“In and Out Diffusion” Hypothesis of Exponential Multilayer Film

COMMENT pubs.acs.org/Langmuir

“In and Out Diffusion” Hypothesis of Exponential Multilayer Film Buildup Revisited Donald T. Haynie,*,†,‡ Eunhee Cho,†,‡ and Pradeep Waduge† †

Nanomedicine and Nanobiotechnology Laboratory and ‡Center for Integrated Functional Materials, Department of Physics, University of South Florida, Tampa, Florida 33620, United States ABSTRACT: A hypothesis concerning the exponential buildup of polyelectrolyte multilayer films prepared by layer-by-layer assembly has become widely accepted in the scientific community. This model was first introduced with experimental data in Langmuir. It was subsequently described in Proceedings of the National Academy of Sciences and extended and amended in papers in Langmuir and other journals. According to the “in and out diffusion” hypothesis, as it is called, or “common rule” of exponential multilayer film buildup, as it is widely regarded, “a diffusion-based buildup mechanism ... explains most of the exponential-like growth process of polyelectrolyte multilayers reported in the literature.” The present work offers an alternative viewpoint to specific elements of the hypothesis and the model as a whole.

’ INTRODUCTION One of the more significant subtopics of the now vast body of polyelectrolyte multilayer film studies has been the mechanism of film buildup.1
12 It is generally agreed that polyelectrolyte adsorption is both driven and limited by Coulombic interactions and entropy increase. Charged polymers in solution will bind electrostatically to oppositely charged polymers on the film surface, releasing partially solvated counterions from both types of polyelectrolyte. Adsorption will not continue indefinitely because newly bound polymers on the film surface repel likecharged polymers in solution. A provocative viewpoint concerning the film buildup mechanism has been the “in and out diffusion” hypothesis. Developed to explain the experimental finding that polyelectrolyte molecules can migrate in a multilayer film that grows exponentially, the hypothesis posits polymer diffusion as the molecular cause of exponential buildup.13
17 The core idea of the hypothesis is that polymer diffusion implies exponential growth and vice versa. The “in and out diffusion” model has become widely accepted in the scientific community. Here, we summarize the “in and out diffusion” hypothesis, present experimental multilayer film data, compare the data to predictions based on the hypothesis, comment on other aspects of the hypothesis, and draw attention to alternative models that are apparently no less explanatory. The discussion is aimed at clarifying what the hypothesis says about the mechanism of exponential multilayer film buildup, and whether the hypothesis is any more valid an explanation of exponential buildup than alternative models. ’ THE “IN AND OUT DIFFUSION” MODEL The migration of polyelectrolytes in multilayer films is said to occur during film fabrication and afterward.13
17 The film systems in the indicated works grow exponentially or pass r 2011 American Chemical Society

through an exponential phase before entering a linear phase during the buildup process. “Exponential buildup” means that an appropriate physically observable quantity increases exponentially as the number of polyelectrolyte adsorption steps increases, for instance, film thickness (or volume). The entities hypothesized to diffuse “in and out” are the same polyelectrolytes used to make the film. Inward diffusion of polyelectrolytes is said to occur during a polymer adsorption step and in some cases afterward, for example, during the following rinse step. The inwardly diffused polymers then contribute to the formation of a “reservoir” in the film, if not of “free” polymers,13,14 then of “uncompensated” charged groups on the polymers.15
17 Outward diffusion is said to occur during the next polymer adsorption step, though some may occur during the rinsing step immediately following adsorption. Let the reservoir contain “free” polyelectrolytes.13
15 Following outward diffusion and on reaching the film surface, these molecules will bind newly adsorbing polymers, which have the opposite sign of charge, forming insoluble complexes. Most adsorbing polymers will thus be trapped at the film surface. Some, however, will continue to diffuse inward, depending on the polymer.13
17 In any case, the amount of polymer that becomes associated with the film during an adsorption step is supposed to compensate not only for the charges on the film surface, but also for the polymers diffusing outward. Because the amount of polymer or uncompensated charge on polymers in the hypothetical reservoir will increase with each additional bilayer of film, buildup will be exponential, as is readily shown by elementary mathematics.13
15 Received: November 12, 2010 Published: April 05, 2011 5700

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COMMENT

thereof (uncompensated charges) are said to enable polymer diffusion, which in turn causes exponential buildup.

Figure 1. Spectroscopic analysis of multilayer film buildup. All films were prepared from polymer solutions at 2 mg/mL in 10 mM phosphate buffer, pH 7.4. No film was dried during fabrication or analysis. The duration of polymer deposition steps was 15 min. All films were rinsed extensively in buffer after each adsorption step to remove any loosely bound polymer. All peptides were from Sigma (USA). (A) Far-UV analysis of PLL/PLGA films. Squares, M/M. Circles, M/L. Hexagons, L/M. Triangles, M/S. Diamonds, S/M. The solid lines are fits of an exponential model to experimental data, A = ReβN, where A is absorbance and N is the number of adsorption steps. The fitting parameter β is shown for each data set. All data are presented; where data points do not appear, no data were collected for logistical reasons. The molecular masses of the polymers were S = 4
15 kDa, M = 30
70 kDa, and L e 150 kDa for PLL; and S = 3
15 kDa, M = 15
50 kDa, and L = 50
100 kDa for PLGA. The buffer was sodium phosphate. (B) Near-UV analysis of PLLT/PLGA and PLLT/PDGA films. The solid lines are fits of an exponential model to all experimental data points collected. The best-fit values of β were 0.185 for PLLT/ PLGA and 0.144 for PLLT/PDGA. The molecules masses of the polymers were g20 kDa for PLLT, 15
50 kDa for PLGA, and 15
25 kDa for PDGA. The buffer was potassium phosphate.

The behavior of polymers in the “in and out diffusion” model is further said to depend on the formation of a free energy barrier at the film/solution interface, and on a difference in electrochemical potential between “free” polyelectrolytes in the film and polyelectrolytes in solution.15 The difference in electrochemical potential will drive polyelectrolytes from solution into the film. The accumulation of polyelectrolytes on the film surface will then set up a free energy barrier, trapping “free” polyelectrolyte molecules or parts thereof in the film interior. The trapped molecules may or may not diffuse out during rinsing, and they may influence the distribution of ions across the film/solution interface by the Donnan effect. The trapped molecules or parts

’ RESULTS AND DISCUSSION The diffusion constant of a polymer varies inversely with chain length. A short polymer will diffuse more rapidly than a long one of the same species, in solution or in a multilayer film. If diffusion causes exponential film buildup, as the “in and out” mechanism hypothesizes, short polyelectrolytes will be expected to show exponential buildup at a greater rate than long polyelectrolytes of the same species. If there is no such expectation, what the hypothesis will predict is unclear. This ambiguity and the results of the experiments described below prompted the present reconsideration of the “in and out diffusion” hypothesis. In earlier work, we found that building up a film from poly(Llysine) (PLL) and poly(L-glutamic acid) (PLGA) was difficult if the degree of polymerization was low but not if it was high.18 One of these polymers (PLL labeled with FITC) was used to build the films discussed in refs 13 and 14; the polyanion was hyaluronic acid. Figure 1A shows film buildup results for different combinations of PLL and PLGA of different degrees of polymerization, “small” (S), “medium” (M), and “large” (L). The data sets are for M/M, L/M, M/L, S/M, and M/S, where in each case the polycation is given first; for instance, M/L stands for medium molecular mass PLL and large molecular mass PLGA. Absorbance at 220 nm, which is due to the peptide bond, is plotted as a function of adsorption steps. Further experimental details are provided in the figure legend. Buildup is clearly exponential in several of these cases. It must be assumed that the average diffusion constant of the polymers used to obtain the data in Figure 1A varied as S > M > L, in solution and in the films; the actual quantities are not essential here. If it is supposed that neither the polycation nor the polyanion diffused during buildup, the “in and out diffusion” hypothesis does not predict several instances of exponential buildup in the absence of diffusion. Alternatively, if either PLL or PLGA diffused during buildup (or both, as in ref 15), then the underlying reason for a buildup rate so much lower for S/M and M/S than M/M is unclear. If the “in and out diffusion” hypothesis says that polymer diffusion causes exponential buildup, but the hypothesis does not predict how the relative magnitude of diffusion constants will translate into relative rates of film buildup, it is not clear to us how the hypothesized role of diffusion in film buildup is directly testable. Of course, this is not to say that polymer diffusion does not occur. Nor does it have to mean that diffusion plays no role in exponential buildup. It simply is not clear how the “in and out diffusion” hypothesis could predict the results presented here. The “in and out diffusion” hypothesis has been said to predict that if “none of the polyelectrolytes diffuse within the multilayer ... the film grows linearly”.15 The data in Figure 1 suggest that the statement cannot be generally valid; M/S appears to be an instance of linear growth. The point is made more emphatically, however, by data in refs 16 and 17, which show that polymer diffusion occurs in some films that grow linearly (after passing through a phase of exponential growth). If all these data are supposed to be consistent, then they seem to say that the “in and out diffusion” hypothesis makes no definite prediction regarding the relationship between diffusion constant, polymer diffusion, and shape of the film growth curve. Instead, a likely explanation for the behavior of S/M and M/S in Figure 1A is soluble complex 5701

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Langmuir formation at the film/solution interface; relatively short-chain polyelectrolytes will more readily form soluble complexes than long-chain polyelectrolytes.19 It seems the “in and out diffusion” hypothesis explains nothing here. We have tested whether the results in Figure 1A are polymer system-dependent. Figure 1B shows buildup data obtained with poly(L-Lys, Trp) (PLLT) as polycation and PLGA or poly(Dglutamic acid) (PDGA) as polyanion. The copolymer has 4 lysine residues for each tryptophan residue. The aromaticity of the tryptophan side chain is convenient for spectroscopic analysis in the near UV. The peptide bond makes essentially no contribution to absorbance above 250 nm.20 Both PLLT/PLGA and PLLT/PDGA showed exponential buildup. The values of the fitting parameter β were similar to those obtained for the PLL/ PLGA films in Figure 1A. The rate of buildup was greater for PLGA than PDGA, even though the average molecular weight of PDGA was smaller than for PLGA. We had thought the opposite would be predicted by the “in and out diffusion” hypothesis. If the hypothesis makes no such prediction, it is unclear to us what the hypothesis predicts with regard to polymer diffusion and film buildup. The data in Figure 1 can be compared with the data in ref 17, which describes diffusing polymers in films that show a linear increase in thickness. A spraying method was used to make the films, and films were dried before ellipsometry measurements. We assume that spraying will give qualitatively similar films to dipping21 and that structural changes following dehydration of polypeptide-based multilayer films will be largely reversible.22 It was found that film thickness increased as the molecular weight of PLL increased for a fixed molecular weight of hyaluronic acid.17 The result resembles the data in Figure 1A and in ref 19. PLL, according to ref 14, diffuses in a multilayer film, but not hyaluronic acid. The diffusion constant for PLL is expected to be greatest in the film system involving the lowest molecular weight molecules of PLL. If polymer diffusion causes exponential buildup, it seems that the “in and out diffusion” hypothesis would predict the thickest film for the lowest molecular weight PLL molecules. Just the opposite was found experimentally. Ref 17 also reports that, when the molecular weight of PLL was fixed, increasing the chain length of hyaluronic acid decreased film thickness. It is not clear to us how this outcome could have been predicted by the “in and out diffusion” hypothesis. Whether the chain length of PLL or hyaluronic acid was fixed, film growth became linear after some number of layers were deposited, even though all films contained diffusing PLL. It is not clear to us how this outcome could have been predicted by the “in and out diffusion” hypothesis.13
17

’ COMMENT Certain elements of the “in and out diffusion” hypothesis are convincing. We do not dispute whether some polyelectrolyte molecules migrate in some multilayer films. Nor do we dispute whether a subpopulation of a species of polyelectrolyte can display net migration from the film surface toward the substrate (“inward diffusion”) under some circumstances, or from within the film volume toward the surface (“outward diffusion”) under others. Nor do we dispute the likelihood that some ionized side chains in films will be compensated by counterions. Instead, the apparent inconsistencies between the “in and out diffusion” hypothesis13
17 and the results presented above have led us to question whether the hypothesis makes any testable

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predictions regarding the character and role of molecules in the film buildup process. Consider a film made of oppositely charged polyelectrolytes A and B. Let there be two classes of identical species A in the film, A1 and A2. Let A2 signify the molecules known from experiments, say fluorescence studies, to migrate inward and not simply to form a complex on the film surface. The “in and out diffusion” hypothesis says that A1 and A2 are qualitatively different with regard to character or role in film buildup; that one is (partially) free in the film reservoir and the other is (mostly) bound in complexes with B; that one class differs from the other in various ways, for instance, charge compensation. We question whether it has ever been shown that A1 and A2 are different and whether it is possible to show they are different, if they are different. We assume that the addition of a small number of fluorescent labels to a polymer does not result in an essential change of character or role. We do not question whether the hypothesis that “a diffusionbased buildup mechanism ... explains most of the exponential-like growth processes of polyelectrolyte multilayers reported in the literature”14 is consistent with certain experimental observations. Examples include evident polymer migration in film system P, evident exponential growth in film system P, lack of evident polymer migration in film system Q, and lack of evident exponential growth in film system Q. If it is assumed, however, that the behavior of system P implies that polymer diffusion causes exponential growth,13
17 then by elementary logic the behavior of system Q does not imply that polymer diffusion causes exponential growth. It seems suggestions to the contrary should be supported by evidence. A claim completely irrelevant to a hypothesis will be consistent with the hypothesis simply by not being contradictory. Is an example known in which the polyelectrolyte constituents of a multilayer film diffuse in and out but film buildup is linear? If so, the “in and out diffusion” hypothesis will be weakened. The answer is yes, as noted above.16,17 (The same film systems displayed exponential growth at an early stage of the buildup process.) An instance of exponential film growth in the demonstrated absence of polyelectrolyte diffusion will further weaken the hypothesis. We do not suppose our lack of awareness of any such example can be taken to imply that polymer diffusion causes exponential growth. We do not question the Debye
H€uckel law or the Donnan effect in general terms. Nor do we doubt the relevance of Debye
H€uckel law to polyelectrolytes in solutions or films or the possible relevance of the Donnan effect to some aspects of multilayer films. We question whether the association of the Debye
H€uckel law or the Donnan effect with the “in and out diffusion” hypothesis15 can provide it with a firm theoretical underpinning, lead to the prediction of any subsequently observed film property, enable the evaluation of the electrostatic potential of polyelectrolytes within films, or enable the demonstration of “free” polyelectrolyte molecules trapped in films known to grow exponentially. We question whether the “in and out diffusion” hypothesis explains exponential film buildup. Ref 14, for example, describes the hypothesis as a “common rule” of exponential multilayer film buildup, “a diffusion-based buildup mechanism that explains most of the exponential-like growth process of polyelectrolyte multilayers reported in the literature”. Ref 15 says, “the existence of the inward and outward diffusion mechanism seems to be of primary importance in the attempt to understand the formation 5702

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Figure 2. Models of film buildup. The substrate for film fabrication is shown as a horizontal slab. The film resulting from successive polymer adsorption steps is shown in red, green, cyan, orange, and navy. (A) Island model. (B) Dendritic model. (C) Approximate growth of material deposited.

of an exponentially growing multilayer,” and “the mechanism responsible for ... exponential growth cannot be explained solely the interaction of polyelectrolytes from the solution with those that constitute the other layer of the film.” We question these statements, not the data presented in the same references. From what we can tell, certain ambiguities complicate testing the “in and out diffusion” hypothesis. We note four of them. One, it is not clear how tightly polyelectrolytes in the hypothesized reservoir will bind the film. Ref 15 describes polymers in the reservoir as “free”. If instead only uncompensated charges are “free”, how do these charges differ qualitatively from any other charged groups on polymers in a film? It is implied that “free” is not merely the ability of a polymer to change position, which will occur in response to thermal fluctuations, especially for relatively weakly bound polymers, but a qualitative difference in binding relative to other molecules of the same species in the film. Has this difference been demonstrated? Has it been demonstrated that polymer or charge in excess of that required for charge compensation is present in a film that grows exponentially? And does this charge excess vanish when a film goes from exponential growth to linear, as in refs 16 and 17? Two, it is not clear whether the polymers (or uncompensated charges) said to form the reservoir will be solvated as in bulk solution. Three, there is lack of clarity concerning the meaning of “outward diffusion”. Is this the net migration of free polymers toward the film surface and that alone, or the exchange of previously adsorbed polymers for more recently adsorbed ones in the absence of a change in the amount of free polymer in the film, or the migration of polymers out of the film and into the surrounding solution, or some combination of these? Four, there is ambiguity regarding the forces whereby free polymers are supposed to be driven into the film and remain there. “Free” polyelectrolytes in the film will be electrostatically repelled by an electrostatic barrier on the film surface, which now has the same sign as the polymers hypothesized to have diffused in? Ref 15 claims the electrostatic barrier is a prediction of the “in and out diffusion” hypothesis. Does the “in and out diffusion” hypothesis provide unique and testable predictions? It is unclear to us what they are. If a film grows exponentially, then at least one polymer diffuses? Refs 13
15 make much broader claims. The “in and out diffusion” hypothesis presumes a causal link between independent observations (polymer migration and exponential growth) when the causal link is what should be established. Ref 14 claims that polymer diffusion provides a molecular explanation of exponential film

COMMENT

buildup. Ref 15 says that “the mechanism responsible for ... exponential growth cannot be explained solely by the interaction of polyelectrolytes from the solution with those that constitute the outer layer of the film.” As far as we know, it has not been demonstrated or convincingly argued that exponential growth cannot be explained solely by the interaction of polyelectrolytes from solution with the film surface. Nor is it not clear to us how the argument is any different if the Debye
H€uckel law and the Donnan effect are taken into account, or if hypothetical “noncompensated fixed charges within the film” are substituted for hypothetical “free” polyelectrolytes in the film.15 Two alternative models that seem no less consistent with exponential growth than the “in and out diffusion” model and are fully compatible with polymer diffusion are an island model and a dendritic model. Both are illustrated in Figure 2. Ironically, the island model (Figure 2A) was discussed in ref 13 but considered less mechanistically explanatory than the “in and out diffusion” model, which was then more fully elaborated in refs 14
17. It may also have been assumed that the island model was consistent with the “in and out diffusion” hypothesis, that the latter filled gaps left by the former, that there was no obvious contradiction. The island model can be considered a variant of the surface roughness model, which has been discussed in ref 23. The substrate is presumed to become only partly coated by the first polymer adsorption step. The second adsorption step results in more extensive coverage of the substrate and film buildup. It is assumed for the sake of illustration that the film thickness increment is a constant. Early in the fabrication process, the amount of material deposited increases with successive adsorption steps, because the radius and height of each island increase incrementally. The average rate of growth over the entire film is exponential. Later, when the islands begin to coalesce, the rate of deposition will decrease and become constant—if the substrate surface available for fabrication is fixed and the thickness increment is approximately constant. This model seems consistent with the data of refs 16 and 17, in which films that grow exponentially at an early stage of fabrication show linear growth later, with the AFM film surface morphology data presented in ref 24, and with the data presented in Figure 1. Refs 16 and 17 seek to account for the observed behavior by adding details to the “in and out diffusion” hypothesis of refs 13
15. Figure 2B depicts the dendritic model. Experimental support for this view is found in the well-known tendency of linear polyelectrolytes to form brush-like structures on an oppositely charged surface.25,26 Tethering in the present case will be due to electrostatic attraction and van der Waals interactions. If the degree of polymerization is high enough, the same polymer can bind to a surface at more than one location along the chain, and more than one location can be surrounded by counterions as in solution. Starting with the second polyelectrolyte deposition step of a film buildup process, one or more adsorbing polymer molecules may bind to the same adsorbed polymer molecule on the film surface, yielding branch-like structures. Such structures will form if they represent an entropy maximum for the quasi-equilibrium state reached by the end of a polymer adsorption step or the subsequent rinse step. In Figure 2B, a Y-shaped object has been selected for convenience to represent a bound polyelectrolyte molecule for which multiple locations are surrounded by counterions. As with the island model, there is a degree of similarity to the film roughness model discussed in ref 23. In any case, the amount of material deposited will increase with successive adsorption steps and will continue to do so in the 5703

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Langmuir absence of steric hindrance, because the number of polymer binding sites, the number of incompletely adsorbed regions after a polymer deposition step, keeps increasing. The buildup of film material averaged over the entire surface of the film will be an exponential function of adsorption steps, at least initially. The film surface area and therefore the number of binding sites will not grow without bound: there will be a leveling off of the rate of buildup. This model too seems consistent with the data of refs 16 and 17 and the data in Figure 1. Figure 2C shows a schematic of film buildup for the two models just described. Data points in the shaded region of the graph may show an exponential rise for an increase in the number of adsorption steps. Both models will show a decrease in the rate of material deposited as the fabrication process proceeds under the constraints mentioned here, as in refs 16 and 17. Neither the island model nor the dendritic model makes any particular assumption concerning polymer diffusion. Of course, this need not mean that diffusion will not occur in the island model or the dendritic model. It may provide further reasons to doubt, however, whether the diffusion of hypothetically “free” polymers or parts thereof in a polyelectrolyte film causes exponential buildup.

’ CONCLUSION The “in and out diffusion” hypothesis posits not merely that polymer diffusion occurs in multilayer film that grow exponentially but that diffusion causes exponential buildup. To the best of our knowledge, the causal relationship has not been demonstrated. Specific predictions of the hypothesis, for instance, that an increased rate of diffusion will lead to an increased rate of film buildup, are contradicted by experimental results. There are numerous ambiguities as to what the hypothesis predicts and whether predictions made by the hypothesis can be tested. Nevertheless, the model is widely accepted. The “in and out diffusion” model may provide a less accurate description of exponential film buildup than the island model or the dendritic model. We believe further study will clarify the mechanism of polyelectrolyte multilayer film buildup.

COMMENT

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’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ1 (813) 974.7793. Fax: þ1 (813) 974.5813.

’ ACKNOWLEDGMENT We thank Pierre Schaaf for comments on the manuscript. E.C. was supported by a U.S. Army Medical Research and Materiel Command award to the Center for Integrated Functional Materials (W81XWH-07-1-0708). The project was also supported in part by a Faculty Research and Development Grant to DTH. ’ REFERENCES (1) Decher, G.; Schlenoff, J. B. Multilayer Thin Films: Sequential Assembly of Nanocomposite Materials; Wiley-VCH: Weinheim, Germany, 2003. (2) Schlenoff, J. B.; Dubas, S. T.; Farhat, T. Langmuir 2000, 16, 9968–9969. (3) von Klitzing, R.; M€ohwald, H. Langmuir 1995, 11, 3554–3559. (4) Mendelsohn, J. D.; Barrett, C. J.; Chan, V. V.; Pal, A. J.; Mayes, A. M.; Rubner, M. F. Langmuir 2000, 16, 5017–5023. 5704

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