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Langmuir 1989,5, 1230-1234
Effects of Surfactant Concentration on Polymer-Surfactant Interactions in Dilute Solutions: A Computer Model Anna C. Balazs* and Jenny Y. Hu Materials Science and Engineering Department, 848 Benedum Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 Received February 17, 1989. In Final Form: April 27, 1989 We have developed a computer simulationto model the aggregation of associating polymers and Surfactants in a dilute solution. Using this model, we examine the effect of the polymer-to-surfactantconcentration ratio on the size of polymeHurfactant aggregates. Our results predict that for polymeHurfactant association there exists a critical surfactant concentration. Below this critical value, an increase in surfactant concentration promotes association; above this value, an increase in the concentration of surfactant causes a decrease in the total aggregation number. This effect is most pronounced for short-tailed surfactants. We discuss the consequencesof these results on the determination of an appropriate polymer-to-surfactant ratio to be used for enhancing the viscosity of a solution.
Introduction Aqueous solutions containing a mixture of polymer and surfactant molecules are used in a wide variety of industrial applications, including enhanced oil recovery.' One reason for the widespread utility of this mixture is its role as a viscosity modifier. In water, the strong interactions between amphiphilic or charged polymer and surfactant molecules cause these species to aggregate and form clusters. The presence of these clusters, in turn, modifies the viscosity of the host fluid. Despite its technological significance, the science of combining polymer, surfactant, and water to achieve the desired solution viscosity remains a black art. Furthermore, what progress has been made in this field has come about primarily through an Edisonian or trial and error appr0ach.l Consequently, developing models to examine the nature of polymer-surfactant interactions in solutions constitutes a significant priority. In a recent article, we developed a computer simulation to model the aggregation of polymers and surfactants in solution.2 Here, the polymers contained two associating sites or "stickers", one a t each chain end. Examples of such polymers include telechelic ionomers and hydrophilic chains that contain terminal hydrophobic segments. The surfactants, on the other hand, had a sticker on only one end. This architecture represents a surfactant with a short hydrophobic "head" and a longer hydrophilic "tail". Using this model, we investigated the effect of surfactant tail length on the aggregation behavior. Our results predided that there is a critical tail length: below this value, the surfactants promote aggregation between the polymer chains and, thus, enhance the viscosity of the solution. Above this value, the long surfactant tails sterically hinder association. We concluded that a solution containing the long-tailed surfactants will have a lower viscosity than a pure polymer-water solution. Our predictions coincided with recent experimental evidence3 and, thus, helped demonstrate the utility of using these simulations in correlating microstructure to macroscopic behavior. In this article, we use the same model to investigate the effect of surfactant concentration on the size and shape of the polymer-surfactant aggregates. Our interest is specifically focused on the nature of the polymer-surfactant versus polymer-polymer interactions and how these interactions are affected by varying the polymer-to-sur-
factant concentration ratio. Consequently, we do not examine the self-associationof free surfactants into isolated micelles. In our model, a surfactant in solution can interact with another surfactant only if the latter is already bound to the polymer chain. Clearly, below the critical micelle concentration (cmc) of the surfactant, polymer-surfactant and polymer-polymer associations represent the primary interactions between the species in solution. For concentrations above the surfactant cmc, these again represent the primary interactions if the attractive energy between the polymer binding site and the surfactant is greater than the mutual attraction between the surfactant molecule^.^ Outside of these conditions, neglecting the presence of isolated micelles will modify any quantitative predictions. However, the qualitative aspects of our observations on the nature of polymer-surfactant interactions will still be valid. Finally, it is assumed that the stickers bind irreversibly. While a reversible-binding model more accurately corresponds to experimental reality, our results accurately illustrate the significant role geometric contraints play in controlling polymer-surfactant interactions.
(1) Surfactants; Tadros, F. Th., Ed.; Academic Press: London, 1983. (2) Balazs, A. C.; Hu, J. Y . Langmuir, in press. ( 3 ) Bassett, D., private communication.
(4) Nagarajan, R. Polym. P r e p . 1981,22, 33. (5) Balazs, A. C.; Anderson, C.; Muthukumar, M. Macromolecules
The Model The general procedure for the simulation has been described in detail elsewhere;@thus, we will give only a brief description here. The algorithm is similar to that used to study the diffusion-limited aggregation (DLA) of polymer chains! By using a DLA model, we are assuming that the interaction between the associating sites on the various species is sufficiently strong that the reaction is limited only by the ability of these reactants to "find one another". This is a valid assumption for dilute solutions of molecules that bind through strong electrostatic attractions or via the hydrophobic effect. The simulation is started by placing a seed chain of specified length and self-avoiding random configuration at the center of a three-dimensional lattice. The location of the first chain remains fixed. The fist chain represents an associating polymer: the last lattice bond on 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. By varying the probability that a polymer or surfactant is introduced, we can
1987, 20, 1999.
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Langmuir, Vol. 5, No. 5, 1989 1231
Polymer-Surfactant Interactions: A Computer Model
alter the polymer-to-surfactant concentration ratio. Specifically, we define q as the probability that a surfactant will be added to the lattice and 1 - q as the probability that a polymer will be introduced. We specify a value of q that can range from 0 (no surfactant) to 1(no polymer). Next, a random number generator is invoked to produce a number between 0 and 1. If the resulting number is greater than q, a polymer is introduced; if the number is less than or equal to q, we introduce a surfactant. The surfactant molecule has a “sticky” lattice bond on only one end of the chain. It too is of random self-avoiding configuration and specified length. The chosen molecule is placed a large distance from the fixed seed chain and allowed to execute a self-avoiding random walk. Note that whether a solution is dilute or concentrated can be modeled by varying the placement of the new, incoming chains.6 For the dilute solution, the chains start their walk from a point on the circumference of a circle whose radius is large compared to the radius of gyration of the cluster, R,. (For the more concentrated solution, the chains are introduced from a distance closer to, but still greater than, the value of Rg.)Here, we are only concerned with the case of dilute solutions. The random walk executed by these molecules consists of a translation and “wiggling” motion. The translation is accomplished by moving the entire chain one lattice site, in a direction to be picked a t random. The “Wiggling” motion models the chain dynamics by using the VerdierStockmayer algorithm6 with the corrections suggested by Hilhorst and Deutch.’ The walk continues until a sticker on the diffusing species is parallel and adjacent to a sticker on the seed chain. The aligning sticker remains stuck at this position; however, the remaining portion of the chain is free to wiggle. A chain with one stationary sticker is referred to as “partially frozen”. Next, a third chain is picked and added to the lattice: depending on the specified probability, it can be either a polymer or surfactant. At this point, a chain is picked a t random from the available chains on the lattice. If the free chain (number 3) is picked, it can translate and wiggle. If chain 2 is picked, it is allowed to wiggle. If this second chain represents a polymer, two stickers can interact to form a self-loop. After each move, a check is made for any new sticker-sticker pairings. If no new pairings are found, the above procedure is repeated. If a new pairing is found, another chain (either polymer or surfactant) is added, and the above steps are repeated. Note that all moves obey the excluded volume criteria. When both ends of the polymer become paired, the chain becomes less mobile. Consequently, the entire chain remains fixed in position and is referred to as “frozen”. There are two types of sticker-sticker interactions for a frozen polymer: interchain or intrachain. In the f i t , both ends of the 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, as noted above. Since the surfactant can have a t most one stationary end, all attached surfactants are partially frozen. Thus, their tails can continue to wiggle throughout the simulation. 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 (6) Verdier, P. H.; Stockmayer, W. H. J. Chem. Phys. 1962,36, 227. (7) Hilhorst, H. J.; Deutch, J. M. J. Chem. Phys. 1975, 63, 5153.
4
0.0
!0.0
1.0
2.0
3.0
4.0
E.0
5.0
I 7.0
8.0
SURFACTANT TAIL LENGTH Figure 1. Aggregation number versus surfactant tail length for q = 0.1 (o),0.3 (o),and 0.5 (0). The curves through the data points are drawn as a guide for the eye. Table I. Aggregation Number versus q
surfactant tail length
w 1 2
3 4 5 6 8
q = 0.1
30f2 291 f 16 167 f 68 201 f 37 194 f 63 178 f 48 87 f 43 58 f 54
aggregation number q = 0.2 Q = 0.3 30f2 30f2 213 f 45 220 f 44 127 f 24 123 f 32 71 f 14 50 f 14 36 f 14 43 f 32 48 30 35 f 6 42 f 23 25 f 7 35 15 59 f 39
*
*
q = 0.5
30f2 99 f 49 96 f 38 44 f 18 41 & 10 33 f 15 24 f 7 17 f 7
This represents the case where no surfactants are present and the cluster is composed entirely of associating polymers.
simulation comes to a halt when a specified number of chains have been incorporated into the cluster or a specified number of time steps have been executed.
Results and Discussion In the simulations, the length of the polymer was held constant a t 20 lattice sites. The length of the surfactant was varied from 3 to 10 lattice sites. The following values of q were examined 0, 0.1,0.2,0.3, and 0.5. Note that the last case implies that the polymer and surfactant have an equal probability of being introduced onto the lattice. Three independent runs were performed for each specified value of q and surfactant length. The simulations were run for 600000 time steps. The significant effect of varying the polymer-to-surfactant ratio is graphically displayed in Figure 1, where the aggregation number versus surfactant tail length is plotted for q = 0.1, 0.3, and 0.5. The actual values (including those for q = 0.2), with the error bars, are included in Table I. The point on the Y-axis corresponds to the case where no surfactants are present and the cluster is composed entirely of associating polymers. The aggregation number is the total number of species in the cluster (polymer and surfactant). In a previous article,2 we discussed the effect of varying the surfactant tail length a t q = 0.5. Here, we are interested in understanding the difference in the aggregation behavior at fixed surfactant tail length but different values of q. As Figure 1and Table I indicate, the lower surfactant/polymer ratios are more effective in promoting aggregation than the case where the surfactant/polymer ratio is equal to 1, Le., q = 0.5. Moreover, the data indicate that the difference between the various q values is most significant for the shortest surfactant tail length.
1232 Langmuir, Vol. 5, No. 5, 1989
Balazs and Hu
b
a
2 d"
0
C
d
I
r
L
Figure 2. (a)State of a cluster after 'ZOO00 computer time steps. Here, q = 0;consequently, the cluster is composed entirely of associating polymers. The stickers are drawn with a thicker line to distinguish them from the rest of the chain. The frozen stickers are drawn with the thickest lines. There is a total of 10 chains in the cluster. (b) Here q = 0.1. There are 10 chains and 1 surfactant in the cluster. This configuration was obtained after 39235 computer time steps. (c) Again, q = 0.1. The f i e shows the state of the cluster after 250000 time steps. There are 50 chains and 2 surfactants in the cluster. (d) Here, q = 0.5. The figure shows that state of the cluster after 250000 time steps. There are 7 chains and 15 surfactants in the cluster.
To understand the difference between the q = 0.1 and q = 0.5 cases, it is helpful to first appreciate the difference in behavior between q = 0.1 and q = 0, the case of no surfactant. In a dilute solution, the number of free polymer chains in the vicinity of a cluster is fairly low. This has two important consequences. First, the number of intrachain interactions or self-loopsin a cluster is relatively high. Second, the remaining binding sites form multichain loops through a series of interchain interactions. Both cases lead to a situation where there are very few or no available binding sites to which a new incoming chain can attach. This situation is clearly seen in Figure 2a, where no surfactant has been added to the solution. This figure is generated by performing the simulation in two dimensions. Since figures in 2-D are easier to interpret than 3-D images, the results obtained from these 2-D calculations yield significant insight into the reaction mechanism. Most importantly, the figure shows that when chains interact via multichain loops or self-loops the availability of binding sites is decreased since the nonbonding segments of the polymer can sterically hinder other chains from joining the cluster. Though more pronounced in 2-D, this effect will also contribute to limiting the size of the cluster in 3-D. Now, we add a small amount of surfactant: q = 0.1. In Figure 2b, there are 10 chains in the cluster (as in Figure
2a) and one short surfactant (taillength equals one lattice bond; thus, the total surfactant length equals three lattice sites). As can be seen, the presence of the one surfactant molecule has dramatically changed the structure of the cluster! (Note too that it took significantly fewer computer time steps to generate this cluster.) The short surfactant, with only one sticky end, can (1)sterically inhibit self-loops or intermolecular loops (see Figure 2b) and (2) promote intermolecular bridging by binding to more than one p ~ l y m e r . ~Through .~ these mechanisms, the surfactants effectively act to free up ends that were previously lost by reacting with each other to form loops. This results in the more open structure seen in Figure 2b, where there are several available binding sites to which a new chain could attach. For the 3-D cases, the dramatic decrease of selfloops in the presence of surfactants is documented in Table 11. (We comment further on these values later in the discussion.) Finally, in Figure 2c, we see the state of the q = 0.1 cluster after 250000 time steps, the number of steps it took to generate the image in Figure 2a, where q = 0, i.e., no surfactant. Note the dramatic difference in aggregation number: in part a there are 10 polymers in the cluster, while in part c there are 52 molecules (50 polymers, 2 surfactants) in the cluster! Furthermore, the later cluster again displays a significant number of available binding
Langmuir, Vol. 5, No. 5, 1989 1233
Polymer-Surfactant Interactions: A Computer Model Table 11. Fraction of Self-Loops versus q surfactant tail length 0' 1 2 3 4 5 6 8
% self-loopsb
a = 0.1 0.61 f 0.15 0.308 f 0.009 0.310 f 0.035 0.276 f 0.034 0.281 f 0.034 0.343 f 0.063 0.352 f 0.078 0.42 f 0.13
9
= 0.2
9 = 0.3
0.61 f 0.15 0.315 f 0.014 0.259 f 0.042 0.425 f 0.007 0.307 f 0.085 0.382 f 0.053 0.25 f 0.19 0.51 f 0.12
0.61 f 0.15 0.284 f 0.026 0.273 f 0.041 0.316 f 0.025 0.35 f 0.11 0.352 0.026 0.47 f 0.17 0.423 f 0.079
9
= 0.5
0.61 f 0.15 0.30 f 0.10 0.40 f 0.13 0.30 f 0.13 0.378 0.088 0.40 f 0.13 0.55 f 0.13 0.556 f 0.032
"This represents the case where no surfactants are present and the cluster is composed entirely of associating polymers. bThis value was obtained by dividing the number of self-loops in the cluster by the total number of chains in this cluster.
sites. This clearly demonstrates that a small concentration of surfactant promotes aggregation. However, the dramatic effect of increasing the surfactant-to-polymer ratio is seen in Figure 2d, where q = 0.5. Now the surfactant and polymer are introduced onto the lattice with equal probability. Again, the short surfactant is three lattice sites in length, and the figure represents the state of the cluster after 250000 time steps, as in parts a and c of Figure 2. Here, the aggregation number is 22 (7 chains, 15 surfactants). This is significantly less than the value obtained for q = 0.1. (See Table I for the values in 3-D, which were obtained from calculations that were run for longer times and averaged over three separate simulations.) The behavior described above can be understood if we draw an analogy between these interactions and reactions in step-growth polymerization. For this reason, we will consider the surfactants as monofunctional monomers, with the reactive species at only one end of the chain, and the polymers as bifunctional monomers, with reactive groups on both ends. In step-growth polymerization, a small amount of monofunctional monomer is added to terminate the polymerization reaction and, thus, achieve the desired molecular weight.s The monofunctional unit limits the polymerization of bifunctional monomers because its reaction with the growing polymer yields chain ends devoid of a functional group and, therefore, incapable of further reaction. On completion of this reaction, the degree of polymerization, X,,is given by8
8, = 1 + N/NM where NMis the number of monofunctional monomers and N is the number of bifunctional molecules. As can be seen, the ratio of bifunctional to monofunctional units has a tremendous effect on the degree of polymerization or, as in our case, the aggregation number. This simple model is only qualitatively similar to our situation. It is quantitatively different since a functional group in our model can bind to more than one species, as is common when electrostatice or hydrophobiclo interactions are involved. Furthermore, eq 1does not take into account the formation of loops. Consequently, it cannot explain the initial increase in aggregation number seen in Figure 1. (In order to understand this behavior, we refer to eq 2 below.) However, the mechanisms that limit growth a t higher surfactant or N M concentration are the same in both situations. (8)Odian, G. Principles of Polymerization, 2nd ed.;Wiley New York, 1970; pp 86, 114. (9) Hegedus, R. D. Ph.D. Dissertation, University of Massachusetts, Amherst, 1985. (10) For examples, see such references as: Eicke, H.-F. In Micelles; Dewar, M. J. S.,Ed.; Springer-Verlag: Berlin, 1980; p 85.
Another way of summarizing our results is to recall a modified form of eq 1,the Carothers equation for polymerizations where we have again assumed that the reaction has been carried to completion. The term fa, is the average monomer functionality in the system. we have shown, a small fraction of surfactants actually increases the average functionality by exposing ends that would otherwise be consumed in loops. However, as the fraction of surfactants (monofunctional units) is increased, the average functionality of the system decreases. Consequently, the aggregation number of the cluster also decreases. Our results indicate that for polymer-surfactant association there exists a critical surfactant concentration. Below this critical value, an increase in surfactant concentration promotes association; above this value, an increase in the concentration of surfactant causes a decrease in the total aggregation number. This effect is particularly pronounced for short-tailed surfactants. We are currently attempting to develop an analytical model with which we could calculate an exact value for this critical concentration under a variety of reaction conditions. As the error bars in Table I indicate, this effect is less pronounced for longer surfactant tail lengths. Here, as we demonstrated in a previous report,2 the effects of steric hinderance dominate the behavior of the cluster: the long, flexible tails sterically prohibit new chains from joining the cluster. However, our results do show that a t lower surfactant concentrations longer tail lengths are needed to reduce the value of the aggregation number below that obtained in the pure polymer case. This notion seems intuitively correct: since there are fewer surfactants, longer tail lengths are required to produce the same steric effects. Further examination of Table I1 reveals that for fixed q the fraction of self-loops reaches a minimum value as the surfactant tail length continues to increase. This effect is again due to steric hinderance: since the long tails prohibit new chains from approaching the aggregate, most of the available binding sites belong to chains already in the cluster. To conclude, we note that these predications have important consequences in determining the appropriate polymer-to-surfactant ratio to be used for enhancing the viscosity of a solution. As previously noted,2J1 a larger aggregation number corresponds to a higher solution viscosity. Thus, adding surfactant up to the critical concentration described above will enhance the viscosity of the solution beyond that obtained when polymer alone is used. However, addition of surfactant beyond this value will decrease the viscosity from its optimal value. This
AS
(11) Jenkins, R.; Silebi, C. M.; El-Aasser, M. S., preprint.
Langmuir 1989, 5, 1234-1241
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effect is most pronounced when short-tailed surfactants are used. In furture work, we will examine to what extent the introduction of reversible binding between the stickers affects these calculations. Currently, there are no reports in the literature of a systematic study of the effect of surfactant concentration on the association behavior of polymersurfactant aggregates. We hope the predictions presented above will en-
courage experimentalists to explore this behavior.
Acknowledgment. A.C.B. acknowledges financial support from the donors of the Petroleum Research Fund, administered by the American Chemical Society, the NSF through Grant DMR-8718899, and the Union Carbide Corp. We thank Dr. Chris Lantman for several helpful discussions.
Spinning of Partially Engulfed Drops Y. Shaot and T. G. M. van de Ven* Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Quebec, Canada H 3 A 2A7 Received October 14, 1988. I n Final Form: April 24, 1989
The deformation of spinning partially engulfed drops has been studied both theoretically and experimentally. Two drops of different liquids are suspended in a third liquid and brought into contact. Provided that one drop does not engulf the other, the drops will form a partially engulfed pair, the shape of which is determined by the Laplace equation and the Neumann triangle boundary condition at the three-phase contact line. The shapes of quasi-stationarydrop pairs were found to be fully consistent with theoretical predictions. Slight discrepancies can be ascribed to small amounts of impurities accumulating at interfaces. By subjecting a drop pair to a rotation about its axis of revolution, it will deform to a degree depending on the relative density differences, the interfacial tensions, the drop volumes, and the angular velocity. It is predicted that the total length of the drop pair increases with increasing angular velocity, while the radius of the contact area decreases. Experiments were in good agreement with these theoretical predictions.
Introduction When three immiscible phases make contact with each other, the equilibrium shapes of the drops or bubbles are governed by the Laplace equation,1,2subject to the condition that a t the three-phase contact line (TPL) the interfacial tensions of these three phases form a closed Neumann triangle.3 Using the Laplace equation, Huh and Scriven4calculated the shape of an axisymmetric fluid interface in which a solid object is positioned. Princen et al.23”7 investigated the mechanical equilibrium of solid axisymmetric particles floating in a horizontal liquidffluid interface. Ivanov et a1.8 measured the equilibrium shape of fluid and solid particles at a fluid interface and calculated the line tension a t the TPL. Having found a theoretical solution for the shape of a liquid drop spinning about a horizontal axis, Princen et al.’ improved the method for measuring interfacial tensions with a spinning drop apparatus proposed by V ~ n n e g u t . ~The calculation and measurement of shapes of deformable interfaces are important in many fields related to colloid and interface science, such as petroleum recovery, mineral flotation, etc. The validity of the Neumann triangle has been investigated experimentally with partial success by some authors. In this paper, we propose a method whereby the validity of the Neumann triangle can be tested by studying the shape of a spinning liquid drop pair. The shapes of partially engulfed pairs are also relevant to the concept of line tension, which affects these shapes for small drops. By means of a bispinner,l2l3we can produce partially engulfed drop pairs suspended in a third immiscible phase. When a drop pair spins horizontally about its axis of rotation, ‘Present address: Department of Textile Chemistry, China Textile University, 1882 West Yan An Road, Shanghai, P.R. China.
0743-7463/89/2405-1234$01.50/0
its shape changes to a degree dependent upon the boundary conditions at the TPL. The experimental resulta were compared with those obtained numerically from the Laplace equation. The calculations show that the shape of drop pairs and the radius of the TPL change only by a few percent when the angular velocity was increased to 10 000 rpm. Experiments confirm the general predicted trends, but small deviations were observed which were attributed to the presence of impurities. To investigate these effects further, additional experiments must be performed at higher angular velocities which, because of speed limitations of the bispinner, are presently not possible with our experimental technique.
Theory As is well-known, the equilibrium shapes of drops or bubbles are based on the Laplace equation:
AP = P1 - Pz =
.(;+ $)
where P1and P2 are the pressures inside and outside the drop, respectively. l / R 1 and 1 / R 2 are the principal curvatures of the interface, and u is the interfacial tension. (1) Princen, H. M.; Zia, I. Y. Z.; Mason, S.G. J. Colloid Interface Sci. 1967, 23, 99. (2) Princen, H. M. Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, p 2. (3) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977,66, 5464. (4) Huh, C.;Scriven, L. E. J. Colloid Interface Sci. 1969, 30, 323. (5) Princen, H.M.; Mason, S. G. J. Colloid Sci. 1965, 20, 246. (6) Princen, H.M.; Mason, S. G . J. Colloid Sci. 1965, 20, 156. (7)Huh, C.;Mason, S. G. J. Colloid Interface Sci. 1974, 47, 271. (8) Ivanov, I. B.; Kralchevski, P. A.; Nikolov, A. D. J.Colloid Interface Sci. 1986, 112, 97. (9) Vonnegut, B. Reo. Sci. Instrum. 1942, 13, 6. (10) Fox, W. J. Am. Chem. SOC.1945,67,700. (11) Miller, N. F.J. Phys. Chem. 1941,45, 1025. (12) Hollemever, S. W.: Mar, A. “The Bispinner: A Novel Liquid Micro-manupulator” (to appear). (13) Shao, Y.;van de Ven, T. G. M. Langmuir 1988,4, 1173.
0 1989 American Chemical Society
