Chapter 4 Diﬀusion with a Chemical Concentration Gradient Up to this point, we have studied diﬀusion in mixtures of chemically identical species and in dilute alloys. We have seen that impurity and host atoms can have greatly diﬀerent diﬀusivities which can depend on the concentration of the impurity. We have not yet considered the eﬀect of large chemical concentration gradients. We now consider the general situation, allowing for the presence of large concentration gradients of chemically dissimilar species. We see that for ideal or dilute solutions, the diﬀusive ﬂux is driven by the entropy increase associated with mixing of the species, while for non-ideal solutions, the diﬀusive ﬂux is driven by the enthalpy of mixing as well. We ﬁnd that we are able to describe the intermixing of two species using Fick’s laws with a single diﬀusion coeﬃcient D̃, which depends on concentration and the chemical nature of the two species. In order to do so, we examine the intermixing in a diﬀusion couple formed by joining two rods of diﬀerent composition of two mutually soluble elements. There are two eﬀects which are important in this situation. The ﬁrst, known as the Kirkendall eﬀect, is motion of the atomic planes due to the diﬀerence in diﬀusivities of the two constituents. The second is the eﬀect of the chemical driving force on the diﬀusion. 99 100 4.1 CHAPTER 4. DIFFUSION WITH A CHEMICAL . . . Kirkendall Eﬀect For the general case of a diﬀusion couple of two mutually soluble but chemically diﬀerent elements, the diﬀerence in diﬀusivities of the two species can give rise to motion of atomic planes in the interface region where diﬀusion is occurring. For example, if in a A-B diﬀusion couple, the diﬀusivity of A is greater than that of B, the ﬂux of A across a given lattice plane will be greater than the ﬂux of B in the opposite direction. Thus there will be an accumulation of atoms on one side of the reference plane, resulting in its motion. In order to examine this phenomena, we deﬁne a position coordinate z which is measured from a region in the sample where no diﬀusion is occurring. There will be two contributions to the atomic ﬂux relative to this ﬁxed coordinate system. • Flux due to diﬀusive motion of atoms relative to the moving atomic planes. • Flux due to the motion of the atomic planes. The total ﬂux of component A relative to a stationary observer is: JA = −D̃A ∂cA + vcA ∂z (4.1) where v is the velocity of the atomic planes relative to the ﬁxed reference frame z , and D̃A is the diﬀusivity of A in the presence of the concentration gradient1 . This situation is analogous to the ﬂux of ink in a moving stream. If we travel in a canoe moving with the water in the stream, then we will observe a drop of ink spreading in response to a diﬀusive ﬂux: −D̃A ∂cA ∂z where the coordinate z is measured relative to the boat. But since z and z are related by z = z + vt and the composition gradient is independent of the origin of the coordinate system, the diﬀusive ﬂux is also given by: −D̃A 1 ∂cA ∂z This diﬀusivity is sometimes called the intrinsic or chemical diﬀusion coeﬃcient. We will see that D̃A reﬂects the chemical nature of the intermixing of the constituents. 4.1. KIRKENDALL EFFECT 101 An observer sitting on the bank will see this ﬂux as well. However, he will also see the ink moving by with the velocity of the water. This is an additional ﬂux vcA due to motion of the water.2 We now wish to ﬁnd the velocity and see how the diﬀusion equation is aﬀected by this extra ﬂux in Eqn. 4.1. We assume that the molar volume is independent of composition so that: c = cB + cA = constant (4.2) therefore: ∂cA ∂cB ∂c = + =0 ∂t ∂t ∂t To ﬁnd how the compositions cA and cB change with time, we apply the conservation equation to both species: ∂JB ∂cB = − ∂t ∂z ∂cA ∂JA = − ∂t ∂z and insert the ﬂux from Eqn. 4.1: (4.3) ∂cA ∂cB ∂ ∂c = D̃A − vcA + D̃B − vcB ∂t ∂z ∂z ∂z ∂cA ∂cB ∂ D̃A + D̃B − cv = 0 = ∂z ∂z ∂z Therefore we have that: ∂cA ∂cB + D̃B − vc = constant (4.4) ∂z ∂z To ﬁnd the value of this constant, we recognize that in a region far away from the interface, where no diﬀusion occurs, we know that: D̃A ∂cB ∂cA = =0 ∂z ∂z And, since there is no diﬀusion in this region, v = 0 also. Therefore the constant in Eqn. 4.4 is zero. Solving Eqn: 4.4 for v, we ﬁnd: 1 ∂cA ∂cB v= D̃A + D̃B c ∂z ∂z 2 In fact, we have already seen this type of ﬂux in the moving interface problem. 102 CHAPTER 4. DIFFUSION WITH A CHEMICAL . . . Furthermore, from Eqn. 4.2 cA = c − cB so that: ∂cA ∂cB =− ∂z ∂z and the velocity is just: v = D̃A − D̃B 1 ∂cA c ∂z (4.5) Thus we see that a diﬀerence in diﬀusivities of the two constituents will result in a motion of atomic planes in the region where diﬀusion is occurring. We can ﬁnd the diﬀusion equation for this case by applying the conservation equation to the ﬂux equation, just as we did in deriving Fick’s second law. We insert the velocity from Eqn. 4.5 into the ﬂux equation (Eqn. 4.1) and apply the conservation equation (Eqn. 4.3) to ﬁnd: ∂cA ∂cA ∂ = − −D̃A + vcA ∂t ∂z ∂z c ∂c ∂cA ∂ A A = − −D̃A + D̃A − D̃B ∂z ∂z c ∂z ∂ = − ∂z ∂ = ∂z −cA D̃A − cB D̃A + cA D̃A − cA D̃B c cB D̃A + cA D̃B c ∂cA ∂z This is just Fick’s Second Law: ∂cA ∂cA ∂ = D̃ ∂t ∂z ∂z with: cB D̃A + cA D̃B c = xA D̃B + xB D̃A D̃ = where xi = ci /c is the atomic fraction i. ∂cA ∂z 4.2. CHEMICAL DRIVING FORCE 4.2 103 Chemical Driving Force A second eﬀect of a chemical concentration gradient on diﬀusion is to change the nature of the driving force. This is because diﬀusion changes the bonding in a solid. If, for example, two constituents in a diﬀusion couple have a preference for bonding with unlike neighbors, that is, they have a negative heat of mixing, then the decrease in free energy associated with diﬀusional mixing will have an enthalpy contribution as well as the mixing entropy contribution characteristic of ideal or dilute solutions. This added enthalpy contribution will act as a driving force to increase intermixing. Conversely, if the mixing enthalpy is positive, then the diﬀusional rate will be lower than that for an ideal or dilute solution. In order to examine this eﬀect we must generalize Fick’s ﬁrst law, the ﬂux equation, by realizing that ﬂux occurs as a result of a system’s drive to approach thermodynamic equilibrium. With this treatment, generally attributed to Darken, we can describe the mixing of chemically dissimilar materials with a diﬀusion coeﬃcient which is a function of the chemical nature of the solution. 4.2.1 Generalized Flux Equations Thermodynamic equilibrium is characterized by the absence of spatial or temporal variations in temperature T , pressure P , external potentials φ, and chemical potentials of the components µi . This condition does not always mean the absence of concentration gradients. Hence it is more reasonable to assert that the rate of return to equilibrium, that is, the ﬂux of atoms, energy, and defects, is proportional the deviation from equilibrium. Hence, to ﬁrst order, the ﬂux will be proportional to gradients in temperature, pressure, potential, and chemical potential, rather than just to composition gradients. The ﬂux of the ith component is given by: Ji = − Mik ∇µk − MiT ∇T − MiP ∇P − Miφ ∇φ (4.6) k where the Mij ’s are the coupling coeﬃcients between ﬂuxes in i and gradients in k. These coeﬃcients reﬂect the strength driving the ﬂux and the mobility of the species in responding with movement. In the case of a one-dimensional, isothermal, isobaric diﬀusion with no external potential gradients of an alloy of two components with a vacancy 104 CHAPTER 4. DIFFUSION WITH A CHEMICAL . . . mechanism we can write for the ﬂux of the components: ∂µA ∂µB ∂µV − MAB − MAV ∂z ∂z ∂z ∂µA ∂µB ∂µV = −MBA − MBB − MBV ∂z ∂z ∂z ∂µA ∂µB ∂µV = −MV A − MV B − MV V ∂z ∂z ∂z JA = −MAA JB JV (4.7) Vacancies can only be created or destroyed at sources or sinks such as surfaces or defects. Hence, throughout most of the crystal, the number of lattice sites is conserved, so that the ﬂuxes of the three species which can reside on a lattice site are related by: JA + J B + J V = 0 If this is to be true for arbitrary gradients, the sum of the coeﬃcients must be zero, i.e.: MAA + MBA + MV A = 0 MAB + MBB + MV B = 0 MAV + MBV + MV V = 0 In addition there is a set of reciprocity relations, known as the Onsager relations, which state that Mij = Mji . Combining these with Eqn. 4.7 we ﬁnd: ∂ ∂ (µA − µV ) − MAB (µB − µV ) ∂z ∂z ∂ ∂ = −MBA (µA − µV ) − MBB (µB − µV ) ∂z ∂z JA = −MAA JB 4.2.2 Darken’s Flux Equation To arrive at Darken’s ﬂux equation, we must make the further assumptions that the vacancies are in thermal equilibrium everywhere, so that µV = 0, and that the oﬀ diagonal terms are negligible. The ﬂux for a given component then reduces to: ∂µi Ji = −Mii ∂z 4.2. CHEMICAL DRIVING FORCE 105 The chemical potential for a given component can be written: µi = µ0 (T, P ) + kB T ln ai = µ0 + kB T (ln xi + ln γi ) (4.8) where ai is the activity of component i, and γi is the activity coeﬃcient of i, deﬁned as: ai γi = xi where xi is the atomic fraction of the ith component (xi = ci /c). The term kB T ln x1 represents the ideal mixing entropy contribution, while the term kB T ln γi deals with the non-ideality of the solution. For example, in considering the chemical potential of a system of vacancies, impurities, and vacancy-impurity pairs, we only considered the ideal mixing entropy term and found: µideal = µ0 + kB T ln xi i In this treatment, we are interested in deviations from ideality, and so must use the more general expression for chemical potential Eqn. 4.8. Our expression for the ﬂux is then: ∂µi ∂z ∂ ln xi ∂ ln γi = −Mii kB T + ∂z ∂z ∂ ln xi ∂ ln γi ∂ ln xi = −Mii kB T + ∂z ∂ ln xi ∂z −Mii kB T ∂ ln γi ∂ci = 1+ ci ∂ ln xi ∂z Ji = −Mii (4.9) (4.10) Equation 4.10 relates the ﬂux of a component to its concentration gradient, and as such is a generalization of Fick’s ﬁrst law. In order to examine the relationship between Eqn. 4.10 and Fick’s ﬁrst law, we consider the case of an ideal solution where ai = xi , or the case of a dilute solution where the activity follows Henry’s Law, that is ai = γi0 xi where γi0 = constant. In either case, ∂ ln γi /∂ ln xi = 0 so that ﬂux will be given by: Ji = −Di Mii kB T ∂ci ∂ci =− ∂z ci ∂z 106 CHAPTER 4. DIFFUSION WITH A CHEMICAL . . . where Di is the tracer diﬀusivity of component i, and Di dci /dz is just the Fick’s Law ﬂux. The tracer diﬀusivity Di is the diﬀusion coeﬃcient for the constituent i which would be measured in a homogeneous alloy where the only concentration gradients were in the relative concentration of i and a chemically identical but distinguishable tracer i∗ . Hence, although Di is aﬀected by the alloying eﬀects discussed in chapter 3, and as such can be a function of composition, it does not reﬂect the presence of concentration gradients of chemically dissimilar materials. In a pure material, the tracer and self diﬀusivities are only diﬀerent by the correlation factor. We see that the tracer diﬀusivity is related to the mobility by: Di = kB T Mii ci We also note that Fick’s ﬁrst law with the tracer diﬀusivity results from considering only the ideal mixing entropy term in the chemical potential. 4.2.3 Relationship Between Tracer and Intrinsic Diffusivities If we now return to the more general case of a nonideal, nondilute solution, we can write: ∂ci Mii kB T Ji = −D̃i =− ∂z ci From this we can ﬁnd that: D̃i Mii kB T = ci ∂ ln γi 1+ ∂ ln xi ∂ ln γi 1+ ∂ ln xi ∂ ln γi = Di 1 + ∂ ln xi ∂ci ∂z (4.11) This gives us a relationship between the tracer diﬀusivity, Di which is measured in dilute solution or by tracer diﬀusion in a otherwise homogeneous alloy, and the intrinsic diﬀusivity, D̃i , which takes into account the eﬀects of a concentration gradient and nonideality of the solution. If we write our expression for the ﬂux in terms of the tracer diﬀusivities: Ji = −Di ∂ci ∂ ln γi ∂ci − Di ∂z ∂ ln xi ∂z 4.2. CHEMICAL DRIVING FORCE 107 we see that the ﬁrst term in this expression comes from the concentration driving force arising from the ideal entropy of mixing, and the second term arises from the non-ideality of the solution. 4.2.4 Chemical Diﬀusion Coeﬃcient By combining the eﬀect of the thermodynamic biasing with the results we found by examining the Kirkendall eﬀect of the moving atomic planes we can ﬁnd the chemical interdiﬀusion coeﬃcient D̃ with which we can describe the intermixing of two constituents which form a nonideal solution. Recall our expression for D̃: D̃ = xA D̃B + xB D̃A where D̃A and D̃B are the intrinsic diﬀusivites of the two components. We can ﬁnd a relationship between the thermodynamic biasing of these diﬀusivities by using the Gibbs-Duhem relation: xA dµA + xB dµB = 0 (4.12) Looking at our expression for µ: µA = µ0 + kB T ln aA = µ0 + kB T (ln xA + ln γA ) we ﬁnd: xA dµA = kB T (dxA + xA d ln γA ) ∂ ln γA = kB T 1 + dxA ∂ ln xA Plugging this into the Gibbs-Duhem relation (Eqn 4.12) we ﬁnd: ∂ ln γA kB T 1 + ∂ ln xA ∂ ln γB dxA + kB T 1 + ∂ ln xB And since dxA = −dxB we can ﬁnd that: ∂ ln γB ∂ ln γA = ∂ ln xA ∂ ln xB dxB = 0 108 CHAPTER 4. DIFFUSION WITH A CHEMICAL . . . Plugging Eqn. 4.11 into our expression for the chemical diﬀusivity, D̃, and using the above relation, we ﬁnd: D̃ = D̃A xB + D̃B xA ∂ ln γA ∂ ln γB = DA x B 1 + + DB xA 1 + ∂ ln xA ∂ ln xB ∂ ln γA = (DA xB + DB xA ) 1 + ∂ ln xA This is our ﬁnal expression relating the chemical diﬀusivity, D̃, which is a measure of how a diﬀusion couple intermixes and is deﬁned by Fick’s laws, and the tracer diﬀusivities, Di , which measure the interdiﬀusion of dilute or ideal solutions, and the non-ideality of the solution represented by the activity coeﬃcient, γi . 4.2.5 Regular Solution Example As an example of the thermodynamic driving force for diﬀusion, we consider a regular solution of N atoms, where the entropy of mixing is given by the ideal solution mixing entropy: ∆Smix = −kB N (xA ln xA + xB ln xB ) and the non-ideality of the solution is represented by the enthalpy of mixing, which in the quasichemical approximation is given by: ∆Hmix = xA xB nΩRS where ΩRS is a measure of the strength of unlike bonds, and is given by: Ω RS ! 1 = z HAB − (HAA + HBB ) 2 where z is the number of nearest neighbors, n is the total number of moles of atoms, and Hij is the bond enthalpy per mole for i − j bonds. Here the bonding enthalpy is negative for a stable bond, so the enthalpy of mixing ∆Hmix is negative for systems where the A-B bond is stable relative to A-A and B-B bonds.