Architecture-Controlled Solution Properties of Hydrophobically

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Downloaded by UNIV LAVAL on April 6, 2016 | http://pubs.acs.org Publication Date: January 15, 1996 | doi: 10.1021/ba-1996-0248.ch011

Architecture-Controlled Solution Properties of Hydrophobically Associating Copolymers Arlette R. C. Baljon and Thomas A. Witten 1,2

2

Department of Physics and Astronomy, Johns Hopkins University, Baltimore, M D 21218 The James Franck Institute, The University of Chicago, 1

2

Chicago, IL 60637

Monte Carlo computer simulations predict that dilute-solution properties of water-soluble polymers that contain a small number of hydrophobic groups are dependent on the placement of these groups. We explore a simple model in which hydrophobic associations are assumed to be strong and each associating group or sticker is assumed to be constrained to be adjacent to one other. Thus we determined how the placement of the stickers controls the mutual second virial coefficient of identical chains containing two stickers apiece in dilute solution. The net interaction passes from repulsive to attractive when the ratio of the distance between the stickers over the chain length exceeds 80%. Likewise, for chains with alternating short and long intervals between stickers, the placement asymmetry controls the swelling of the chain. Our simulations show that virtually all placements result in swelling rather than collapse behavior in the chain.

M

O N T E C A R L O C O M P U T E R S I M U L A T I O N S for studying the influence of hydrophobe placement on the statistical properties of hydrophobically modified water-soluble polymers (HMWSPs) in dilute solution were developed (1,2). Because of the tendency of hydrophobes to associate

0065-2393/96/0248-0181$12.00/0 © 1996 American Chemical Society

Glass; Hydrophilic Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1996.


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HYDROPHILIC POLYMERS

or stick together in aqueous media, these HMWSPs exhibit interesting solution properties (3). For example, the intermolecular associations give rise to enhanced viscosification, so these copolymers are sometimes called associative thickeners. Their particular properties under shear make them potentially useful as aqueous viscosity modifiers. Some of the possible applications are coating and oil recovery processes (3). A problem, however, is that enhanced viscosity very often leads to insolubility (4). In this work we explore the possibility of obtaining the desirable rheological properties as well as good solubility by controlling the placement of hydrophobes. Although from what we know the chain architecture cannot yet be fully controlled experimentally, such control may soon be possible. Several experiments (4, 5) in which the relative composition of the copolymers is varied show that because of changes in the balance between inter- and intrachain hydrophobic associations, the rheology and phase behavior of these copolymers depend strongly on their composition.

Model The distinctive molecular feature that appears to give rise to the aforementioned solution properties is essentially geometric: It is embodied in the strong tendency of hydrophobes to stay together. Therefore understanding the effect of this geometric constraint in isolation is important. A simplified model (6) for polymers can aid in understanding the effect. Each polymer chain is treated as a lattice self-avoiding random walk with a small number of hydrophobic groups or "stickers" placed alongside. Each sticker is constrained to be adjacent to one other sticker, though stickers are free to change partners. The equilibrium state of the solution consists of all configurations of chains that obey these sticker constraints as well as configurations of mutual and selfavoidance. The aforementioned Monte Carlo simulations were employed to study these equilibrium configurations. As we shall see, both the interaction between two chains containing two stickers each and the conformation of a single chain containing many stickers depend strongly on where the stickers are placed along the chain backbone. The results presented in the next couple of paragraphs were published in an extended form in references I and 2. Here we emphasize the applicability of these results to hydrophobically associating polymers. A discussion of some new results for randomly placed stickers on a polymer ring is added.


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Architecture-Controlled Interaction Between Two Polymers Consider two chains with two stickers each. First, in a good solvent the interaction between two self-paired chains (Figure la) is repulsive. The range of the repulsion is the size of each chain (e.g., its radius of gyration). When brought to a separation comparable to their size, the chains may exchange stickers (Figure lb). This freedom to exchange partners results in an increased number of configurations and amounts to an entropie attraction of the order kT. Since at such separations the repulsive free energy is also of the order kT (7), the free energies of repulsion and attraction are comparable. Alterations of the sticker placements can alter the balance between repulsion and attraction. Theoretical work (6) shows that near dimension d = 4, the net interaction is attractive if all stickers are on the ends of the chains. Since the interaction is repulsive when the stickers are next to each other, the repulsive and attractive effects just cancel each other out at critical placements. In the limit of long chains, the critical fractional separation between the stickers is a fixed universal fraction of the chain that is independent of molecular weight and the excluded volume of two monomers (6).

a

b

Figure 1. Two self-paired (a) and two cross-paired (b) chains. The stickers are shown as black dots. (Reproduced from reference 1. Copyright 1992 American Chemical Society.)



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Simulation Strategy. Our simulations check these theoretical predictions. A configuration consists of two random-walk chains on a simple three-dimensional cubic lattice and has the following proper ties: (1) Each step of a random walk goes arbitrarily to a neighbor, second-neighbor, or third-neighbor site. (2) No two steps of the walks occupy the same lattice site. (3) Two particular monomers at specific points along the chain are designated as stickers. Parameter x, defined as the ratio of the sticker distance to the chain length, controls the chain architecture. Stickers are placed either symmetrically or maximally asymmetrically (Figure 2). (4) Each sticker is adjacent to one other sticker. A Monte Carlo algorithm is used to study all possible configu rations of this two-chain system. The details of this algorithm can be

a

b

Ν

xN Figure 2. Symmetrical (a) and asymmetrical (b) sticker placements. The stickers are shown as black dots. (Reproduced from reference 1. Copy right 1992 American Chemical Society.)


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found elsewhere (I). The free energies of repulsion and attraction are obtained from the number of forbidden configurations because of chain crossings and the number of additionally allowed cross-paired configurations that are due to sticker exchange. The net interaction energy is expressed as a second virial coeffi cient (B2). If the interaction between the two chains is repulsive, the second virial coefficient is positive; if the interaction is attractive, the coefficient is negative. In general, the second virial coefficient mea sures how the interaction between many chains in the dilute region affects thermodynamic quantities. For example, the osmotic pressure Π is given by

^

= c + Bc 2

2

+ 0(c )

(1)

3

Here c is the number of polymers per unit volume, ke is the Boltzmann factor, and Τ is the temperature. The second term on the right-hand side is a correction to the pressure to take account of interactions. In the dilute region we can neglect the 0(c ) terms. 3

C r i t i c a l P l a c e m e n t . Figure 3 shows the normalized second vir ial coefficient B2 for two polymer chains in three dimensions with symmetrically placed stickers. For reasons explained later in this chap ter, we plot B2 versus y = x , where χ is the ratio of the sticker distance to the chain length. The data for the different chain sizes (molecular weights 34, 66, and 130) are normalized so that Ê (y) = 1 at y = 0. We checked that the unnormalized excluded-volume data scale with chain length as B ~ N ~ Ν · . At y = 0 we found that B ]V . The error bars shown in Figure 3 are obtained from the statistical error in the run: the value of the sum in a full run minus that in a half run divided by the square root of 2. The data are within error bars independent of the chain length. This superposition of the data for different chain sizes confirms the expected asymptotic scaling. Moreover, the B versus y data in Figure 3 show that as y increases, the chains cross from self-repelling to self-attractive behavior as antici pated. According to reference 8, at low x, Ê (x) - B (0) « x* >. Here Θ is an exponent describing certain interior correlations within a self-avoiding chain. Its value was measured as approximately 0.67 (8). Considering that ν « 0.588 in d = 3 dimensions, we obtain v(d + 02) ~ 2.16. Evidently the B versus y data in Figure 3 are consistent with this prediction, since B shows linear behavior for small y. In order to calculate the critical sticker placement t/ , for which there is no net mutual interaction between the chains ( B - 0), we fit a poly2 1 6

2

3v

2

2

552

1

8

( 1 8 0 ± 0 0 5 )

2

2

2

2

2

2

c

2


d+d2

186



1.0 ft

y

Figure 3. Normalized second virial coefficient B2 versus sticker place ment for symmetrically placed stickers, y = x , where χ is the distance between the stickers divided by the length of one chain. The values for the different chain lengths (34, 66, and 130) are scaled so that they are unity at χ = 0. The solid line shows the result of a third-order polyno mial fit to the data; the dotted line shows the result for maximally asymmetrically placed stickers. 216

nomial through the data. A third-order polynomial fit gives a minimum value for χ per degree of freedom (1.1). From the coefficients we find that y = 0.58 ± 0.02 or x = 0.776 ± 0.012. According to universality principles, B , and this critical placement in particular, should be in sensitive to details of the simulation, providing chains are long enough. In order to test this, we altered the x = 1 random walks to allow only steps of length < y/2. The normalized second virial re mained unaltered, a result that confirmed the universality of our re sults. The dotted curve in Figure 3 shows the results for maximally asymmetrical placed stickers. Here the critical placement is shifted to x = 0.892 ± 0.009. 2

c

c

2

c

Conformations of a Single Chain Containing Many Stickers Next we studied the equilibrium behavior of a single long chain made by joining our two sticker segments (Figure 4) head to tail. The result-



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Figure 4. Chain architecture. Each chain consists of S equal segments. Each segment contains two symmetrically placed stickers. Parameter x, defined as the distance between the stickers over the segment length, defines the chain architecture. The stickers are shown as black dots. Each segment consists of η monomers; the total number of bonds Ν equals Sn - 1. (Reproduced from reference 2. Copyright 1993 American Chemical Society.) ing chain has alternating short and long intervals between the associat ing groups. A parameter x, defined as the length of the short interval divided by the sums of the lengths of the short and long intervals between stickers, defines the chain architecture. In the χ = 0 limit, the chain behaves as an ordinary polymer chain and has a swollen configuration. Increasing x, however, may result in architectures for which the chain collapses. That case also shows a critical placement, x , with which a chain shows the statistical behavior of an ideal poly mer chain in the asymptotic (large N) limit. c

Influence of Sticker Placement. For details of the simulations the reader is referred to reference 2. The results presented in the next two paragraphs were published in reference 2. Jn three dimensions we obtain the radius of gyration R for S = 2, 4, and 8 and η = 34. Since the total number of monomers equals the number of segments S times the number of monomers per segment n, the total number of bonds Ν varies from 67 to 271. Each run lasts at least 1000 equilibration times. Figure 5 shows the results for five different chain architectures. The swelling factor a is defined as the measured radius of gyration divided by the expected radius of gyration of an ideal Gaussian chain with the same number of monomers and average squared bond length (9). For swollen chains, a increases with increasing chain length N, whereas for collapsed chains, a decreases with Ν in the asymptotic large-2V limit. Since the swelling factor is defined as the ratio of the radius of gyration of a chain with a sticker to that of an ideal chain g


188



0

50

100

200

500

Ν Figure 5. Log-log plot of the swelling factor a versus total chain length for different sticker placements in three dimensions. The swelling factor is defined as the average radius of gyration divided by the radius of gyration of an ideal Gaussian chain with the same length and average squared bond length. The solid lines are fits to equation 2. The dotted line is a fit to equation 5. The dashed lines indicate the expected be havior for swollen and θ chains. (Reproduced from reference 2. Copy right 1993 American Chemical Society.)

without stickers, this factor is smaller than one for a chain with criti cally spaced stickers would be (x = x ). The error bars shown in Figure 5 are the statistical errors and are estimated by averaging over different parts of the run. The fits through the data will be explained shortly. The dashed lines indicate the expected behavior for swollen and θ chains (note the log-log scale). At first sight the data at small χ seem to be in accordance with this swollen behavior, but for χ = 0.5 the match is not as good: Although the swelling factor increases with N, the asymptotic regime is clearly not reached. Predicting the scaling behavior of the chains at high Ν from the data at low Ν is a tricky business that is complicated by the lack of theories for associating polymers on which to rely. For small x, though, we expect the chains to behave similarly to ordinary homopolymer chains. The asymptotic Ν dependence of the radius of gyration of a swollen homopolymer chain is c

R

g

« kN (l v

+

CN-™)


(2)

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Table I. Results for k and C from a Fit to Equation 2 k

X


0.147 0.265 0.382 0.441 0.500

0.004 0.470 0.003 0.437 0.3873 ± 0.0012 0.340 0.002 0.002 0.305

0.08 0.13 0.47 1.22 1.86

C

X

± 0.09

0.4 2 0.8 0.9 5

0.06 ± 0.02 ± 0.05 ± 0.06

2

The first term on the right-hand side gives the expected scaling in the asymptotic limit. The second term is a correction for the fact that we deal with finite-size chains, ν and ω are obtained from renormalization group theory (10): ν = 0.5880 ± 0.0010, and ω = 0.850 ± 0.015. Fits to equation 2 are made for each architecture. They are drawn as solid lines in Figure 5. Table I shows the obtained values of k and C . The value of C increases with increasing x. In fact, C = 0 within error bars for the lowest χ value. This means that for low x, no scaling correction is necessary and the sampled chains exhibit asymptotic behavior. One might expect this result if the excluded volume is much higher than the entropie attraction. This result is in accordance with previous re sults on two sticker chains (1), for which the entropie attraction in creases with increasing x. As a goodness-of-fit measure, the values of χ are given in Table I (11). Since we have three data points and two variables tofit,thefitis acceptable if ^ « 1. Clearly, for χ - 0.5 the data are not wholly in accordance with the swollen-chain behavior, but for all other χ values they are. For other χ values (i.e., χ Φ 0.5), nothing clearly indicates that the chains will collapse on larger length scales. The sticking is predominantly local with associations between chemically nearby stickers (2). 1

Search for a Collapsed Conformation. A scaling argument is used to predict the scaling behavior for χ > 0.441. Table I indicates that the value of k decreases with increased x. Parameter χ will reach its critical value, at which the chain behaves as an ideal chain as k becomes 0. The functional dependence of k on χ is obtained as follows: For homopolymers it is known from scaling theory (12) that /

R ~N™^l- \jj s

d

/ \(4-d)/2v(2vN

l)/(4-d)

J

(3)

Here β is the excluded volume parameter, d is the dimension of space, and I is the bond length. In our case the parameter β is the effective excluded volume between two chain segments. This effective ex-


190


eluded volume contains the monomer repulsion as well as the sticker attraction. The parameter controlling the net effective excluded vol ume is x. Since β measures a local interaction, it is natural to expect it to vary smoothly with χ as χ passes through x . Thus near χ = x , β ~~ (x — x ). Then equation 3 results in c

c

c

- Xc) ""

k-(x

(4)

1

Figure 6 shows a plot of k ~ versus x. For comparison, the value at χ = 0.5 is indicated, although in this case the fit is not accepted. The other data square with our expectation: Remarkably, the anticipated linearity in χ extends over the whole range of χ studied. A linear leastsquares fit through these data gives x = 0.496 ± 0.004 (χ = 0.2). This critical point is very close to χ = 0.5. Could the chain with equally spaced stickers be ideal in the asymptotic limit? If it is, the first thing to expect is that this ideal behavior is related to the symmetric sticker l/(2v


2

l)

1

c

0.0

0.2

0.4

0.6

χ Figure 6. In order to test our scaling argument (in three dimensions), ^i/(2v-d i pi tt i versus x. From a linear least-squares fit we obtain Xc = 0.496 ± 0.004. Although the data point atx = 0.5 was not accepted (and not used for the fit), it is indicated for comparison. (Reproduced from reference 2. Copyright 1993 American Chemical Society.) s

0

ec



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placement, that is, to the fact that there is no distinction between long and short intervals. If this lack of distinction is the cause for the ob served ideal behavior, we expect these chains to show ideal behavior in two dimensions as well. In two dimensions, swollen-chain confor mations exist for all χ values. In two dimensions, however, many con figurations in which the sticking is not local are prohibited because the chain is not allowed to cross itself. This fact could explain the swollen behavior in two dimensions. Our scaling argument indicates that in three dimensions the chain with equally spaced stickers is marginally collapsed. The data in Fig ure 5 show marginally swollen behavior, though. Indeed, if we try to treat the interaction as a small perturbation and fit the data to (13), then R

g

-

kN (l

+

05

(5)

CN ' ) 0 5

The resulting fit, shown in Figure 5 as a dotted line, is quite good (χ = 0.2). For ordinary homopolymer chains this fit would indicate a swollen asymptotic conformation. For associating polymers, it is still possible that at high Ν the data will decrease or become Ν inde pendent. 1

Randomly Placed Stickers on a Polymer Ring.

Next we

wanted to know what happens if stickers are placed randomly on a polymer chain. In order to circumvent end effects, we placed 4 to 16 stickers randomly on a closed loop. A loop with 2S stickers contains a total of Ν = nS monomers, where η = 34. For two different loop sizes (S = 2 and 8), the radius of gyration is obtained for 16 different random sticker placements. We observed that the R for a specific random placement increases with an increase in the unevenness in sticker distribution. In Figure 7 this unevenness is shown for S = 8. We plot R versus the amount of randomness R. R is defined as the average variation of the distances between two stickers from the sticker distance in a chain with evenly spaced stickers: g

g

2S

(6)

where N is the number of bonds in a chain segment (i) between two stickers. The increase in R with increasing R indicates that local stick ing becomes more important at higher randomness R. Figure 8 shows the expansion factor a for chains with randomly spaced stickers. In addition, data for regularly spaced stickers (x = 0.147 and χ = 0.382) t

g



192


0.20

Figure 7. Dependence of the radius of gyration (R ) on the amount of randomness R defined in equation 6. g

are shown. Data for chains with randomly spaced stickers are not far off those obtained for a chain with χ = 0.382. The randomness R = 0.118 for χ = 0.382 is much lower than the average R for the 16 configu rations of Figure 7. The fact that both chains (random stickers and χ = 0.382) nonetheless have the same amount of swelling shows that all regularity in the sticker placement, such as a regular alternation of short and long segments, enhances swelling.

Conclusion Using a simple topological model, we showed that the exact placement of stickers on an associative polymer chain has huge consequences for dilute-solution properties of those chains. We found that in equilib rium, the mutual interaction between two two-sticker chains passes from repulsive to attractive with increasing distance between stickers. This finding may result in an interesting new way to attain the theta state for a dilute polymer solution. The required balance is achieved by using geometrical constraints that depend only on the chain archi tecture. For chains with alternating long and short intervals between stickers, the amount of swelling is controlled by sticker placement. Phase separation was observed in none of the cases we studied. A l -



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log(N)

Figure 8. Swelling factor α for three-dimensional polymer loops. The squares represent the data obtained for loops on which the stickers are placed randomly. though two two-sticker chains with stickers at the ends attract each other, we estimated that this attraction by itself is not strong enough to result in phase separation. Moreover, from the asymptotic swollen scaling behavior for single chains, we predict that for almost all archi tectures studied, the second virial coefficient of many chains in solu tion will be positive, indicating good solubility at low concentrations. We found that chains on which the stickers are placed randomly also swell. For chains with associating groups that stick in bigger multi plets, collapsed conformations and phase instability are expected for more architectures. Simulations suggest that solubility can in such cases also be controlled by sticker placement. We expect our results to hold for real chains in a real solvent if the sticking energy is high enough, so that the hydrophobic groups are dissociated only a small fraction of the time. This fraction will depend on the binding energy of stickers as well as on sticker concen tration, that is, on the average distance between two dissociated stick ers. Since the interesting rheological effects are observed at high stick ing energy, we studied only conformations in which all stickers are stuck. However, our simulations are easily extendable to include con formations in which some stickers are dissociated. Testing our findings experimentally would be interesting. One thing to measure would be Glass; Hydrophilic Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1996.

194


Ê2 as a function of sticker placement resulting from, for example, osmotic pressure versus concentration measurements in the dilute regime for two sticker chains. Such tests using polymers synthesized with living polymerization techniques may soon be possible (14).


Acknowledgments We thank Michael Murat for his help with the initial program code. This work was supported by the Material Research Laboratory at The University of Chicago under National Science Foundation Grant D M R 88-19860 and by a grant from I B M Corporation. Part of the research was conducted using the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulations in Science and Engineering (Cornell Theory Center), which receives major funding from the National Science Foundation and I B M Corporation and additional support from New York State and members of the Corporate Research Institute.

References 1. Baljon, A . R.C.;Witten, T. A . Macromolecules 1992, 25, 2969. 2. Baljon, A . R. C. Macromolecules 1993, 26, 4339. 3. Water-Soluble Polymers: Beauty with Performance; Glass, J. E., Ed.; ACS Symposium Series 213; American Chemical Society: Washington, D C , 1986. 4. Hill, Α.; Candau, F.; Selb, J. Prog. Colloid Polym. Sci. 1991, 84, 61. 5. Kaczmarski, J. P.; Glass, J. E. Polym. Mater. Sci. Eng. 1992, 67, 284. 6. Cates, M. E.; Witten, T. A. Macromolecules 1986, 19, 732. 7. Pincus, P. Α.; Witten, T. A. Macromolecules 1986,19,2509. 8. des Cloizeaux, J.; Jannink, G. Polymers in Solution: Their Modelling and Structure; Oxford University: Oxford, England, 1990; pp 575-582. 9. Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Claridon: Oxford, England, 1986. 10. Le Guillou, J. C.; Zinn-Justin, J. Phys. Rev. Lett. 1977, 39, 95. 11. Press, W. H.; Flannery, B. P.; Teukolsky, S. Α.; Vetterling, W. T. Numeri cal Recipes; Cambridge University: Cambridge, England, 1988; p 502. 12. Freed, K. F. Renormalization Group Theory of Macromolecules; Wiley: New York, 1987; pp 81-87. 13. Yamakawa, H. Modem Theory of Polymer Solutions; Harper and Row: New York, 1971. 14. Storey, R. F. University of Southern Mississippi, personal communication. RECEIVED

30, 1994.

for review February 14, 1994. A C C E P T E D revised manuscript June


Architecture-Controlled Solution Properties of Hydrophobically

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