Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 2 ppt

The first step Chapter 2 is the derivation of the general coupling between these forces and fluxes using the methods of irreversible thermodynamics.. 24 CHAPTER 2 : IRREVERSIBLE THERMO

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

necessarily have real eigenvalues

distance from the point (1,1,1)

(a) The direction o f maximum rate o f change is along the gradient vector Vc given

(1.38) (1.39)

(b) The maximum rate of change o f c is then

the origin vanishes

dent of the sphere radius

Solution

The problem is most easily solved using the divergence theorem:

L J i L d A = L V fd V (1.42) Consider first the divergence o f 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 o f T i n Eq 1.42 is zero if the singularity at

r = 0 is avoided Therefore, if the closed surface does not include the origin,

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 o f radius R,

(1.46) Therefore the total flux is independent o f R and equal t o 47r

1.3 Suppose that the flux of some substance i is given by the vector field

where A = constant Find the rate, Mi, at which i flows through the hemi- spherical surface of the unit sphere

Also, the integral may be converted t o an integral over the projection of the hemisphere

on the (x, y) plane (denoted by P ) by noting that

M,= JJ

Converting t o 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

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)

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 d E i j (where aij and ~ i j are the stress and strain tensors), which can be further separated into hydrostatic and deviatoric terms as dw = P d v - xi xj 6ij d z i j (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 = - 4 d q (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 j t h generalized intensive quantity and <j represents its con- jugate extensive quantity densitye2 Therefore,

C$j d<j = - P d u + 4 d q + 6 k i dgkl + y d A + d6+ E dp’

+ 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 ~ y s t e r n ~ 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 d V For an isolated system, dS/dt = Jv Cr d V 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

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

2.1: 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 t e m p e r a t ~ r e ~ The mobility, Mi, is defined as the velocity of component i induced by a unit force

Table 2.1:

Force-Flux Laws Selected Conjugate Forces, Fluxes, and Empirical for Systems with Unconstrained Components, i

Extensive Quantity Flux Conjugate Force Empirical Force-Flux Law*

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,

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

Combining Eqs 2.15 and 2.21 results in a relation for the entropy production that applies near equilibrium:

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 ( - T V T ) 1

(2.24) Comparison with Eq 2.17 shows that the thermal conductivity K is related to the direct coefficient LQQ by

6Note that the fluxes and forces are written a: scalars,+cons@te@ with the assumption that the material is isotropic Otherwise, terms like JQ = ( ~ J Q / ~ F Q ) F Q 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-

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

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

Y d N , u = 0

i = l

(2.30)

2.2: LINEAR IRREVERSIBLE THERMODYNAMICS 31 Solving Eq 2.30 for d N N C and putting the result into Eq 2.3 yields

N,-l Tds = du + dw - C (pi - PN,) d c i

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

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

Eqs 2.33 and 2.34 imply that

(2.34)

LAA + LAB + LAC = 0

LBA + LBB + LBC = 0 LCA + LCB + L c C = 0

32 CHAPTER 2 : IRREVERSIBLE THERMODYNAMICS: COUPLED FORCES AND FLUXES

If the lattice network defines the coordinate system in which the fluxes are mea- sured, the network constraint requires that

i=l and this imposes the further condition on the Lij that

The conjugate forces and fluxes that are obtained when the only constraint is a

network constraint are listed in Table 2.2 However, there are many cases where

further constraints between the extensive quantities exist For example, suppose that component 1 is a nonuniformly distributed ionic species that has no network constraint Each ion will experience an electrostatic force due to the local electric field, as well as a force due to the gradient in its chemical potential This may be

demonstrated in a formal manner with Eq 2.5, noting that dq in this case is not

independent of dcl but, instead, dq = q l d q , where q1 is the electrical charge per ion assuming that all electric current is carried by ions Thus dq and dcl can be

combined in Eq 2.5 into a single term ( p 1 +ql@)dcl, and when this term is carried

through the process leading to Eq 2.15, the ion flux, A, is found to be conjugate

to an ionic force

The potential that appears in the total force expression is the sum of the chem- ical potential and the electropotential of the charged ion This total potential is generally called the electrochemical potential

Additional forces would be added to the chemical potential force if, for example, the particle possessed a magnetic moment and a magnetic field were present As will be seen, many possibilities for total forces exist depending upon the types of components and fields present

Table 2.2:

Network-Constrained Components, Conjugate Forces and Fluxes for Systems with i

Component i -V ( p z - p N , ) = -vQ.,

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 The

2 2, LINEAR IRREVERSIBLE THERMODYNAMICS 33

sum of all such supplemental potentials, including the chemical potential, appears

as the total conjugate force for a diffusing component and is called the diffusion potential for that component and is represented by the symbol @.’ The conjugate force for the flux of component 1 will always have the form

-

and thus for the special case leading to Eq 2.39,

2.2.4 Onsager’s Symmetry Principle

Three postulates were utilized to derive the relations between forces and fluxes:

0 The rate of entropy change and the local rate of entropy production can be in- ferred by invoking equilibrium thermodynamic variations and the assumption

of local equilibrium

0 The entropy production is nonnegative

0 Each flux depends linearly on all the driving forces

These postulates do not follow from statements of the first and second laws of thermodynamics

Onsager’s principle supplements these postulates and follows from the statisti- cal theory of reversible fluctuations [5] Onsager’s principle states that when the

forces and fluxes are chosen so that they are conjugate, the coupling coefficients are symmetric:

7The potential is an aggregate of all reversible work terms that can be transported with the species i Using Lagrange multipliers, Cahn and Larch6 derive a potential that is a sum of the diffusant’s elastic energy and its chemical potential [4] Cahn and Larch4 coined the term diffusion potential to describe this sum Our use of the term is consistent with theirs

sPositive definite means that the matrix when left- and right-multiplied by an arbitrary vector will yield a nonnegative scalar If the matrix multiplied by a vector composed of forces is proportional

to a flux, it implies that the flux always has a positive projection on the force vector Technically, one should say that Lap is nonnegative definite but the meaning is clear

34 CHAPTER 2 IRREVERSIBLE THERMODYNAMICS COUPLED FORCES AND FLUXES

The statistical-mechanics derivation of Onsager's symmetry principle is based

on microscopic reversibility for systems near equilibrium That is, the time average

of a correlation between a driving force of type Q and the fluctuations of quantity /3 is identical with respect to switching ct and /3 [6]

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 [7,8] Each isomer can be converted into either of the other two, without

any change in composition Assuming a closed system containing these molecules at constant temperature and pressure, the rate of conversion of one type into another

is proportional to its number, with the constant of proportionality being a rate constant, K (Fig 2.1) The rates at which the numbers of A, B, and C change are then

- - dNA dt - - (KAC + KAB)NA + KBANB + KCANC

- dNB dt = KAB NA - ( KBC + KBA)NB + KCB Nc dNc

- dt =KACNA + KBCNB - (KCA + KCB)NC

(2.44)

At equilibrium, the time derivatives in Eq 2.44 vanish Solving for equilibrium

in a closed system (NA + NB + NC = Ntot) yields

K7 Ntot Neq = K , Ntot Neq - Kp Ntot Neq -

A K , + K ~ + K , - K , + K ~ + K , - K , + K ~ + K ~

(2.45)

where

K, E KBAKCA + KBAKCB + KCAKBC

Kp KCBKAB + KCBKAC + KABKCA (2.46)

K7 E KACKBC + KACKBA + KBCKAB

For the system near equilibrium, let YA be the difference between the number of

A and its equilibrium value, YA = NA - N:q Introducing this relationship and similar ones for B and C into Eq 2.44,

2.2: LINEAR IRREVERSIBLE THERMODYNAMICS 35

with similar expressions for B and C

chemical potential (Eq 2.2) near equilibrium (small If Henry's law is obeyed, the activity coefficient is constant and expanding the Y A / N ~ ~ ) yields

(2.48)

Substituting Eq 2.48 into Eq 2.47 and carrying out similar procedures for B and

C ,

These constitute a set of linear relationships between the potential differences pz -

p:q, which drive the Y, toward equilibrium and their corresponding rates, dY,/dt

In terms of the Onsager coefficients, they have the form

= LAAFA + LABFB + LACFC

3 - di - LCAFA + LCBFB + LCCFC 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 Microscopic reversibility therefore requires that the forward and backward reaction fluxes in Fig 2.1 be equal, so that

K B A - N i p = Ka - KBAKCA + KBAKCB + KCAKBC

KAB NEq Kp KCBKAB + KCBKAC + KABKCA

-KCB - -NB = Kp = KCBKAB + KCBKAC + KABKCA

KBC Nzq K-, KACKBC + KACKBA + KBCKAB

- KAC - -N~ - = K-r = KACKBC + KACKBA + KBCKAB

KCA N i p Ka KBAKCA + KBAKCB + KCAKBC

Comparison of Eq 2.50 with Eqs 2.49 and 2.51 shows that L,, = L,, and therefore demonstrates the role of microscopic reversibility in the symmetry of the Onsager coefficients More demonstrations of the Onsager principle are described in Lifshitz and Pitaerskii [6] and in Yourgrau et al [8]

Solving Exercise 2.5 shows that the products of the forces and reaction rates in

Eq 2.49 appear in the expression for the entropy production rate for the chemical reactions The forces and reaction rates are therefore conjugate, as expected

36 CHAPTER 2 : IRREVERSIBLE THERMODYNAMICS COUPLED FORCES AND FLUXES

2 F.C Larch6 and J.W Cahn A nonlinear theory of thermomechanical equilibrium of

solids under stress Acta Metall., 26(1):53-60, 1978

3 J.W Cahn and F.C Larch& An invariant formulation of multicomponent diffusion in crystals Scripta Metall., 17(7):927-937, 1983

4 F.C Larch6 and J.W Cahn The interactions of composition and stress in crystalline solids Acta Metall., 33(3):331-367, 1984

5 L Onsager Reciprocal relations in irreversible processes 11 Phys Rev., 38( 12):2265-

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

Using Ohm's law, & = -pV4,

TU = p 10c$l2 (2.53) Because U 2 0 and lVc$12 is positive, p must be positive

2.2 An isolated bar of a good electrical insulator contains a rapidly diffusing unconstrained solute (i.e., component 1) Impose a constant thermal gradient along the bar, and find an expression for its thermal conductivity when the system reaches a steady state Assume that no solute enters or leaves the ends of the bar Express your result in terms of any of the Lap coefficients in

Eq 2.21 that are required

Solution Using a similar method as the development that led t o Eq 2.29, the relevant linear force-flux relations are

2.3 A common device used to measure temperature differences is the thermocou-

ple in Fig 2.2 Wires of metals A and B are connected with their common