pubs.acs.org/Langmuir © 2009 American Chemical Society
Simulation of Thin Film Membranes Formed by Interfacial Polymerization Rachel Oizerovich-Honig, Vladimir Raim, and Simcha Srebnik* Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel 32000 Received July 10, 2009. Revised Manuscript Received September 26, 2009 Interfacial polymerization is widely used today for the production of ultrathin films for encapsulation, chemical separations, and desalination. Polyamide films, in particular, are employed in manufacturing of reverse osmosis and nanofiltration membranes. While these materials show excellent salt rejection, they have rather low water permeability, both properties that apparently stem from the rigid cross-linked structure. An increasing amount of experimental research on membranes of different chemistries and membrane characterization suggests the importance of other factors (such as unreacted functional groups and surface roughness) in determining membrane performance. We developed a molecular simulation model to qualitatively study the effects of various synthesis conditions on membrane performance, in terms of its estimated porosity and permeability. The model is of an interfacial aggregation process of two types of functional monomers. Film growth with time and structural characteristics of the final film are compared with predictions of existing theories and experimental observations.
1. Introduction Interfacial polymerization (IP) is a procedure used for rapid preparation of high-molecular weight polymer thin films at room temperature. The polymerization takes place at the interface between two immiscible phases upon contact. To provide stability to the thin film, IP is frequently conducted at the surface of a microporous substrate, by first saturating the support with a water-based reagent and then bringing it into contact with an organic phase (e.g., refs 1-6). Such a thin film composite (TCM) was first introduced by Cadotte and Peterson7 for the reaction of a diamine and a triacid and is still the main type of membrane used in reverse osmosis (RO) and nanofiltration (NF). While IP films are most commonly used as the rejection layer in separation membranes, other uses include sensors8 and encapsulation9 for applications such as drug delivery. IP proceeds through irreversible polymerization of two fast reacting intermediates at the interface between two immiscible liquid phases.10 The film tends to form and grow in the organic phase10 because of the low solubility of the acid in water and relatively good solubility of the amine in the organic phase.11 The continued polymerization leads to the formation of a dense layer that hinders diffusion of the amines across the film,12 and hence, *To whom correspondence should be addressed. E-mail: simchas@ technion.ac.il. (1) Wamser, C. C.; Bard, R. R.; Senthilathipan, V.; Anderson, V. C.; Yates, J. A.; Lonsdale, H. K.; Rayfield, G. W.; Friesen, D. T.; Lorenz, D. A.; Stangle, G. C.; Vaneikeren, P.; Baer, D. R.; Ransdell, R. A.; Golbeck, J. H.; Babcock, W. C.; Sandberg, J. J.; Clarke, S. E. J. Am. Chem. Soc. 1989, 111(22), 8485–8491. (2) Ahmad, A. L.; Ooi, B. S.; Choudhury, J. P. Desalination 2003, 158(1-3), 101–108. (3) Roh, I. J. J. Appl. Polym. Sci. 2003, 87(3), 569–576. (4) Rao, A. P.; Joshi, S. V.; Trivedi, J. J.; Devmurari, C. V.; Shah, V. J. J. Membr. Sci. 2003, 211(1), 13–24. (5) Zhang, W.; He, G. H.; Gao, P.; Chen, G. H. Sep. Purif. Technol. 2003, 30(1), 27–35. (6) Freger, V. Langmuir 2003, 19(11), 4791–4797. (7) Peterson, R. J.; Cadotte, J. E. Thin film composite reverse osmosis membranes; Noyes Publications: Park Ridge, NJ, 1990. (8) Huang, J.; Virji, S.; Weiller, B. H.; Kaner, R. B. Chem.;Eur. J. 2004, 10(6), 1315–1319. (9) Couvreur, P.; Barratt, G.; Fattal, E.; Legrand, P.; Vauthier, C. Crit. Rev. Ther. Drug Carrier Syst. 2002, 19(2), 99–134. (10) Morgan, P. W.; Kwolek, S. L. J. Polym. Sci. 1959, 40, 299–327. (11) Sundet, S. A. J. Membr. Sci. 1993, 76(2-3), 175–183. (12) Chai, G. Y.; Krantz, W. B. J. Membr. Sci. 1994, 93(2), 175–192.
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IP films are typically very thin. The thickness of the formed film varies with the type of reactants, solvents, concentration (especially in the organic phase12), and reaction time, ranging from 10 nm to several micrometers.6,10,13-15 Early experiments show that there is an optimum ratio between reactant concentration in the organic phase and reactant concentration in the aqueous phase for the production of a high-molecular weight polymer.16 The optimum ratio is affected by the properties of the reactants, the organic solvent, agitation, and additives in the aqueous phase,10,17-19 that influence the diffusion rates of the amines and the location of the polymerization zone. The nonequilibrium conditions and the extremely fast kinetics of the polymerization reaction, which make it difficult to analyze IP formation experimentally, have led to significant theoretical developments (see ref 20 for a recent review) in describing interfacial film formation and characterizing kinetics21-31 and structure,23,26,32 as well as transport properties of the formed membrane.33,34 From these studies, we learn that factors such as (13) Ji, J.; Dickson, J. M.; Childs, R. F.; McCarry, B. E. Macromolecules 2000, 33, 624–633. (14) Kim, C. K.; Kim, J. H.; Roh, I. J.; Kim, J. J. J. Membr. Sci. 2000, 165(2), 189–199. (15) Petersen, R. J. J. Membr. Sci. 1993, 83(1), 81–150. (16) Arthur, S. D. J. Membr. Sci. 1989, 46(2-3), 243–260. (17) Wittbecker, E. L.; Morgan, P. W. J. Polym. Sci. 1959, 40, 289–297. (18) Jegal, J.; Min, S. G.; Lee, K. H. J. Appl. Polym. Sci. 2002, 86(11), 2781– 2787. (19) Buch, P. R.; Mohan, D. J.; Reddy, A. V. R. Polym. Int. 2006, 55(4), 391– 398. (20) Berezkin, A. V.; Khokhlov, A. R. J. Polym. Sci., Part B: Polym. Phys. 2006, 44(18), 2698–2724. (21) Enkelmann, V.; Wegner, G. Appl. Polym. Symp. 1975, 26, 365–372. (22) Enkelmann, V.; Wegner, G. Makromol. Chem. 1976, 177(11), 3177–3189. (23) Freger, V.; Srebnik, S. J. Appl. Polym. Sci. 2003, 88(5), 1162–1169. (24) Janssen, L.; te Nijenhuis, K. J. Membr. Sci. 1992, 65(1-2), 69–75. (25) Janssen, L.; Tenijenhuis, K. J. Membr. Sci. 1992, 65(1-2), 59–68. (26) Yashin, V. V.; Balazs, A. C. J. Chem. Phys. 2004, 121(22), 11440–11454. (27) Freger, V. Langmuir 2005, 21(5), 1884–1894. (28) Yadav, S. K.; Khilar, K. C.; Suresh, A. K. AIChE J. 1996, 42(9), 2616–2626. (29) Yadav, S. K.; Suresh, A. K.; Khilar, K. C. AIChE J. 1990, 36(3), 431–438. (30) Bouchemal, K.; Couenne, F.; Briancon, S.; Fessi, H.; Tayakout, M. AIChE J. 2006, 52(6), 2161–2170. (31) Kubo, M.; Harada, Y.; Kawakatsu, T.; Yonemoto, T. J. Chem. Eng. Jpn. 2001, 34(12), 1506–1515. (32) Dhumal, S. S.; Wagh, S. J.; Suresh, A. K. J. Membr. Sci. 2008, 325(2), 758– 771. (33) Mason, E. A.; Lonsdale, H. K. J. Membr. Sci. 1990, 51(1-2), 1–81. (34) Mehdizadeh, H.; Dickson, J. M. J. Membr. Sci. 1989, 42(1-2), 119–145.
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decreasing diffusivity, accumulation of reactive end groups, and increasing resistance of the film lead to shrinking of the reaction zone and to the formation of a dense barrier layer inside the loose developing film.23,27 Furthermore, the dense core separates the film into two distinct regions, in each of which one type of monomer and end group is in large excess over the other.6,23 Models in which film formation is attributed to phase separation and precipitation of the forming polymer to spinodal decomposition have also been proposed for prediction of film thickness and molecular weight properties of the polymer32,35,36 as well as crystallinity.32 While all these studies focus on the kinetics of film formation, several important structural characteristics of the formed film, such as pore size distribution, permeability, and surface roughness, have not been analyzed. These features are expected to play a key role in determining membrane performance,14,37 since fabrication conditions affect the barrier layer thickness and hence permeation properties.33,34,38-42 Unlike earlier models of IP films that assumed a uniform density distribution,13,21,22 it is now believed that the film is not spatially or chemically homogeneous across its thickness.6,20 The highest film density, which presents the selective barrier of the membrane,23,27 is located about the reaction zone.20 Freger and Srebnik23 developed the first model that characterized the progression of the polymerization reaction, taking into account the resistance of the forming polymer to monomer diffusion. Their model predicted a symmetric density distribution centered around the reaction zone, but with a fine asymmetric distribution of unreacted functional groups within the film. Their model suggests that beyond the initial stages of film growth, film thickness becomes essentially independent of time, and further polymerization serves only to increase the density of the existing film. Clearly, the nonequilibrium conditions and high reaction rates complicate modeling, requiring the use of a large number of simplifying assumptions.20 Nonetheless, while theoretical models have received much attention, no attempts to simulate the process have been reported. We recently developed a modified cluster cluster aggregation (CCA) algorithm to simulate IP film formation.43 The simulated results revealed qualitative agreement with both experimental observations and theoretical models of the kinetics of film formation. In this work, we improve on this model and focus on structural analysis of the formed film. We discuss the implications of the observed porous structure on membrane performance.
Our coarse-grained representation of the polymerization process utilizes the cluster cluster aggregation (CCA) model,44 which was developed to describe porous materials. In the original CCA model, particles are initially randomly distributed within the simulation box. Particle movement occurs through Brownian motion, with mobility proportional to some power of cluster size.45 Particles and clusters aggregate upon contact, forming
larger and larger clusters until a single cluster or network is obtained. Subsequent models also examined the effect of monomers of varying functionality on cluster formation.46-49 Common IP reactions involve a trifunctional acid halide (O) dissolved in an organic solvent and a bifunctional amine (A) in an aqueous solution.10 This choice of functionality allows for the creation of a moderately cross-linked porous membrane.37 The polymerization reaction results in the release of an acid halide that diffuses back to the aqueous phase. To simulate such a system, we make several assumptions that allow us to focus on general characteristics of IP thin films and their formation. First, the reaction is assumed be unaffected by the liberated acid halide, which is of relatively small molecular size, though it has been suggested that the release of the acid halide upon reaction and its diffusion back to the aqueous phase affect film porosity.13 Second, we model the monomers as spherical particles with uniform size but with differing functionality, although it is known that the chemical structure of the monomers can strongly influence the performance50-52 and structure16,52 of the membranes. Third, we allow for random positioning of the functional groups, though the position of the side group in the aromatic ring is known to affect membrane flux and salt rejection.37 Fourth, solvent molecules are not included explicitly, but only through their effect on the partition coefficients. However, it is known that solvent choice (especially organic solvent) is important since it affects several other polymerization factors such as the diffusion of reactants, reaction rate, and solubility.17 In addition, solvent polarity has recently been suggested to be a determining factor in its effect on reaction kinetics.53 Other factors that have not been included in the present model are the presence of molecular forces (e.g., electrostatic and van der Waals) that would result in slower diffusion of particles, which in turn would lead to the formation of thinner and less porous films. In addition, we assume a rigid membrane. However, preliminary simulations conducted by our group investigating the effect of bond potential (i.e., spring constant) of the membrane on solute permeability suggest that membrane flexibility affects solute mobility, which is further discussed below. The simulation is initiated by randomly placing the monomers in separate halves (representing the organic and aqueous phases) of a three-dimensional asymmetric box (dimensions of L L LX). We consider periodic boundaries in the axes (y and z) parallel to the interface between the two phases and reflective boundaries in the transversal x axis. This axis is taken to be 3 times longer than the y and z axes (i.e., LX = 3L) to reduce the influence of box dimensions on the thickness of the forming film and the location of the reaction zone. The monomers are modeled as soft spheres similar in size, σ, with a maximum allowable overlap of 0.9σ. The trifunctional O monomers can bind only to A’s, and the A’s only to O’s. Apart for the differing monomer functionalities, the two phases are distinguished by the partitioning of each of the monomers in the opposite phase,10,13,25 taken to be 1.0 for the amine and 0.01 for the acid. Since the reaction is diffusion-controlled in the organic phase, maximal polymerization is achieved when the acid concentration is in excess.20 Therefore, we consider a total of 3000 particles, of which 1200 are A’s and 1800 are O’s. Different initial monomer
(35) Karode, S. K.; Kulkarni, S. S.; Suresh, A. K.; Mashelkar, R. A. Chem. Eng. Sci. 1997, 52(19), 3243–3255. (36) Karode, S. K.; Kulkarni, S. S.; Suresh, A. K.; Mashelkar, R. A. Chem. Eng. Sci. 1998, 53(15), 2649–2663. (37) Kwak, S. Y. Polymer 1999, 40(23), 6361–6368. (38) Mehdizadeh, H.; Dickson, J. M.; Eriksson, P. K. Ind. Eng. Chem. Res. 1989, 28(6), 814–824. (39) Pusch, W. Desalination 1986, 59, 105–198. (40) Soltanieh, M.; Gill, W. N. Chem. Eng. Commun. 1981, 12(4-6), 279–363. (41) Kimura, S.; Souriraj, S AIChE J. 1967, 13(3), 497. (42) Lonsdale, H. K.; Merten, U.; Riley, R. L. J. Appl. Polym. Sci. 1965, 9(4), 1341. (43) Nadler, R.; Srebnik, S. J. Membr. Sci. 2008, 315(1-2), 100–105. (44) Meakin, P.; Deutch, J. M. J. Chem. Phys. 1984, 80(5), 2115–2122. (45) Meakin, P.; Vicsek, T.; Family, F. Phys. Rev. B 1985, 31(1), 564–569.
(46) Meakin, P.; Djordjevic, Z. B. J. Phys. A: Math. Gen. 1986, 19(11), 2137– 2153. (47) Meakin, P.; Miyazima, S. J. Phys. Soc. Jpn. 1988, 57(12), 4439–4449. (48) Ohno, K.; Kawazoe, Y. Comput. Theor. Polym. Sci. 2000, 10(3-4), 269– 274. (49) Miwa, K.; Deguchi, T. J. Phys. Soc. Jpn. 2003, 72(5), 976–978. (50) Kwak, S. Y.; Yeom, M. O.; Roh, I. J.; Kim, D. Y.; Kim, J. J. J. Membr. Sci. 1997, 132(2), 183–191. (51) Kwak, S. Y.; Kim, C. K.; Kim, J. J. J. Polym. Sci., Part B: Polym. Phys. 1996, 34(13), 2201–2208. (52) Verissimo, S.; Peinemann, K. V.; Bordado, J. J. Membr. Sci. 2006, 279 (1-2), 266–275. (53) Wagh, S. J.; Dhumal, S. S.; Suresh, A. K. J. Membr. Sci. 2009, 328(1-2), 246–256.
2. Model of IP Formation
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concentrations were considered by varying the simulation box dimensions, L and LX. In each iteration, a random cluster (composed of one or more particles) is chosen and a random direction for movement is set with a maximum step length of 2.4σ. This step length corresponds to an approximately 50% acceptance rate at initial stages of the simulation for the range of concentrations considered. The maximum step length is assumed to be inversely proportional to cluster size.45 The mobility of the growing polymer is naturally reduced during the course of the simulation because of the increasing degree of overlap of the larger clusters. A trial move is accepted if the maximum allowed overlap between any particles making up the cluster and all other particles in the simulation box is not exceeded. If overlap occurs, than the step size is reduced accordingly to prevent overlap. In addition, if the trial move positions the particle in an area with a finite local density of particles of opposite phase, it is accepted with a probability proportional to the partition coefficient of the moving particle. For simplicity, we assume a linear relationship between the partition coefficient of the particle and its local concentration. This simplification assumes that the interaction of the different solution components behaves additively, and therefore, the partition coefficient of the solute is proportional to the concentration of the components.54,55 The probability is calculated as pβi, R ¼
X j ¼R, β
Kij fi
ð1Þ
β wherep i,R is the acceptanceprobability for translation ofparticlei from solvent type R (A or O) into environment β (A or O), Kji is the partition coefficient of particle i in solvent j, and fi is the fraction of particles of type i in the local environment. Once a trial move is accepted, clustering of two or more particles is accepted if two particles of opposite type, each with at least one functional group available for bonding, are in contact. Furthermore, bonding is accepted with a probability equal to a fixed particle sticking probability, ps, which is set to unity unless otherwise stated. Once bonded, the particles remain part of the same cluster for the remainder of the simulation; i.e., the reaction is irreversible. The simulation proceeds with attempted movement of random clusters for up to 2 107 cycles, when the film has already stopped growing and the rate of polymerization approaches zero. We previously reported on our results using a similar algorithm to model film formation, where the total number of particles remained fixed for the duration of the simulation.56 Clearly, such a model leads to significant depletion of particles from the bulk as the polymer film forms. To account for the continuous repletion of particles in the bulk, the addition of new particles to their respective phases is attempted every 104 iterations. New particles are added at random positions to the outer edges of the simulation box (thickness of LX/18) until the concentration of particles at the box edge equals the initial concentration of particles. In this manner, the bulk concentrations are kept approximately constant at the initial values. Averages are calculated over at least 10 independent simulations for each of the initial monomer concentrations studied.
3. Results and Discussion Film Formation. As the simulation proceeds, a finite film forms very quickly at the interface between the two immiscible phases. Assuming an average diffusion coefficient, D0, of 10-5 cm2/s,23 we can approximate the time elapsed for film formation (54) Schmidt, T. C.; Kleinert, P.; Stengel, C.; Goss, K. U.; Haderlein, S. B. Environ. Sci. Technol. 2002, 36(19), 4074–4080. (55) Schmidt, T. C.; Kleinert, P.; Stengel, C.; Goss, K. U.; Haderlein, S. B. Environ. Sci. Technol. 2003, 37(4), 815–815. (56) Nadler, R.; Srebnik, S. J. Membr. Sci. 2009, (in press) .
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Figure 1. Side view (left) and top view (right) of a sample polymer film. The blue particles represent the trifunctional acids, and the cyan particles represent the difunctional amines (t=2 ms, and c0 = 2.3 mol/L).
Figure 2. Fraction of monomers belonging to percolating clusters making up the film, φpc, as a function of time. Percolation is measured in the direction perpendicular to the interface (y and z axes).
by comparing D0 to the calculated average particle diffusivity in solution, ÆDæ n P
ÆDæ ¼
i ¼1
Δri 2
6nΔt
ð2Þ
where the summation is over n particles, and Δri corresponds to the displacement of particle i in time period Δt. We obtain a ÆDæ of =0.9 in units of (particle size)2/iteration, a value that is relatively insensitive to the concentrations considered in this work. Assuming a particle size of approximately 0.5 nm, we find that each iteration equals 0.1 ns, and a simulation of 107 iterations characterizes film growth within the period of 1 ms. In our model system, the extent of film growth is for the most part greatly reduced beyond 1 ms. Thus, most of our measurements below present averages conducted with films simulated over this approximate time period. Figure 1 shows a sample film obtained after 2 ms. Within approximately 20 μs, a percolating film forms, which is indicated by the fraction of monomers that make up the largest percolating cluster of the film, φpc, in Figure 2. At our simulation time limit of 2 ms, the film is made up of a single percolating unit. Since, initially, the polymerization reaction is much faster than diffusion of the monomers within the rapidly growing film, the film is essentially static (with diffusivity approximately proportional to Np-1, where Np is the number of monomers making up DOI: 10.1021/la9024684
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Figure 3. (a) Reaction frequency and average polymer concentration (moles per liter) and (b) location of the reaction front (average position of reacting groups) as a function of simulation time (c0 = A O A 2.3 mol/L). KO A = 1 = 100KO (black curve), and KA = 0.1 = 10KO (gray curve).
the polymer film) and grows around the original barrier layer. This observation supports the assumption of a static film used in the IP model of ref 23. In Figure 3a, we plot the average reaction frequency (number of reacted groups per unit time) and the average molar concentration of the polymer film as a function of simulation time. Figure 3a indicates that even beyond approximately 106 iterations (approximately 100 μs), reaction and film growth rates are quite small, as the increasingly more dense polymer film becomes a barrier to further growth, especially near the interface between the two phases where the functional groups are depleted.12 The initial stages of the reaction are characterized by a sharp decrease in the reaction rate corresponding to a sharp increase in the amount of polymer formed. The continuous decrease in reaction rate that follows is due to the low diffusivity across the barrier layer and the exhaustion of monomer O in the diffusion boundary layer.20 This observed behavior has been attributed to the limited partitioning of monomer O in the aqueous phase and the rather good solubility of monomer A in the organic phase and the growing film.10,11,22 Consequently, as the reaction progresses, the reaction zone (average position of reacting groups) shifts deeper within the organic phase (Figure 3b, black curve), so that the greater part of the polymer is concentrated in the region between the reaction zone (just within the organic phase) and the interface. However, the partitioning of the aqueous monomer strongly 302 DOI: 10.1021/la9024684
Figure 4. (a) Polymer concentration, (b) concentration of unreacted functional groups, and (c) number of unreacted functional groups per monomer as a function of distance along the film at different simulation times. Black curves represent data for organic monomers and gray curves data for aqueous monomers (c0 = 2.3 mol/L).
affects the location of the forming film, which can be seen by the gray curve in Figure 3b, where the partition coefficient of the aqueous monomer in the organic phase was reduced 10-fold for this experiment. Distributions of the polymer and unreacted groups in the transversal direction to film growth at different polymerization times are shown in Figure 4. Figure 4a shows a relatively symmetric polymer distribution, with the barrier layer (maximum concentration) slowly shifting from the interface at short polymerization times to the organic phase at longer times. The dense barrier layer is characterized by low concentrations of unreacted functional groups (Figure 4b). The film edges, however, are characterized by monomers that are attached through a single group (Figure 4c), hence forming a rather loose exterior. The presence of unreacted groups (e.g., the carboxylic and amine groups) is an important feature of polyamide films, since they are believed to play a crucial role in the rejection and resistance to fouling. pH can be used to control protonation and deprotonation of the functional groups such that the resulting Langmuir 2010, 26(1), 299–306
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(usually negatively) charged membrane can be tailored for solute rejection.6,15,57,58 The large fraction of unreacted functional groups at the outer layers of the film and in the organic phase in particular, seen in Figure 4c, indeed indicates that the film can be deprotonated to produce a negatively charged membrane or charged layers in the film.6,16,37 The surface of IP membranes is relatively rough.11,59-62 Quantification and control of surface roughness are important since it is believed to be responsible for the increased flux seen in high-flux IP membranes due to the increased surface area.61 On the other hand, roughness is one of the leading causes of fouling and the formation of a cake layer,63,64 which ultimately reduces flux. Our simulation snapshots of the film, shown in Figure 1, clearly reveal a rough nonuniform morphology and a nonhomogeneous distribution of pore sizes across the surface of the film. This morphology may be explained by the polymerization process during the course of the simulation, which is initiated through the formation of small clusters of various sizes that agglomerate to form the initial active layer of the film. While the bulk of polymerization proceeds through reaction of monomers and very small clusters (made up of several monomers) with the growing film, larger clusters grow in the vicinity of the film, especially within the organic phase. These clusters aggregate with the film at later stages, giving rise to a rough and porous surface layer. This mechanism of film formation has been proposed previously6,11 but could not be substantiated with previous models. Film Thickness. From distributions such as those shown in Figure 4a, we can calculate the average thickness, δ, of the forming film. The final film was considered as the largest percolating cluster, and its thickness was calculated as the maximum distance between monomers along the x axis, averaged over six equal slices in the z direction. During the early stages of the simulation, the forming film was taken to be the ensemble of clusters that are made up of 20 or more particles. Assuming a monomer size of approximately 0.5 nm, in this short simulated period of film formation, we observe film thicknesses on the order of 2-10 nm for the initial monomer concentrations studied, which fall in the range of experimental measurements of very thin films.4,14 The change in thickness of the polymer film with time is shown in Figure 5. Interestingly, we observe four distinct regions of film growth. The incipient fast stage is characterized by δ ∼ t2 (as previously reported43) until the formation of a dense barrier layer,27 followed by slow growth limited by diffusion across the dense polymer barrier, which scales approximately as δ ∼ t1/2, and then δ ∼ t1/3. Finally, film thickness appears to grow very little with time, with δ ∼ t0. Similar scaling is seen for other initial monomer concentrations; however, the onset of each of the regions occurs earlier with increasing concentrations. In the inset of Figure 5, we show that an overall square root dependence of film thickness with time for the incipient film provides a good approximation, supporting the well-known predicted δ ∼ t1/2 dependence,13,21,22,24,25,27 which has also been observed in a series (57) Childress, A. E.; Elimelech, M. Environ. Sci. Technol. 2000, 34(17), 3710– 3716. (58) Schaep, J.; Vandecasteele, C. J. Membr. Sci. 2001, 188(1), 129–136. (59) Kwak, S. Y.; Ihm, D. W. J. Membr. Sci. 1999, 158(1-2), 143–153. (60) Bowen, W. R.; Doneva, T. A.; Stoton, J. A. G. Colloids Surf., A 2002, 201 (1-3), 73–83. (61) Kwak, S. Y.; Jung, S. G.; Kim, S. H. Environ. Sci. Technol. 2001, 35(21), 4334–4340. (62) Vrijenhoek, E. M.; Hong, S.; Elimelech, M. J. Membr. Sci. 2001, 188(1), 115–128. (63) Costa, A. R.; de Pinho, M. N.; Elimelech, M. J. Membr. Sci. 2006, 281 (1-2), 716–725. (64) Elimelech, M.; Zhu, X. H.; Childress, A. E.; Hong, S. K. J. Membr. Sci. 1997, 127(1), 101–109.
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Figure 5. Film thickness as a function of reaction time (c0 = 0.96 mol/L). The inset shows the approximate square root dependence during the initial stages of film formation for various concentrations: 0.96 (O), 2.3 (0), and 3.2 mol/L (4).
of experimental kinetic studies of interfacial synthesis of linear aliphatic21,22 and cross-linked aliphatic-aromatic polyamides.24,25 However, the intermediate regime, believed by some to be characterized by diffusion-limited growth,20 appears to be short-lived in our simulation, representing the transition from the incipient stages where δ ∼ t2 and later stages where δ ∼ t1/3. That is, growth is slower than particle diffusion, because of the decreasing polymer porosity as time proceeds. The earliest stages of reaction are characterized by the formation of the dense barrier layer. Freger27 conducted a scaling analysis of the different stages of film growth, predicting the maximal polymer concentration to scale as t2 for the incipient stages of reactions when the reaction accelerates due to accumulation of reactive end groups. Film thickness is proportional to polymer concentration δ ¼
N p vp L2 ð1 - εÞ
ð3Þ
where vp is the particle volume and ε is the porosity and therefore would also be expected to grow as t2 at first. The initial concentration of the monomers is known to influence the thickness of the final film.22 In Figure 6, we plot the average concentration and thickness of the film as a function of initial monomer concentration, c0. As expected, the average polymer concentration increases with c0 (Figure 6a) because of the presence of higher concentrations of functional groups at the reaction zone. However, an asymptotic polymer concentration is reached at sufficiently high monomer concentrations. The rate of formation and structure of the incipient barrier layer will ultimately determine the final film thickness. High monomer concentrations result in fast formation of a dense barrier layer due to the presence of a large number of monomers at the vicinity of the interface, and the film is expected to be thinner and more homogeneous. On the other hand, a low c0 leads to initial formation of rather porous smaller clusters, which will diffuse and aggregate in time to form a loosely packed thick film. Further polymerization leads to densification of the film. The relationship between film thickness and c0 is shown in Figure 6b. Film thickness is found to decrease with increasing initial concentrations. This nearly inverse relation may be anticipated since higher initial concentrations result in the formation of a denser (lower ε) film in the incipient stages of reaction. Since from eq 3, δ (1 - ε)-1, we expect that δ should decrease with a decrease in porosity or an increase in concentration. Freger and Srebnik23 predict a weaker dependence of film thickness on initial concentration. The difference DOI: 10.1021/la9024684
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Figure 7. Concentration at the dense (1.5 nm thick) barrier layer as a function of initial monomer concentration and polymerization time.
Figure 6. (a) Average polymer concentration and (b) average film thickness as a function of initial monomer concentration (t=2 ms).
in scaling could be manifested in a number of assumptions used in their model, including the rate of decrease of monomer diffusivity within the film, reaction rate constants, and reaction kinetics within the film. In Figure 7, we show the evolution of the concentration in the densest layer, c*, as a function of initial monomer concentration and time. In our calculation, c* corresponds to the densest 1.5 nm thick layer of the film. The film begins to form after approximately 0.2-0.5 μs, depending on the initial concentration. At this point, we observe percolation of the polymer in the plane perpendicular to film growth. At short polymerization times, c* increases rapidly with monomer concentration merely due to the presence of a larger number of reactants near the interface. At long polymerization, times a plateau is reached, indicating a limit to the concentration of the barrier layer. Once this limit is reached, the film is limited to densification through reaction of monomers or small clusters with unreacted functional groups. Porosity, Permeability, and Surface Roughness. A critical assumption in this model is that the membrane is rigid; i.e., conformational fluctuations are suppressed by assuming infinitely rigid bonds and angles of the polymer. However, allowing for polymer flexibility influences monomer mobility, and hence its flux, through the membrane. To test this effect, we conducted molecular dynamics simulations of 270 Lennard-Jones (LJ) solute particles of diameter σ and ε=1kBT (ε is the LJ energy parameter) penetrating a 5 5σ2 cross section of the central region of the membrane (simulation details will be presented in a future communication focusing on flux studies). A harmonic spring constant was introduced between the bonded membrane particles, 304 DOI: 10.1021/la9024684
Figure 8. Flux as a function of polymer bond constant (c0 = 2.3 mol/L).
with the following contribution to the energy: Eb ¼ kr2
ð4Þ
where k is the harmonic bond constant, in units of ε/σ2, and r is the distance between two bonded particles. For each value of k, the flux was calculated from the average number of particles penetrating the membrane per unit simulation time, divided by the cross-sectional area of the membrane. Each value was averaged over 20 independent simulation runs. Figure 8 presents the results of solute flux as a function of polymer bond constant. These results suggest an order of magnitude increase in flux in a flexible membrane (kσ2/ε = 1) compared with a very rigid one. Nonetheless, our results below are for the rigid bonds. The initial monomer concentration also affects membrane performance and structure. Figure 6, which shows the decrease in film thickness and the increase in average film density with an increase in concentration, suggests that high initial concentrations lead to the formation of a thin dense film with relatively low porosity, while low initial concentrations yield a looser thick film.13 We can gain further insight into the porosity and internal structure of the film from the surface area to volume ratio (av). We calculated av by projecting the film onto a grid and counting the average number of void nearest neighbors per occupied site. These calculations reveal that the Langmuir 2010, 26(1), 299–306
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barrier layer is nonporous, characterized by an avσ of =1.6 (compare with 6 for an isolated spherical particle) for the range of concentrations studied. Thus, what apparently determines the thickness of the film is the rapidity with which the barrier layer is formed. Clearly, as c0 increases, the dense core will form more quickly. Using av and δ, we can approximate the pore size distribution and permeability. Darcy’s law describes the flow of a fluid through a porous medium, by providing a relation of fluid flux q to applied pressure gradient ΔP through the permeability (κ = qμδ/ΔP, where μ is the viscosity of the fluid (water) and δ is the film thickness). The flux is related to the pressure gradient from the Hagen-Poiseuille equation for flux in capillaries (q = (ΔPdp2ε)/(32μδτ), where dp is the mean pore diameter and τ is the tortuosity). Using the Carman-Kozeny (CK) equation (κ = ε3/[8τ(1 - ε)2av2]), we relate the permeability to the porosity of the material to obtain an estimate of the mean pore size sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ε3 ð5Þ dp ¼ ð1 - εÞ2 av 2 where the porosity ε is estimated as the free volume within the film from eq 3. The permeability is obtained from the CK relation, and assuming a simple inverse relation between tortuosity and diffusivity,65 D*=ε/τ (where D* is the diffusivity through the final film normalized by the self-diffusivity of the particles) K¼
Dε2 2
8ð1 - εÞ av 2
ð6Þ
The use of the CK equation is appropriate for viscous flow through porous media or laminar flow through straight channels. Indeed, for complex porous media, it has been shown that the CK equation does not capture the dependence of permeability on porosity, especially at small porosities,66 and on the complex microstructure.67 Thus, the permeability cannot be characterized by simple empirical equations, but the CK assumption gives us a qualitative estimate of its dependence on the various parameters and evaluated membrane properties. Figure 9 shows the dependence of membrane permeability and mean pore diameter on initial monomer concentration. Consistent with the observations that increasing concentrations lead to the formation of an overall denser and less porous film, we observe that the mean pore diameter and membrane permeability decrease with an increase in monomer concentration, with an integral power law relation. In the inset of Figure 9b, we show the pore size distribution across the polymer film for different monomer concentrations. The dense core is characterized by an average pore diameter of 1 nm, approximately twice the diameter of the monomers, which agrees with the mean pore size of manufactured NF membranes63,68 of ∼1-10 nm. The small pore size and their random organization make this layer essentially impenetrable to the monomers and block further film growth in our simulation. However, in real films, fluctuations of the cross-linked chains are likely to allow for further permeation and growth, as shown in Figure 8. Moreover, the porous structure of the film appears relatively symmetric across the dense core (Figure 9b), and especially for films formed from lower solution concentrations. Nonetheless, (65) Vieth, W.; Wuerth, W. F. J. Appl. Polym. Sci. 1969, 13(4), 685. (66) Valdes-Parada, F. J.; Ochoa-Tapia, J. A.; Alvarez-Ramirez, J. Phys. A (Amsterdam, Neth.) 2009, 388(6), 789–798. (67) Jourde, H.; Fenart, P.; Vinches, M.; Pistre, S.; Vayssade, B. J. Hydrol. 2007, 337(1-2), 117–132. (68) Chowdhury, S. R.; Keizer, K.; ten Elshof, J. E.; Blank, D. H. A. Langmuir 2004, 20(11), 4548–4552.
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Figure 9. (a) Film permeability and (b) mean pore diameter as a function of initial monomer concentration at t = 2 ms. The inset shows pore size distributions across the film for different initial concentrations [3.9 (;), 2.3 (---), and 0.96 mol/L (---)], and the dotted line represents the interface.
the aqueous side appears to be characterized by somewhat larger pores. Indeed, it has been observed that the organic side of IP membranes has a nonporous structure whereas the aqueous side has a loose more porous structure.12 Diffusion. Monomer diffusivity is intimately connected with the volume fraction of the polymer.20 The forming polymer, and the dense core in particular, significantly reduce the mobility of the monomers across the film. To calculate the average monomer diffusivity through the film, ÆDfæ, during the course of the simulation, we limit the summation in eq 2 to unreacted monomers diffusing through the polymer film, averaged over a fixed period of simulation time. ÆDfæ is shown in Figure 10 as a function of simulation time and polymer volume fraction at the dense core. It is seen that the incipient stages of film growth are characterized by a sharp drop in diffusivity, due to the fast formation of the polymer barrier layer.13 As the simulation proceeds, the porosity of the dense core decreases. A common empirical fit to the change in diffusivity with polymer volume fraction is given by D ¼ D0 φR
ð7Þ
where φ is the polymer concentration and R takes values between 1 and 3.23,69,70 In Figure 10b, we show a fit of eq 7 to our simulation results for two different concentrations. The dense core is defined by the densest 2 nm thick layer of the film, with a porosity ε*. In addition, we also calculated the average diffusivity of monomers of size dp in a cluster formed from a random distribution of particles (also size dp) for various particle volume fractions. An excellent fit to the diffusion in random clusters is (69) Freger, V.; Korin, E.; Wisniak, J.; Korngold, E.; Ise, M.; Kreuer, K. D. J. Membr. Sci. 1999, 160(2), 213–224. (70) Muhr, A. H.; Blanshard, J. M. V. Polymer 1982, 23(7), 1012–1026.
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Figure 11. Effect of sticking probability on film thickness for t = 2 ms and c0 = 2.3 mol/L. Solid lines serve to guide the eye.
Figure 10. Diffusivity of unreacted monomers in the densest 2 nm layer of the film with a porosity ε* as a function of (a) time (c0 = 2.3 mol/L) and (b) polymer volume fraction for c0 =2.3 mol/L (0) and c0 = 0.96 mol/L (2); the circles show diffusivity through a random structure at the given void fraction. Curves show fits to eq 6 with R values of 0.4 ( 3 3 3 ), 1.2 (---), and 1.6 (;).
obtained with an R of 1.6. However, for the simulated IP film, R depends on c0 and may acquire values below unity. This result strongly suggests that the fine internal structure of the film depends on monomer concentration. That is, while the void fraction of IP films prepared in different ways may be the same, features such as pore size distribution and tortuosity must be different. Sticking Probability. The sticking probability is simply the tendency for two particles to aggregate upon contact and at its two extremes distinguishes between diffusion-limited (ps f 1) and reaction-limited (ps f 0) regimes.48 Factors such as steric hindrance and positioning of the functional groups may be distinguished by ps values below unity. The effect of varying ps on the thickness of the formed film is shown in Figure 11. A clear tendency toward thicker (and overall looser film) is indicated with decreasing values of ps. That the film becomes larger and less dense with a decreasing sticking probability is contrary to the usual observations of a standard reaction-limited aggregation process.48 This contradiction may be explained by the increase in monomer diffusivity within films formed with low values of ps, allowing the monomers to traverse across the film and react at the other end, where the concentration of the opposite type of monomers is high. We note, however, that at low ps values the aggregation process takes a very long time until it reaches its final stages,71 resulting in films that occupy the entire simulations box as time progresses for the small-scale simulations conducted in (71) Kudoh, M.; Xiao, H.; Ohno, K.; Kawazoe, Y. J. Cryst. Growth 1993, 128 (1-4), 1162–1165.
306 DOI: 10.1021/la9024684
this work. Nonetheless, the strong dependence of δ on ps suggests that the physical nature of the monomer may be a critical factor determining film properties. We note that the Smoluchowski rate equation provides a mean field approach to aggregation kinetics in general. The equation defines the rate of aggregation as a sum of two terms, the rate of formation of clusters of size ni from two smaller clusters and the rate of disappearance of clusters of size ni through collision, summed over all cluster sizes. A homogeneity parameter can be used to express the tendency of a cluster to coalesce with another cluster,72,73 which allows us to classify an aggregation process as gelling or nongelling. λ allows comparison with experimental measurements since it can be related to the average hydrodynamic radius of the clusters,74 and thereby related to the reaction rate RH ðtÞ ¼ RH ð0Þez=df
ð8Þ
where z = 1/(1 - λ) is a kinetic parameter and df is the fractal dimension of the clusters. Thus, measurements of the hydrodynamic radius can be compared with calculations for quantitative comparison of the reaction rate constant and can provide an estimate of the sticking probability. The homogeneity parameter is directly related to the sticking probability, ps, through λ=1 - ps.
4. Concluding Remarks Over the years, a number of theoretical models for describing film properties and formation kinetics have been developed. However, the complex reaction-diffusion mechanism necessitates the use of many assumptions, which leads to incongruous predictions about the scaling of film properties with time and concentration. Thus, we developed a molecular simulation for thin film formation through interfacial polymerization that allows us to move beyond mean field approximation predictions of past theories. The simulation allows for estimation of fine structural parameters that are difficult to obtain using analytical models. Furthermore, the method can be used as a qualitative tool for predicting, e.g., membrane performance and membrane fouling, which depend on pore size, membrane charge, and roughness. Extension to atomistic models should allow for quantitative comparison with experimental data and systemspecific predictions. Acknowledgment. This research was supported, in part, by the Israel Science Foundation. (72) Vandongen, P. G. J.; Ernst, M. H. J. Stat. Phys. 1988, 50(1-2), 295–329. (73) Vandongen, P. G. J.; Ernst, M. H. Phys. Rev. Lett. 1985, 54(13), 1396–1399. (74) Olivier, B. J.; Sorensen, C. M. Phys. Rev. A 1990, 41(4), 2093–2100.
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