10.1.1.509.9728

8/18/2019 10.1.1.509.9728

1/8

Mathematical Model of Charge and Density Distributionsin Interfacial Polymerization of Thin Films

V. Freger,1 S. Srebnik2

1

Institutes for Applied Research, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel2Department of Chemical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel

Received 19 November 2001; accepted 13 June 2002Published online 18 February 2003 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/app.11716

ABSTRACT: A general model for interfacial polymeriza-tion is proposed and solved numerically. The model takesinto account diffusion and reaction of monomers, presenceof unreacted functional groups on the growing polymer, andsolubility effects. The formation of a polyamide film in com-posite separation membranes is taken as an example. Theevolution of the concentrations of the polymer and unre-acted moieties are followed explicitly, thus enabling the

calculations of the limiting thickness and the asymmetric

distribution of density and charge in the resulting film. Suchknowledge is important for the prediction of rejection andtransport properties of the film. The effects of reaction ki-netics, monomer concentrations, and hydrodynamic condi-tions on the properties of the film are analyzed, and anumber of analytical correlations are developed. © 2003 WileyPeriodicals, Inc. J Appl Polym Sci 88: 1162–1169, 2003

Key words: membranes; morphology; polyamides; films

INTRODUCTION

Interfacial polymerization (IP) as a method of prepa-ration of thin film composite (TFC) membranes has

been widely used and studied since Cadotte first in-troduced the method.1, 2 Applications of the TFCmembranes include desalination and purification of water, separation of industrial effluents, and wastetreatment. IP is also highly suitable for manufacturing

polymeric films, such as polyamides for packaging orencapsulation.3–7 The general IP procedure used tofabricate TFC membranes is as follows.1, 2 Difunc-tional amines in a support membrane are brought incontact with an organic phase containing trifunctionalacid chlorides. Subsequent fast polymerization resultsin the formation of a rejection layer. With a properchoice of monomers [such as piperazine or m-phenyl-enediamine (MPD) and trimesoylchloride (TMC)] andreaction conditions, a thin and dense layer is quicklyformed that creates a barrier to further polymeriza-tion.

It has been shown that the formation of a thin filmoccurs very quickly and often results in an asymmetricdensity distribution. Based on small-angle X-ray scat-tering (SAXS) results, Sundet8 suggested the followingscheme. An extremely fast reaction between aminesand trifunctional acid chlorides occurs just within theorganic phase and forms a highly branched network.

Polymerization proceeds to high molecular weights ata thin reaction zone where the amines interact withthe acids. It is further believed that the reaction islikely to resemble the diffusion-limited aggregation(DLA) process,9 and first proceeds through the forma-tion of functional colloidal particles that subsequentlyaggregate to a lower density structure.8

An important feature of the polyamide films is the

presence of unreacted charged moieties (carboxylicand amine groups), which are believed to play a cru-cial role in the rejection and resistance to fouling.However, little is known about the actual content anddistribution of these groups across the film, or themechanism that leads to the asymmetric morphologyof the thin film. It would be informative to know theeffects of variables such as reactant concentration, dif-fusion, and reaction rates on the final film morphol-ogy.

Several efforts have focused on modeling the kinet-ics of film growth. Enkelman and Wegner10, 11 consid-ered the effects of diffusion of amine and waterthrough the film and the competition between theamidation and hydrolysis reactions. This model, un-like the later models, was capable of predicting thelimiting thickness due to the hydrolysis reaction andexplicitly introduced asymmetry to the film. However,the kinetic equation for this model poorly predictedthe film growth, particularly at initial stages of theprocess. Janssen and te Nijenhuis5, 6 modeled thegrowth of a thin encapsulating film assuming that theamines diffuse through the film and then react instan-taneously with the acid chloride. This model predictedan unlimited growth of the film thickness scaling as

Correspondence to: V. Freger ([email protected]) orS. Srebnik ([email protected]).

Journal of Applied Polymer Science, Vol. 88, 1162–1169 (2003)© 2003 Wiley Periodicals, Inc.

8/18/2019 10.1.1.509.9728

2/8

square root of time. Ji et al.12 generalized this ap-proach by introducing a finite reaction zone near theorganic face of the growing film and limiting theamount of amine in the aqueous phase, which resultedin a finite film thickness at infinite time. However, thisapproach also implies that an isotropic film forms and

does not account for the concentration distributions of the polymer and unreacted functional groups withinthe film that lead to pore and charge asymmetries.Thus, the information obtained through this model isclearly insuf ficient for characterizing and quantifyingthe rejection properties of the film.

Another common drawback of the models just de-scribed is the questionable assumption that the film (of zero thickness) already exists at the interface separat-ing the phases where the respective monomers areuniformly distributed. In this respect, the model of Mikos and Kiparissides13 offered a significant im-provement by considering the reaction, diffusion, andsolution thermodynamics explicitly from the very firstmoment. However, the system and reactions consid-ered by Mikos and Kiparissides were very differentfrom our system, so their conclusions cannot fullyapply to our system.

We developed a simple model for the formation of asymmetric thin films that explicitly considers the re-action and diffusion throughout the IP process. Itsessential feature is the explicit account of the concen-tration of unreacted functional groups within the film.Comparison of our results with experimental observa-tions and conjectures show qualitative agreement.

MODEL DEVELOPMENT

We present a mathematical model for the formation of a thin-film membrane under nonsteady-state condi-tions. The two-dimensional surface of the film is as-sumed infinite (practically, it is determined by theavailable surface of the amine phase). Thus, we areonly concerned with the profiles within the thicknessof the thin film. The model is general and can beapplied to many interfacial polymerization reactionsthat conform to the model formulation described later.However, it is especially relevant to the formation of

polyamide desalination and separation membraneswhere the presence of functional groups within themembrane plays an important role in the separationprocess.2 A schematic of the problem is shown inFigure 1.

We assume a constant concentration for the bifunc-tional reactant A (amines in support material) at the

boundary of x 0. This concentration is, in effect, theratio of the concentration of amine in the aqueousphase and its partition coef ficient. The data presented

by Morgan3 suggest that for the pertinent diaminesand solvents, the latter should be significantly 1,allowing us to disregard the amine diffusion in the

aqueous phase. By disregarding the amine diffusion,we can justify the assumption of a constant N A at x 0, where N A is the local amine concentration in theorganic phase.

Initially, the concentration N B of the trifunctionalreactant B (acid chloride) is assumed to be uniformthroughout the organic phase. It is known that theforming polyamide in its pure state is almost imper-meable to the acids because of the large difference inpolarity and thus very low solubility. Furthermore,once the polymer has been formed, the reaction islimited by diffusion of the amines through the poly-mer.3–5, 10 Therefore, we approximate the activity co-ef ficient of B in the polymer-containing organic sol-vent as B (1 N C )

1, where N C is the volume

fraction occupied by the polymer and is inaccessiblefor B. We assume the activity coef ficient of amine(equal to unity) to be unaffected by the polymer. Thissimplification is likely to incorrectly predict the per-meability of the developed film to amine, which willaffect the rate of growth of the film at late stages, butshould not change any of our principal conclusionsabout the resulting film structure.

Several works have considered the competing reac-tion of the amines with acid (HCl) resulting fromamidation and hydrolysis of B with water carried byamines.6,11,12 However, the heterogeneous hydrolysisof the relevant acid chlorides appears to be very slow

due to their low solubility in water.3

The presence of significant amounts of water in the organic phase ispossible only after accumulation of rather large quan-tities of polymer. As our interest is mostly directed tothe stages of incipience and early growth of the film, itis reasonable to neglect these additional reactions andassume the effective amine concentration in the ab-sence of the acid acceptor to be reduced accordingly.

Finally, we assume that the polymer and aggregatesof all sizes (included in the polymer concentration N C)are immobile; that is, have zero diffusivity. This as-sumption is certainly unrealistic for small oligomers,yet the error should be significant only at the very

Figure 1 Schematic depiction of thin film growth by inter-facial polymerization . Reactants A (aqueous phase) and B(organic phase) react to form a polymer film C .

MATHEMATICAL MODEL FOR INTERFACIAL POLYMERIZATION 1163

8/18/2019 10.1.1.509.9728

3/8

early stages of film growth because the relative frac-tion of the oligomer population rapidly decreases withincreasing polymerization.

We follow the volume fraction profiles of five vari-ables within the film: N A, N B, N C, nA, and nB. The lattertwo are unreacted end groups of the growing polymer

having amine ( A) and acid chloride (B) functions,respectively. For consistency, all five variables are ex-pressed as molar concentrations (mol/m3) multiplied

by the molar volume M 104 m3/mol, which isassumed to be the same for the amine, acid chloride,and their respective monomer units. This approxima-tion is well justified for MPD/TMC and piperazine/TMC pairs. For N A, N B, and N C, these units are equiv-alent to volume fractions. The appearance and disap-pearance of these variables within the thin film as thepolymerization reaction proceeds obey the followinggeneral equations:

N At

x DAN Ax 2kN AnB 3N B (1)

N Bt

x DBN Bx

N B1N C

N Cx 3kN BnA 2N A

(2)

N Ct 2kN AnB 3N B 3k nA 2N AN B (3)

nA

t

2kN AnB

3N B

3knAN B

k nAnB (4)

nBt 6k nA 2N AN B 2kN AnB k nAnB (5)

where DA and DB are the diffusion coef ficients of reactants A and B, respectively, which depend onpolymer concentration. Their functional form is dis-cussed below. The parameter k is the reaction rateconstant of all amine– chloride group– group reac-tions. The intrapolymer (cross-linking) reaction rateconstant is k ’. For the concentrations to be expressed as

volume fractions, the values of k and k ’ in m3

/(mol s)should be divided by M to be used in eqs. 1–5. Equa-tions 1 and 2 describe the diffusion of A and B throughthe membrane and subsequent reaction with B and Afunctional groups. Equation 3 describes the formationof the polyamide film. Equations 4 and 5 describe thekinetic appearance and disappearance of functionalgroups on the formed aggregates. The integer coef fi-cients preceding the kinetic terms correspond to thenumber of functional groups per reactant (2 for A and3 for B). For instance, the coef ficient “6” in eq. 5 is theproduct of f B 3 (the number of reacting groups in B)and f B 1 2 (the number of fixed acid groups added

upon reaction). Equations 1–5 are subject to the fol-lowing boundary conditions:

N Ax0 N A0 (6a)

N BxL N B0 (6b)

and the following initial conditions:

0 x L: N Ct0 nAt0 nBt0 0 (7a)

0 x L: N At0 0 (7b)

0 x L: N Bt0 N B0 (7c)

Equations 6 and 7 amount to assuming that the reac-tion takes place inside a unidimensional “ box” of fi-nite thickness L attached to the interface at the organic

side, with concentrations of the monomers fixed at the boundaries. The validity of this assumption is dis-cussed later.

Naturally, diffusion of the reactants will decrease asthe volume fraction occupied by the formed polymerincreases because of the geometric constraints im-posed by the polymer. The “effective” diffusion coef-ficients of low molecular species in polymer gels andsolutions have been shown to be satisfactorily de-scribed by the following relation:6,14,15

DN C D01 N C (8)

where D0 is the effective diffusion coef ficient of thereactant in the pure solvent. Because the molecularsizes of the monomers under consideration (MPD orpiperazine and TMC) are similar, we assumed equalD0 values for both monomers (10

5 cm2/s). Variousworks have suggested that the exponent is between1 and 3, depending on the polymer and solute.6, 14 Theparameter , introduced here, is slightly smaller thanunity and allows for finite solute diffusivity in thepolymer.15 The value 0.99 used in the presentwork gives an amine diffusivity in the polyamideclose to that reported by Ji et al.12

The diffusivities in eq. 8 are, in effect, self-diffusivi-ties. These values replace regular diffusivities if thechemical potential gradient, RT ln( N ), ratherthan the concentration gradient, N , is to be used asthe driving force.16 This approach is essential for the Bmonomer, which, as a result of the effect of B dis-cussed earlier, can move against its own concentrationgradient while being excluded from the film.

The rate constant k is known to be between 102 and106 L/(mols) for the reactions between acid chloridesand amines employed in IP synthesis,3, 17 and shouldnot be lower for monomers of higher functionality. Wetherefore primarily used a relatively high value

1164 FREGER AND SREBNIK

8/18/2019 10.1.1.509.9728

4/8

[105 L/(mol s), divided by M if volume fractions areto be used for concentrations]. In general, a number of different rate constants should be considered as thereaction proceeds because the reactivity of the groupsin multifunctional monomers is known to decrease(typically by factors of 1.5–5)17, 18 after some groups

have reacted. Nevertheless, the difference is onlymoderate, and we therefore neglect this complication

by assuming that the reaction is significantly sup-pressed only when both reacting groups belong to apolymer moiety. In the latter case, the mobility and,thus, the probability of contact is severely restrictedfor both reacting species; that is, for the cross-linkingreaction, the reaction rate constant k was generallytaken to be 10 –100 times lower than k .

Normally, the film formation is considered an IPprocess without stirring.1–3, 19 For such conditions, theconcentration of B in the organic phase in our modelshould be fixed at infinite distance. As will be shownlater, the dense film emerges when the reaction zonereaches the location where the amine and acid chlo-ride flows become stoichiometrically equal. It may beshown16, 20 that in the case of an infinite box, thiscondition can never be met. Indeed, assuming thereaction zone to be suf ficiently narrow, which is thecase for the fast reactions considered here, the amineand chloride fluxes will both decrease as (time)1/2 or1/X , where X is the distance from the interface to thereaction zone, and we arrive at the paradoxical con-clusion that the fluxes will never equalize and the filmwill never form.

In contrast, in a finite box, such a location alwaysexists. When a dense film emerges, uniform profiles of the monomers are quickly established, with most of the resistance falling inside the film and further reac-tion proceeding independent of the box size. When thesystem is stirred, the boundary layer constitutes afinite box. During the formation of a membrane on anamine-soaked porous support, relative motion be-tween the organic and aqueous phases is inevitablycreated for a time that might be long enough to initiatethe irreversible film formation. The boundary layerthickness is roughly given by L ( y/U )1/2 Pr1/3,where is the kinematic viscosity of the organic phase,

y is the distance from the leading edge of the sample,U is the relative velocity, and Pr /D0.

21 Taking 106 m2/s, Pr 103 and ymax/U 1 s (this isapproximately the time taken to fully immerse thesupport ) and averaging over y, we estimate the boxsize to be of the order of 50 m . Moreover, as is wellknown by electrochemists,22 experiments on solidelectrodes in quiescent solutions cannot last more than10 –50 s because of weak natural convection. There-fore, even in an “unstirred” solution adjacent to a flatsolid surface there exists a “ boundary layer” that is(10 109)0.5 m 100 m thick. Obviously, thisnumber should be substantially lower for a liquid–

liquid interface. Therefore, most of the simulationswere performed using a 50-m box, but in an addi-tional series of simulations, the box size was taken asa parameter and varied to assess the possible effect of

hydrodynamics on the resulting film.

RESULTS AND DISCUSSION

In Figure 2 we present numerical results from eqs. 1– 8for the progression in time of the concentration pro-files of the reactants (Figure 2a), total polymer (Figure2b), and unreacted functional groups in the polymer(Figure 2c) obtained numerically for box size L 50m. Within seconds, nA, nB, and N C emerge. A thinfilm develops quickly 2.5 m within the organicphase. For the particular set of parameters shown, thefinal polymer concentration profile, which nearly fully

Figure 2 Concentration profiles of (a) monomers (amine,dashed line; acid, solid line), (b) polymer, and (c) unchargedgroups (amine, dashed line; acid, solid line). The numbersdenote the reaction time in seconds. [k 100 m3/(mols),k /k 0.01, N A0 10

4, N B0 103, L 50 m].


8/18/2019 10.1.1.509.9728

5/8

develops in 20 s, has an asymmetric unimodal curvewith a tail that extends towards the organic phase.This tail is likely to cause the well-known rough mor-

phology of the surface of the TFC composites.

23

We can see from Figure 2c that an even more prom-inent asymmetry develops in the distribution of thecharged groups. The minor peculiar features (e.g.,small local maxima in the profiles) apparently reflectsome narrowing of the reaction zone as a result of thepolymer formation so that unreacted groups formedat the earliest stage could not be later reached by theopposite monomer. As mentioned earlier, informationon the fixed charges is highly important in connectionto the ion rejection by the membrane. Due to thedifferent solubility of the monomers in the polymer-rich solvent, the acid groups occupy a relatively thin

layer at the organic face of the film, whereas the aminogroups dominate throughout the rest of the film.

This qualitative behavior of the charge distributionis preserved for larger ratios of the rate constants, k /k ,as seen from Figure 3a. However, as expected, highercrosslinking reaction rates decrease the content of thecharged groups in the membrane. On the other hand,Figure 3b shows that the thickness in this case is larger

because the monomer addition is slower due to thecompeting crosslinking reaction, which is equivalentto effective reduction of k and widening of the reactionzone (vide infra). Recently, Schaep and Vande-casteele23 determined the fixed charge content of some

nanofiltration TFC membranes. For the NF-40 poly-amide composite, they obtained surface charges of theorder 106 and 105 mol/m2 for positive and negativegroups, respectively. For the thickness of the skin of the order 107 m, these concentrations corresponds tonA 10

3 and nB 102 in the units of Figures 2c and

3a, which provides some justification for the choice of k /k between 10 and 100.

Interestingly, the observed charge distributions ex-plain the contradiction encountered while trying todetermine the charge of TFC polyamide membranesusing various experimental methods. These mem-

branes are often found to be negatively charged basedon the results obtained using the streaming potential(SP) technique.24 This finding is also supported byX-ray photoelectron spectroscopy (XPS or ESCA) dataindicating that nB for TMC/MPD composites is 13%of N C.

24 Both methods, however, reflect the very thinoutermost layer of the skin (1 nm) and are in fullagreement with our results that indicate a high nega-tive charge for that layer of the skin. However, nuclearmagnetic resonance (NMR)25, 26 or Fourier transforminfrared (FTIR) spectra,27 which give a picture aver-aged over the whole skin, show very few or no acidgroups. More importantly, the selectivity of the mem-

brane toward ions of various charge indicates that themembrane charge is very small.28 It has been sug-gested recently that the actual selectivity mechanismin such membranes might be the dielectric exclusionand not the Donnan exclusion as has been long be-lieved (most clearly it is seen for TFC nanofiltration

membranes).

29

Our results show that the averagecharge of the skin that determines the selectivity to-wards ions of different charge can be quite low and,on average, the skin might indeed behave just as adielectric medium, with the charge repulsion being of secondary importance.

We attempted to obtain direct visualization of thecharge distribution in the polyamide layer of compos-ite nanofiltration membranes using TEM combinedwith amine- and acid-selective staining. A cross-sec-tion of a membrane treated with uranyl salt (acid-selective stain) is shown in Figure 4.30 In agreementwith the calculated profiles, it clearly shows a thin and

dark acid-rich layer separated from the dark support by a bright amine-dominated region.

The time evolution of the film thickness is shown inFigure 5. We define the thickness, , as the film widthwhere N C is 0.2, which roughly corresponds to thepercolation threshold for many continuous systems.Because cannot be determined for early times, in thisparticular plot we used the thickness M, which wasmeasured as the total volume of the polymer formedper unit interface area (M N Cdx) and would beequivalent to the thickness of the polymer if it allprecipitated onto the interface. For a developed film,M is well correlated with but changes in a continu-

Figure 3 Concentration profiles of (a) unreacted groups(amine, dashed line; acid, solid line) and (b) polymer for k /k 0.01 and 0.1, for reaction time 200 s [k 100 m3/(mols),N A0 10

4, N B0 103].


8/18/2019 10.1.1.509.9728

6/8

ous manner for all times. The thickness–time curveshown in Figure 5 differs significantly from the squareroot law derived in the earlier models. The initialincrease is fast and linear and is followed by aslow growth phase (plateau). Only the latter, slowgrowth follows the mechanism proposed by earliermodels.5,6,10,12 Inspection of the curves presented byEnkelman and Wegner10,11 and Ji et al.12and the lightreflection data of Chai and Krantz19 shows that our

results might provide a much better fit to the experi-mental results, particularly concerning the early stagesof the process.

The effect of the reaction rate constant on the thick-ness of the film is shown Figure 6a. The simulationssuggest that scales approximately as k 1/3. A simplescaling analysis also predicts this behavior. Assumingthat the film thickness is determined by the width of the reaction zone, (this width is determined by the

condition that the diffusion of A and B to the zone andthe rate of disappearance due to reaction are stoichio-metrically related when the polymer peak emerges),then the average concentration of the amine groups of A in the reaction zone is roughly N A* f A. Similarly,N B* f B J B/D0 f A J A/D0, where f A and f B are thefunctionalities of A and B, respectively. The rate of

group consumption by the reaction per unit interfacearea may then be formulated as f A J A kN A*N B* k ( f A J A/D0)

2. By noting that J A D0N A0/X and J B

D0N B0/(L X ), we find that, in agreement with sim-ulations, X LN A0 f A/(N A0 f A N B0 f B), then f A J A D0(N A0 f A N B0 f B)/L and, finally,

D0L/k f AN A0 f BN B01/3 (9)

where L is the simulated box size or film thickness.The –k dependence predicted by our simulations is inagreement with eq. 9. Equation 9 also predicts thedependence of the final thickness on the concentrationof monomers and on the box size (hydrodynamic con-ditions). The variation of the thickness with the mono-mer concentrations for k 102 m3/(mol s), k /k

Figure 4 A TEM image of the cross-section of an uranyl-stained NF-270 membrane (Dow-Filmtec).

Figure 5 Film thickness (M) as a function of time for k 100 m3/(mols), k /k 0.01, N A0 10

4, N B0 103.

Figure 6 Film thickness () at plateau onset as a function of (a) reaction rate constant k and (b) initial monomer concen-trations in respective phases, one (A or B) fixed at the value

shown in the legend and the other varied.


8/18/2019 10.1.1.509.9728

7/8

0.01, and L 50 m is shown in Figure 6b. Theresults are presented as at the plateau onset versusN 0 N A0 3/2N B0. The results are in excellent agree-ment with eq. 9, where the only degree of freedom isthe proportionality constant (0.75).

Equation 9 suggests that the film thickness should

decrease with monomer concentration. This conclu-sion contradicts the results of Enkelman and Weg-ner10, 11 and of Janssen and te Nijenhuis6 who, inagreement with their models, observed a square rootdependence of the thickness on concentration. Appar-ently, the film growth in those cases was determined

by the slow stage following the plateau formation,presumably due to the higher permeability of the filmto amine and/or higher hydrophobicity of the ali-phatic monomers used that could effectively preventhydrolysis. In contrast, the tendency observed by Chaiand Krantz19 for the MPD/TMC system is in qualita-tive agreement with eq. 9. In this case, the thickness of the denser and more hydrophilic MPD/TMC films isprobably governed primarily by the earlier stages of the IP process described by the present model, afterwhich the reaction is effectively terminated by hydro-lysis and/or very slow diffusion of amine in the film.

Finally, in Figure 7 we present the effect of thehydrodynamic conditions (i.e., box size) on the filmthickness and charge distribution. Clearly, a largerreaction zone allows for a thicker film to form, as isseen in Figure 7a. This process, however, requires alonger time period to reach the plateau region. Thesimulation results seem to predict a somewhat stron-

ger dependence of on box size (Figure 7b) than eq. 9at short time periods (200 s), which decreases, how-ever, at longer periods of time. It is worth noting that,despite our a priori assumption that 50 m seems to bea plausible value for the box size, the time and thick-ness in the actual IP process employed in fabricatingthe TFC membranes1, 2 corresponds much closer to a5-m box. Whether such a small box is indeed created

by the hydrodynamic conditions in the process orthere are other effects present is still a question to beanswered.

CONCLUSIONS

We present a mathematical model for interfacial po-lymerization that explains phenomena observed inpolyamide membranes. Our model accounts for thecharge and density distributions within the film,which agree with and clarify experimental observa-tions. Film growth is predicted using independentlymeasurable quantities, such as reactant diffusivitiesand reaction rate constants. Parameters that cannot bemeasured directly (such as permeability) are moreimportant for late stages of film growth and haveminor impact on our results. Our model better de-scribes the kinetics of early stages of film growth than

the other models previously proposed. However, twoshortcomings of our model are its strong dependenceon box size and the unlimited growth with time of thefilm. The latter could probably be controlled throughinclusion of a competing hydrolysis reaction of theamines.

10, 11

Finally, a more quantitative model shouldinclude the diffusion of oligomers, which might leadto their adsorption onto the interface, thus reducingthe dependence of the model on box size and resultingin faster termination of the reaction, thinner film, anda shift of its position closer to the interface.

References

1. Cadotte, J.E. U.S. Pat. 4, 277, 344 (July 7, 1981).2. Petersen, R.J. J Membr Sci 1993, 83, 81.3. Morgan, P.W. Condensation Polymers: By Interfacial and Solu-

tion Methods; Interscience Publishers: New York, 1965.4. Mathiowitz, E.; Cohen, M.D. J Membr Sci 1989, 40, 1.5. Janssen, L.J.J.M.; te Nijenhuis, K. J Membr Sci 1992, 65, 59.6. Janssen, L.J.J.M.; te Nijenhuis, K. J Membr Sci 1992, 65, 69.7. Janssen, L.J.J.M.; te Nijenhuis, K. J Membr Sci 1993, 79, 59.8. Sundet, S.A. J Membr Sci 1993, 76, 175.9. Meakin, P. In The Fractal Approach to Heterogeneous Chemis-

try; Avnir, D., Ed., Wiley: Chichester, 1989; pp.131–160.

Figure 7 Effect of hydrodynamic conditions (box size L) on(a) the kinetics of film growth and (b) the thickness of the

film at plateau onset.


8/18/2019 10.1.1.509.9728

8/8

10. Enkelman, V.; Wegner, G. App Polym Symp 1975, 26, 365.11. Enkelman, V.;Wegner, G. Makromol Chem 1976, 177, 3177.12. Ji, J.; Dickson, J.M.; Childs, R.F.; McCarry, B.E. Macromolecules

2000, 33, 624.13. Mikos, A.G.; Kiparissides, C. J Membr Sci 1991, 59, 205.14. (a) Muhr, A.H.; Blanshard, J.M.V. Polymer 1982, 23, 1012. (b)

Freger, V.; Korin, E.; Wisniak, J.; Korngold, E.; Ise, M.; Kreuer,

K.D. J Membr Sci 1999, 160, 213.15. The parameter was not necessary in previous works,6, 14 where

the system contained an appreciable amount of solvent. For close to unity, the diffusivities predicted by eq. 9 stray from thatreported in refs 6 and 14 only at very high polymer volumefractions. However, the qualitative predictions of our modelhold for late stages of film growth.

16. Crank, J.; Park, G.S. Diffusion in Polymers; Academic Press:London, 1975.

17. Nikonov, V.Z.; Savinov, V.M. In Interfacial Synthesis; Millich, F;Carraher Jr., Eds., Marcel Dekker: New York, 1977; Vol. 2, pp.157–207.

18. Kuritzyn, L.V.; Sokolov, L.B. Vysokomol Soed 1972, A14, 2028.

19. Chai, G.-Y. Krantz, W.B. J Membr Sci 1994, 93, 175.20. Dankwerts, P.V. Gas-Liquid Reactions; McGraw-Hill: New

York, 1970.21. Levich, V.G. Physicochemical Hydrodynamics; Prentice-Hall,

New Jersey, 1962.22. Gileadi, E. Electrode Kinetics for Chemists, Chemical Engineers

and Material Scientists: VCH Publishers: New York, 1993.

23. Schaep, J.; Vandecasteele, C. J Membr Sci 2001, 188, 129.24. Elimelech, M.; Zhu, X.; Childress, A.E.; Hong, S. J Membr Sci

1997, 127, 101.25. Koo, J.-Y.; Petersen, R.J.; Cadotte, J.E. Pol Preprints Pol Chem

1986, 27, 391.26. Kwak, S. Y. Polymer 1999, 40, 6361.27. Belfer, S.; Purinson, Y., private communication.28. Peeters, J. M. M.; Boom, J. P.; Mulder, M. H. V.; Strathmann, H.

J Membr Sci 1998, 145, 199.29. Yaroshchuk, A.E. Adv Colloid Interface Sci 2000, 85, 193.30. Freger, V.; Dihlajamaki, A.; Shabtai, Y.; Gilron, J. In Proceedings

of International Congress on Membranes and Membrane Tech-nology ICOM, Toulouse, 7–12 July, 2002.


10.1.1.509.9728

Documents