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Coil to Globule Collapse Transitions of Polymers: The Effect of Adsorbed Surfactants G. G. Pereira,*,†,‡ D. R. M. Williams,† and D. H. Napper‡ Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra A.C.T. 0200, Australia, and School of Chemistry, University of Sydney, NSW 2006, Australia Received June 26, 1998. In Final Form: December 4, 1998 Recent experimental studies of the coil-to-globule transition of polymer chains in the presence of adsorbing anionic surfactants have shown a variety of novel transition curves. In this paper we theoretically model this system using a modified mean-field model. We show that as the surfactant concentration is increased, the transition becomes a strong first-order transition and occurs in the worse than θ regime. This is due to the adsorbed surfactants abruptly moving off the chain. We also implement a nonconstant density model in an attempt to understand the gradual collapse of these chains before the first-order transition.
One of the most important and fundamental phenomena in polymer science is the coil-to-globule transition which occurs when a polymer is quenched from a good solvent into a poor solvent.1-3 In a good solvent the binary interactions between monomers along a long chain molecule are repulsive and so tend to swell the chain’s conformations. Conversely in a very poor solvent these interactions are attractive, and so the coil conformation is the space-filling globular form. The binary interactions are dependent on the temperature, and so the θ temperature is defined as the temperature at which these interactions are zero. The transition from an expanded coil to a contracted globule is continuous and occurs close to the θ temperature. This picture seems to be fairly accurate for flexible polymers dissolved in organic solvents. In many practical scenarios, polymers are mixed with surfactants and these surfactants often adsorb onto the polymer.4 One particular system of interest was recently investigated by Zhu and Napper.5 This involved poly(Nisopropylacrylamide) (PNIPAM), which has been extensively studied recently.6 At room temperature (≈ 27 °C) PNIPAM chains dissolve in water to form expanded coils. On increasing the temperature, the chains undergo a coilto-globule transition at around 31-34 °C. In a solution with many chains, attractive binary interactions occur not only between monomers along the same chain but also between monomers along different chains. As a result the chains tend to aggregate, thus making the coil-toglobule transition difficult to observe. To prevent this aggregation, Zhu and Napper5 added a small amount of the anionic surfactant sodium dodecyl sulfate (SDS). The amount of surfactant added varied between zero and the critical micellar concentration. The shapes of the resulting size versus temperature transition curves were found to be quite novel and dependent on the SDS concentration. At low or zero SDS * To whom correspondence should be addressed. † Australian National University. ‡ University of Sydney. (1) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: London, 1979. (2) des Cloizeaux, J.; Jannink, G. Polymers in Solution; Clarendon Press: Oxford, 1990. (3) Grosberg, A. Yu.; Khokolov A. R. Statistical Physics of Macromolecules; Nauka Publishers: Moscow, 1989. English translation: AIP Press: New York, 1994. (4) Borisov, O. V.; Halperin, A. Macromol. Symp. 1997, 117, 99. (5) Zhu, P. W.; Napper, D. H. Langmuir 1996, 12, 5992. (6) Shibayama, M.; Tanaka, T. Adv. Polym. Sci. 1993, 109, 1.
concentration, the transition proceeded gradually; that is, there was a gradual collapse. At higher SDS concentrations the transition became a sharp, first-order transition. Interestingly this transition occurred above the θ temperature, that is, for water soluble polymers in worse than θ conditions. Finally, at the highest SDS concentrations, the transition proceeded through a gradual collapse beginning at the θ temperature of the PNIPAM chains, before a first-order collapse deep in the worse than θ regime. The appearance of two quite different collapse transitions, one continuous, in the better than θ regime, and one first-order, in the worse than θ regime, suggests the overall collapse of the chain is a complex process. Our present theoretical study is motivated by these curious experimental results.5 We are also interested in seeing how accurately the experiments can be modeled using a Flory mean-field model in the presence of surfactants. We initially assume a model with a constant monomer density profile throughout the polymer. We do this not only as an introduction to the more complex nonconstant density model but also to see if the experimental results require a more sophisticated model. We find that a more sophisticated model is required, and so we proceed to a modified nonconstant density profile model. We consider a dilute polymer solution with no overlap between monomers of different chains and consequently no interaction between monomers of different chains. As a result we consider one polymer chain having Nm monomers of size b. Also in solution are Ts surfactant molecules in a system of volume V. The free energy of the system is the sum of five terms:
F ) Fel + Fentb + Fint + Fads + Fentf
(1)
The first term in this expression, Fel, is the elastic free energy of a polymer chain given by the usual Gaussian form 3kBTR2/2b2Nm, where kB is Boltzmann’s constant, T is the temperature, and R is the size of the chain. The second term, Fentb, is the entropy of surfactant molecules adsorbed to the chain. It is given by the number of arrangements of the Ns adsorbed surfactant molecules along a chain with Nm monomers, and since the fraction Ns/Nm may be anywhere between 0 and 1, we write this entropy as Ns ln(Ns/Nm) + (Nm - Ns) ln(1 - Ns/Nm). We have assumed Ns < Nm, since only one surfactant molecule can adsorb to each monomer site. Experimentally, Ns is
10.1021/la980769d CCC: $18.00 © 1999 American Chemical Society Published on Web 01/13/1999
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probably much less than Nm. In addition we assume the surfactant molecules do not form micelles along the polymer chain, since, experimentally, Zhu and Napper5 always use SDS concentrations less than the critical micellar concentration. The third term, Fint, represents the interaction energy between monomers along the chain and adsorbed surfactant molecules. We include both binary and ternary interactions. Thus Fint is given by
Fint ) kBT[v(T)φ2m + 2vmsφms + vssφ2s ]b3R3 + kBT[wmmmφ3m + 3wmmsφ2mφs + 3wmssφ2s φm + wsssφ3s ]b6R3 (2) where v(T) is the temperature dependent binary (monomer-monomer) interaction parameter, while wmmm is the corresponding ternary interaction parameter (between three monomers) and is assumed to be independent of temperature. We assume v(T) > 0 for T < Tθ, where Tθ is the θ temperature. The parameters vms and vss are the binary monomer-surfactant and surfactant-surfactant interactions, while wmms, wmss, and wsss are the corresponding ternary interactions. Zhu and Napper5 postulate the surfactant interactions may be of an electrostatic origin, in which case these interactions can be several times larger than the monomer-monomer interactions. The values of the v’s and w’s are not known directly from experiment. In our results, we have tried to use a variety of different sizes to see their effect. The variables φm and φs represent the monomer density and the adsorbed surfactant density, respectively. These may be written as φm ) Nm/R3 and φs ) Ns/R3. The fourth term in eq 1 is the energy gained by the Ns adsorbed surfactant molecules and is given by -Ns, while the fifth term is the entropy of surfactant molecules that are free in solution. Since we are in the dilute solution regime, this may be written as (Ts - Ns) ln[(Ts - Ns)/V]. Finally we find the total free energy F of the system as a function of the two variables R and Ns. To obtain the equilibrium conformation of the polymer, we minimize F with respect to these variables for a given set of values of the various parameters. This is done numerically. The numerical minimization process is done with care to find the global energy minima. As there may exist a number of local energy minima in which our numerical process may become trapped, we initially randomly sample the free energy landscape to find coordinates that are approximations to the global energy minima. Using these approximations as initial guesses to the solutions, we then implement a quasi-Newton algorithm to find the coordinates corresponding to the global energy minima. Figure 1a shows our results for a system of chain density (number of chains per unit volume) Fchain ) Ncb3/V ) 10-12 with each chain consisting of Nm ) 50 000 monomers. The volume V of the system is 1012, the adsorption energy is one kBT, and vms ) 102, vss ) 104, wmmm ) 1, wmms ) 102, wmss ) 104, and wsss ) 106. The surfactant volumes here are much larger than the monomer volumes, and the origin for these large values may be electrostatic.5 Figure 1a displays the computed radius of gyration R as a function of v(T) for a variety of surfactant concentrations. At low surfactant concentration (Fs ) 10-5), the difference between the zero-surfactant case and low-surfactant case is negligible. This is as one may expect, since there would be insufficient surfactant present to significantly alter the chain conformation. This transition is continuous. As the surfactant concentration is increased to Fs ) 10-4, we initially observe that the temperature at which the polymer chain collapses moves into the worse than θ
Figure 1. (a, top) Coil-to-globule transition [radius vs v(T)] for Nm ) 50 000, V ) 1012, ) kBT, and vms ) 102, vss ) 104, wmmm ) 1, wmms ) 102, wmss ) 104, and wsss ) 106. The surfactant concentrations are (smallest polymer dimensions to largest) 10-5, 10-4, 10-3, 2 × 10-3, and 3 × 10-3. Note that the curve for zero surfactant concentration lies on top of the curve with surfactant concentration 10-5, on this scale. (b, bottom) Coilto-globule transition for Nm ) 106, V ) 1012, ) kBT, and vms ) 1, vss ) 1, wmmm ) 1, wmms ) 1, wmss ) 1, and wsss ) 1. The surfactant concentrations are (smallest polymer dimensions to largest) 0.0, 0.01, 0.1, 0.2, 0.3, 0.4, and 0.5.
regime. This is because the energy gained by the system when surfactant molecules adsorb to the polymer chain offset the entropic losses of moving out of solution. Eventually the solvent quality is sufficiently poor so that the combined loss of the entropy of the surfactant molecules and of the monomer-monomer interaction energies is too large to be overcome by the extra adsorption energy, so that the surfactant molecules move back into solution. The chain collapses to the globular state. The transition now becomes first-order, with a discontinuous jump in the polymer’s dimensions. We have studied the free energy surface and have verified two equal minima exist at this first-order transition. The minima correspond to the expanded and contracted chain configurations. After the collapse, the minima of the system are at small R. Increasing Fs beyond 10-4 leads to a larger jump discontinuity in the first-order transition, since now many more surfactant molecules are adsorbed per chain. Note the difference between the various transition curves before the major collapse. For low surfactant densities the absolute value of the gradient of the curve in this region decreases on increasing surfactant density. At the highest surfactant densities, there is almost no decrease in the
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polymer’s dimensions; that is, the gradient is almost zero. Having many surfactants adsorbed to the chain prevents monomers from coming close together and hence prohibits any decrease in polymer dimensions. This behavior is precisely what is observed experimentally.5 We have studied the system using this model for a wide variety of parameters, for example, chain length, surfactant-to-monomer size ratio, and adsorption energies. The structure of the collapse transition curve is always similar to that seen in Figure 1a. For example, in Figure 1b, we show the collapse for Fchain ) Ncb3/V ) 10-12, Nm ) 106, V ) 1012, ) kBT, and vms ) 1, vss ) 1, wmmm ) 1, wmms ) 1, wmss ) 1, and wsss ) 1. The main difference between parts a and b of Figure 1 is that a much larger surfactant concentration is needed to observe a first-order collapse in the chain. For example, in Figure 1a, we have a surfactant concentration of roughly 10-4 when a firstorder transition occurs initially, while, in Figure 1b, this concentration is roughly 10-1. When surfactants adsorb to the chain, they expand the chain; the amount the chain expands depends on the surfactant size. As the quality of solvent is reduced, eventually these surfactants must be “squeezed” off the chain. If the surfactant density is relatively low, and the surfactant size is relatively small, this does not affect the transition drastically. However, as the surfactant density is increased, a first-order collapse occurs, corresponding to all surfactants moving off the chain. On close inspection of the transition curves in Figure 1, it appears that the decrease in chain size before the final collapse occurs gradually and continuously. However, the experimental results of Zhu and Napper5 exhibited two distinct regions over which the collapse proceeded. In the first region just beyond the θ temperature, a gradual collapse is observed. The size of this collapse increases as the surfactant density decreases. In the second region, they observe the sharper first-order type transition. Zhu and Napper5 postulate that the gradual collapse just beyond the θ temperature, where the binary interactions become attractive, is due to the formation of ion pairs between the charges on the PNIPAM-SDS complexes and their counterions. At present, the precise reason for this gradual collapse is unclear. However, we believe the gradual collapse may still be driven by the attractive (monomer-monomer) binary interactions. On a qualitative level, this could occur in the following manner. Near the θ temperature many surfactants are adsorbed to the chain. These surfactants actually prevent the chain from decreasing in size. However, since the binary interactions are attractive, the monomers, themselves, would actually prefer to be close together. If in a small region of the chain a few monomers, which do not have surfactant molecules attached, come close together, they would preferably form a small dense core. As the binary interactions become more attractive, this core should increase in size and lead to an overall decrease in the polymer’s dimensions. To quantitatively model such a scenario, we must assume a nonuniform polymer density profile, that is, an inner dense core surrounded by an outer less dense corona.7,8 Clearly, for a constant-density model, the density throughout the chain may not vary. Thus, we modify our simpler Flory mean-field model by assuming a step-function density profile for the polymer. That is, we write (7) de Gennes, P. G. C. R. Acad. Sci. Paris II 1991, 313, 1117. (8) Wagner, M.; Brochard-Wyart, F.; de Gennes, P. G. Colloid Polym. Sci. 1993, 271, 64.
Letters out out φm(R) ) (φin m - φm )Θ(R - Rin) + φm
(3)
where Θ is the step function
{
1 for R < R Θ(R - Rin) ) 0 for R > Rin in
(4)
In the above, R represents the polymer size, while Rin denotes the inner core radius. In the inner core, that is, R < Rin, we would expect a high-density phase to exist, with a density of at least that of a θ chain, that is, N-1/2. However, we do not assume this a priori but rather obtain the inner core density naturally as part of the minimal free energy solution. By assuming a nonconstant density profile for the monomers, we must also assume a nonconstant density profile for the surfactant molecules along the polymer chain. In doing this, we allow many surfactant molecules to still adsorb to the chain in the outer corona but have fewer surfactants in the inner core, thus allowing the attractive monomer-monomer interactions to reduce the inner core dimensions and so gradually reduce the overall dimensions of the chain. However, once again, we do not assume anything a priori about the surfactant density in the inner core and the outer corona but rather find these naturally as part of the minimal free energy solution. We now write the elastic energy in eq 1 as a sum of the stretching energy of the inner dense core, 3kBTR2in/ 2(fNm)b2, and that of the outer, less dense corona, 3[R Rin]2/2[Nm - fNm]b2, where f is the fraction of monomers in the inner dense core. The entropy of surfactant molecules adsorbing to the chain, Fentb, is now different from before, since we are dividing the chain into two sections. It is Nsin ln[Nsin/fNm] + (fNm - Nsin) ln[1 - Nsin/ fNm] + Nsout ln[Nsout/(Nm - fNm)] + (Nm - fNm - Nsout) ln[1 - Nsout/(Nm - fNm)], where Nsin and Nsout are the number of surfactant molecules in the inner core and outer corona, respectively. All the interaction energy terms, that is, eq 2, must be rewritten with respect to the inner core and outer corona notation. The last two contributions in eq 1 are the same as before if we note that the total number of surfactants along the chain is just Nsin + Nsout. The total free energy of the system is now a function of five independent variables: f, R, Rin, Nsin, and Nsout. Minimization of the free energy with respect to these five variables, to obtain the equilibrium polymer conformation, is once again done numerically. Figure 2a plots the results of this minimization for the same conditions as in Figure 1a and a surfactant concentration of 10-5. We compare them with the uniform density model. It is evident that just above v(T) ) 0, where Zhu and Napper5 observed a significant decrease in polymer size, we do not observe the same decrease. In fact the two models are indistinguishable. Only just before the actual first-order collapse do we observe any difference in our modified model. The modified model was proposed so that an inner dense core could initiate a gradual collapse in the polymer dimensions, just after the θ temperature, even though many surfactants are still adsorbed on the chain. The term in the free energy preventing the build up of an inner core is the bound surfactant entropy, Fentb. Although when v(T) < 0, the core would like to be as large as possible, that is, containing Nm - Ns monomers, where Ns is the equilibrium number of surfactants attached to the chain for v(T) < 0, Fentb prohibits this. In fact if we use Fentb from the uniform-density model, this is precisely what happens and the decrease in the polymer dimensions just above the θ temperature is quite drastic. In Figure 2b we
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Figure 2. (a, top) Plot of the radius of gyration versus v(T) using the modified Flory-Huggins model for the same parameters as in Figure 1a and with surfactant concentration 10-5. The dashed line corresponds to the constant-density model, while the line with filled circles represents the nonconstant density model. (b, bottom) Plot of the number of monomers in the inner core versus v(T).
plot the number of monomers in the inner core as a function of v(T). Although it can be seen that the number of monomers in the inner core does increase as v(T) increases, this increase is not sufficient to significantly change the overall polymer dimensions. Again, we have studied this model over a wide variety of parameters. In all cases we do not observe any significant decrease in the polymer size for v(T) slightly larger than zero. This is in contrast (9) Sevick, E. M. Macromolecules 1998, 31, 3361. (10) Bekiranov, S.; Bruinsma, R.; Pincus, P. Europhys. Lett. 1993, 24, 183. (11) Bekiranov, S.; Bruinsma, R.; Pincus, P. Phys. Rev. E 1997, 55, 577.
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to the experimental results of Zhu and Napper.5 Basically the number of monomers in the inner core is too small to cause a significant change in the polymer size. Only when we are actually quite close to the first-order collapse temperature does the number of monomers in the inner core become significant. These results point to an additional interaction which drives the collapse just above the θ temperature. This interaction may well be the formation of ion pairs between PNIPAM-SDS complexes and their counterions, as has been postulated by Zhu and Napper.5 However a quantitative analysis of such a scenario is beyond the scope of the present study. In conclusion, we have attempted to theoretically model the coil-to-globule transition of polymer chains in the presence of surfactants which may adsorb to the chain. Experimental observations of this system5 have shown a variety of novel transition curves which depend on the surfactant concentration. Our model, which is based on a Flory mean-field theory, does produce many of the experimentally observed results. We have seen for low surfactant densities the transition is unaffected, compared with the collapse in the absence of surfactants. As the surfactant concentration is increased, we see that the collapse temperature moves into the worse than θ regime and the collapse becomes first-order. This collapse may be attributed to many surfactant molecules moving from the polymer chain into solution. We also find, on increasing surfactant concentration, the collapse in better than θ conditions decreases. This may be attributed to the surfactant molecules, whose large excluded volume prevents monomers from approaching each other. To model the gradual collapse of the chain just above the θ temperature, we introduced a nonconstant density model. This model was proposed so that an inner dense core of monomers could form, just above the θ temperature, and so decrease the overall polymer size. Although we did observe such a core forming, the size of this core was never large enough to significantly change the overall dimensions of the polymer. This result may point to an additional driving interaction, such as ion pair formations, which help to drive the collapse. We finish with a brief remark on the fact that PNIPAM is water soluble. Water soluble polymers in general are known to behave in nonstandard ways, and there have been several attempts recently to account for this.7-11 As part of this study, we also implemented the “n-cluster” model proposed by de Gennes7,8 for water-soluble polymers. In this model binary interactions are repulsive, but ternary or higher interactions can be attractive. We found that the inclusion of attractive ternary clusters can change the transition temperature but has no qualitative effect on the transition curves. Acknowledgment. The authors gratefully acknowledge the financial support of an ARC Large Grant, and D.R.M.W. is supported by an ARC QEII. LA980769D
