ANOMALOUS DIFFUSION OF LIQUIDS IN GLASSY POLYMERS
COREMetadata, citation and similar papers at core.ac.ukProvided by Elsevier - Publisher ConnectorMarhemaricolModelling,Vol. 4, pp. 535-543,Printed in the USA. All nghts reserved.1983Copyright0270-0255183 3.00 .tHl0 1984 Pergamon Press Ltd.ANOMALOUSDIFFUSION OF LIQUIDSIN GLASSY POLYMERSLEVENT YILMAZ,isMAiL TOSUN*,T RKERG RKANDepartment of Chemical EngineeringandULGEN G LCATDepartment of Engineering SciencesMiddle East Technical UniversityAnkara, TurkeyAbstract-Anomalousdiffusion of liquid penetrants in glassy polymers is formulated as amoving boundary problem. The concept of volume average velocity is utilized in mathematical modelling. The model equation is solved using a method which is a combination ofthe method of lines and the method of invariant imbedding. The results are shown to be inagreement with the existing experimental observations.INTRODUCTIONSolid polymers can be in two different states. Below the glass transitiontemperaturethepolymer is in a glassy state; it is a hard and brittle material. The glass transitiontemperatureis the temperatureat which a polymer changes from a glassy material to a softer, flexiblerubbery material. In addition to stiffness, many other properties,such as refractive index,thermal conductivity,specific volume, dielectric constant, diffusivity, etc., differ considerablyfor a polymer in its two different states. The difference of these physical properties in rubberyand glassy polymer states is due to the difference of chain mobility in the two states.All theories of diffision in polymers, in one way or another, are based on chain mobility atthe molecular level. For example, when a polymer absorbs an organic penetrant the polymermolecules rearrange toward a new equilibrium conformation.In rubbery polymers, polymerchain segments are sufficiently active to achieve this structural equilibriuminstantaneouslyandthe diffusion mechanism is Fickian and unique. However, in glassy polymers equilibrium chainconformationis not instantaneousbecause of the limited mobility of the polymer segments. Inthe latter case, different patterns of segmental motion under different conditionsgive rise todifferent diffusion mechanisms.Hence, diffusion mechanisms in glassy polymers are classifiedas follows [l, 21:1. Case I (Fickian) diffusion2. Case II diffusion3. Anomalousdiffusion*Author to whom correspondenceshould be addressed.535
L. YILMAZ, i. TOSUN, T. GUKKAN. AND u. GUL(;AT536Table1. Characteristicsof Case I and Case II diffusionCase II diffusionCase I olsmobilityrelaxationInitial weight gainproportionalto JIInitial weight gainproportionalto fConcentrationvariationacross the swollen shellSwollenuniformLow activationSharp boundary*movementa constant velocityenergyHighrubbery shell in aequilibriumconcentrationactivationwithenergy*When liquid penetrant diffuses into a polymer, a sharp advancing boundaryis observed between the inner glassy region and the outer swollen rubbery layer.General characteristics of the two limiting types of diffusion mechanisms, Case I and CaseII, are summarized in Table 1.The diffusion mechanism in the anomalous case can be considered to be the resultant ofthe sum of the Case I and Case II mechanisms. For this case, experimental observations ofweight gain M and boundary movement s can be expressed as a function of time in thefollowing manner:M k,t”s k2tnOS n l.Oor0.5 - zn 1.0orM a,& b,ts a, J t b,t(1)(4Although existing models [3-71 explain Case I and Case II diffusion mechanisms, they failto predict the behavior in the so-called anomalous diffusion. The purpose of this paper is todevelop a rigorous mathematical model for the anomalous diffusion mechanism and to obtainsolutions to the model equation.DERIVATIONOF THE DIFFUSIONEQUATIONConsider a polymer-penetrant system in which A represents the penetrant and B representsthe polymer. For the penetrant, the conservation of chemical species can be written asaPA7g !T ’ PAZ A rAwhere pa denote the mass density of species A and rA denote the rate of production of speciesA per unit volume by homogeneous chemical reaction. The mass flux of species A with respectto the volume average velocity, co, is represented byj; pA(!?A-!?“ICombining (4) with (3) and considering that r, 0 (i.e., no homogeneous(4)reaction) yields
Anomalousdiffusionof liquidsin glassy polymers(R)537For a binary system, Fick’s first law takes the form [8]1; where D is the diffusivity. Substitution-D\ P,(6)of (6) into (5) gives2 V.pAgo V. (DVp,)(7)Ytlmaz [9] showed that the volume average velocity is either a constant or zero in many liquiddiffusion processes. With this constraint (7) becomes! lie. VP, V. (DVp,)which is the governing differential equation for the diffusion of an organic penetrant into apolymer.PROBLEMSTATEMENTDiffusion takes place into a polymer sheet which is initially penetrant-free. At t 0 thepenetrant starts to diffuse and the surface is maintained at a constant mass concentration pAO.For diffusion of liquid penetrants with good swelling powers, the polymer can stay glassy onlyfor very small concentrations. Therefore, the amount of penetrant in the glassy region isneglected. For a coordinate system shown in Fig. 1, the governing differential equation, (8),together with the initial and boundary conditions take the formapADdt BoundaryConditiona2pA o aPAax2-v-x axpa 0att 0, for all xpa pAOatx 0,PA atxat the MovingPA3 s(t),for t 0for t 0InterfaceWhen (9) is multiplied by dx and integrated from 0 to s(t), the result isl-xFig. 1.(9)(10)(11)(12)
L. YILMAZ.538Applicationof Leibnitz’sThe use of the equationi.Tosu ,T. G RKAN,ANDu. GI L ATrule to the left side of (13) givesfor the conservationof mass in the form-sddtsimplifiesSC0o(14) toDg PACT- u:pAs 0 at x s(t),which is the boundaryconditionat the movingSOLUTIONIntroductionof the dimensionlessOF -PAO' (17)(18)(19)andp !!p(20)sreduces(9), (11)(12) and (16) toa0 a0 ia*8- - -7,aT aq 4aq8 O,at0 1,q Oat q q3i ae1 dqs 1jx j 4 ,atq q,(21)(22)(23)(24)where(25)
Anomalousdiffusionof liquidsin glassypolymers539(R)Hitherto,no known analyticalsolution to (21) with the boundaryconditionsdefined by(22-24) exists. Since the glassy core is assumed to be completely unpenetratedat any instant,a discontinuousjump in the concentrationprofile occurs at the moving boundaryand thiscauses an additionaldifficulty.A method suggested by Meyer [lo, 1l] which is a combinationof the method of lines andthe method of invariant embedding is modified and adopted for the solution of the problem.The method of lines converts a partial differential equation to a set of ordinary differentialequationsby discretizingthe time coordinate.The method of invariantembedding,on theother hand, converts a boundaryvalue problem to an initial value problem without therequirementof shooting. Therefore, when these two methods are combined and applied toa partial differential equation, a set of ordinary differential equations are obtained as initialvalue problems and these can be simultaneouslyintegrated with any one of the well-knownmethods such as the fourth-orderRunge-Kuttamethod.The governingdifferentialequation,(21), is discretizedin time using the backwarddifference0,-e, ,de, dq ;p.AZ1 d28,(26)Meyer [lo, 1 l] states that the time-discretizeddifferential equation can be expressed as a setof two first-order ordinary differentialequations with variable coefficients in the formde, 4k)en ff,(q)w mddq(27)-and(28)where w is a dummy dependentvariable.For the specific problem at hand, (27) and (28) take the formde2 48, 4w,dq---endwen ,& ArNote that (29) and (30) satisfy (26).The Riccati transformationis appliedfunctions(30)AZ’to the dependente wwwhere R and f are dummyin (29) leads to(29variable8 in the form f(d,(31)of the dimensionlessdistance,q. Use of (30) and (3 1)dR !? 4&4dqAZwhich is an identityin terms of the dependent(32)variablew. Hence, the coefficientsof the powers
L.YILMAz, .TosuN,T.G RKAN,540of w must vanishANDij.G L ATto givedR--&-R2 4R 4At(33)and 3-,-/) 4/In orderbecomesto solve the aboveequations,(34)initialconditionse(O) R(O)w(O) f(O)(35) must be valid whateverthe functionalshouldbe known.At q 0, (31) 0.(35)form of w is. This is satisfiedifR(0) 0(36)f(0)(37)and e(0) 0.Substitutionof (29) into the time-discretizedboundaryasw(q) form of (24) gives the conditionA (4Jn- (dn-1“BUse of this resultalongwith the Riccatitransformation,that (33) is in the form of a RiccatiR (38)(31), leads to1-1- 1 f(qJ-equationc2 ewhq)evhq)1/I’-fNote---AZat the moving1 o.and the solution(39)iscl ew(c2q)hIc2)(40)expbq)where(41)Hence, only (34) is solved numericallyusing the fourth-orderRunge-Kuttamethod.Once R and f are determined,the position of the moving boundaryqs, is computedfirst positive root of (39). For this purpose the modified bisection method is used.To calculate the concentrationprofile, (31) is rearrangedin the formh-fW R.as the
Anomalousdiffusionof liquidsin glassy polymers541(R)Use of (43) in (29) yields(44)The initial conditionfor this equationis given by (23). The dimensionlessconcentrationprofile 0, is obtained as a function of the dimensionlessdistance q by the integrationof (44)from qs to 0 using the fourth-orderRunge-Kuttamethod.The weight gain per unit cross-sectionalarea M, can be determined from the integrationof the concentrationprofile over the thickness of the rubbery region asss(l)M 0In terms of the dimensionlessquantities,Padx.(45)(45) takes the form(46)where the dimensionlessweightgain, M*, is definedasVlM* M4CP.40- PAD1(47).The second term on the right side of (46) is numericallycalculatedby the 15pointGauss-Legendrequadrature.Calculationsare carried out by using the Burroughs B-6900 computer. The processing timefor Aq A /20 and 15 time steps is approximatelyfour minutes. Details of the calculationsand the computer program are given elsewhere [9].RESULTSThe magnitudevft/D which in turnSince vz 4 D andpossible to predictvalues of z.The weight gainANDCONCLUSIONSof the dimensionlesstime 7, defined as (v!)‘f/4D, is strongly dependent ondepends on the characteristicsof the specific polymer-penetrantsystem.vz D for Case I and Case II diffusion mechanisms,respectively,it isthe diffusion behavior from the solution of the model equation for differentand the boundaryTable2. Regressionpositionresultsdata are obtainedof dimensionlessM* x10-410-x10-z10-1weightgainfrom numericalfor p 1.0M* a,,/; s
542L. YILMAZ, 1.TOSUN, T. GORKAN, AND U. Gii q rTable3. Regressionresultsof dimensionless1111xxxx10-4-15lo- or B 681.3820.5740.564M* a,& b,,k,n1.0280.8830.8080.9645boundaryand fitted to the expressions represented byM* k,z”orqs kzznorqS a2fi(48) b,,(49)using a nonlinear regression algorithm. The regression results of the weight gain and theboundary position for /3 1 are shown in Tables 2 and 3 respectively.The following observations, for the equation of the form kz”, are deduced from anexamination of Tables 2 and 3:1. The exponent of T is always between 0.5 and 1.0.2. An increase in T causes an increase in exponent n (Note that n 0.5 for Case I diffusion and n 1for Case II diffusion).3. The exponents of T in the weight gain and in the boundaryposition expressions are the same.The above conclusions are in agreement with experimental observations, as can be seen fromTable 4.To determine the effect of j? on the results, calculations are repeated for two more j? valuesand the results are shown in Table 5.As /? decreases, i.e., the concentration at the moving boundary (pAs)approaches the surfaceconcentration (pAO),one can expect the diffusion mechanism to approach Case II and thevalue of the exponent becomes equal to 1.0. The results shown in Table 5 fit theseexpectations.In a practical case, the predictions of the present model can be put into use in the followingmanner: for several polymer-penetrant systems pAsis correlated in terms of the glass transitiontemperature Tg. Thus pas can be predicted from a known value of T,, uz can be obtained fromrate of swelling data, and D from diffusivity data. With the known values of pA,,,pAs, V andTable4. Experimentaldiffusiondatafor olymerMedium crosslinkedepoxyHigh crosslinkedepoxyMedium crosslinkedepoxyMedium crosslinkedepoxyMedium crosslinkedepoxyP.M.M.A.P.M.M.A.Note:N.R.standsfor 5of nFor0.51.01.00.580.661.00.65qrRef.12121212121313
AnomalousTable5. Calculatedvaluesdiffusionof liquidsof the exponentin glassypolymersof r in the equation1111XXXX10 3kr” using regressionanalysisn for M*n for 4,T(R)p 0.2/I 1.0p 5.0ji 0.2p 1.0b 7510.5350.5490.5880.683D one can calculate /I and z for various lengths of processing time. The weightboundaryposition values can then be calculated from (48, 49) using the predicteda, h, k, and n.gain andvalues ofREFERENCESI. T. Alfrey, E. F. Gumee, and W. G. Lloyd, Diffusion in glassy polymers. J. P&n.Sci. (C) 12, 2499261 (1966).2. H. B. Hopfenbergand H. L. Frisch, Transportof organic micromoleculesin amorphouspolymers. J. Polym.Sci. (B) 7, 405409 (1969).3. J. Crank, Mufhematics of Diffusion, 1st ed., ClarendonPress, Oxford (1956).4. H. L. Frisch, T. T. Wang, and T. K. Kwei, Diffusion in glassy polymers II. J. Polym. Sci. (A-2) 7, 879-887(I 969).5. A. Peterlin, Diffusion in a network with discontinuousswelling. J. Polym. Sci. (B) 3, 1083-1087 (1965).6. G. Astarita and G. C. Sarti, A class of mathematicalmodels for sorption of swelling solvents in glassy polymers.Polym. Eng. Sci. 18, 388395(1978).7. N. L. Thomas and A. H. Windle, A theory of case II diffusion. Polymer 23, 529-542 (1982).8. R. B. Bird, W. E. Stewart, and E. N. Lightfoot,Transport Phenomena, Wiley, New York (1960).9. L. Ytlmaz, Model&ofliquiddiffuusion in glassypo/ymers. M. S. Thesis, Middle East Technical University, Ankara(1982).10. G. H. Meyer, Heat transfer during fluidized-bed coating. Int. J. Num. Meth. Eng. 9, 669-678 (1975).11. G. H. Meyer, An alternating direction method for multi-dimensionalparabolic free surface problems. Int. J. Num.Meth. Erzg. 11, 741 (1977).12. T. K. Kwei and H. M. Zupko, Diffusion in glassy polymers I. J. Polym. Sci. (A-2) 7, 867-877 (1969).13. N. L. Thomas and A. H. Windle, Transportof methanol in poly (methylmethacrylate).Polymer 19, 255-265(1978).
LEVENT YILMAZ, isMAiL TOSUN*, T RKER G RKAN Department of Chemical Engineering and ULGEN G LCAT Department of Engineering Sciences Middle East Technical University Ankara, Turkey . matical modelling. The model equation is solved using a method which is a combination of the m
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