Mass and Heat Diffusion in Ternary Polymer Solutions: A Classical Irreversible Thermodynamics Approach

Transcription

1 Mass and Heat ffuson n Ternary Polymer Solutons: A Classcal Irreversble Thermodynamcs Approach S. Shams Es-hagh and M. Cakmak * epartment of Polymer Engneerng, 5 S. Forge St., The Unversty of Akron, Akron, Oho, , USA ABSTRACT Governng equatons for evoluton of concentraton and temperature n three-component systems were derved n the framework of classcal rreversble thermodynamcs usng Onsager s varatonal prncple and were presented for solvent/solvent/polymer and solvent/polymer/polymer systems. The dervaton was developed from the Gbbs equaton of equlbrum thermodynamcs usng the local equlbrum hypothess, Onsager recprocal relatons and Prgogne s theorem for systems n mechancal equlbrum. It was shown that the detals of mass and heat dffuson phenomena n a ternary system are completely expressed by a 3 3 matrx whose entres are mass dffuson coeffcents (4 entres), thermal dffuson coeffcents ( entres) and three entres that descrbe the evoluton of heat n the system. The entres of the dffuson matrx are related to the elements of Onsager matrx that are bounded by some constrants to satsfy the postve defnteness of entropy producton n the system. All the elements of dffuson matrx were expressed n terms of dervatves of exchange chemcal potentals of the components wth respect to concentraton and temperature. The spnodal curves of ternary polymer solutons were derved from the governng equatons and ther correctness was checked by the Hessan of free energy densty. Moreover, t was proved that settng crossdffuson coeffcents to zero results n a contradcton, and the governng equatons wthout cross-dffuson coeffcents do not express the actual phase behavor of the system. *Correspondng author: cakmak@uakron.edu

2 I. INTROUCTION Mass and heat dffuson n polymer solutons s of great mportance n the technologes related to pantng, coatng, nkjet prntng, plastc flms and producton of electronc devces []. ffuson processes also play a key role n self-assembly [] and mportant areas of study n soft matter physcs [3]. A proper understandng of dffuson phenomena n polymer solutons s requred n order to have a control on the processes nvolvng polymer solutons. Polymer solutons commonly used for ndustral purposes are generally mult-component systems,.e., ternary polymer solutons of one polymer n two solvents or two polymers n one solvent [4]. Most of the expermental and theoretcal studes thus far, have been about dffuson n bnary systems, and despte the broad range of applcatons nvolve ternary polymer solutons [4-4] there s no theoretcal work that formulates thermodynamcally consstent governng equatons for mass and heat dffuson n ternary polymer solutons n a way that all the mass and thermal dffuson coeffcents necessary to descrbe the transport phenomena are derved n terms of thermodynamc varables. By ncreasng the number of components n a mxture and emergence of cross-dffuson effects, the transport phenomena become very complex. As a result, a large number of mass dffuson coeffcents are requred to descrbe the mass transport []. Ths complexty wll ncrease by addng a spatally varyng temperature feld to the system, snce the effect of thermal dffuson and contrbutons of mass fluxes to the heat flow must also be consdered. Mass and heat dffuson are rreversble phenomena and they contrbute to entropy producton n the system. Hence, the governng equatons for mass and heat evoluton n ternary polymer solutons must be derved n the framework of classcal rreversble thermodynamcs n whch the balance of entropy plays a crucal role. An overvew of the governng equatons formulated for mass and

3 heat dffuson n ternary polymer solutons ndcates that these equatons were derved wthout consderng the balance of entropy n the system. Vrentas et al. [5] derved governng equatons for sothermal mass dffuson n ternary polymer solutons based on a theory due to Bearman [6]. They consdered a ternary polymer system contanng one polymer and two solvents. In that system, the concentraton of solvents was much lower than that of the polymer. They set the cross-dffuson coeffcents to zero and estmated the prncpal mass dffuson coeffcents by the self-dffuson coeffcents of the solvents. Usng the same approach based on the statstcal theory of Bearman [6], Shojae et al. [7] developed a model for non-sothermal mass dffuson n a ternary polymer soluton contanng a polymer and two solvents. The heat transfer n ther model was descrbed by the Fourer s law of heat conducton. Alsoy and uda [8] studed the dryng of ternary polymer solutons usng a model n whch the mass dffuson coeffcents were expressed n terms of self-dffuson coeffcents of the solvents. In that model they assumed temperature to be temporally dependent and spatally ndependent. abral et al. [4] modeled the dryng process of a ternary polymer soluton as an sothermal mass dffuson problem. ue to the lack of relable expermental data and the absence of a sutable predctve theory, they set the cross-dffuson coeffcents to zero. In the above-mentoned formulatons, there s no crteron that guarantees the governng equatons are thermodynamcally consstent. An analyss of these models developed for dffuson n mult-component systems ndcates that some of them are not consstent wth the Onsager recprocal relatons [9]. Moreover, n these studes, contrbuton of temperature gradent to mass dffuson n ternary polymer solutons was neglected. Furthermore, the heat transfer was modeled usng the Fourer s law of heat conducton. Ths formulaton does not take nto account the effect of mass dffuson on the heat transfer and the couplng between mass and heat transfer s only due to the temperature dependency of mass dffuson coeffcents. 3

4 Recently, we developed a thermodynamcally consstent model of mass and heat dffuson n bnary polymer solutons []. In ths paper, we focus on generalzaton of ths model to ternary polymer solutons. Although ths study ams to develop a theoretcal understandng of dffuson n ternary polymer solutons, the governng equatons derved heren can be used for any ternary system. Snce, the governng equatons have been formulated for the general case of non-sothermal three-component systems, the effect of temperature gradent on the evoluton of concentraton of the components was shown by dervaton of thermal dffuson coeffcents n terms of phenomenologcal coeffcents and dervatves of exchange chemcal potentals of the components wth respect to temperature. Moreover, t was proved that crossdffuson coeffcents play a crucal role n expressng the phase behavor of a ternary system and settng the cross-dffuson coeffcents to zero leads to a contradcton. II. THEORY We consder a non-reactng ternary system n whch non-convectve mass dffuson and heat conducton occur. The system s consdered to be n one phase, far from the crtcal regon of phase separaton. We also assume the system s n mechancal equlbrum. The hypothess of local equlbrum allows the fundamental equaton of classcal thermodynamcs to be vald for every volume element n the system, although the whole system s not n equlbrum []. Usng ths hypothess and Prgogne s theorem for systems n mechancal equlbrum, the rate of change of entropy per unt volume for a three-component system such as solvent/solvent/polymer or solvent/polymer/polymer s gven by (see Appendx A) 4

5 ds Jq,3 J,3J dt T T J J J J J q,3,3,3,3 T T T, () where s, J q, J, J and T are entropy per unt volume, heat flux, mass flux of component, mass flux of component and absolute temperature, respectvely.,3 and,3 are exchange chemcal potentals of components and 3, and and 3, respectvely. Equaton () represents the rate of change of entropy per unt volume of the mxture n terms of dvergence of entropy flux and rate of entropy producton per unt volume of the system, whch can be presented n the blnear form of thermodynamcs forces X,3,,3, T T T T and the conjugated fluxes J J, J, Jq,3 J,3J, T J J J J J. () q,3,3,3,3 T T T As one can see due to a constrant for the fluxes, n a ternary system the mass flux of two components are expressed explctly n the rate of entropy producton. In effect, for a general case of an n -component system, n equaton are requred to descrbe the mass dffuson n the system. The governng equatons can be found usng lnear flux-force relatonshps consderng the Onsager recprocal relatons and also they can be derved from Onsager s varatonal prncple [] by maxmzng J, the dfference between rate of entropy producton n the system and a dsspaton functon presented n terms of thermodynamc forces n the system 3 ( X, X, X 3) Lk X X k, (3) k, 5

6 here L k are phenomenologcal coeffcents. Usng lnear flux-force relatonshps, the fluxes can be wrtten n terms of the forces as shown n Eqs. (4) - (6) whch n matrx presentaton would yeld Eq. (7) J l,3 l,3 l 3 T, T T T J l,3 l,3 l 3 T, T T T Jq,3 J,3J l3,3 l3,3 l33 T, T T T,3 J T l l l3 J l l l. 3,3 T Jq,3 J,3J l3 l3 l 33 T T (4) (5) (6) (7) Entres of matrx L ( lj ) 33 are Onsager s coeffcents and based on Onsager s recprocty relatons, matrx L s a symmetrc matrx [3]. In condtons for whch lnear flux-force relatons are vald, rate of entropy producton takes the quadratc form l,3 l,3 l33 T T T T l,3,3 l3,3 T T T T T l3,3 T. T T (8) Matrx L ( lj ) 33 that satsfes Eq. (8) should be postve defnte and to be so, ts entres should satsfy the condtons 6

7 l l l3 l, ll l, det l l l3. l3 l3 l 33 (9) Snce L s a symmetrc postve defnte matrx, all of ts dagonal entres, l are postve. By consderng l l l l T T T T 3 3 (where = or or 3),,,, () and replacng,3 and,3 wth the rght hand sdes of Eqs. () and (), respectvely, knowng the fact that chemcal potental s a functon of concentraton and temperature, T,,3,3,3,3 T, T T,, () T,,3,3,3,3 T, T T,, () we can recast Eqs. (4) - (6) n the forms J,3,3 T, T,,3,3 T, T,,3,3 T, T T T,, (3) 7

8 J,3,3 T, T,,3,3 T, T,,3,3 T, T T T,, (4),3,3 Jq,3 J,3J T, T,,3,3 T, T,,3,3 3 T. T T T,, (5) be satsfed: In order to preserve the postve defnteness of matrx L, the followng condtons must,,3,. 3 3 (6) After dervng the heat and mass fluxes, usng the defnton of the tme dervatve of enthalpy per unt volume (see Eq. A.4) and mass concentraton for component (see Eq. A.5), we obtan the governng equatons from the dvergence of the fluxes 8

9 d,3,3 dt T, T,,3,3 T, T, T T T,,,3,3 T, (7) d,3,3 dt T, T,,3,3 T, T,,3,3 T, T T T,, (8) dt,3,3 cp,3 J,3J dt T, T,,3,3 T, T,,3,3 3 T, T T T,, (9) where and c P are the mass densty and sobarc specfc heat capacty of the system, respectvely. Usng Eqs. (7) and (), one can express the equaton of heat flux n the form (see Appendx B) J k T, () q 9

10 where J q and k are reduced heat flux and thermal conductvty and are gven by Jq J q J J,3,3, () k T 3. () Equaton () shows thermal conductvty as a functon of phenomenologcal coeffcents and the constrant gven by Eq. (6) guarantees that the thermal conductvty s a postve quantty. There s another way of expressng the equaton of heat dffuson. Ths can be done by replacng the mass fluxes n Eq. (9) usng Eqs. (3) and (4). ong so, the equaton of heat dffuson wll be dt,3,3 cp (,3,3 ) (,3,3 ) dt T, T,,3,3 (,3,3 ) (,3,3 ) T, T,,3,3 (,3,3 ) (,3,3 ) T T T,, (,3,3 3) T. T (3) Equatons (7), (8) and (3) can be wrtten n a compact form as 3 d. dt 3 cpt T (4)

11 We call the 3 3 matrx n Eq. (4) dffuson matrx whose entres j (see Appendx C) descrbe the detals of mass and heat dffuson n the system. Mass dffuson matrx s a matrx wth entres,, and. and are cross-dffuson coeffcents. 3 and 3 are thermal dffuson coeffcents. The three entres n the thrd row of the dffuson matrx descrbe the heat conducton n the system. III. RESULTS AN ISCUSSION In secton, the mass and heat dffuson coeffcents were derved as part of the model n terms of dervatves of exchange chemcal potentals of the components wth respect to concentraton and temperature. Thus, n the model developed heren, there s no need for defnton of dffuson coeffcents. The approach adopted n ths paper for dervng the governng equatons n ternary systems can be compared wth the model developed by Curtss and Brd [4] who derved the generalzed Maxwell-Stefan equatons for the mult-component dffuson. In that model the dffuson coeffcents were not derved. Curtss and Brd [4] consdered two dfferent defntons for mass dffuson coeffcents; zero-dagonal and symmetrc dffusvty defntons and based on these defntons they derved generalzed Maxwell-Stephan equatons. The 3 governng equatons derved n secton can be used for any ternary mxture. The entres of the dffuson matrx wll be dfferent dependng on the chemcal potentals of the components n a partcular system. In order to express the governng equatons for ternary polymer solutons, one needs to have the chemcal potentals of the components. We consder two general ternary polymer solutons; A soluton of one polymer n two solvents,

12 solvent()/solvent()/polymer(3) and a soluton of two polymers n one solvent, solvent()/polymer()/polymer(3). The numbers n parenthess wll be used as subscrpts n equatons to represent components n the system. The chemcal potentals of the components n a ternary polymer soluton can be derved by extenson of Flory-Huggns theory to ternary polymer soluton [5,6]. Then, the exchange chemcal potentals of the components can be derved usng Eqs. (A.) and (A.). A. ffuson matrx for solvent()/solvent()/polymer(3) systems In case of solvent()/solvent()/polymer(3) systems, the exchange chemcal potentals of the components wll be (see Appendx ) M(,3,3 ) ln ln RT N3 N3, M (,3,3 ) ln ln RT N3 N3, (5) (6) where,3 and,3 are,3 3, M N 3 (7),3 3. M N 3 (8) Here, M, M, N 3 and j are chemcal potental of component n ts pure lqud state, molecular weght of component, molecular weght of component, degree of polymerzaton

13 of component 3 (polymer) and the Flory-Huggns nteracton parameter between components and j. Usng Eqs. (5) (8) the dffuson matrx for solvent()/solvent()/polymer(3) systems can be derved (see Appendx E). B. ffuson matrx for solvent()/polymer()/polymer(3) systems In case of solvent()/polymer()/polymer(3) systems, the exchange chemcal potentals of the components wll be (see Appendx ) M(,3,3 ) ln ln RT N3 N3, m (,3,3) ln ln RT N N3 N N3, (9) (3) where m and N are the molecular weght of the repeatng unt of the component (polymer ) and ts degree of polymerzaton, respectvely and,3 and,3 are gven by,3 3, M N 3 (3) N,3 3. M N 3 (3) Usng Eqs. (9) (3) the dffuson matrx for solvent()/polymer()/polymer(3) systems can be derved (see Appendx F). C. Mass and heat dffuson n an n-component system The governng equatons derved for ternary mxtures can easly be generalzed to descrbe mass and heat dffuson n an n -component system. 3

14 gven by The governng equatons for mass and heat dffuson n an n -component system are ( n ) n d. dt ( n ) ( n ) ( n) ( n)( n) ( n) n ( n) ct P n n ( n ) nn T (33) The ( n) ( n ) submatrx shown nsde a box n Eq. (33) s the mass dffuson matrx and the components of the vector n the last column of dffuson matrx are thermal dffuson coeffcents. The Onsager matrx assocated wth the dffuson matrx s an n n symmetrc postve defnte matrx whose entres must satsfy l l l3 l l n l, ll l, det l l l3,, det. l l l l l n nn (34) Ths concept of generalzng governng equatons to an n -component system may become mportant when a soluton of several polymers s prepared n one or several solvents. Moreover, knowng the fact that polymers have a dstrbuton of molecular weght, a soluton of one polymer n one solvent s a mult-component system of polymer chans wth dfferent molecular weghts n that solvent [8].. ervaton of spnodal curve from the governng equatons. Important nformaton about phase behavor of a system can be extracted from the governng equatons for mass dffuson n the system. The spnodal hypersurface for an n - components system s gven by [9] 4

15 ( n) det. ( n) ( n)( n) (35) In effect, the spnodal curve for a system s the hypersurface on whch the mass dffuson matrx becomes sngular. Therefore, for a ternary mxture the spnodal curve s gven by,3,3,3,3 T, T, T, T, det.,3,3,3,3 T, T, T, T, (36) ue to the constrant, the determnant of the mass dffuson matrx wll be,3,3,3,3 T, T, T, T,. (37) Equaton (37) gves the spnodal curve for a general ternary system. Usng the equatons of exchange chemcal potentals of the components, the spnodal curve for solvent()/solvent()/polymer(3) and solvent()/polymer()/polymer(3) systems wll be 3 3 N3( 3) N3( 3) 3 3, N3( 3) (38) 3 3 N3( 3) N N3( 3) 3 3, N3( 3) (39) 5

16 respectvely. The results derved heren can be checked wth a well-establshed alternatve way n that the spnodal hypersurface of an n -component system can be derved from the determnant of the Hessan matrx of free energy densty of the system [3,3] f f ( n) det. f f ( n) ( n) (4) Therefore, for a ternary mxture the spnodal curve wll be f f f f. (4) Free energy densty functons of solvent()/solvent()/polymer(3) and solvent()/polymer()/polymer(3) systems are gven by [3] f f ln ln ln, (4) N3 ln ln ln, (43) N N3 respectvely. The free energy densty functons are presented n dmensonless form and accordng to Eq. (4) ths does not affect the result of the problem. It s easy to show that the spnodal curves derved from Eq. (4) usng Eqs. (4) and (43) are dentcal wth the spnodal curves derved from the governng equatons. E. The consequence of settng cross-dffuson coeffcents to zero Accordng to the lterature (see for example Ref. 3), due to the lack of data or n order to smplfy the problem the cross-dffuson coeffcents are set to zero. However, t has been observed that droppng cross-dffuson coeffcents resulted n sgnfcant errors n predctons of 6

17 dryng of ternary polymer solutons [3]. In ths secton we dscuss the consequence of gnorng cross-dffuson coeffcents n a ternary system. By droppng the cross-dffuson coeffcent,..e., the mass dffuson matrx for a ternary system wll be presented ether n the form,3 T,,3 T,, (44) or,3 T,,3 T,. (45) ue to the constrant, the spnodal curve of the system wll be,3,3 T, T,, (46) or,3,3 T, T,, (47) respectvely. It s clear that the spnodal curve demonstrated by the governng equatons after reducng the cross-dffuson coeffcents to zero s not the actual spnodal curve of the system. Moreover, Eqs. (46) and (47) lead to a contradcton n that the exchange chemcal potentals of the components,3 and,3 must be a constant or be only a functon of or so that Eqs. 7

18 (46) and (47) are vald. Therefore, one can see that settng the off-dagonal entres of mass dffuson matrx to zero can result n a serous problem n that not only the actual phase behavor of the system wll be altered, but also a contradcton arses. Therefore, one s not elgble to gnore the cross-dffuson coeffcents n the system. It s worth mentonng that the proof provded heren s for a general case and can be consdered for any three-component system. IV. SUMMARY AN CONCLUSIONS 3 governng equatons for mass and heat dffuson n three-component systems were derved n the framework of classcal rreversble thermodynamcs. The formulaton of the governng equatons was developed from the Gbbs equaton usng Onsager s varaton prncple. It was shown that the detals of mass and heat dffuson n a three-component mxture are descrbed by a 3 3 dffuson matrx whose entres are mass dffuson coeffcents, thermal dffuson coeffcents and three entres that descrbe the evoluton of temperature feld n the system. Mass dffuson coeffcents are the entres of mass dffuson matrx, a submatrx whch remans after elmnaton of the thrd row and thrd column of the dffuson matrx. The entres of the dffuson matrx were expressed n terms of phenomenologcal coeffcents and dervatves of exchange chemcal potentals of the components wth respect to concentraton and temperature. The model was formulated for the general case of non-sothermal mass dffuson n three-component systems and can be used for any three-component system, when the chemcal potentals of the components are known as functons of concentraton and temperature. The entres of the dffuson matrx were derved for partcular cases of ternary polymer solutons, solvent/solvent/polymer and solvent/polymer/polymer systems by explotng the Flory-Huggns theory to derve the chemcal potentals of the components. It was shown that the model can be 8

19 easly generalzed to descrbe mass and heat dffuson n an n -component system. In ths general case dffuson matrx has n entres where ( n ) entres are mass dffuson coeffcents, ( n ) entres are thermal dffuson coeffcents and n entres descrbe the evoluton of temperature feld n the system. The spnodal curves of ternary polymer solutons were derved from the entres of the mass dffuson matrx and ther correctness was checked usng the determnant of the Hessan matrx of free energy densty functons of the systems. Moreover, for general case of ternary systems, t was shown that cross-dffuson coeffcents play a crucal role n expressng the phase behavor of the system. In effect, t was proved that settng the cross-dffuson coeffcents to zero leads to a contradcton. ACKNOWLEGMENT We are grateful for fundng enttled Polymde research and Commercalzaton Grant from The State of Oho Thrd fronter program. 9

20 Appendx A. Rate of change of entropy per unt volume of a ternary system Equaton (A.) states the fundamental equaton per unt volume of a mxture; k de Tds dn, (A.) where T, s, e, and n are absolute temperature, entropy per unt volume, enthalpy per unt volume, chemcal potental of component and moles of component per unt volume, respectvely. One can rewrte Eq. (A.), usng the mass concentraton of the components and take a dervatve wth respect to tme to fnd k ds de dc T, dt dt (A.) M dt where c and M are the mass concentraton and molecular weght of component, respectvely. Tme dervatves n Eq. (A.) are substantal tme dervatves gven by d dt v. t (A.3) Here, v s the mean velocty of the components. The tme dervatves of e and c n Eq. (A.) are related to the dvergence of heat and non- convectve mass fluxes, de J, q dt (A.4) dc J. dt (A.5)

21 Accordng to Prgogne s theorem, for systems n mechancal equlbrum an arbtrary frame of reference can be chosen []. The non-convectve mass flux of the component n ths frame of reference whch moves wth the mean velocty v s gven by J v v, (A.6) where, and v are mass densty, volume fracton and the velocty of component, respectvely and the mean velocty v s expressed as k v v. (A.7) It follows from Eqs. (A.6) and (A.7) that the fluxes are not ndependent and ther dependency s gven by k J. (A.8) Rewrtng Eq. (A.) for a three-component system and replacng de dt and dc dt wth the dvergences of the assocated fluxes gven by Eqs. (A.4) and (A.5), we obtan ds T Jq J J J (A.9) dt M M M The subscrpts, and 3 are attrbuted to component, component and component 3 n the system, respectvely. Equaton (A.9) can be reduced after applyng the constrant for fluxes gven by Eq. (A.8) to the form ds T J, q,3 J,3 J (A.) dt where,3 and,3 are the exchange chemcal potentals of the components and are gven by

22 3 3,3, (A.) M M3 3 3,3, (A.) M M3 respectvely. We can rewrte Eq. (A.) by replacng the rght hand sde wth an equvalent form and dvdng both sdes by T, ds Jq,3 J,3J J,3 J,3. (A.3) dt T T T The frst term on the rght hand sde of Eq. (A.3) can be manpulated to an equvalent form so that ds Jq,3 J,3J dt T T J J J J J q,3,3,3,3 T T T. (A.4)

23 Appendx B. ervaton of reduced heat flux Usng Eq. (), and knowng that l l T, l l = T, and 3 3 l3 l3 T the lnear flux-force relatonshps gven by Eq. (7) can be wrtten as J J (B.) T,3,3 T, (B.) T,3,3 T, 3 Jq,3 J,3J,3,3 T, (B.3) T Usng Eqs. (B.) and (B.),,3 and,3 are gven by,3 J J T, T (B.4),3 J J T. T (B.5) By substtutng for,3 and,3 n Eq. (B.3), the heat flux wll be Jq,3 J,3 J T 3 T. (B.6) Equaton (B.6) can be expressed n terms of the reduced heat flux that s gven by Jq J q J J,3,3, (B.7) Therefore, we obtan J q 3 T. T 3 (B.8)

24 Appendx C. Entres of dffuson matrx,3,3 T, T,,,3,3 T, T,,, T T T,3,3 3,,,3,3 T, T,,,3,3 T, T,,, T T T,3,3 3,, ( ) ( ),,3,3 3,3,3,3,3 T, T, ( ) ( ),,3,3 3,3,3,3,3 T, T, ( ) ( ) ( ). T T T,3,3,3,3 3 33,3,3,3,3,, 4

25 Appendx. Exchange chemcal potentals of components n a ternary polymer soluton Exchange chemcal potentals,3 and,3 can be derved usng Eqs. (A.) and (A.) whch can be wrtten n the forms V V,3 3,,3 3, M V3 M V3 (.) where V / V s the rato of molar volumes of the components and wll be consdered equal to the j rato of ther degrees of polymerzaton. Chemcal potentals of the components n a ternary polymer soluton such as solvent()/solvent()/polymer(3) wth respect to ther pure states are gven by RT RT ln , N3 ln , N3 3 3 ln3 N3 N3 RT N N N, (.) (.3) (.4) where, R, N 3 and j are chemcal potentals of component n ts pure lqud state, gas constant, degree of polymerzaton of component 3 (polymer) and the Flory-Huggns nteracton parameter of components and j. Therefore, the exchange chemcal potentals for the solvent()/solvent()/polymer(3) systems wll be M(,3,3) ln ln RT N3 N3, (.5) 5

26 M (,3,3) ln ln RT N3 N3, (.6) where,3 and,3 are,3 3, M N 3 (.7),3 3. M N 3 (.8) Chemcal potentals of the components n a ternary polymer soluton such as solvent()/polymer()/polymer(3) wth respect to ther pure states are gven by ln 3 RT N N3, N ln N 3 RT N3 N N N, N 3 ln3 N3 RT N N N N (.9) (.) (.) Therefore, the exchange chemcal potentals for the solvent()/polymer()/polymer(3) systems wll be M(,3,3 ) ln ln RT N3 N3, (.) 6

27 m (,3,3) ln ln RT N N3 N N3, (.3) where m and N are the molecular weght of the repeatng unt of the component (polymer ) and ts degree of polymerzaton, respectvely and,3 and,3 are gven by,3 3, M N 3 (.4) N,3 3. M N 3 (.5) 7

28 Appendx E. Entres of dffuson matrx for solvent()/solvent()/polymer(3) systems Snce the chemcal potentals of pure components are only functons of temperature [7], the dervatve of wth respect to the concentraton vanshes. Moreover, the nteracton parameters were treated as constants. In the case where a temperature dependency s consdered for nteracton parameters the terms contanng dervatves wth respect to the temperature must be modfed. RT RT, M N3( 3) M N3( 3) RT RT, M N3( 3) M N3( 3) d,3 (,3,3) d,3 (,3,3), dt T dt T T 3 RT RT, M N3( 3) M N3( 3) RT RT, M N3( 3) M N3( 3) d ( ) d ( ), dt T dt T T,3,3,3,3,3,3 3 RT ( ),3,3 3 3 M N3( 3) RT (,3,3 ) 3 3, M N3( 3) 8

29 RT ( ),3, M N3( 3) RT (,3,3 ) 3, M N3( 3) d,3 (,3,3 ) ( ) dt T 33,3,3 d,3 (,3,3) (,3,3 3) (,3,3 ). dt T T 9

30 Appendx F. Entres of dffuson matrx for solvent()/polymer()/polymer(3) systems Snce the chemcal potentals of pure components are only functons of temperature [7], the dervatve of wth respect to the concentraton vanshes. Moreover, the nteracton parameters were treated as constants. In the case where a temperature dependency s consdered for nteracton parameters the terms contanng dervatves wth respect to the temperature must be modfed. RT RT, M N3( 3) m N3( 3) RT RT, M N3( 3) m N N3( 3) d,3 (,3,3) d,3 (,3,3), dt T dt T T 3 RT RT, M N3( 3) m N3( 3) RT RT, M N3( 3) m N N3( 3) d ( ) d ( ), dt T dt T T,3,3,3,3,3,3 3 RT ( ),3,3 3 3 M N3( 3) RT (,3,3 ) 3 3, m N3( 3) 3

31 RT ( ),3, M N3( 3) RT (,3,3 ) 3, m N N3( 3) d,3 (,3,3 ) ( ) dt T 33,3,3 d,3 (,3,3) (,3,3 3) (,3,3 ). dt T T 3

32 e T s n t M c J q J NOTATION enthalpy per unt volume absolute temperature entropy per unt volume chemcal potental of component chemcal potental of component n ts pure state moles of component per unt volume tme molecular weght of component mass concentraton of component heat flux non-convectve mass flux of component mass densty of component volume fracton of component v velocty of component v mean velocty exchange chemcal potental of components and j, j rate of entropy producton per unt volume dsspaton functon L Onsager matrx X vector of thermodynamc forces J vector of thermodynamc fluxes,,, phenomenologcal coeffcents mass densty of the system sobarc specfc heat capacty of the system c P J q k j j R f N reduced heat flux thermal conductvty entres of dffuson matrx Flory-Huggns nteracton parameter of components and j gas constant free energy densty functon degree of polymerzaton of component m molecular weght of repeatng unt of component V molar volume of component 3

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