies. Van Voorst, London, 1867. J.W. Gibbs. On the equilibrium of heterogeneous substances (1876). In Collected Works, volume 1. Longmans, Green, and Co., New York, 1928. L. Onsager. Reciprocal relations in irreversible processes. 11. Phys. Rev., 38( 12):2265- 2279, 1931. W.C. Carter, J.E. Taylor, and J.W. Cahn. Variational methods for microstructural evolution. JOM, 49(12):30-36, 1997. J.F. Nye. Physical Properties of Crystals. Oxford University Press, Oxford, 1985. EXERCISES 1.1 The concentration at any point in space is given by c = A (zy + yz + ZX) (1.37) where A = constant. (a) Find the cosines of the direction in which c changes most rapidly with (b) Determine the maximum rate of change of concentration at that point. Solution. distance from the point (1,1,1). (a) The direction of maximum rate of change is along the gradient vector Vc given (1.38) (1.39) by Vc = A [(y + ~)i + (Z + ~)j + (Z + y)2] Vc( 1,1,1) = 2A (i + j + i) Therefore, and the direction cosines are [l/&, l/& l/G. (b) The maximum rate of change of c is then IVc(l,1, 1)1 = 2Ah 1.2 Consider the radially symmetric flux field -r' r3 J=- EXERCISES 17 (1.40) (1.41) where r' = xi + yj + zk. (a) Show that the total flux through any closed surface that does not enclose (b) Show that the flux through any sphere centered at the origin is indepen- the origin vanishes. dent of the sphere radius. Solution. The problem is most easily solved using the divergence theorem: LJ.iLdA = LV. fdV (1.42) Consider first the divergence of radially symmetric vector fields of a general form, including the present field as a special case, i.e., For such fields (1.43) (1.44) In this case, n = 3 and the divergence of Tin Eq. 1.42 is zero if the singularity at r = 0 is avoided. Therefore, if the closed surface does not include the origin, V. JdV = 0 (1.45) and the total flux through the surface, s, f. AdA, is also zero. When the closed surface does enclose the origin, the total flux through the surface does not vanish. For a sphere of radius R, (1.46) Therefore the total flux is independent of R and equal to 47r. 1.3 Suppose that the flux of some substance i is given by the vector field = A (xi + vj) (1.47) where A = constant. Find the rate, Mi, at which i flows through the hemi- spherical surface of the unit sphere x2 + y2 + z2 = 1 (1.48) 18 CHAPTER 1: INTRODUCTION which lies above the (x,y) plane where z 2 0. Solution. Mi = J;.dA=/ J:.AdA (1.49) For the hemisphere, A = xi + yj + zi (1.50) Also, the integral may be converted to an integral over the projection of the hemisphere on the (x, y) plane (denoted by P) by noting that J hemi hemi k.AdA=dxdy (1.51) so that JJ x2+y2 dxdy (1.52) - dxdy J,.A-=A z JC-Fjp M,= JJ P P Converting to polar coordinates and integrating over P, 1.4 The matrix A is given by (1.53) (1.54) (a) Find the eigenvalues and corresponding eigenvectors of A. (b) Find matrices p and p-' such that p-'AP is a diagonal matrix. Note: The tedium of completing such exercises, as well as following many derivations in this book, is reduced by the use of symbolic mathematical software. We recommend that students gain a working familiarity with at least one package such as MathematicaB, MATLAB@, Mathcadco, or the public-domain package MAXIMA. Solution. (a) The characteristic equation of ,cl is given by Eq. 1.35 as X3 - 16X2 + 72X - 68 = 0 (1.55) The eigenvalues are solutions to the characteristic equation, giving Xi = 8.36258 Xz = 6.35861 XB = 1.27881 (1.56) The eigenvectors corresponding to the eigenvalues are -1.27252 -0.871722 0.144238 01 = [ -0.5:8613 ] 212 = [ -2.413084 ] w3 = [ -0.03105511 ] (1.57) Note that these eigenvectors are of arbitrary length. EXERCISES 19 (b) From Eq. 1.36 it is seen that the desired matrix is 3 x 3 and has the three eigenvectors as its columns: (1.58) 1 1 1 -1.27252 -0.871722 0.144238 1 1 1 - P= [ -0.538613 -2.43084 -0.0305511 The inverse of may be calculated as (1.59) -0.832149 0.352221 0.130788 P-' = 0.176139 -0.491171 -0.0404117 -[ 0.65601 0.13895 0.909624 By substitution it is readily verified that Eq. 1.36 is obeyed: Xi = 8.36258 0 0 Xz = 6.35861 0 (1.60) 0 As = 1.27881 - P-lM = PART I MOTION OF ATOMS AND MOLECULES BY DIFFUSION There are two arenas for describing diffusion in materials, macroscopic and mi- croscopic. Theories of macroscopic diffusion provide a framework to understand particle fluxes and concentration profiles in terms of phenomenological coefficients and driving forces. Microscopic diffusion theories provide a framework to under- stand the physical basis of the phenomenological coefficients in terms of atomic mechanisms and particle jump frequencies. We start with the macroscopic aspects of diffusion. The components in a system out of equilibrium will generally experience net forces that can generate correspond- ing fluxes of the components (diffusion fluxes) as the system tries to reach equilib- rium. The first step (Chapter 2) is the derivation of the general coupling between these forces and fluxes using the methods of irreversible thermodynamics. From general results derived from irreversible thermodynamics, specific driving forces and fluxes in various systems of importance in materials science are obtained in Chap- ter 3. These forces and fluxes are used to derive the differential equations that govern the evolution of the concentration fields produced by these fluxes (Chap- ter 4). Mathematical methods to solve these equations in various systems under specified boundary and initial conditions are explored in greater depth in Chapter 5. Finally, diffusion in multicomponent systems is treated in Chapter 6. 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$j& i Substituting Eq. 2.13 into Eq. 2.11 then yields (2.13) (2.14) 4Here, all the extensive densities are treated as conserved quantities. This is not the general case. For example, polarization and magnetization density are not conserved. It can be shown that for nonconserved quantities, additional terms will appear on the right-hand side of Eq. 2.11. [...]... from Eqs 2. 21 and 2. 32: JF = L i i F i + LizF2 d = -L1 1- 8% -LIZ d @2 = -L11 d(P1 - Pv) - L 12 ( P 2 - PV) Jg + = -L2 1- 8% + dX dX dX dX =~ 5 2 1 ~ 1 L22F2 3 3 2 - L2 2- dX = -L21 d(P1 - P V ) dX dX - L 22 d(P2 (3.7) - Pv) dX and J$ = - ( 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... correspond to i = 1, 2, and 3, respectively If component C is the N,th component, Eqs 2. 21 and 2. 32 yield - PC) - LABV(PB - PC) f A = -LAAv(PA -~ JB = - L B A v ( P A f c = -LCAV(PA - ~ c-) B B v ( P B L c-) C B V ( P B L -~ c ) -~ c ) (2. 33) On the other hand, if B is the N,th component, JL = - L A A ~ ( P-AP B ) - L A C ~ ( P-C B ) P J A = - L B A ~ ( P-A B ) - L B C ~ ( P-C B ) P P + + & = -LCAV(PA Because... 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... force LqQFQ F4 - (2. 28) LW 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- 4, LQqLqQ pQ = - [% 1- TL4, VT= -KVT (2. 29) 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... 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... (3 .20 ) [Rlc1w,R and is the flux of volume passing through a plane in the R-frame There is a particular R-frame, called the V-frame, for which f ( t ) = 0 In this frame, according to Eq 3 .20 , RiJ,V+R2J,V=O I[ (3 .21 ) 6For the derivation of Eq 3.16, cv has been assumed t o be negligible; the exact expression on the - D ~ f l v % Since the distance over which the species right-hand side is ( D l - D2)R1%... 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 . L. Onsager. Reciprocal relations in irreversible processes. 11. Phys. Rev., 38( 12) :22 6 5- 22 79, 1931. W. C. Carter, J.E. Taylor, and J .W. Cahn. Variational methods for microstructural evolution 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. Onsager. Reciprocal relations in irreversible processes. 11. Phys. Rev., 38( 12) :22 6 5- 22 79, 1931. 6. E.M. Lifshitz and L.P. Pitaevskii. Statistical Physics, Part 1. Pergamon Press,