4.2 Chemical Driving Force

Transcription

1 4.2. CHEMICL DRIVING FORCE Chemcal Drvng Force second effect of a chemcal concentraton gradent on dffuson s to change the nature of the drvng force. Ths s because dffuson changes the bondng n a sold. If, for example, two consttuents n a dffuson couple have a preference for bondng wth unlke neghbors, that s, they have a negatve heat of mxng, then the decrease n free energy assocated wth dffusonal mxng wll have an enthalpy contrbuton as well as the mxng entropy contrbuton characterstc of deal or dlute solutons. Ths added enthalpy contrbuton wll act as a drvng force to ncrease ntermxng. Conversely, f the mxng enthalpy s postve, then the dffusonal rate wll be lower than that for an deal or dlute soluton. In order to examne ths effect we must generalze Fck s frst law, the flux equaton, by realzng that flux occurs as a result of a system s drve to approach thermodynamc equlbrum. Wth ths treatment, generally attrbuted to Darken, we can descrbe the mxng of chemcally dssmlar materals wth a dffuson coeffcent whch s a functon of the chemcal nature of the soluton Generalzed Flux Equatons Thermodynamc equlbrum s characterzed by the absence of spatal or temporal varatons n temperature T, pressure P, external potentals φ, and chemcal potentals of the components µ. Ths condton does not always mean the absence of concentraton gradents. Hence t s more reasonable to assert that the rate of return to equlbrum, that s, the flux of atoms, energy, and defects, s proportonal the devaton from equlbrum. Hence, to frst order, the flux wll be proportonal to gradents n temperature, pressure, potental, and chemcal potental, rather than just to composton gradents. The flux of the th component s gven by: J = k M k µ k M T T M P P M φ φ 4.6 where the M j s are the couplng coeffcents between fluxes n and gradents n k. These coeffcents reflect the strength drvng the flux and the moblty of the speces n respondng wth movement. In the case of a one-dmensonal, sothermal, sobarc dffuson wth no external potental gradents of an alloy of two components wth a vacancy

2 104 CHPTER 4. DIFFUSION WITH CHEMICL... mechansm we can wrte for the flux of the components: µ J = M z M µ B B z M µ V V z µ J B = M B z M µ B BB z M µ V BV z J V = M V µ z M VB µ B z M VV µ V z 4.7 Vacances can only be created or destroyed at sources or snks such as surfaces or defects. Hence, throughout most of the crystal, the number of lattce stes s conserved, so that the fluxes of the three speces whch can resde on a lattce ste are related by: J + J B + J V =0 If ths s to be true for arbtrary gradents, the sum of the coeffcents must be zero,.e.: M + M B + M V = 0 M B + M BB + M VB = 0 M V + M BV + M VV = 0 In addton there s a set of recprocty relatons, known as the Onsager relatons, whch state that M j = M j. Combnng these wth Eqn. 4.7 we fnd: J = M z µ µ V M B z µ B µ V J B = M B z µ µ V M BB z µ B µ V Darken s Flux Equaton To arrve at Darken s flux equaton, we must make the further assumptons that the vacances are n thermal equlbrum everywhere, so that µ V = 0, and that the off dagonal terms are neglgble. The flux for a gven component then reduces to: J = M µ z

3 4.2. CHEMICL DRIVING FORCE 105 The chemcal potental for a gven component can be wrtten: µ = µ 0 T,P+k B T ln a = µ 0 + k B T ln x + ln γ 4.8 where a s the actvty of component, and γ s the actvty coeffcent of, defned as: γ = a x where x s the atomc fracton of the th component x = c /c. The term k B T ln x 1 represents the deal mxng entropy contrbuton, whle the term k B T ln γ deals wth the non-dealty of the soluton. For example, n consderng the chemcal potental of a system of vacances, mpurtes, and vacancy-mpurty pars, we only consdered the deal mxng entropy term and found: µ deal = µ 0 + k B T ln x In ths treatment, we are nterested n devatons from dealty, and so must use the more general expresson for chemcal potental Eqn Our expresson for the flux s then: µ J = M z ln x = M k B T + ln γ z z ln x = M k B T + ln γ ln x z ln x z = M k B T c c ln x z Equaton 4.10 relates the flux of a component to ts concentraton gradent, and as such s a generalzaton of Fck s frst law. In order to examne the relatonshp between Eqn and Fck s frst law, we consder the case of an deal soluton where a = x,or the case of a dlute soluton where the actvty follows Henry s Law, that s a = γ 0 x where γ 0 = constant. In ether case, ln γ / ln x =0so that flux wll be gven by: c J = D z = M k B T c c z

4 106 CHPTER 4. DIFFUSION WITH CHEMICL... where D s the tracer dffusvty of component, and D dc /dz s just the Fck s Law flux. The tracer dffusvty D s the dffuson coeffcent for the consttuent whch would be measured n a homogeneous alloy where the only concentraton gradents were n the relatve concentraton of and a chemcally dentcal but dstngushable tracer. Hence, although D s affected by the alloyng effects dscussed n chapter 3, and as such can be a functon of composton, t does not reflect the presence of concentraton gradents of chemcally dssmlar materals. In a pure materal, the tracer and self dffusvtes are only dfferent by the correlaton factor. We see that the tracer dffusvty s related to the moblty by: D = k BTM c We also note that Fck s frst law wth the tracer dffusvty results from consderng only the deal mxng entropy term n the chemcal potental Relatonshp Between Tracer and Intrnsc Dffusvtes If we now return to the more general case of a nondeal, nondlute soluton, we can wrte: J = D c z = M k B T c c ln x z From ths we can fnd that: D = M k B T c ln x = D ln x 4.11 Ths gves us a relatonshp between the tracer dffusvty, D whch s measured n dlute soluton or by tracer dffuson n a otherwse homogeneous alloy, and the ntrnsc dffusvty, D, whch takes nto account the effects of a concentraton gradent and nondealty of the soluton. If we wrte our expresson for the flux n terms of the tracer dffusvtes: c J = D z D ln γ c ln x z

5 4.2. CHEMICL DRIVING FORCE 107 we see that the frst term n ths expresson comes from the concentraton drvng force arsng from the deal entropy of mxng, and the second term arses from the non-dealty of the soluton Chemcal Dffuson Coeffcent By combnng the effect of the thermodynamc basng wth the results we found by examnng the Krkendall effect of the movng atomc planes we can fnd the chemcal nterdffuson coeffcent D wth whch we can descrbe the ntermxng of two consttuents whch form a nondeal soluton. Recall our expresson for D: D = x DB + x B D where D and D B are the ntrnsc dffusvtes of the two components. We can fnd a relatonshp between the thermodynamc basng of these dffusvtes by usng the Gbbs-Duhem relaton: Lookng at our expresson for µ: we fnd: x dµ + x B dµ B = µ = µ 0 + k B T ln a = µ 0 + k B T ln x + ln γ x dµ = k B T dx + x d ln γ = k B T dx Pluggng ths nto the Gbbs-Duhem relaton Eqn 4.12 we fnd: k B T dx + k B T nd snce dx = dx B we can fnd that: ln γ = ln γ B 1+ ln γ B dx B =0

6 108 CHPTER 4. DIFFUSION WITH CHEMICL... Pluggng Eqn nto our expresson for the chemcal dffusvty, D, and usng the above relaton, we fnd: D = D x B + D B x = D x B 1+ ln γ = D x B + D B x + D B x 1+ ln γ 1+ ln γ B Ths s our fnal expresson relatng the chemcal dffusvty, D, whch s a measure of how a dffuson couple ntermxes and s defned by Fck s laws, and the tracer dffusvtes, D, whch measure the nterdffuson of dlute or deal solutons, and the non-dealty of the soluton represented by the actvty coeffcent, γ Regular Soluton Example s an example of the thermodynamc drvng force for dffuson, we consder a regular soluton of N atoms, where the entropy of mxng s gven by the deal soluton mxng entropy: S mx = k B N x ln x + x B ln x B and the non-dealty of the soluton s represented by the enthalpy of mxng, whch n the quaschemcal approxmaton s gven by: H mx = x x B nω RS where Ω RS s a measure of the strength of unlke bonds, and s gven by: Ω RS = z [ H B 1 ] 2 H + H BB where z s the number of nearest neghbors, n s the total number of moles of atoms, and H j s the bond enthalpy per mole for j bonds. Here the bondng enthalpy s negatve for a stable bond, so the enthalpy of mxng H mx s negatve for systems where the -B bond s stable relatve to - and B-B bonds.

7 4.2. CHEMICL DRIVING FORCE 107 we see that the frst term n ths expresson comes from the concentraton drvng force arsng from the deal entropy of mxng, and the second term arses from the non-dealty of the soluton Chemcal Dffuson Coeffcent By combnng the effect of the thermodynamc basng wth the results we found by examnng the Krkendall effect of the movng atomc planes we can fnd the chemcal nterdffuson coeffcent D wth whch we can descrbe the ntermxng of two consttuents whch form a nondeal soluton. Recall our expresson for D: D = x DB + x B D where D and D B are the ntrnsc dffusvtes of the two components. We can fnd a relatonshp between the thermodynamc basng of these dffusvtes by usng the Gbbs-Duhem relaton: Lookng at our expresson for µ: we fnd: x dµ + x B dµ B = µ = µ 0 + k B T ln a = µ 0 + k B T ln x + ln γ x dµ = k B T dx + x d ln γ = k B T dx Pluggng ths nto the Gbbs-Duhem relaton Eqn 4.12 we fnd: k B T dx + k B T nd snce dx = dx B we can fnd that: ln γ = ln γ B 1+ ln γ B dx B =0