Langmuir 1989,5, 1253-1255
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Letters A Computer Model for the Effect of Surfactants on the Aggregation of Associating Polymers Anna C. Balazs* and Jenny Y. Hu Materials Science and Engineering Department, 848 Benedum Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 Received January 10, 1989. I n Final Form: April 19, 1989 We have developed a computer simulationto model the aggregation of associating polymers and Surfactants in solution. Using this model, we investigated the effect of surfactant tail length on the size and geometry of the resulting aggregate. These results show that there is a critical tail length: below this value, the surfactants promote aggregation; above this value, the long surfactant tails sterically hinder the growth of clusters. Consequently, we predict that short surfactants will enhance the viscosity of these solutions, while solutions containing the long-tailedsurfactants will have a lower Viscosity than that of the pure polymer solution. We cite recent experimental observations that confirm these predictions. Finally, we hypothesize that the value for the critical tail length will depend on the surfactant-to-polymer concentration ratio.
Introduction Associating polymers are flexible macromolecules that contain a number of sites that strongly attract each other. In solution, the strong interaction between these sites leads the chains to aggregate and form clusters. The presence of these clusters alters the rheological properties of the medium and, thus, these polymers have found wide applications as viscosity modifiers. Of particular interest are polymers in which the associating sites or “stickers” are located at both ends of the chain. Examples of such polymers include telechelic ionomers and hydrophilic chains that contain terminal hydrophobic segments. In the case of these water-soluble polymers, nonionic surfactants are frequently added to further control the viscosity of the solution. The surfactants are composed of a short hydrophobic “head” and a longer hydrophilic “tail”. Here, the hydrophobic interactions’ drive all the different species to self-assemble. The solution viscosity has been observed to vary widely with surfactant tail length,2 but no fundamental understanding of the role of surfactant geometry has yet emerged. In this letter, we present computer simulations in two and three dimensions for the aggregation of associating polymers in the presence of surfactants. The model will be used to develop an understanding of the effect of surfactant tail length on the microscopic association phenomena and, consequently, the macroscopic rheological behavior. In the simulation, the polymer chains contain two stickers, one at each chain end. The surfactants, on the other hand, have a sticker on only one end. Of particular interest is the case where the stickers on the polymer and surfactant are of equal block length. We assume that the molecules aggregate through the attractive sticker-sticker interactions. Attention is focused on the polymer-surfactant interaction. Specifically,a surfactant in solution can interact with another surfactant only if the latter is already bound to the polymer chain. We do not examine the self-associationof free surfactants into isolated (1) Tanford, C. The Hydrophobic Effect; Wiley Press: New York,
1973. (2) Bassett, D., private communication.
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micelles, since our primary objective is to investigate how the polymer-surfactant interaction can contribute to the experimental behavior described above. Furthermore, in dilute solutions that contain surfactants below the critical micelle concentration (cmc), no micelles will be formed. The simulations will be particularly relevant to this regime. Finally, it is assumed that the stickers bind irreversibly. Though a reversible simulation is experimentally more realistic, we believe this model will illustrate how variations in such geometric factors as tail length affect the overall aggregation behavior.
The Model The general procedure for the simulation has been described in detail e l s e ~ h e r ethus, ; ~ we will give only a brief description here. The algorithm is similar to that used to study diffusion-limited aggregation. The simulation is started by placing a seed chain of specified length, but with self-avoiding random configuration, at the center of a three-dimensional lattice. The configuration of the first chain remains fixed. This first chain represents an associating polymer: the last lattice bond at both ends of the chain is designated as a “sticker”. Next, a decision is made whether to introduce a surfactant or another associating polymer onto the lattice; both events are given equal pr~bability.~ The surfactant has a “sticky” lattice bond only on one end of the chain. It too is of random selfavoiding configuration and specified length. The chosen molecule is placed at a large distance from the fixed chain and allowed to execute a self-avoiding random walk. The random walk consists of a translation and a “wiggling” motion. The translation is accomplished by moving the entire chain one lattice site, in a direction to be picked at random. The “wiggling” motion simulates the chain dynamics by using the Verdier-Stockmayer algorithm,5 with the corrections suggested by Hilhorst and (3) Balazs, A. C.; Anderson, C.; Muthukumar, M. MacromoZeLiles 1987,20,1999. (4) By altering the probability that a polymer or surfactant is intro-
duced, we can vary the polymer-to-surfactant ratio. Consequently, we can also investigate how this ratio affecta the aggregation process. This will be the topic of future work.
0 1989 American Chemical Society
1254 Langmuir, Vol. 5, No. 5, 1989
Deutch.6 The walk continues until a sticker on the diffusing species is parallel and adjacent to a sticker on the seed chain. The alligning sticker on chain 2 remains stuck at this position; however, the remaining portion of the chain is free to wiggle. Now a third chain is picked and added to the lattice: with equal probability, it can be either a polymer or a surfactant. At this point, one of the chains (number 2 or 3) is selected. If the free chain (number 3) is picked, it is allowed to translate and wiggle. If a chain already in the cluster is picked (chain 2), it is only allowed to wiggle. A check is made to see if these motions have resulted in any new sticker-sticker pairings. If no new pairings occur, the above steps are repeated. If a new pairing is found, another chain (either polymer or surfactant) is added, and the above steps are repeated. A polymer with one stationary end is referred to as "partially frozen". When both ends of the polymer become paired in the manner described above, the chain becomes less mobile. Consequently, the entire chain remains fixed in position and is referred to as "frozen". For a polymer chain that is frozen, there are two types of sticker-sticker interactions: interchain or intrachain. In the first, both ends of a given chain are paired with stickers belonging to other chains. In the second, the end of a given chain is paired with its own end to form a self loop. Since the surfactant can have at most only one stationary sticker, all attached surfactants are partially frozen, and their tails can continue to wiggle. A sticker on a newly introduced chain is only permitted to pair with a sticker on a partially frozen or frozen species. In this way, new chains bind to chains already in the cluster, and the cluster keeps growing. Note that we focus our attention only on the formation of a single cluster. The simulation comes to a halt when a specified number of time steps have been executed or a specified number of chains have been incorporated into the cluster.
Results and Discussion In order to investigate the influence of surfactant tail length on the morphology of the cluster, the length of this tail was systemically varied. In three dimensions, the tail length was altered from one to eight sites, and simulations were run for 600000 time steps. The length of the surfactant head was held constant at two lattice sites (one lattice bond), and the polymer chain length was also held constant at 20 lattice sites. Five independent runs were executed for each specified value of the surfactant length. After the specified number of time steps was executed, the number of each species (polymer and surfactant) and the total number of species in the cluster (the aggregation number) were calculated. In addition, the number of surfactant-polymer, polymer-polymer, and surfactantsurfactant bindings was tabulated. The radius of gyration for the cluster was also obtained. The significant effect of tail length on the size of the cluster is clearly evident in Figure 1. Here, the aggregation number is plotted versus total surfactant length for the simulations in 3-D. The point on the y-axis (surfactant length = 0) corresponds to the case where no surfactants are present and the cluster is composed entirely of associating polymers. With a similar simulation, the behavior of such surfactant-free clusters has been studied previ0us1y.~As can be seen from the curve, short surfactants actually promote aggregation: the aggregation number is greater than the pure polymer case for surfactant lengths (5) Verdier, P. H.; Stockmayer, W. H. J . Chen.Phys. 1962,36, 227. (6)Hilhorst, H. J., Deutch, J. M. J. Chem.Phys. 1975,63,5153.
Letters
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Figure 1. Plot of aggregation number (total number of species in the cluster) versus total surfactant length. The point on the y-axis corresponds to a cluster composed entirely of associating polymers. The results shown are for the 3-D simulations. Each point represents an average over five independent simulations, which were run for 600000 time steps each.
8.0 0.6
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Figure 2. Plot of In N (aggregation number) versus In L (total surfactant length) for L 5 3. The slope of the line is -1.55. of 6 and less. Beyond a surfactant length of 6, the aggregation number is actually lower than obtained for the pure polymer case. Plotting the natural logarithm of N (aggregation number) versus the natural logarithm of L (surfactant length) for L I3 in Figure 2 clearly shows that N decreases with L through a power law dependence. Specifically, by measuring the slope of the curve, we find that N L-1.55. Since polymers and surfactants are introduced with equal probability, approximately half the species in the cluster are polymers and half are surfactants. This ratio remains fairly constant for all examined values of the surfactant length. However, while the number of surfactant-surfactant and polymer-surfactant interactions also remains relatively uniform with changes in tail length, the number of polymer-polymer bindings increases as the surfactant length is increased. (To account for variation in cluster size, the absolute number of bindings between two species is divided by the total number of molecules in the cluster to yield an appropriately normalized value.) These observations can readily be explained by examining the frames in Figure 3, where results from similar simulations in two dimensions are shown. These 2-D images are easier to analyze than figures from the 3-D simulations and, thus, provide valuable visual insight into the effect of the surfactant tail. As can be seen in parts a and
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Langmuir, Vol. 5, No. 5, 1989 1255
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