13 PolymerDi¡usion 13.1INTRODUCTION Inengineeringpractice,weroutinelyencounterthediffusionofsmallmolecules throughsolidpolymers,thediffusionofpolymermoleculesindiluteorconcen- tratedsolution,andthetransportofmacromoleculesthroughpolymermelts.We cameacrossdiffusionindilutesolutionwhenwedevelopedthetheoryofthe ultracentrifugeinChapter8asamethodofdeterminingpolymermolecular weight. Similarly, solution polymerization involves diffusion in a concentrated solution. The reverse situation of mass transfer of small molecules through polymers has great technological importance. Thus, anisotropic cellulose acetate membranes can be used for desalination of water by reverse osmosis [1], and ethyl cellulose membranes can be used to separate gas mixtures such as air to yield oxygen [2]. Other common situations include the drying of polymeric coatings [3] and the removal of the monomer and other unwanted volatiles from finished polymer by the process of devolatilization [4]. In the field of medicine, polymeric drug delivery systems have become a reality [5]. For example, there is now a commercially available implant for glaucoma therapy consisting of a membrane-controlled reservoir system made from an ethylene–vinyl acetate copolymer [6]. This implant is placed in the lower eyelid’s conjunctival cul-de- sac, and it delivers the drug pilocarpine continuously over a 1-week period; normally, patients would receive eyedrops of this drug four times each day. A few other examples involving diffusion through polymers are biomedical devices such 526 Copyright © 2003 Marcel Dekker, Inc. as blood oxygenators and artificial kidneys. Finally, polymer diffusion in polymer melts is relevant to the self-adhesion of polymer layers. Polyimide layers, for instance, are used as insulators in electronic packaging, and the peel strength of such a bilayer is found to correlate with the interdiffusion distance [7]. A fundamental study of diffusion of and through polymers is clearly necessitated by all the applications just cited. For a rational design of devices employing polymer diffusion, it is necessary to know the mechanism of diffusion and how the rate of diffusion is affected by variables such as temperature, concentration, and molecular weight. Another extremely important variable is polymer physical structure. Because the transport of small molecules through a nonporous polymer is a solution-diffusion process [8], the flux depends on both the solubility and the diffusivity. Because of the small amount of free volume in a glassy polymer, diffusion coefficients are low and decrease rapidly with increas- ing molecular size of the diffusing species. This fact can be used to advantage in separating small molecules of air, for example, from larger organic vapors. Conversely, because solubility increases with increasing condensability, rubbery polymers permit easier transport of larger molecules because these are more condensable [9]. Polymer structure can affect properties in more subtle ways as well. Small changes in crystallinity and polymer chain orientation can alter the diffusion path and adversely affect the dyeability of knitted and woven fabrics and lead to color nonuniformities known as barre [10]. Such mechanistic information is also useful for testing molecular theories of polymer behavior, especially because transport properties such as diffusivity and viscosity are closely inter- related. In this chapter, therefore, we define the various diffusion coefficients, show their relevance, discuss methods of measuring the mutual diffusion coefficient, present typical data, and see how these might be explained by available theories. 13.2 FUNDAMENTALS OF MASS TRANSFER When concentration gradients exist in a multicomponent system, there is a natural tendency for the concentration differences to be reduced and, ultimately, elimi- nated by mass transfer. This is the process of diffusion, and mass transfer occurs by molecular means. Thus, water evaporates from an open dish and increases the humidity of the air. However, the rate of mass transfer can be increased by blowing air past the dish. This is called convective mass transfer or mass transfer due to flow. The basic equation governing the rate of mass transfer of component A in a binary mixture of A and B is (in one dimension) given by J A;z ¼ÀD AB c dx A dz ð13:2:1Þ Polymer Di¡usion 527 Copyright © 2003 Marcel Dekker, Inc. which is often known as Fick’s first law. Here, J A;z is the flux of A in the z direction in units of moles per unit time per unit area, D AB (assumed constant) is the mutual or interdiffusion coefficient, c is the molar concentration of the mixture, and x A is the mole fraction of A. Clearly, dx A =dz is the mole fraction gradient, and diffusion occurs due to the presence of this quantity. Although Eq. (13.2.1) is similar in form to Fourier’s law of heat conduction and Newton’s law of viscosity, the similarity is somewhat superficial. This is because the flux given by Eq. (13.2.1) is not relative to a set of axes that are fixed in space, but are relative to the molar average velocity; that is, J A;z ¼ cx A ðv A;z À V z Þð13:2:2Þ where the molar average velocity is defined as V z ¼ x A v A;z þ x B v B;z ð13:2:3Þ in which v A;z and v B;z are the velocities in the z direction of the two components relative to a fixed coordinate system. Because N A;z , the molar flux of A relative to the fixed axes, is cx A v A;z ,this quantity must also equal the flux of A due to the mixture (molar) average velocity plus the flux of A relative to this average velocity; that is, N A;z ¼ cx A V z À D AB c dx A dz ð13:2:4Þ and we can write similar equations for fluxes in the x and y directions as well. If we choose to work in terms of mass units, the corresponding form of Eq. (13.2.1) is as follows [11]: j A;z ¼ÀD AB r dw A dz ð13:2:5Þ where r is the mixture density and w A is the mass fraction of A. Now the mass flux j A;z is given relative to the mass average velocity, v z ¼ w A v A;z þ w B v B;z ð13:2:6Þ which then leads to the following expression for the mass flux n A;z relative to fixed axes: n A;z ¼ÀD AB r dw A dz þ r A v z ð13:2:7Þ where r A is the mass concentration of A and equals rw A . Note that the diffusion coefficient D AB appearing in Eqs. (13.2.4) and (13.2.7) is the same quantity and that Eq. (13.2.4) can be obtained by dividing both sides of Eq. (13.2.7) by M A ,the molecular weight of A. The binary diffusion coefficient is not the only kind of diffusion coefficient that we can define. If we label some molecules of a pure material and follow their 528 Chapter 13 Copyright © 2003 Marcel Dekker, Inc. motionthroughtheunlabeledmolecules,theforegoingequationswouldstill apply,butthediffusioncoefficientwouldbecalledaself-diffusioncoefficient.A similarexperimentcanbeconductedbylabelingsomemoleculesofone componentinauniformmixtureoftwocomponents.Themotionofthelabeled moleculeswouldgivetheintradiffusioncoefficientofthisspeciesinthemixture [12].Stillotherdiffusioncoefficientscanbedefinedformasstransferin multicomponentsystems[13]. Example13.1:Showthatcx A V z inEq.(13.2.4)canalsobewrittenas x A ðN A;z þN B;z Þ,whereN B;z isthefluxofBrelativetofixedaxes. Solution:Usingthedefinitionofthemolaraveragevelocityandthemole fractiongivesthefollowing: cx A V z ¼cx A ðx A v A;z þx B v B;z Þ¼cx A c A c v A;z þ c B c v B;z ¼x A ðc A v A;z þc B v B;z Þ ¼x A ðN A;z þN B;z Þ Anexaminationoftheforegoingequationsshowsthatweneedthe diffusioncoefficientandtheconcentrationprofilebeforewecandeterminethe fluxofanyspeciesinamixture.If,forthemoment,weassumethatweknowthe interdiffusioncoefficient,weobtaintheconcentrationprofilebysolvingthe differentialmassbalanceforthecomponentwhosefluxisdesired.Thegeneral massbalanceequationitselfcanbederivedinastraightforwardway,asfollows. IfweconsiderthemasstransportofspeciesAthroughtherectangular parallelepipedshowninFigure13.1,thenitisobviousthatthemassof component A inside the parallelepiped at time t þ Dt equals the mass of A that was present at time t plus the mass of A that entered during time interval Dt minus the mass that left during time interval Dt. In mathematical terms, therefore, r A DxDyDzj tþDt ¼ r A DxDyDzj t þ n A;x DyDzDtj x À n A;x DyDzDtj xþDx þ n A;y DxDzDtj y À n A;y DxDzDtj yþDy þ n A;z DxDyDtj z À n A;z DxDyDtj zþDz ð13:2:8Þ Dividing the above equation by DxDyDz Dt, rearranging, and taking limits yields @r A @t þ @ @x n A;x þ @ @y n A;y þ @ @z n A;z ¼ 0 ð13:2:9Þ Polymer Di¡usion 529 Copyright © 2003 Marcel Dekker, Inc. Replacing the flux components with expressions of the type given by Eq. (13.2.7) and noting that the overall mass balance is HH Áv ¼ 0 for incompressible materials [11] we have @r A @t þ v x @r A @x þ v y @r A @y þ v z @r A @z ¼ D AB @ 2 r A @x 2 þ @ 2 r A @y 2 þ @ 2 r A @z 2 ð13:2:10Þ where it has been assumed that both r and D AB are constant. Also, rw A is equal to r A . If there is no bulk fluid motion, the mass average velocity components are zero and @r A @t ¼ D AB H 2 r A ð13:2:11Þ which is often called Fick’s second law. The term H 2 is Laplace’s operator ð@ 2 =@x 2 þ @ 2 =@y 2 þ @ 2 =@z 2 Þ in rectangular coordinates. At steady state, H 2 r A ¼ 0 ð13:2:12Þ and the equivalent forms of Eqs. (13.2.9)–(13.2.12) in mole units are obtained by dividing these equations by the molecular weight of A. Similar expressions in curvilinear coordinates are available in standard textbooks [11,13]. In any given situation of practical interest, the concentration profile is obtained by solving the appropriate form of Eq. (13.2.9). For diffusion through solids and liquids (this is the situation of interest to us here), the flow terms are always small in comparison to the other terms. Consequently, we solve Eq. FIGURE 13.1 Coordinate system used for deriving the mass balance equation. 530 Chapter 13 Copyright © 2003 Marcel Dekker, Inc. (13.2.11)todeterminetheconcentrationprofileifthediffusioncoefficientis known.Alternately,thesolutionofEq.(13.2.11)canbeusedinconjunctionwith experimentaldatatoobtainthemutualdiffusioncoefficient.Becauseanew solutionisgeneratedeachtimeaboundaryconditionischanged,averylarge numberofmethodsexistfortheexperimentaldeterminationofthediffusivity. Someofthesemethodsarediscussedinthenextsection.Notethatifthe diffusivityisnotconstantbutdependsonthemixturecomposition,themass transfersituationistermednon-Fickian;thisisexaminedlaterinthechapterin thediscussionoftheoreticalpredictionsofthemeasureddiffusioncoefficients. 13.3DIFFUSIONCOEFFICIENTMEASUREMENT Inthissection,weexaminesomecommonexperimentaltechniquesformeasur- ingthediffusioncoefficientinliquidsandsolidsunderisothermalconditions. Bothsteady-stateandtransientconditionsareencounteredinthesemethods. EitherEq.(13.2.1)or(13.2.5)isemployedfortheformersituation,whereasEq. (13.2.11)isusedinthelattercase,becausetheflowtermsareeitheridentically zeroornegligible. 13.3.1Di¡usionintheLiquidPhase Thesimplestmeansofobtainingbinarymutualdiffusioncoefficients,especially forliquidmixturesoflow-molecular-weightmaterials,isthroughtheuseofa diaphragmcell,shownschematicallyinFigure13.2.Thismethodwasintroduced originally by Northrop and Anson and consists of two compartments separated by a porous diaphragm made of glass or stainless steel [14]. A concentrated solution is placed in the lower compartment of volume V 2 and a dilute solution is kept in the upper compartment of volume V 1 . Both solutions are mechanically stirred to eliminate concentration gradients within the respective compartments, and diffu- sion is allowed to occur through the channels in the diaphragm. If the initial solute concentration in the upper chamber is c 1 ð0Þ and that in the lower chamber is c 2 ð0Þ, these concentrations will change with time at a rate (assuming quasi- steady-state conditions) given by the following: V 1 dc 1 dt ¼ JðtÞA ð13:3:1Þ V 2 dc 2 dt ¼ÀJ ðtÞA ð13:3:2Þ in which A is the total area of the channels in the diaphragm and J ðtÞ is the solute flux across the diaphragm given by Eq. (13.2.1), and this flux varies with time Polymer Di¡usion 531 Copyright © 2003 Marcel Dekker, Inc. because the concentration gradient varies with time. This time dependence is given by J ðtÞ¼À D 12 ðc 1 À c 2 Þ L ð13:3:3Þ where D 12 is the mutual diffusion coefficient and L is the diaphragm thickness. Combining the three preceding equations gives À d dt lnðc 2 À c 1 Þ¼D 12 A 1 V 1 þ 1 V 2 L À1 ð13:3:4Þ Integrating from t ¼ 0tot gives ln c 2 ð0ÞÀc 1 ð0Þ c 2 À c 1 ¼ D 12 At 1 V 1 þ 1 V 2 L À1 ð13:3:5Þ which allows for the determination of D 12 from experimentally measurable quantities. If the diffusion coefficient varies with concentration, the procedure just illustrated will yield an average value. Details of specific cell designs and operating procedures are available in standard books on the topic [12]. Example 13.2: Northrop and Anson examined the diffusion of HCl in water by means of a diaphragm cell fitted with a porous aluminum membrane; this membrane separated pure water from 0.1 N HCl. Over a period of 30 min, the amount of acid that diffused through the membrane was equivalent to 0:26 cm 3 of 0.1 N HCl. What is the value of the membrane constant L=A? It is known that D 12 equals 2:14 Â10 À5 cm 2 =sec. Assume that c 1 and c 2 remain unchanged over the course of the experiment. FIGURE 13.2 Schematic diagram of the diaphragm cell. 532 Chapter 13 Copyright © 2003 Marcel Dekker, Inc. Solution:UsingEq.(13.3.3)andnotingthatc 1 ¼0,wefindthefollowing: Amountdiffused¼ D 12 c 2 tA L or L A ¼ D 12 c 2 t Amountdiffused ¼ 2:14Â10 À5 Âc 2 Â30Â60 c 2 Â0:26 ¼0:148cm À1 Althoughthediaphragmcellhasbeenusedbysomeresearcherstomeasure diffusioncoefficientsinpolymersolutions,theresultsarelikelytobeinfluenced bythemechanicalstirringofthesolutions,whichcancauseextensionand orientationofthepolymermolecules.Polymermoleculescanalsoadsorbonthe membrane.Asaconsequence,itispreferabletousemethodsthatdonotrequire thefluidtobesubjectedtoanyshearstress.Thiscanbeachievedinfree-diffusion experiments[15]. If,inthesituationshowninFigure13.2,thetwocompartmentswere infinitely long and the barrier separating them infinitely thin, then initially c would equal c 1 ð0Þ for z > 0 and c would equal c 2 ð0Þ for z < 0; here, z is measured from the plane separating the two solutions and taken to be positive in the upward direction. If the barrier were instantly removed at t ¼ 0, there would be interdiffusion and the time dependence of the concentration would be given by the solution of Eq. (13.2.11). This situation is termed free diffusion because the solute concentration remains unchanged at the two ends of the cell. Thus, c equals c 1 ð0Þ at z ¼1and c equals c 2 ð0Þ at z ¼À1for all times. To obtain cðz; tÞ,we use a combination of variables as the new independent variable [16]: x ¼ z ffiffiffiffiffiffiffiffiffiffiffiffi 4D AB t p ð13:3:6Þ so that Eq. (13.2.11) becomes the following, in molar units: d 2 c dx 2 þ 2x dc dx ¼ 0 ð13:3:7Þ subject to cð1Þ ¼ c 1 ð0Þ and cðÀ1Þ ¼ c 2 ð0Þ. The solution to Eq. (13.3.7) is given as follows [17,18]: c À cc c 1 ð0ÞÀ cc ¼ erf z ffiffiffiffiffiffiffiffiffiffiffiffi 4D AB t p ð13:3:8Þ in which cc ¼½c 1 ð0Þþc 2 ð0Þ=2 and erf is the error function, erf x ¼ 2 ffiffiffi p p ð x 0 e Àu 2 du ð13:3:9Þ Polymer Di¡usion 533 Copyright © 2003 Marcel Dekker, Inc. andD AB isobtainedbycomparingthemeasuredconcentrationprofilewithEq. (13.3.8). Example13.3:Inafree-diffusionexperiment,ifc 1 ð0Þ¼0,howdoesthefluxof thespeciesdiffusingacrosstheplanethatinitiallyseparatedthetwosolutions varywithtime? Solution:Becausec 1 ð0Þ¼0,Eq.(13.3.8)becomes c c 2 ð0Þ ¼ 1 2 1Àerf z ffiffiffiffiffiffiffiffiffiffiffiffi 4D AB t p Thefluxacrosstheplaneatz¼0isgivenbyEq.(13.2.1)asfollows: J¼ÀD AB dc dz z¼0 Carryingoutthedifferentiationgives dc dz ¼À c 2 ð0Þ ffiffiffi p p 1 ffiffiffiffiffiffiffiffiffiffiffiffi 4D AB t p expÀ z 2 4D AB t Evaluatingthisexpressionatz¼0andintroducingtheresultintheequationfor thefluxgivesthefollowing: J¼ c 2 ð0Þ 2 ffiffiffiffiffiffiffiffi D AB pt r Alargenumberofopticalmethodsareavailableformeasuringthetime- dependentconcentrationprofilesfordiffusioninsolutions[19];theaccuracyof interferometrictechniquesisverygood;datacanbeobtainedwithaprecisionof 0.1%orbetter[19].Fortheinterdiffusionofpolymermelts,however,thenumber oftechniquesisverylimitedbecausediffusioncoefficientscanbeaslowas 10 À15 cm 2 =sec;thismeansthatthedepthofpenetrationmeasuredfromthe interfaceisverysmalleveniftheexperimentisallowedtorunforseveraldays. Analysismethodsthathavethenecessaryresolutioncapabilityincludeinfrared microdensitometry,[20],forwardrecoilspectrometry,[21],andmarkerdisplace- mentusedinconjunctionwithRutherfordbackscatteringspectrometry[22].Ifthe diffusioncoefficientislarge,wecanalsouseradioactivelabelingandnuclear magneticresonance[20].Clearly,makingmeasurementsofdiffusioncoefficients inpolymermeltsisanontrivialexercise. Athirdpopularmethodofmeasuringdiffusioncoefficientsintheliquid state(especiallyself-diffusioncoefficientsofionsindilutesolution)istheopen- endedcapillaryofAndersonandSaddington[23].Here,asshowninFigure13.3, a capillary of length L, closed at the bottom, is filled with solution of a known and 534 Chapter 13 Copyright © 2003 Marcel Dekker, Inc. uniform concentration, c 0 , and immersed in a large tank of pure solvent. Because of the concentration difference, solute diffuses out of the capillary and into the tank, but the tank concentration remains essentially unchanged at zero. After diffusion has taken place for a few hours to a few days, the capillary is withdrawn and the average solute concentration, cc, determined. This measured quantity is then related to the diffusion coefficient. An expression for cc=c 0 is derived by solving Eq. (13.2.11) by the method of separation of variables [19,24,25] subject to the initial and boundary conditions, c ¼ c 0 for 0 < z < L when t ¼ 0 @c @z ¼ 0 for z ¼ 0 c ¼ 0 for z ¼ L The solution is as follows [24,25]: c ¼ 4c 0 p P 1 n¼0 ðÀ1Þ n 2n þ1 exp À ð2n þ1Þ 2 p 2 4L 2 D AB t ! cos ð2n þ1Þpz 2L ð13:3:10Þ which can be integrated over the length of the capillary to yield cc c 0 ¼ 8 p 2 P 1 n¼0 1 ð2n þ1Þ 2 exp Àp 2 ð2n þ1Þ 2 D AB t 4L 2 ð13:3:11Þ FIGURE 13.3 The open-ended capillary. Polymer Di¡usion 535 Copyright © 2003 Marcel Dekker, Inc. [...]... for diffusion in polymer solutions, polymeric gels, blends of polymer melts, and structured solid polymers 13.7 GAS DIFFUSION IN GLASSY POLYMERS The diffusion behavior of simple gases (i.e., gases above the critical temperature) in glassy polymers is often quite different from the behavior of the same gases in the same polymers but above the polymer glass transition temperature [64] In particular, gas... attractive forces x ¼ ratio of critical volume of solvent to that of polymer segment ^0 V1 ð0Þ ¼ occupied volume per gram of solvent ^0 V2 ð0Þ ¼ occupied volume per gram of polymer ^ VFH ¼ hole free volume per gram of mixture g; D0 ¼ constant Finally, following Bearman [57], Duda et al [58] have shown that when the self-diffusion coefficient of the solvent greatly exceeds that of the polymer (this is generally... use of Stokes’ law requires that the spheres be rigid and the flow regime be one of creeping flow In addition, the size of the spheres has to be large in comparison to the size of the molecules of the suspending liquid Under these conditions, D is typically of the order of 10À5 cm2 =sec Equation (13.4.3) does a fair job of predicting not only the absolute value of D but also the temperature dependence of. .. the merging of two streams of the same polymer (see Chap 15) If polymer molecules in a polymer melt moved in an unhindered manner, we would expect that the self-diffusion coefficient would be given by D¼ kT f ð13:4:6Þ with f replaced by Zz, where n is the number of monomers in the polymer and z is the friction experienced by each monomer (see Chap 14) Because n equals M =M0 , the ratio of polymer to... (13.6.11) is insensitive to polymer molecular weight, and there is, therefore, no influence of polydispersity Furthermore, for semicrystalline polymers above the glass transition temperature, the polymer may be considered to be made up of two phases—one of which has a zero diffusivity [51] Thus, if the volume fraction of the crystalline phase is f, the effective diffusivity of the polymer is Df Finally,... © 2003 Marcel Dekker, Inc Polymer Di¡usion 549 F IGURE 13.11 Illustration of the division of the specific volume of an amorphous polymer (From Vrentas, J S., and J L Duda: ‘‘Molecular Diffusion in Polymer Solutions,’’ AIChE J., vol 25, 1–24 Reproduced with the permission of the American Institute of Chemical Engineers Copyright # 1979 AIChE All rights reserved.) mixture of solute and solvent, the hole... comprehensive work of Kolospiros et al for details [77] 13.9 POLYMER^ POLYMER DIFFUSION The diffusion coefficient relevant for diffusion in polymer melts is the selfdiffusion coefficient [78] It influences the kinetics of mass-transfer-controlled Copyright © 2003 Marcel Dekker, Inc Polymer Di¡usion 561 bulk polymerization reactions, and in injection molding it determines the extent of healing of weld lines,... ð13:6:1Þ where T 0 is the boiling point of the permeate and T is the temperature of interest Furthermore, the diffusivities of a number of gases in a variety of polymers can be found in the literature as a function of temperature [49] These factors make it easy to estimate, using limited experimental data, the diffusional properties of a gas in a given rubbery polymer [50] For organic vapors, however,... more concentrated polymer solutions (where the solution viscosity differs significantly from the solvent viscosity), the bulk or macroscopic viscosity of the surrounding medium is not indicative of the actual flow resistance experienced by the solute This is because a solute sees a local environment of a sea of solvent with polymer molecules serving merely to obstruct the motion of this particle in a minor... predicted from a limited amount of experimental data, because both the diffusivity and permeability obey the Arrhenius relation [28] The process of predicting the permeability is also helped by the observation that, for any given pair of gases, the ratio of the permeabilities is constant, independent of the type of polymer used [29] This is also found to be true for the ratio of the permeability activation . 13 PolymerDi¡usion 13.1INTRODUCTION Inengineeringpractice,weroutinelyencounterthediffusionofsmallmolecules throughsolidpolymers,thediffusionofpolymermoleculesindiluteorconcen- tratedsolution,andthetransportofmacromoleculesthroughpolymermelts.We cameacrossdiffusionindilutesolutionwhenwedevelopedthetheoryofthe ultracentrifugeinChapter8asamethodofdeterminingpolymermolecular weight and Anson and consists of two compartments separated by a porous diaphragm made of glass or stainless steel [14] . A concentrated solution is placed in the lower compartment of volume V 2 and a dilute. properties of polymer films used in packaging applications. A very simple method of accomplishing this is to use the film of interest to seal the mouth of a dish containing a desiccant. The rate of moisture