Fickian and Non-Fickian Diffusion in High Polymer SystemsBY G. REHAGE, 0. ERNST AND J. FUHRMANNPhysikalisch-Chemisches Institut der Technischen Universitat Clausthal, B. R.D.Received 12th June, 1970The normal and the anomalous diffusion in binary polymer solutions are considered. After asurvey of the theory of diffusion it is demonstrated which criteria must be fulfilled so that the Fickiandiffusion occurs. For transport between the solvent and a solution, or between two solutions,normal diffusion is found. Experimental results with polystyrene+solvent systems are given. Withgood solvents the diffusion coefficient increases strongly with increasing polymer concentration,passes through a maximum at medium concentrations and decreases by several decades at highpolymer concentrations.This concentration dependence arises because the diffusion coefficient isnot only a transport coefficient, but also contains a thermodynamic factor. With poor solvents,which show a phase separation of the polymer at low temperatures, the concentration dependenceof the diffusion coefficient is even more complicated. With increasing concentration of polymer thediffusion coefficient first decreases, passes through a minimum and then increases again. In thiscase there is also a maximum at medium concentrations and a strong decrease at high polymerconcentrations. The minimum for a binary system is located at the critical point of the system.Normally In D is a linear function of 1/T. Deviations from the linear course can be explainedby the temperature dependence of the thermodynamic factor.For anomalous diffusion the l/t-relations are no longer valid, since the diffusion coefficient dependsnot only on concentration, but also explicitly on time.The anomalous diffusion is due to the factthat superimposed on the normal diffusion is another process. If a solvent penetrates intoa glassy polymer, then relaxation processes are superimposed on the diffusion process. The polymerchanges from the glassy state into a state of internal thermodynamic equilibrium. Experimentallythe continuous range between the pure glassy polymer and the pure solvent can be divided into threeparts of an open system. In the first and the third part only diffusion occurs, while in the secondpart diffusion and structural relaxation are superimposed.Diffusion is mass transport caused by concentration gradients, or more exactly,by gradients of the chemical potential. For binary non-electrolyte systems, withwhich we are concerned in the following, we can write, if we consider diffusion in onedirection only,Wl dP2 J1 = -all--aa Z 12- aZ4% aP2a Z a2J 2 = -a21--a22--whereJ1,J2 = mass fluxes of the components 1 and 2,p1,p2 = chemical potentials of the components 1 and 2,afk = phenomenological coefficients,z = space coordinate.Since we consider only the diffusion in one direction, we have not to allow for thevector character of the flux.The Gibbs-Duhem-equation is20G . REHAGE, 0. ERNST AND J . FUHRMANN 209where p,, p 2 are the mass concentrations of the two components, e.g., in g ~ r n - ~ .From the eqn (1) and (2) it follows thata11 a12 a , = -- - 3 P l .cz ' P1 P2J1 = -alp,-dPu, a 2 2 a21a2 P2 P1J 2 = -a2p2-; a2 = ---.Further we can write, for liquid mixtures, which are only considered here,J1 = P1 (Vl-w),J 2 = P2 @ 2 - 4 ,(34where v l , and v2 are the mean particle velocities and w is the mean volume velocity.For w, we may writewhere q1 and 'p2 are the volume fractions of the two components.(4) it follows thatw = CPlVl +P2V2 (5)From eqn (3) anda1 = (01 - W M - a P l l a z )a2 = (02 - w ) f c - 3P2PZ)( 6 4(6b)Eqn (6a) and (6b) imply that the coefficients al and a, are the relative velocities of theparticle species 1 and 2, and are related to the driving forces, i.e., negative to thegradients of the chemical potentials.A mean particle velocity, related to the drivingforce, is defined as mobility. It follows thatwhere u1 and u2 are the mobilities of the two particle species related to the meanvolume velocity.The first Fickian law for diffusion isJ , = -D,ap,/aZ,J~ = - ~ ~ a p ~ / a z .Dl = ~ l ( P l a P l / a P l ) ,D2 = U2(P2aP2/aP2).By comparing the coefficients with eqn ( 7 ) one finds thatFurther for a binary system where D1 = D2,1 thenD = u l ( P l ~ P l / a P , ) = %(P2aP2/aP2). (10)Eqn (10) shows that the diffusion coefficient contains a hydrodynamic or transportfactor u1 or u2 and a thermodynamic factor p l a p l / a p l or pzap2/dp2. To obtainmobilities therefore, the diffusion coefficient and the concentration dependence of thechemical potentials must be known.Instead of eqn (10) one can writ210 FICKIAN AND NON-FICKIAN DIFFUSIONwhere fl and f 2 are the friction coefficients.For infinite dilution there are well-known relationships, e.g., the Einstein-Stokes eqn for spherical particles.2 By meansof the thermodynamics of irreversible processes one can derive the following equivalentrelation :where s1 and s2 are the sedimentation coefficients, i.e., the mean particle velocitiesdivided by the gravitational or centrifugal acceleration, M1 and M2 the molar masses, vl and v’. the partial specific volumes of the two components and p the density of thesolution. At infinite dilution one has the Svedberg eqn, which is of great importancefor measurements with the ultracentrifuge.From eqn (lo)-( 12), the followingrelations between the transport coefficients are obtained :u2 = llfi = s2/M2(1-F2p). (13b)The quantities s1 and s2 can be measured directly; so one can evaluate ul, u2 oi-fl,f 2 respectively from eqn (13). If u1 and u2 and the diffusion coefficients D are known,one can evaluate from eqn (10) the concentration dependence of the chemicalpotential.FICKIAN DIFFUSION(i) For normal diffusion the Fickian equations are valid. Eqn (8) defines thediffusion coefficient when it is independent of the concentration gradient. Fromthe local mass balance it follows, in absence of convection and chemical reactions,thatIn eqn (14) D may depend on the concentration p1 or p 2 respectively.But thediffusion coefficient is not allowed to depend explicitly on the time t. If D is indepen-dent of the concentration, the second Fickian eqn is valid, i.e.,For the evaluation of D one measures the concentration pi and its dependence ont and z and integrates the partial differential equation, with specific initial andboundary conditions, and obtains close expressions with constant values of D.But in polymer systems, the diffusion coefficient generally depends strongly on theconcentration, especially over a large concentration range. Boltzmann has shownhow one can then proceed for a system that is infinitely large and when D onlydepends on concentration and not explicitly on time34* 35* 38. So, by introduction of anew variable, 11 = z / Jt, one obtains an ordinary differential equation from the partialdifferential eqn (15), with the initial conditions expressed by q.The latter is onlypossible for q = i- 00. The initial conditions are, if the concentration of the solvent p1is considered,11 = +a, = 0 ; dplldv = 0, (1 6 4(1 6b) 11 = -a, P1 = Pol;where pol is the density of the pure solvent.dPIld11 = 0G . REHAGE, 0. ERNST AND J . FUHRMANN 21 1After substitution,-and after integration and resubstitution, the diffusion coefficient D at the concentra-tion p1 is given byOne must know the concentration-distance curve and must evaluate integrals andtangents, as can be seen from eqn (1 8). This graphical method, gives the concentra-tion dependence of the diffusion coefficient.We now have to find criteria for normaldiffusion, i.e., when the Ficks eqn are valid. At u] = 00, apl/dq vanishes, but Dhas a finite value in the whole concentration range; it therefore follows from eqn(17) thatP i =Po1qdp, = O . s P l = oThis is the Matano condition4.(ii) From the relation p1 = p l ( z / J t ) , it is evident that at a given concentrationp1 with diffusion proceeding the space coordinate will change such that Z-J Jt.This is the Jt-law, which is always observed for unperturbed diffusion. It is validalso, e.g., for solvent absorption by a foil in the early stage. For one-dimensionaldiffusion it follows for a constant surface q for the absorbed amount of solvent in ahalf-infinitely extended body thatfa, +oo0 0m1= 41 pldz = 41 Jtdu].(19)This is valid at any time.per surface unit from the formula,From this one obtains the amount of solvent absorbedThe inlegral is a constant, so thatFor swelling of a crosslinked polymer one can characterize the constant of propor-tionality in more detail.5(iii) If one evaluates the concentration-distance diagram at different times, oneshould obtain within the accuracy of the measurements the same curve for the plotIf these criteria are fulfilled, the diffusion is normal. As yet, volume change onmixing has not been considered ; in general it is very small (order of magnitude 1 %)and does not affect appreciably the results for the accuracy of measurements withliquid mixtures.33 The absence of convection is also presumed.This means inpractice that the mean volume velocity equals zer0.lP1 = Pl(Y).*EXPERIMENTALThe diffusion process for a pure solvent and a dilute solution or for two solutions ofdifferent concentrations, always follows a normal course. We performed such measurementson polystyrene (M, = 180 OOO) with a number of solvents in the temperature range 20-60°C.*The special form of these criteria, which must be fulfilled in vitreous solidified systems arepublished el~ewhere.21 2 FICKIAN AND NON-FICKIAN DIFFUSIONThe concentration range was kept as large as possible. The highest concentration, at whichmeasurements could be done yet, was about 40 % polymer. The mass concentration is thenabout 400 g I-'. The measurements were performed in an interferometer, the opticalarrangement being that of Jamin.6 Into the measuring and comparison cells of the diffusioncuvette two solutions differing in concentration about 2 % were placed.From the inter-ferograms one can evaluate at a given initial concentration the plot of concentration at acertain time.The analysis was not made according to the procedure of Matano, which is laboriousand has some disadvantages. Secondly,the Matano plane, which usually is the initial dividing plane, must be known exactly. Forthe two solutions in layers one usually does not get initially a sharp dividing line, so that thefixing of this plane is uncertain.We used a new procedure, based on the existence of the so-called " common points " ofstoke^.^-^ He found empirically that for all systems, for a diffusion coefficient which wasindependent of the concentration or a linear function of it, all concentration-distance curvespass through two points, equidistant from the Matano plane.In fig.1 (C- CII)/CI- CII) is plotted against q* = z / 2 2 / 5 . C is the mol concentrationof one of the components, z the space coordinate, t the time and 5 the diffusion coefficientfor C = (CI+ C11)/2. CI and CII are the initial concentrations of the two solutions. The curvesare valid for different values of &/&I, where DI and DII are the diffusion coefficients at theconcentrations CI and CJI. The coordinates of the common points are according to Stokes :First, drawing a tangent to a curve is not exact.q* = + 0.66 ; (C- CII)/(CI- CII) = 0.176 ;q* = -0.66 ; (C- CII)/(C~- C~I) = 0.824.The existence of the common points has been confirmed by Sauer and Freise for a linearconcentration dependence of the diffusion coefficient.Deviations from the linearity must beconsiderable to cause invalidity.lO- 2.0FIG. l.-(C-C'~~)/(C~--C~~) against r)* at constant D or for a linear concentration dependence of D.z = space coordinate; t , time; A, B, common points; D is the diffusion coefficient at C =(C1+C11)/2. @, DI/DII = 7.11 ; @, DI/DII = 1 ; @, DI/DII = 0.141.For fig. 2, starting from the dividing line, one evaluates the distances Z" and z', at which(C-CI~/(CI-CII) = 0.176 and 0.824 respectively. From the definition of q* and thecoordinates of the " common points " one obtains the mean diffusion coefficient 5 at thepoint (CI+ C11)/2 according to the relation :5 = ~ ' / 0 .6 6 ~ X 4twithz = ( I 2' I + I 2' I )/2.The method avoids the difficulties of fixing the Mataiio plane and the evaluation of thG. REHAGE, 0. ERNST AND J. FUHRMANN 213tangent. Because the method gives the average value, one replaces the real concentration-distance curve by a steplike curve. The agreement is naturally better the smaller thedifferences between the initial concentrations CI and CII.2' 0 2" + zFIG. 2.-Concentration-distance curve (schematically). A and B are the common points.RESULTS AND DISCUSSIONIn the above manner numerous diffusion measurements were 9 nWith good solvents the diffusion coefficient increases strongly with increasing ploymerconcentration and passes through a maximum (see also ref. (28)).In fig. 3 this is2E20nIv, 150 "EuI. 2Q 10X5CFIG. 3.-Concentration dependence of the diffusion coefficient for the system polystyrene+ethyl-benzene at different temperatures. xz = segment mol fraction of the polymer. x: = mz/(nl +nz) ;nl, n2 = mol numbers of solvent and polymer ; r = degree of polymerization214 FICKIAN AND NON-FICKIAN DIFFUSIONshown for the system PST + ethylbenzene. The initial large increase of the diffusion co-efficient is unexpected, because the viscosity also increases considerably with increasingpolymer concentration. Thus, diffusion and viscosity are not reciprocally related,as is usually assumed.The maximum value of the diffusion coefficient is locatedat concentrations at which the solutions are already highly viscous. Measurementswith polystyrene (PST) in a different interferometer, in which the whole concentrationrange from the pure solvent to the pure polymer could be investigated, showed thatD at polymer concentrations of 60-70 % strongly decreased.12 In the glassy rangethe diffusion coefficient decreases to values of about 10-l2 cm2 s-l, while it can reachvalues of order cm2 s-' at the maximum. Thus, there is a change of severaldecades in the same system. The strong decrease at large polymer concentrationsis related to the vitreous solidification. The maximum at medium concentrationsdenotes that the diffusion coefficient, as eqn (10) shows, is not a mere transportcoefficient, but also contains a thermodynamic factor.We evaluated the latter fromosmotic measurements on PST-solutions and separated it from the transportcoefficient, i.e., the mobility.6* l310080PI LY 8v) 60c(OI4rl 0X 3" LO200 0- I 0 *2 0.3x;FIG. 4.-Concentration dependence of the mobility u2 of the polymer component in the systempolystyrene + chlorobenzene.In fig. 4 is given the concentration dependence of the mobility u2 of the polymermolecules in the system PST + chlorobenzene. As expected, the mobility is smaller,the more concentrated the solution. In contrast, the thermodynamic factorp2(i?p2/i?p2) generally increases with increasing polymer concentration. Thecooperative effect of these two factors can lead to extreme values in the diffusion-concentration curve.The reciprocal value of the mobility is the molar frictionCoefficient. The friction coefficient f2 of the polymer component increases witG. REHAGE, 0. ERNST AND J . FUHRMANN 215increasing polynier concentration inore rapidly than linearly. The concentrationdependence of the friction coefficient is similar to that of the viscosity. In simplecases proportionality may exist, as is so for Stokes' law. Thus, there is a directrelation between the viscosity and the friction coefficient (or the reciprocal value ofthe mobility) but not between the viscosity and the diffusion coefficients. The latteris related to the viscosity in a complicated manner through the thermodynamic factor.The relation between the diffusion coefficient and the viscosity is simple if the thermo-dynamic factor equals RT (R = gas constant).From the diffusion coefficients, thethermodynamic factors and the densities we evaluated the sedimentation coefficientsby eqn (12). The concentration dependence of the sedimentation coefficient of thepolymer was similar to that of the mobility. The calculated values were in goodagreement with those measured directly. **3.0!AN 20EWk 40 O C30 "C28OCv25.20 -0.1 0.2~ ~~0 0: I 0.2xz*FIG. 5.-Phase separation curve (phase equilibrium curve) and concentration dependence of thediffusion coefficient for the system polystyrene+cyclohexane.For systems with poor solvents which are inclined to phase separation, the con-centration dependence of the diffusion coefficient is even more complicated.Inthis case one finds at temperatures immediately above the critical temperature adistinct minimum in the diffusion-concentration curve at smaller concentrations.The maximum again appears at medium concentrations. In a binary system thecondition ap2/dp2 = 0 must be valid at a critical phase separation point. Becausethe mobility u2 cannot become infinite, however, according to eqn (10) the diffusioncoefficient has to vanish at the critical point. Diffusion measurements at the criticalpoint are not feasible because of the critical opalescence. But somewhat above thecritical point the diffusion-concentration curve should show a minimum.This isindeed the case, as is shown in fig. 5 for the system PST+cyclohexane216 FICKIAN AND NON-FICKIAN DIFFUSIONThe minima are located near the critical concentration. In the hoinogeneousstate polymer solutions can be considered as binary systems despite the fact thatthere is a distribution function of the polymer component, consequently eqn (10)-(12)are applicable. However, the ‘‘ quasi-binary ” polymer systems have now to beconsidered as multicomponent systems. This problem has been investigated,following the work of Tompa and Stockmayer, especially by Koningsveld andStaverman and by We only state here that there exists not only one, but afamily of phase equilibrium curves. The curve in fig. 5 is the “ closed phase separa-tion curve ” of the system, the maximum of which equals the critical point.The temperature dependence of the diffusion coefficient is usually described bythe relationD = D , exp (-AD/RT), (21)where AD is the activation energy of the diffusion and D, the diffusion coefficient atan intinitely high temperature. But often the temperature dependence is morecomplicated.This is due to the fact that the diffusion coefficient is not a meretransport coefficient. Because the mobility represents solely a transport coefficient,however, one can assume that its temperature dependence can be expressed by therelationu2 = u2, exp (- A,/RT).A, is the activation energy of the mobility and u200 the mobility at an infinitely hightemperature. From eqn (10) and (22) one then obtains the relationship :D = U2a3 exp (-Au/RT)P2aP2/aP2, (23)orIn D = In u2, - (AJRT) + In (p,ap2/ap2).So the temperature dependence of the diffusion coefficient depends not only on theactivation energy A,, but also on the temperature dependence of the thermodynamicfactor.25 For ideal dilute solutions, ideal mixtures and athermal solutions one canwrite eqn (24) in the form :In D = const - (Au/RT) + In T (25)and in this temperature range In T virtually decreases linearly with 1/T.Therefore,a term is added to the activation energy A,, which comes solely from the temperaturedependence of InT. It is about 20-30 % of AD. The diffusion activation energyAD contains this term, as eqn (21) and (25) show. Therefore, this quantity is a“ virtual ” activation energy.According to swelling measurements,2 the systemPST+ethylbenzene is nearly athermal. One would expect therefore a linear plot ofIn D with l/Ty and this has been found e~perimentally.~~If mixing enthalpy and entropy depend strongly on the temperature, as withphase-separating systems, then the temperature dependence of the thermodynamicfactor has a great influence on the activation energy of the diffusion. AD is tempera-ture dependent and In D no more a linear function of 1/T. This has been measuredalso for the system PST + cyclohexane.Fig. 6 and 7 show that the temperature dependence of the diffusion coefficient isvery similar to that of the thermodynamic factor, consequently the temperaturedependence of the thermodynamic factor may be the reason for a temperaturedependence of the activation energy of the diffusionG .REHAGE, 0. ERNST AND J . FUHRMANN30 7r(I v1N E < 2.0-24XIsO-21 7% = 0.15Y2' 0.01I I IFIG. system polystyrene+polymer.1/Tx lo3FIG. 7.-Logarithm of the thermodynamic factor Cz (dpz/dC2) as a function of 1/T for the systempolystyrene + cyclohexane. Cz = mol concentration of PST218 FICKIAN AND NON-FICKIAN DIFFUSIONNON-FICKIAN DIFFUSIONIf the diffusion coefficient at constant temperature depends not only on theconcentration, but also explicitly on time, then the simple Jt-relations are no longervalid. The time dependence of e.g., solvent absorption by, or desorption from afoil, is then more c~mplicated.~* 3 2 9 3 6 * 37 For a detailed investigation of theprocesses,one has to measure the concentration-distance diagrams. We did this for PST+solvent systems, using good and poor solvents.A micro-interferometer wasdeveloped, which works in a similar way to the instrument of Robinson and Crankon the Fabry-Perot-principle. 2* 27-29 We investigated the diffusion of solvents intoglassy PST and measured the concentration-distance curves. Anomalies are causedby the fact that the normal diffusion processes have superimposed on it relaxationprocesses. The relaxation phenomena arise because the polymer after penetrationof the solvent tends to reach a state of internal thermodynamic equilibrium, if theconcentration, which limits the glassy state, is exceeded.29Z / ~ O - ~ cmFIG.8.-Subdivision of the continuous system of the mixed phase between the pure glassy polymer(xp = 0) and the pure solvent ( x r = 1). In the concentration interval xyG1 < x: Q X ~ E , the enlarge-ment of which is shown on the right, the diffusion has superimposed on it the relaxation from theglassy state into a state of internal thermodynamic equilibrium (range II). The neighbouring ranges(I and IU), in which only diffusion processes occur, interact with range 11. z = space coordinate.Fig. 8 shows the concentration profile of the system PST+ toluene at 30°C after50 min diffusion time as a function of the segment mol fraction x: of the solvent.It is useful to distinguish between three domains.30 In range I (glassy mixed phase)in the concentration interval 0 < xT < xTGl the relaxation processes can be neglectedcompared with the diffusion process, so that the diffusion is not influenced by relaxa-tions.In range I1 with xfG,<xf<xTE, both relaxation and diffusion processesoccur at the same time with comparable velocities. At xTGl the molecular motionsof polymer molecules begin, and is the freezing concentration. At this concen-tration all polymer molecules have reached the state of the internal thermodynamicequilibrium. Thus, within range I1 the transition occurs from the glassy state into astate of the internal thermodynamic equilibriumG. REHAGE, 0. ERNST AND J . FUHRMANN 219With glassy, oriented samples at xTGI, the initial change of birefringence occursby rearrangement of the polymer molecules and at xTE the orientation of the moleculesdisappears completely.In range I11 with xTE < xT < 1 the relaxation velocity is highcompared with the diffusion velocity, so that diffusion occurs during internal thermo-dynamic equilibrium. The diffiision is not disturbed by relaxalion. All the parts ofthe system are open, so the entire transport in the whole system is influenced by theprocesses within parts of the system and by the interaction of these parts through theboundaries of these systems. Depending on the quality of the solvents, single partsof the system can become autonomous, i.e., the disturbing influence of the neighbour-ing ranges is negligible. In this case processes occurring in the whole system can bedescribed quantita t i ~ e l y .~I0.3 0-5 0.7 0.9PPST/g/Cm3FIG. 9.-Concentration dependence of the diffusion coefficient D in the system polystyrene+ tolueneat 29.9 "C after different diffusion times: V , 4 min; 0, 16 min; +, 36 min; 0, 50 min. Thefull curve was measured by Rehage and Ernst during internal thermodynamic equilibrium. PPST =mass concentration of the polystyrene.This may be shown with two examples. In the system PST + toluene at 30°C, therange I11 is autonomous. The diffusion in this range is influenced only slightly by therelaxation processes in range 11. The relaxation in range 11 is therefore controlledby diffusion in range 111, i.e., the entire transport process is nearly normal despitethe occurrence of relaxation processes, and the Fickian diffusion laws are approxi-mately valid.In the system PST+toluene the entire transport process occurs, as ifPST were not in the glassy state. Toluene is a good solvent for PST.Fig. 9 shows that the diffusion-concentration curves evaluated at different timesare not greatly different from each other." A curve, obtained fiom measurements* In this case the evaluation was made according to Boltzman and Matano220 FICKIAN AND NON-FICKIAN DIFFUSIONbetween two solutions in a state of internal thermodynamic equilibrium, was notgreatly different from these curves.8*In the system PST+ cyclohexane the behaviour is quite different. Cyclohexaneis a poor solvent. In the temperature range below 30°C separation into two phasesoccurs. It follows from the diffusion measurements that at 7"C, range11 is autonomous,i.e., the diffusion in the range I11 is strongly influenced by the relaxation in range IT.In addition this demonstrated that the concentration profile (i.e., all variables ofstate) are independent of time within range I1 (stationary state).Therefore in thisrange the relaxation velocity is independent of time, i.e., in range I1 a constant amountof polymer relaxes per unit time and this passes through the boundary of range I1into range 111. Therefore the amount of polymer in range I11 increases linearly withthe time, as is shown in fig. 10. This is a typical anomalous behaviour, and it is nolonger possible to evaluate diffusion coefficients.2G100LiL60 I00tlhFIG. 10.-Amount of polymer, which passes per unit area through the boundary between range I1into range 111.The slope of the line gives the stationary relaxation velocity (2x g/cm2 h)for the structural relaxation in the system PST+cyclohexane at 6.9"C. The limits of integration23 and 24 are given by the distance of the boundaries of range TU.These two examples show that subdivision into the three system parts I, I1 andI11 is a useful procedure. In this way it is possible according to the system to separatethe prevailing diffusion from relaxation and the prevailing relaxation from diffusion,if both processes occur at the same time. In other words, depending on the sol-vent, the diffusion can be, despite any superimposed process, normal or anomalous.In general complicated and different anomalous diffusion occurs because of the inter-action of the parts of the open In general, the diffusion seems to be" anomalous " when other time-dependent processes, e.g., relaxation processes, aresuperimposed on the diffusion process.Naturally in this case the Fickian eqn are no longer valid.They have to beextended by means of the thermodynamics of the irreversible processes, if one knowsthe mechanism of s~perirnposing.~G . KEIIAGE, 0. ERNST AND J . FUHRMANN 221R. Haase, Thermodynamik der irreuersiblen Prozesse, (Steinkopff-Verlag, Darmstadt 1963),v. 299 ff.R. Haase, Thermodynamik der Mischphasen, (Springer-Verlag, Berlin-Gottingen-Heidelberg1956), p. 583.C. Boltzmann, Wied. Ann., 1894, 53,959.C.Matano, Japan. J. Phys., 1933, 8, 109.G. Rehage, Symposium uber Makromolekiile in Wiesbaden, 1959, I1 A 15.0. E r s t , Diplomarbeit (Aachen, 1959).0. Ernst, Thesis, (Aachen, 1962).G. Rehage and 0. Ernst, DECHEMA-Monogruphie Band 49, (Verlag Chemie, WeinheimlBergstr., 1964), p. 157.G. Rehage and 0. Ernst, Kolloid-2.2. Polymere, 1964, 21, 64.’ R. H. Stokes, Trans. Faraduy Soc., 1952,48, 887.lo F. Sauer and V. Freise, private communication.I’ J. Fuhrmann, Diplomarbeit, (Aachen, 1963).l3 H. J. Palmen, Thesis, (Aachen, 1965).l4 see ref. (1) p. 182 ; ref. (2), p. 308.l5 H. Tompa, C. R. 2e Reunion Sac. Chim. Phys., (Paris, 1952), p . 163.16H. W. Stockmayer, J. Chem. Phys., 1949,17, 588.I7 G. Rehage, D. Moller and 0. Ernst, Makromol. Chem., 1965, 88, 232.I 8 G. Rehage and D. Moller, J. Polymer Sci. C, 1967, 16, 1787.’O R. Koningsveld and A. J. Staverman, J , Polymer Sci. A-2, 1968, 6,305, 325,349,367, 383. ’’ R. Koningsveld and A. J . Staverman, Kolloid-Z. 2. Polymere, 1967, 218, 114; 220, 31.22 R. Koningsveld, Ado. Colloid Interface Sci., 1968, 2, 151.23 G. Rehage and R. Koningsveld, J. Polymer Sci. By 1968, 6,421.24 G. Rehage and W. Wefers, J. Polymer Sci. A-2, 1968, 6, 1683. ’’ G. Rehage 2. Naturforsch., 1964, 19a, 823.26 G. Rehage, Kolloid-2. Z. Polymere, 1964, 196, 97. ’’ C. Robinson, Proc. Roy. Soc. A, 1950, 204, 339.28 J. Crank and C. Robinson, Proc. Roy. SOC. A, 1951, 204, 549. ’’ J. Fuhrmann, Thesis, (Aachen, 1967).G. Rehage and J. Fuhrmann, 2. phys. Chem., N.F., 1967,56,232.31 J. Fuhrmann and G. Rehage, 2. phys. Chem., N.F., 1969, 67, 291.32 G. Rehage, Kunststoffe, 1963, 53, 605.33 F. Sauer and V. Freise, Ber. Bunsenges. phys. Chem., 1962, 66, 353.34 W. Jost, Diffusion in Solids, Liquids, Gases, (Academic Press, New York, 1960), p. 31.35 J. Crank, The Mathematics of Diflusion, (Clarendon Press, Oxford, 1957), p. 14836 G. S. Park, Diflusion in Polymers, ed. by J. Crank and G. S. Park, (Academic Press, London,37 H. Fujita, Fortschr. Hochpolymeren-Forschung, 1961, 3, 1.38 W. Jost, Diffusion, (Steinkopff-Verlag, Darmstadt, 1957), p. 45.R. Koningsveld, Thesis, (Leiden, 1967).New York, 1958), p. 141