Diffusion with Discontinuous Swelling

Very often a non-solvent diffuses into a glassy polymer with a steep concentration profile proceeding at an almost constant rate v yielding a weight gain proportional to time. Such a diffusion is called type II diffusion in order to distinguish it from the more usual “Fickian” diffusion proceeding without such a constant concentration front and yielding, at least in the beginning, a weight gain proportional to the square root of time. It turns out that the conventional diffusion equation without any special new term but with a diffusion coefficient rapidly increasing with concentration has a series of solutions representing exactly such type II diffusion with v as a completely free parameter which determines the steepness of concentration front. With the usual boundary conditions and infinite medium the diffusion coefficient has to become infinite at the highest penetrant concentration. This case can be considered as an extreme limit which is approached to a high degree in an actual experiment. The finite sample thickness, however, requires only a very large but not an infinite diffusion coefficient. Hence type II diffusion is only a special case of possible diffusion processes compatible with the conventional diffusion equation without any need for new terms if only the diffusion coefficient increases sufficiently fast with penetrant concentration.


. Introduction
In recent yea rs a la rge amount of expe rime ntal [1][2][3][4][5][6][7][8][9][10][11][12] and theore ti ca l [1 3-21] work was done on cl;iffusion of liquids in pol yme r glasses with a lmos t discontinuous s we lling whi c h is now ge ne rally refe rred to as type II diffu sion [2].It is characterized by three basi c conditions: (1) As so lve nt penetrates the polymer, a s harp advanc ing boundary se parates the inner glassy core from the oute r swoll e n and rubbe ry she ll , (2) Behind the solve nt's advancing front, the swolle n gel is in an almost equilibrium s welling s ta te, (3) the boundary between swollen gel and glassy core advances at almost constant rate varying in polystyre ne, de pe nding on tempe rature, penetrant and its activity, between 0.2 and 10 X 10-6 cm/s.
As a consequence the s pecific weight gain pe r unit cross section of th e diffus ing front, W inc reases almost linearly with time as ex pected from the almost constant veloc ity, v, of progress of the swe lling boundary between th e low and high concentration of the pe ne trant.An e ffect of minor importa nce for the diffus ion process itse lf is the partial destruction of polyme r with c raze a nd c rac k formati o n in the swollen region of the polyme r.
As a glassy polymer is coo led the s peed of advanc ing front falls off and a c ritical point is reac hed with ze ro velocity v. Type II diffus ion is re placed by a more or less conventional Fickian diffusion.In this case the polymer has both a glassy shell and glassy core.The swelling by the pe ne trant is not sufficient for the reduc tion of the glass transition point of the swollen material below or to the tempe rature of the experi-I Figures in brackets indicate the literature references a l the e nd of this paper.me nt.According to some a na lyses of the low molec ul a r we ight paraffin diffusion into polys tyre ne, the effec t occ urs at re lative ly hi gh pe ne trant activit y [17] , be tween 0.5 and 1, whil e olde r data on me th yl acetate diffusion into pol y(m e th yl me thac ryla te) a nd o n acetone diffusion in cellul ose nitra te s how the I inear inc rease of W with time mos t c learl y at very low ac tivity [l a].
Since the advanc in g of a s ha rp boundary between low a nd hi gh solvent concentration at a n almost cons tant veloc ity is a most unexpected feature of a diffusion process, a ne w name, type II diffus ion was c reated [2] and the phenome non tried to be expla ined by a new te rm in the diffusion equation depending on the dive rgence of the stress field originating at the boundary be tween the conspic uously swollen and the almost non s wolle n mate rial [15,16].An analysis of the diffusion process in a nonlinear sorption and diffusion range, how ever, shows that the effect is ne ithe r so strange nor so unex pected to need a new name and a new term in the diffusion equation.It follows automatically from the classical formulation of diffusion if the sorption (5) and diffusion (D) coeffic ie nt are strongly dependent on concentration c of diffusant o r its c hemical pote ntial p..
In this pa per it will be demonstrated that the gene ra l diffusion equation in an infinite medium can have a steady state solution with a time indepe nde nt conce ntration profil e progressing uniformly through the polyme r at a con stant velocity.The sole condition for s uc h a solution is that the mobility becomes infinite at th e maximum conce ntrati on Co or maximum che mical potential of the diffusant.Thi s condition is a natural consequence of the geome trical boundari es of th e sample extending from -XJ to +XJ.A constant currentj from the boundary at -XJ can be only sustained if D increases to infinity while (dc/ ax )-oc goes to zero.The more realistic cons ide ration of the finite dime nsions of the sample removes the need for an infinitely large diffusion coefficient but also makes the mathematics a little more complicated than atte mpted in this paper.
For the sake of simplic ity only the linear case will be treated .The swelling boundary has a constant cross-section and proceeds with constant veloc ity in the x -axis direction.
The specific weight gain of the sample pe r unit time, dW/dt , is constant as in the case of type II diffusion if one neglects the trans ie nts at the beginning and the end of the experiment.The transients are of course completely neglected in the cons ideration of the steady state boundary propagation through an infinitely thic k sample exte nding from -XJ to + 00.
In the actual me mbrane the diffusant mobility may increase quite drastically with glass to rubbe r tra nsformation but still remains finite .Yet the changes a re quite substantial , from D = 10-12 cm 2 /s in a glass to 10-6 cm 2 / s in a gel.The above mentioned steady state solution in the infinite medium is therefore the asymptotic approximation to the ac tual diffusion process if the mobility increases by a few orde rs of magnitude while the sorbent concentration inc reases from c = 0 to co.This is indeed the case with a diffusant whi ch transforms the glassy polymer into a rubbe ry gel.The gel must be so much lower in polymer content that the sorbate fl ows through it almost freely and thus eas ily supplies the amount of liquid requested for the swelling at the propagating concent ration front.
Transient from the start at t = 0 with c = 0 throughout the sample to the steady state solution is a combination with the usual type I diffusion with the initial weight gain proportional to the square root of time.During this tra nsition time the steady state of the concentration tail gets established in front of the almost constant concentration profile .The shape of the profile depends on the propagation velocit y [14].The total we ight gain is a sum of two te rms, one proportional to t 112 and one proportional to t [13].The former soon tapers off while the latter remains practi call y unc ha nged.Another transie nt effe c t occurs at the end of the experiment whe n the steadily progressing concentration fronts nearl y meet each other in the cente r of the film if the liquid enters the film from both surfaces.According to Hopfe nberg e t al. [8,9], the spec ific weight gain slightly inc re ases in the case of propagation of npentane into glassy polystyrene just before the diffusion process is compl eted .Such an effect finds a simple explanation in the superposition of the concentration tails in front of the concentration discontinuities as they approach the center plane of the film.He nce the concentration at each point be tween the adva nc ing fronts increases fa ste r than formerly when the fronts were farther apart.As a consequence the critical concentration for the transformati on from glass to gel is reached earlie r.This shows up in an accele ration of s pec ific we ight gain at the end of type II diffusion.
The diffusion equation and the bounda ry conditions do not impose any limitation on v and hence do not determine the ve loc ity of profile propagation .The veloc ity must be conne cted with some independent material pro pe rty.It seems to be a good suggestion that the increase of conce ntration of the penetrant produces by the ensuing membrane swelling a suffic ie ntl y high stress on the polyme r network for the rupture or partial disentanglement of most strained chains .But since the chain rupture or the pulling out of chains is not a n instantaneous process an effect of this type sufficiently large to pe rmit a substantial swelling is only achieved if the stress pers ists for a suffic iently long time .This condition de termines the propagation velocity of the concentration profile .If the condition cannot be met the transport of permeant proceed s in the usual way as type I diffusion , i. e ., without a discontinuous concentration front and with a weight increase whic h is for a long while proportional to the square root of time .

Mathematical Description of Type II Diffusion
All the theore ti cal work up till now is purely descriptive without any serious attempt of explanation.Peterlin [1 3 , 14] based his description on the diffusion and sorbate concentration dependence on time in front of a sorption discontinuity, i.e. , a jump from c * to co , moving through the sample at constant veloc ity v.In this area the Fickia n diffusion equation reads [13] with the diffusion current He rex' = x -vt, D is the constant diffusion coeffi cient, and c* the maximum concentration of liquid in the glass beyond whic h a discontinuous transformation to a gel with concentration Co and so high diffusivity takes place that prac ticall y no driving gradient is needed for the suppl y of liquid to the proceeding discontinuity front.The point x corresponding to a constant x ' moves with the constant velocity v to larger x as does the conce ntration discontinuity.The Fic kian formul ation with the current proportional to the negative gradient of concentration instead of that of chemical potential is full y legitimate because the sorption coeffi cient is assumed constant although much smalle r in front than be hind the concentration discontinuity.
Behind the discontinuity the sorbate concentration is practi call y constant (c o) and hence the curre nt j' in the x ' ,t frame equal to zero.In the laboratory system x, t the diffusion curre nt j is by vc larger than j'.H ence it turns out to be -D fJc/ax in front of discontinuity and vC o behind it.Exac tly speaking the very large value of D in the swollen region permits eve n a higher value of current than vC o in order to supply the liquid needed for the gradual establishment of the steady state concentration tail in front of discontinuity.From these equations one obtains the weight gain pe r unit cross section of the film as fun ction of time (3) for small t and for large t aft e r the concentration profile in front of the discontinuity has reac hed its stationary value.The concentration is measured in g of sorbate per cm 3 of sorbent.
The first term desc ribes the effect of Fic kian diffus ion with cons tant diffus ion coe ffi c ient D in front of the di scontinuity whi c h aft er th e initi a l proportionality to t 1 /2 reach es the constant vaLue c*D /v .The second term describes the weight ga in in the highl y swo llen region behind the discontinuity proportional to th e uniform increase of the volume of the swolle n region as a conseque nce of constant veloc it y v of th e profile propagati on.In these expressions one assumed that the diffus ivity in the gel is so muc h hi ghe r than in the glass that there practically no measurable gradient is needed for the tra nsport of the liquid.A rather similar approac h was used by Crank [22] in hi s desc ription of different cases of diffus ion in so l ids.
In the initial state the coex istence of the sq uare root and linear term in time acco rding to eq (3) yields ove r appreciable time intervals a constant powe r tn relations hip be twee n we ight gain and time, W = Bt III, as can be see n from fi gure 1. where log W is plotted ve rsus log t.At sma ll t one has th e sq ua re root , tn = 0.5 , a nd at hi gh t th e linear, tn = 1, dependence.A rather consta nt tn, i.e., a rather co ns tant s lope, appli es to inte rva ls ex te nding ove r a lmost two decades of time.One ca n guess that within ex pe rimental errors s uc h a theoreti ca l predi cti on may suffi c iently we ll fit the powe r law of we ight ga in observed in so me cases of un conv entiona l so rption [7].

. Bilogarithmic plot of weight gain W over time I according to eq (3).
There is no limita tion about v in eq s 1 to 4 a lthoug h th e time de pe ndence of the concentrations profil e in front of the conc e ntration discontinuity [14] and the asy mptotic s teady state 'value c*D /v depe nd on v.The dime ns ionless time parameter a = (v 2 t/4D)I/2 inc reases and the tota l equilibrium sorption c*D /v in that region dec reases with increasing v.The constant rate progress ion of the conce ntration discontinuity yielding the linear weight gain acco rding to eqs (3) and ( 4) c haracteristic for type II diffusion hence is a distinc t possibility compatible with the classical diffusion equation as long as the supply of the sorbate through the highly swollen section of the film is s uffic ie ntl y hi gh.This flux has to fill continuously with the so lvent a volume increasing in de pth by v per second in spite of the steadil y inc reasing length of the supply route from the out er surface of the film to the s teadily progressing discontinuity.In all practical cases that requires that the diffusion coeffi c ient of the gel (D 2 ) is some ord ers of magnitude higher than in the not swoll en glass (D I)' In the ideal case of a n infinite ex tension of th e film in the direction pe r-pe ndic ular to the front this conditi on a mounts to a n infinite va lue of D2 •  One concl udes that the type II d iffu sion can be suffi c ientl y well app rox imated on the basis of the c lass ical diffusion equation by the limiting case of constant rate propagation of a concentrati on discontinuity.The very beg inning (t = 4Da 2 /v 2 with a ~ 0) with the weight gain proportional to t 1/2 may be ove r in such a short time that it is practi call y unobse rvable.Aft er that a linea r increase of we ight ga in with time is establis hed.In a samp le with a very la rge distance between the borde r whe re th e liq uid ente rs a nd the concentrati on d iscontinuity the weight ga in must show so me inc ipie nt drop as soon as the influence of diffus ion time through the swo ll en region becomes preceptibl e on the so lve nt supply at the conce ntrati on di scontinuity.This does not occur in the above-q uoted ex pe rim e nts by Hopfe nbe rg et a l.[7][8][9] , whe re for a polystyre ne (PS) film of 38/-Lm thi c kness th e diffusion from both su rfaces of the film was completed betw ee n 12 min and 400 hours, de pe nde nt on te mpe rat ur'e, ac ti vit y of npe ntane vapo r, and on polyme r ori e ntati on.The veloc ity of conce ntrati on front pro paga ti on in the sa me cases va ri ed betwee n 1.3 X 10-9 (cast an nea led PS, 25°C, n-penta ne gas ac ti vit y 0.63) and 2.65 X 10-6 c m. s-I (biax ia ll y oriented PS, 35°C, ac ti vit y 1).Since th e maxi mum liquid co ncentration was about 13 g pe r 100 g of polyme r one needs indeed such an extreme ly small grad ie nt for steady supply of so sma ll an amo unt of liquid inc rease pe r unit time that its dec rease would be ha rd to detec t.
He nce the contras t between types I and II diffus ion is one of the conseq ue nces of th e extreme nonid ea lit y of tra nspo rt prope rti es in the latte r case.A di scontinuous s ubsta ntia l increase of sorption a nd diffusion takes place afte r a li miting concentration c* of so rbent is attained in th e film by more or less conventiona l type I diffus ion.The cons ta nt rate propagati on of th e concentration discontinuity represents a pseud ostationary state of diffus ion in such a nonid eal med ium .The veloc ity v, however, is not derivable from the diffus ion equation and boundary conditions a nd must b e determined by some oth e r mechanism of solvent-polymer inte rac tion .
The non-ideality of the medium , i. e., an almost discontinuous increase of diffusion coeffi c ie nt as soon as the pene trant concentration reaches a critical value c* was already suggested by King [l] for explanation of the unconve ntiona l diffusion of alcohol vapor into wool and keratine.Hi s results cannot be directly compared with eq s (1 to 4 ) because he uses an expansion of D in powers of c whic h makes the resu lts more complicated, depe nde nt on the coeffi cients of the powe r ex pans ion .But he actually estimates the e ffec t of rap idl y inc reasing D with conce ntration yielding a steep concentration front propagating with almost constant ve loc ity into the medium.
Frisch et a1.[15,16] assume that the diffusion c urre nt density is caused by the gradient of c hemi ca l potenti a l and the diverge nce of the partial stress tensor of the pe netrant i.e. , if one makes the assumption that the pmtial stress te nsor S of the penetrant at any location and time is proportional to the total uptake of the penetrant up to that location and time.With a constant proportionality fa ctor s one obtains from The " theory" de pe nds on the "natural" assumption formulated in eq (6) that the partial s tress tensor of the liquid S xx in the highl y swolle n section of the sample linearly inc reases with x and he nce goes to infinity in an infinite medium.Since to our know ledge no suc h stress exists one gets the impression that all the naturalness of the above assumption is rather pragmatic based on the success, i.e., on the yie lding of a cons ta nt term in eq (5) one wants to have in order to fit the ex perime ntal da ta.The re is no correlation between any prope rty of the polymer or penetra nt and the coeffi c ie nt s whic h together with B de termines the veloc ity of propaga tion v = Bs of the concentration front.H e nce the derivation of the authors, the introduc tion of the new te rm de pe nding on the s tress gradie nt, a nd the c hoice of stress whic h yields the constant compone nt Bsc of the current can be labelled mathematical d escription but not ex planation of the type II diffu-S IOn .
The differe nce be tween the solution of Frisch e t a!. and that of Pe te rlin is in the s hape of concentration profile and in limiting we ight gain.According to the form e r concept the s hape of concentration front ge ts gradually less steep and finall y becomes pe rfectly fl a t as at the e nd of conventional diffusion.In the latter formulation it rathe r soon reaches a finit e shape which afte r that does not c hange any more .The asymptotic we ight gain is stric tly propOttional to time in the form e r case and still contains a square root te rm in time in the latte r case.
None of the two concepts yields a front propaga tion containing linear and square root terms in time.Both yield a cons tant or almost constant velocity of propagation of conce ntration front if this is taken as the point of fa stest concentra-tion increase, i. e., at c = co/2 (model of Frisch), or at the point where the conce ntration jumps from c * to Co (model of Pete rlin) .H e nce the experime ntal data of Kwe i and Zupko [6] yielding such a combinati on of linear a nd s quare root of time terms cannot be described by any of the two concepts.
A further difficulty for the model of Frisch e t a!.shows up in desorption ex periments [4,8] which correspond to purely Fic kian diffus ion.Peterlin's formulation is not affected because it only describes the diffusion process if a concentration front moves with a constant ve loc ity through the me dium .If the re is none the diffusion is conventiona l.Frisc h's formulation , however, stalts with a la rge S xx which at completed sorption inc reases linearly from 0 at the first boundary, x = 0 , to scc4 at the other boundary, x = d.The gradient as xxi ax would simply continue to pump the liquid th rough the film at the same rate v and in the same direction as during the preceding type II diffusion.There is no end to the process.The way out of this impass is the simultaneous consideration of the identical type II diffusion pumping the liquid at the same rate v and in oppos ite direction from the other boundary as a consequence of S xX<d) starting with value 0 at x = d and inc reasi ng to scc4 at x = o.At the end of sorption the sum of S xX<O)and S xX<d) is constant, scc4, throughout the sample.Its gradient disappears and so does the current corresponding to type II diffusion.This happy end effect has however the di sadva ntage that it yields the same negative result for all previous and later times.The sum of both S xx is a constant as far as x is concerned although it inc reases with time during sorption and decreases during desorption as does the total uptake of pe netrant.H e nce it cannot ge nerate any s tead y flow as observed in type II diffus ion.The modification of S xx in eqs (5) and (6) whic h explains the end of sorption a nd the Fic ki an type of desorption process excludes the expla nation of the unconventional sorption the equations were formulated for.
The formulation according to eq (5) leads to another quite unex pected conseque nce .The coefficient s in the stress te nsor, eq (6), whic h multiplied by B yields the veloci ty of conce ntration front propagation seems to be a cons tant of the penetran t-polymer syste m and hence indepe nden t of c.All the concentration depe ndence is in the factor B which affects equall y the type I (the first te rm) and type II (t he second te rm) diffus ion.H e nce v is proportional to the conve ntional diffus ion coeffi c ie nt.The proportionality fac tor s de pe nds on the penetrant-polymer combination but not on concentration.That leads to the peculiar conseque nce that the normal type of diffusion is not the class ical "Fickian" diffusion but the type II diffusion.
In contrast to that the experiments by Hopfe nberg et a1.[7 , 8] on the de pe ndence of type II diffusion on vapor activity convincingly s how a rapid decrease of v with dec reasing activity a and the comple te cessation of such diffusion below a limiting activity a* -0.3.Hence s cannot be a constant but must be a function of a-a* vanishing at a* and for any a below a*.Although this de pe ndence is the crucial point of explanation of type II diffusion it was never attempted to be de rived from material properties.
Moreover the presence of the term B s in diffusion equation completely c hanges the concentration increase in time ahead of the concentration front as soon as the local concentration c s urpasses the value c* corresponding to the limiting activity a*.This conseque nce may be less disturbing if one assumes that ahead of concentration front c is always smalle r than c* and be hind it larger than or equal to c*.But it still may cause some proble ms in the front itself if the concentration does not jump discontinuously from c I = c* a head of the front to C 2 be hind the front.The variation of propagation ve loc ity with c must c reate a s pec ia l type of front profile whi c h was ne ve r ye t analyzed from this point of vi e w.Wi thout an y more de ta il ed analysis o ne can onl y gues s that a ny concentra tion inc rease in the front will become stee pe r until .it will be almost discontinuous .

Constant Rate Propagation of Concentration Profile
On the basis of the ge ne ral diffusion eq uatio n wi th Sa nd D de pe nde nt on che mical pote ntial one can eas il y formula te the conditions for a consta nt rate propagation of a concentra tion profile The coordina te syste m x / moves to th e ri ght with the same ve loc it y v as the concentration profil e. Hence in thi s syste m the concentra ti on C is consta nt a t eac h x / a nd the c urre nt de nsity j' = O.With these abs umptions the so lutio n of the diffus ion probl e m is trul y conve ntional without a need for a ny additional mo re or less arbitra ry te rm in the diffusion equation.
Accord in g to the the rm od ynami cs of in•eve rsible processes the diffusion c urre nt de ns ity in th e la borato ry fi xed syste m x is proportional to the c he mi cal pote ntial gradie nt of th e diffusant steady s ta te solution a nd th e conditions for a cons ta nt ve loc it y of profil e propagati on.
The the rmod ynamic equilibrium corre la ti on between concentrati on a nd vapo r pressure pe rmits to exp ress th e conce ntration profil e accordin g to eq (9) as a profil e of vapor pressure proceeding with a cons tant ve loc ity v p(x, t) = q(x -VI) (13) because the ma te ri a l pro pe rty S(p) is inde pe nde nt of x a nd t.He nce the v a lu e ~ p(x, t) ca n be ex pressed as fun c ti ons of a single va riable x / = X -vt.Tha t means tha t the parti a l de rivatives in eq (0) can be ex pressed as tota l de ri va ti ves by x /.In suc h a case the diffe re ntiati on on time is eq ui va le nt to diffe re ntiation on locati on aq/at = -v dq/dx ' (14 ) whi c h with consta nt v tra nsforms eq (l0) into the stra ightforwa rdl y integrable total diffe re ntia l equa tio n This formul a ti on ac tu a ll y means tha t the re is ne ithe r a c urre nt nor a c ha nge of concentrati on s ince the coo rdinate syste m x / tra ve ls with th e sa me ve loc it y vas the conce ntrati on or vapo r pressure profile .This yie lds vdx' = -D(q) dq / q (16) j = -(e/J) af..t /ax = -DS ap /ax = -pap /ax ( 10) with the solution whe re f is the fri c tional res is tance yie lding th e diffus ion coeffic ie nt D = RTIf.The c he mi cal po tentia l f..t can be expressed as fun cti on of vapor pressure p(x ,t) in equilibrium with the sorba te at pos ition x at time t f..t = RT tn p(x,t) + f..t0

(ll)
The constant f..t0 is s till a fun ction of T and of a ny othe r parameter indepe ndent of p and x.The concentration c is a produc t of sorption Sip) a nd pressure.The conse rvation of mass yields the concentration variation with time ac/at = a(Sp)/at = -aj/ax = a(pap/ax)/ax.(12) In this well-known class ical one dimensional diffusion equation the sorption S, the diffusion coefficient D and permeability P are fun c tions of p or concentra tion c and not constants as assumed in the ideal Fic kian case .But they a re inde pe ndent of time and locati o n.The membrane is homoge nou s and does not c hange although a t eac h point x its swelling a nd diffusivity vary drasti call y with time.The re is a lso no te mpe rature effect conside red.The drastic c ha nge of sorption in the trans itio n from glass to gel implies quite a substa ntial s we lling of the polyme r causing eventuall y the formati on of mac roscopic c rac ks as observed in the earlie r ex pe rime nts [2 , 3].For simpli city sake, these dime nsiona l c h anges are not at all conside red in the above formulation of the diffus ion equation.Suc h an omission affects the nume rical res ults but not the fun c ti o na l prope rties of the solutions a s for in stance th e ex iste nce of th e J q (x /) -vx/ = D(q)d1n q.
q(O) (17) The bounda ry condi ti on q = p( + :xl) = 0 a t x / = + x is me t a utomatica ll y with a finite , not va ni s hing D at q = o.The finite q = Po at x/ = -00 de mands a n infinite va lue of D(Po).
Eq uation ( 17) is com pl e tely ge ne ra l no t impos ing a ny conditi on on the c hoice a nd continuity of D beyo nd th ose me ntioned in connec tion with th e bou nda ry condition, i. e ., at x/ = -00 .Note that the value of S a nd its de pe nde nce o n p O I' C do not e nte r eqs (16) a nd (17) ex plic itely.One sees that in an infinite medium th e diffusion eq (10) pe rmits solutions with a constant concentration o r th e rmodynamically eq uivalent pressure profil e movi ng with a constant velocity v if only the diffusion coefficie nt goes to infinity at the maximum pressure Po a pplied at th e infinite ly distant negative boundary x / = -:xl of the medium .The profil e q(x /) de pe nds on D(P) and v as s hown in eq (17).For a ve ry s imple de pe nde nce of D on p D = A + Bpo / (Po -p) (18) yielding Do = A + B at p = 0 a nd D = Xl atp = Po , i. e. , a t the maximum pressure of the so rba te at the infinite ly di sta nt film boundary, x / = -Xl, one obtains the profil e en (q(x/) / po) -(B/ Do ) en (1q(x/)/po) shown in figure 2 as function of vx ' /Do = y for different values of B/D o• If the constant is equaled to zero one only displaces the q(x')/po curves horizontally.This does not affect the ir shape which is our primary interest.The larger v and the smaller Do the steeper the true profile if plotted against x' (figure 3).The constant pressure profile q starts at high positive y with a long tail which only s lightly differs from that calculated formerly [14] for a constant Db continues with a sharp rise of pressure and asymptotically approaches the limiting pressure Po at Y = -00.The time independent concentration profile is obtained by multiplication of q by 5ip) which exhibits a large inc rease with p approaching Po if one wishes to d escribe the transition from a glass at p = 0 (5 small) to a highly swollen gel with c = Co at p = Po (5 large).
H ence the concentration profile is ex pected to vary substantially faster than the pressure profile.The s hape of the profile can be varied as freely as the dependence of D and 5 on p.In partic ular, one can choose constant D and 5 in the glass and rubbe r with a discontinuity at p* thus producing a change of slope of P and a step-like concentration profile at p* (figure 4) as already partially treated by CTank [22].Hence the case treated formerly [13,14] with two concentration independent diffusion constants,  9) and (13»).As a conseq ue nce the propagation velocity v has to slow down as soon as x' 0 -vt a pproaches and reaches the outer boundary of the medium by which the penetrant e nters because from that moment on the diffus ion through the gel is unable 10 supply the necessary amount of diffusant.Note that one has ass umed that up to this time the equi li_ hrium pressure a t the entrance to the medium rises continuously and at a constant rate from p* to Po- The boundary between un swollen and swollen regIOn moves with the constant ve loc ity v.The value of v is still complete ly free and is not at all dete rmined by the diffusion equation or the boundary conditions.From the uniform translation of the concentration profile one derives the weight gain w = j(-oo)•t = v5pot = vCot (20) proportional to time in perfect agreement with the observations of the steady state of type II diffusion .
One he nce has the rather unexpected res ult that the same mathe matical formalis m (eq (12)) yields both types of diffusion: Type I for constant or approximately constant and Type II for ex treme ly pressure or concentration depe ndent D, 5, and P. Actually the diffusion constant must be infinite for p = Po in order to satisfy the boundary condition at x' = -00.From a pure mathematical point of view the values of 5 are irrelevant and can be c hosen at will.Type I diffus ion is a very special case confined to ideally linear systems with constant D and 5. Type II diffusion is more or less close to the diffusion in actual polymer-sorbent systems with D and 5 rapidly increasing withp, after the initial transie nt has abated and before the great transport length from the boundary to the moving front starts to reduce perceptibly the ve locity of front propagation.
The diffusion coefficient D of the pene trant or permeability P of an actual medium certainly may become very large but can never assume infinite values.In going from a glass to a swollen rubber or gel the increase in D may be many orders of magnitude, from 10-12 to 10-6 cm 2 /s, so that the steady state One first notices that the concentration c(0) at the oute r boundary has the initial value C*2 which steadil y inc reases with time up to the maximum value Co.If one assumes that during this time the concentrati on front mov es with a constant velocity v one deduces a slight increase of current density and a more than linear inc rease of weight ga in with time The parameter v 2 t/D2 re mains small during the whole ex periment.Its maximum value is reached at the mome nt when the front hits the opposite boundary of the sample or meets the front proceeding from this boundary.In the latter case one derives from film thic kness d and t max = d/2v the maximum value of this parameter (22) In the diffus ion of n-pe ntane into cast annealed PS film [8] the film thic kness was 38 /-Lm = 3.8 X 10-3 cm, t max = 60 hours = 2.16 X l()5s for Po = 550 mmHg at 30°C.From these data one obtains max imum values for the para meter between 1. 7 X 10-4 and 1. 7 X 10-2 if D2 varies be tween 10-7 and 10-9 cm 2 s-\.In biaxially oriented film at penetrant activity 1 and T = 35°C th e values are up to 300 times higher so that a suffic iently small value of the parameter is only obtainable with the higher diffusion coefficient which is indeed more probable than the lower limit.One can be rather certain that in most cases the parameter is so small that the currentj is practically constant, the weight increase Walmost linear with time, and a very small difference c(O) -C*2 required for a constant supply of liquid to the progressing concentration front.The small value of the parameter also tells that the ideal case, eqs ( 9) through (20), is a good approximation of the actual polymer-penetrant systems displaying type II diffusion.
But the finite inc rease in time of penetrant concentration at the sample boundary in contact with the liquid or gas seems to be an important feature of type II diffusion.It is a conseque nce of the fac t that the polymer glass simply cannot expand instantaneously to suc h an extent that the equilibrium concentration of pe netrant co uld be accom modated.Since the polymer is quite inhomogeneous on mol ecu lar scale it exhibits a wide variation of pe netrabilit y in very small regions .The more penetrabl e areas expa nd substantiall y more and faster than the less penetrabl e sec tions .The forces exe rted by the penetrant on taut mol ec ul es connecting the latter sec tions across the form e r regions will e ithe r rupture these c ha ins or pull them out slow ly.Both effects req uire a finite time.As a conseque nce of such e nforced polymer expa nsion the sample crazes and eve n c rac ks.One ge ts convinced that just this relaxa ti on of th e spec ime n under th e os moti c pressure of the pe netrant determines the velocity of propagation of the concentration front which is independe nt of D and of any other parameter of th e diffusion equa ti on.
In a more realistic approach the ina bility of the polymer to expand suffic ientl y for th e acco mmodati on of the eq uilibrium amount of penetrant most lik ely s tarts at a s ubstantially smalle r conce ntration than C*2' let us say c* \.This means that at low ac tivity with e(O) < e* \ the sorption and diffusion are conve ntional without any conce ntration front propagating at constant velocity through the glass.If, however, the ac tivity is so hi gh tha t c(O) > c* \ a finite time is needed for proper polymer expans ion thus c reat ing the c irc umstances observed with type II diffusion.The highe r the driving force c(0) -c* \ as compared with e* I a nd the hi gher th e volume change with the linear expansion of the polyme r the more rapidly the glass adjusts to the s pace req uiremen ts of the penetrant, i.e., the hi ghe r the velocity of concentrati on front propagation in pe rfect agreement with observati ons.
On the other hand , the steady state profile cannot be established instantaneously.Even if a consta nt propagation rate is imposed it tak es a certa in time before the pressure or concentration distribution in the frontal tail assumes time independe nt values.This was explicitly demons trated for a concentration discontinuity moving with a constant velocity [14).The result can be ge ne ralized for any profile .In first approximation the weight gain in this transient is proportional to the squa re root of time.The duration of a substantial contribution of the transient can be unobservably short so that the we ight gain does not exhibit a significant initial compone nt proportional to the square root of time but instead is directly proportional to time.

Conclusions
The steady state solution of the simple diffusion eq ua tion with the constant rate of propagation of the fixed concentration profile according to eq (9) is a good approximation of the pseudo-stationary situation of Type II diffus ion.Its range is between the usual initial trans ie nt with concentration and weight gain dependence on t I /2 and the final stage with the weight gain less than proportional to I because at fixed maximum values of D and P the sorbate has to be transported to the conce ntration front through a steadily increasing de pth of the full y swoll en film.The large ratio between Dmax and D min and the small ma ximum linear exte ns ion f Ir, max of film s through which the mate ri a l transport takes place may make ve ry short th e duration and rather difficult the observation of the initi al and final stages.Therefore, as a rule, the scene is dominated by the pseudo-stationary type II diffusion as formul ated in eq s ( 9) and (19).In the case of thin film with the liquid entering from both plane surfaces the superposition of concentration tails in front of the propagating concentration discontinuity may overcompensate the latter effect and yield a final increase of weight gain rate.
But one also sees that type II diffusion is nothing exceptional requesting any change or modification of diffusion eq uation.It is a s impl e conseque nce of a very rapid change of P , i. e ., of 5 and D with sorbate activity.The inc rease in 5 is only needed for a rapid increase of pe netrant mobility.There is no need for introduc tion of a new mechanism although a new name ma y have some prac ti cal use.The old Fickian formulation in concentration terms is of course rather inadequate for the desc ription of suc h less conv e ntional effects because it is applicable only to perfectly ideal material with activity indepe ndent so rption.But this limitation was already so thoroughly d emon strated [23] that it has no sense to reo pen the subjec t again.
An important result of this investi gation is also the confirmation of the very earl y findin g [1 3] that the veloc ity v of propagation of co ncentration discon tinuity is not in the slightest manner determined or limite d by the diffusion equation and the dependence of D on concentration.Experiments, however, ve ry clearly show a dras ti c increase of v with activity of the permeant and temperature of expe rime nt.There is also a substa ntial depende nce of v on thermodynamic properties of penetrant and polymer and on mechanical and thermal his tory of the polyme r.Hence one will have to consider the molecular effec ts connec ted with swe lling much more thoro ughl y than it was done up till now in order to be able to find the co nnection between the velocity and the mechanical and the rmodynamic properties of the penetrantpolymer syste m.A next pape r will try to discuss some possible approaches to such a molecular theory of unconventional diffusion .

FIGURE 2 ,FIGURE 3 .
FIGURE 2, Pressure profiles according to eq(19) moving to the right with constant velocity v for different values of B/(A + B) as functions of vx' I(A + B).The s mall er the increase of diffusi vily wilhp as measured by BltA + B) the Sleeper is the profile in this representation.

D 4 FIGURE 4 .
FIGURE 4. Pressure and coru:emration profile moving to the right with constant velocity v as furu:tions of vx' ID for a discontinuous change of diffusion constantfrom D, to D2 = 100 D, and of sorptionfrom S, to S2 = 4S, at p = p* = 0.8 fO' Notc the continuity 0 p and the di scontinuity of c at the boundary between glass a nd gel.The constancy of D and S in .heglass (10 the right) and in the gel (to the left) yields p = Po at a finite value x' 0 = D/v instead of at x'o = -Xl as postulated by the steady state solution (eqs (9) and (13»).As a solution with D(po)/D(o) -00 is a very good asymptotic approximation of the actual material transport.But the finite maximum va lue of D imposes a reduction of the permeant supply with inc reas ing trans port le ngth Ltr because the pressure gradient at the outer boundary of the sample, (dp/dx)x=o -(Po -p( etr))/ et , dec reases with this length.That means a slow but steady dec rease of propagation veloc ity of th e concentration profile with increasing distance of the profile from the outer boundary of the sample.In a very thic k sampl e the initially constant velocity propagation of the profile and hence the linear increase of we ight gain with time are expected to show an observable d ecrease .But since the most precise measurements were made on extremely thin films with total liquid path less than 0.1 mm one was never faced with this limitation.A very instructive general picture of the effects caused by the finite although very large diffusion coeffi cient D2 in the highly swolle n materi al ca n be de rived from the very schematic figure4.It is based on a constant D2 , indepe ndent of concentration whi c h varies from C*2 at the concen tration front to the maximum Co in thermodynamic eq uilibrium with th e pe ne tra nt pressure Po at the outer boundary of the sampl e.In order to s implify the matter one ass umes in th at which foll ows that the profile show n in fi gure 4 is established immed iate ly, at time I = 0 , a t the outer boundary, x = 0 , so tha t the initi a l transient effects can be compl etely neglected.At I = 0 one he nce has x' = x -vI = O.
With constan t D and }.L = RT e n c one obtains for diffusion into the half s pace with x ;;,: O.According to thi s solution the steepness of concentration " discontinuity" a t co/2 soon tape rs off in exac tl y the same manner as in conventional diffus ion .The point with c = co/2 proceeds into the me dium at almost constan t veloc ity v .Points with s maller c move fa ster and those with higher c mov e more slowl y so that the maximum slope, located at c = co/2, gradually diminishes and finally becomes zero.The weight gain W /co is initially proportional to t 1/2.Wi th increasing time it approac hes stric t proportionality to t.The assy mptote goes exactly through the origin of W,t coordinate syste m.