Coarse-graining polymers as soft colloids

Physica A 306 (2002) 251 – 261 www.elsevier.com/locate/physa Coarse-graining polymers as soft colloids A.A. Louisa;∗ , P.G. Bolhuisb , R. Finkena , V. Krakoviacka , E.J. Meijer b , J.P. Hansena a Department of Chemistry, University Chemical Laboratories, Lens eld Rd, Cambridge CB2 1EW, UK b Department of Chemical Engineering, University of Amsterdam, Nieuwe Achtergracht 166, NL-1018 WV Amsterdam, The Netherlands Abstract We show how to coarse-grain polymers in a good solvent as single particles, interacting with density-independent or density-dependent interactions. These interactions can be between the centres of mass, the mid- or end-points of the polymers. We also show how to extend these methods to polymers in poor solvents and mixtures of polymers. Treating polymers as soft colloids can greatly speed up the simulation of complex many-polymer systems, including c 2002 Published by Elsevier Science B.V. polymer–colloid mixtures.  PACS: 61.25.H; 61.20.Gy; 82.70.Dd Keywords: Polymer solutions; Colloids; E ective interactions 1. Introduction Binary mixtures of colloidal particles and non-adsorbing polymers have received renewed and growing attention recently, in part because they exhibit complex and interesting structure, phase behaviour, interfacial properties, and rheologyk [1– 6], and in part because they are excellent model systems for the study of large length and time-scale separations in complex uids. Problems with bridging length-scales are immediately apparent in even the simplest models of colloid–polymer mixtures: while the mesoscopic colloidal particles can be modeled as hard convex bodies, the polymers are generally treated at the microscopic (Kuhn) segment level. Thus, even though the average size of the polymer coils may be of the same order of magnitude as that of the colloids, the number of degrees of freedom needed to model the former may be ∗ Corresponding author. Tel.: +44-1223-763872; fax: +44-1223-336362. E-mail address: aal20@cus.cam.ac.uk (A.A. Louis). c 2002 Published by Elsevier Science B.V. 0378-4371/02/$ - see front matter
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 5 0 2 - 2 252 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 Fig. 1. Schematic picture of our coarse-graining scheme: The polymers in a polymer–colloid mixture are treated as single entities, on the same footing as the colloidal particles. several orders of magnitude larger than what is needed for the latter. This naturally provokes the question: Can the polymers also be modelled as single particles? In fact, this is exactly what was done by Asakura and Oosawa (AO) who, in their classic work on colloid–polymer mixtures [7], modelled the polymers as ideal particles with respect to each other, and as hard-spheres with respect to the colloids. This model is strictly speaking only valid for non-interacting polymers, or for interacting polymers in the dilute limit, while many interesting phenomena, such as polymer-induced phase separation, take place at nite concentration of interacting polymers. Our ultimate goal, therefore, is to go well beyond the AO model and describe non-ideal polymers in a good solvent up to semi-dilute concentrations. We recently extended the AO concept to take into account polymer–polymer interactions, rst by rather naively assuming a Gaussian repulsion between polymers [8] to account for the penetrable nature of polymer coils, and then by carrying out a more sophisticated programme which resulted in density-dependent [9 –11] and density-independent [12] interactions between polymer coils (Fig. 1). The next sections will examine these e ective potentials in more detail. 2. Coarse-graining homogeneous polymer solutions Polymers made of L segments are characterized by their radius of gyration, Rg ∼ L , where  ≈ 0:59, i.e., polymers are fractal objects. Much of our understanding of polymer solutions comes from the scaling arguments pioneered by de Gennes [13]. These arguments suggest that the behaviour of a polymer solution di ers in the dilute regime, where the polymer coil density  ¡ ∗ = ( 34 R3g )−1 , from the semi-dilute regime, where =∗ 1. Both these regimes presume that the actual monomer concentration c remains very low. Once c becomes appreciable, one enters the so-called melt regime which will not be treated here, and for which di erent kinds of coarse-graining methods are necessary. A standard scaling argument suggest that the radius of gyration Rg is the only relevant lengthscale for the dilute and semi-dilute regimes of polymers in a good solvent A.A. Louis et al. / Physica A 306 (2002) 251 – 261 253 [13]. This immediately implies that the second-virial coecient should scale as B2 ∼ R3g . If we set x = r=Rg then the second-virial coecient is proportional to
3 (1) B2 ∼ Rg {1 − exp[ − V (x)]} dx ; where V (x) is the interaction between two separate polymers, de ned w.r.t. some yet to be speci ed coordinate. Since this must hold for all Rg , the interaction V (r=Rg ) should not depend on the length L for suciently long L (scaling limit). A more sophisticated version of this argument was put forward by Grosberg et al. [14], see also the review by Likos [6] for a historical overview. To coarse-grain each polymer as a single entity, one must still choose an interaction centre, which may be the centre of mass (CM), the mid-point, the end-points, or some average monomer. For the mid- or end-point representation, V (x) should diverge at the origin, since the actual segments of two di erent polymers cannot overlap. For the CM, we expect a nite value of V (x = 0), since it is possible for two polymers to deform around each other in such a way that their CM coincide without any mutually avoiding monomers overlapping. 2.1. Density-independent polymer–polymer interactions In the description of atomic and molecular liquids and solids it is common to replace the full quantum mechanical treatment of the interactions by a simpli ed e ective potential. Well-known examples include the Lennard Jones pair potential and the Axilrod–Teller three-body potential [15]. Here, we attempt a similar coarse-graining for polymer solutions and choose the constituents to be single polymers, with a speci ed interaction centre for each polymer. Then, following for example Refs. [5,6] or more speci cally [12], the coarse-grained Helmholtz free energy F of a set of N polymers with their interaction centres xed at the coordinates {ri }, in a volume V , can be written as the following expansion: F(N; V; {ri }) = F(0) (N; V ) + N  w(2) (ri1 ; ri2 ) + i1 ¡i2 + · · · + w(N ) (ri1 ; ri2 : : : riN ) ; N  w(3) (ri1 ; ri2 ; ri3 ) i1 ¡i2 ¡i3 (2) where the coordinates of the interaction centres of the polymers, {ri1 ; ri2 : : : rin }, are expressed in units of Rg , the radius of gyration at zero density. Each term in this coarse-grained free energy includes an implicit statistical average over all the internal monomeric degrees of freedom for a xed con guration {ri }. F(0) (N; V ) is the so-called volume term [6], the contribution to the free energy that is independent of the con guration {ri }, and includes the internal free-energy of an isolated polymer. For a homogeneous solution, translational invariance implies that there is no one-body term in the expansion. Each subsequent term w(n) (ri1 ; ri2 : : : rin ) is de ned as the free energy of n polymers with their interaction positions at {ri1 ; ri2 : : : rin }, minus the contributions of all lower order terms. This procedure may in principle be followed to derive higher and higher order interactions, until, for a system with N polymers, the N th 254 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 1 4 0.75 CM endpoints midpoints (2) βr w (r) 2 0.5 2 (2) βw (r) 3 0.25 1 0 0 1 2 r/Rg 3 0 0 1 2 3 r/Rg 4 5 Fig. 2. w(2) (r) and r 2 w(2) (r) for the interaction between two isolated polymers in the CM, end- and mid-point representations. The end and mid-point potentials diverge at the origin, but the CM representation gives a nite value. All three potentials result in the same second virial coecient B2 de ned in Eq. (1). term determines the total coarse-grained free energy. The thermodynamic free energy of the polymer solution nally follows from a statistical average over the interaction coordinates:  F(N; V ) = −ln exp[ − F(N; V; {ri })] : (3) {ri } But in practice, this approach is not often feasible because the number of n-tuple coordinates and related complexity of each higher order term increases rapidly with n, so that the series in Eq. (2) and the full average in Eq. (3) quickly become intractable. Instead, one hopes to show that the series converges fast enough that only a few low-order terms are needed to obtain a desired accuracy. The rst important term in the series expansion is the pair interaction w(2) (r), which can be determined by calculating the logarithm of the probability that two polymers have their interaction centres a distance r apart. Details of our computer simulation technique are described elsewhere [10–12]. 1 In brief, by simulating L = 500 self-avoiding walk (SAW) polymers on a cubic lattice, we determined w(2) (r) for three di erent interaction centres: the end-points, mid-point, and CM of each polymer, as depicted in Fig. 2. The end- and mid-point representations diverge at the origin as limr→0 w(2) (r) ∼ ln(r=Rg ) [6], while the CM representation has a nite value which we estimate to be w(2) (0) = 1:80 ± 0:05 in the scaling limit L → ∞ [10]. By plotting r 2 w2 (r) we see that the CM representation has the shortest range, which is one reason why it is easier to use than the other two representations. In a similar fashion, the higher order interactions can be calculated from higher order probability distributions [5,12]. We calculated the relative strength of the many-body terms up to fth order by computer simulations, and to arbitrary order by a scaling 1 In our earlier work [10–12] we used Rg = 16:5 instead of the correct Rg = 16:83 value for the L = 500 SAW polymers. This means a slight adjustment in the values of Rg and in =∗ quoted in our papers. For example, the values quoted for =∗ should increase by a factor 1:06. 255 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 theory [12]. The simulations and the scaling theory agree well, and suggest that at full overlap the nth order many-body term alternates in sign as (−1)n , and decreases (slowly) in absolute magnitude with increasing n. But, as mentioned before, a description based on three and higher order interactions rapidly becomes too unwieldy to use. Instead, we show in the next section how to derive a pair potential approach which includes these higher order (n ¿ 2) interactions in an average way. 2.2. Density-dependent polymer–polymer interactions An alternative coarse-graining approach to the many-body expansion of the previous section is to nd pair potentials which reproduce known structural information. We are aided in this by a theorem which states that at a given density , there is a one-to-one mapping between the pair distribution function g(r) and a unique pair potential v(r; ) that will exactly reproduce the correct pair correlations [16]. We generate the pair correlations with simulations of SAW polymers, and at each density , use the Ornstein–Zernike equations [15], coupled with the hypernetted-chain closure (HNC) [15], to invert the CM g(r) and nd v(r; ). While the HNC closure is generally not accurate enough for inversions in simple liquids, it is nearly exact for the soft potentials we are investigating here [5,6,17]. Nevertheless, there are a number of subtleties, both in the simulations and in the inversions, which must be carefully examined [10,11]. Density-dependent e ective potentials v(r; ), inverted from the g(r) produced by L = 500 SAW simulations are shown in Fig. 3. The potential changes with increasing density, but approximately retains the shape found at  = 0. We have recently shown that these potentials can be very accurately parameterized for =∗ ¡ 2 by sums of three Gaussians with density-dependent coecients [11]. Within the HNC approximation, the density dependence of an e ective pair potential that reproduces the true g(r) is given to lowest order in  and the w(n) ({ri }) by 1 βv(r) g(r) 0.5 0 ρ/ρ∗ = 0.0 ρ/ρ∗ = 0.58 ρ/ρ∗ = 1.16 ρ/ρ∗ = 2.32 ρ/ρ∗ = 6.62 ρ/ρ∗ = 9.25 2 1 0 1 2 3 0 0 1 2 3 4 5 r/Rg Fig. 3. The e ective polymer pair potentials v(r; ), derived at di erent densities from an HNC inversion of the CM pair distribution functions g(r) of L = 500 SAW polymer coils (from Ref. [12]). 256 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 1000 Z=Π/ρ 100 compres. route EOS for v(r;ρb) L=500 compres. route EOS for v(r;ρb) L=2000 direct L=500 SAW simulation direct L=2000 SAW simulation EOS from RG EOS from v2(r;ρ=0) 10 1 0 1 ρ/ρ∗ 10 100 Fig. 4. EOS for polymers in good solvent. Direct results and the EOS arising from the e ective potentials through the compressibility route are compared for L = 500 and 2000 SAW polymers. van der Hoef and Madden (see e.g. Ref. [18])
(3) (2) v(r12 ; ) = w (r12 ) −  (e− w (r12 ;r13 ;r23 ) − 1)g(r13 ; )g(r23 ; ) dr3 : (4) We found that this expression describes the density dependence quite well for =∗ ¡ 1, and even works qualitatively for higher densities, where we expect higher order  and w(n) ({ri }) e ects to become signi cant [12]. This demonstrates the connection between the density-independent and density-dependent approaches, showing explicitly that the density dependence in the e ective pair-potentials v(r; ) arises from the many-body interactions. One advantage of the structure-based route to the potentials is that one can use the compressibility equation [15] to derive the equation of state (EOS) from the pair correlations. We have done this for both L = 500 and 2000 SAW simulations. We directly measured the EOS and compared this to the EOS derived from the e ective potentials through the compressibility equation [10,11]. The two routes are compared in Fig. 4, where the agreement is shown to be excellent. All three e ective potentials shown in Fig. 2 result in “mean- eld uids” [5,6,17], so named because the EOS takes on the mean- eld form = ∼ 1 + 12 v̂(k = 0) at high enough densities. Here v̂(k) is the Fourier transform (FT) of the potential. This implies that if we only use the  = 0 (Fig. 5) potential, then the EOS at higher density would scale as = ∼  instead of the correct  ∼ 1:3 scaling found for the semi-dilute regime [13]. It is therefore the many-body interactions, expressed through the density dependence of v(r; ), which cause the EOS to be super-linear. We add a caveat here about the route to thermodynamics with density-dependent potentials. The potentials derived here can be used to derive the correct thermodynamics 257 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 2 2 βε=0.0 βε=0.1 βε=0.2 βε=0.25 1 βV(r) 1 L=100,1000 L=200,1000 L=500,1000 L=100,500 L=200,500 0 0 (b) (a) −1 −1 0 1 2 r/R g 3 0 1 2 3 r/R 12 Fig. 5. (a) CM–CM interaction for polymers in a poor solvent, modeled by L = 100 SAW polymers on a cubic lattice with nearest neighbour attractions of strength − . (b) CM–CM interaction for mixtures of di erent length SAW polymers. through the compressibility route. Di erent (but related) density-dependent potentials, which do not reproduce the correct structure, would be needed to derive the correct thermodynamics through the virial route [18]. In other words, there is no unique densitydependent pair potential: when specifying such a potential, one must also specify which route to thermodynamics should be used [19]. 3. Polymers near walls and spheres Polymers form a depletion layer near a hard non-adsorbing wall because the number of possible conformations is restricted there. This is illustrated in Fig. 6 for the CM and for a monomer representation near a planar wall. We have used an inversion method similar to that used in the previous sections to derive e ective wall–polymer potentials for interacting polymers near walls and spheres [9–11]. These potentials are constrained to give the correct density pro le (r) which in turn determines the adsorption . One can show, for example, that the surface tension is completely determined if one knows the EOS and  as a function of  [20]. Our e ective polymer– polymer pair potentials correctly determine the EOS, while the wall–polymer potentials correctly determine , implying that our formulation will reproduce the correct surface thermodynamics. We show this adsorption = in Fig. 7, together with a simple t constrained to give the correct scaling = ≈ () ∼ −0:77 in the semi-dilute regime. Note that the largest relative change in the adsorption is actually in the dilute regime, suggesting that even there descriptions based on the low-density or non-interacting polymer limit rapidly become inadequate. 258 0.5 0.5 0 0 hm(z) h(z) A.A. Louis et al. / Physica A 306 (2002) 251 – 261 −0.5 ρ/ρ∗ = 0.0 ρ/ρ∗ = 0.71 ρ/ρ∗ = 1.36 ρ/ρ∗ = 2.65 ρ/ρ∗ = 6.46 −0.5 (a) −1 0 1 (b) 2 3 −1 0 z/R g 1 2 3 z/R g Fig. 6. (a) The wall-polymer CM pro le h(z) = (z)= − 1 for L = 500 SAW polymers at di erent bulk concentrations. (b) The wall–polymer monomer pro le hm (z) for the same bulk concentrations. Both representations result, by de nition, in the same relative adsorptions =. The straight lines in (b) are a t to the simple form hm (z) = tanh2 (−z=()) − 1 [13]. 0 −0.5 Γ/ρ L=500 SAW simulations analytic fit to data −1 −1.5 0 2 4 ρ/ρ∗ 6 8 10 Fig. 7. Relative adsorption =, in units of Rg of L = 500 SAW polymers near a single hard wall. Circles are direct simulations and the line denotes the simple t with the correct scaling behaviour, given by = = −1:074Rg (1 + 7:63=∗ + 14:56(=∗ )3 )−(0:2565) . 4. Connection with scaling theory Most successful theories of polymers start from a monomer-based description and use scaling or RG approaches to derive properties of polymer solutions [13]. How does our CM-based description compare with these scaling approaches? For example, in the semi-dilute regime, scaling theories predict that the important length-scale is the correlation length (), which decreases with increasing density as () ∼ −0:77 . It is not a priori clear how this lengthscale enters into the g(r) or the v(r) in our A.A. Louis et al. / Physica A 306 (2002) 251 – 261 259 description of homogeneous polymer solutions. The EOS scales as = ∼ −3 = in the semi-dilute regime, a behaviour which is reproduced by our description through the compressibility equation. Since these potentials result in “mean- eld uids”, this  suggests that dr r 2 v(r; ) ∼ −3 =2 in the semi-dilute regime. For inhomogeneous systems in the semi-dilute regime,  enters more directly through the density pro les shown in Fig. 6, but again the direct connection to the potentials is more opaque. So the exact connection with the scaling theory still remains to be worked out. We expect our approach to be most robust in the dilute regime and into the crossover region of the semi-dilute regime. Luckily this is also where much of the interesting physics of the colloid–polymer systems lies. How well our “soft colloids” approach will work deep into the semi-dilute regime still remains to be established. 5. Extensions to poor solvents and mixtures The considerations in the previous sections focused on equal length polymers in a good solvent, where the temperature plays no role. However, the techniques used for polymers in a good solvent should still apply to other types of polymers solutions. 5.1. Poor solvents We rst examine brie y what happens for polymers in a poor solvent, using as a model SAW polymers with a nearest neighbour attraction of strength − . It is known that as the temperature decreases, there is a temperature Tcol below which the polymer collapses into a compact globule and loses its fractal nature [13]. The e ective potentials will then be fundamentally di erent of course. But as long as we stay above this temperature, we expect that the interaction should become less strong with decreasing temperature, as shown in Fig. 5. 5.2. Mixtures Renormalization group (RG) calculations for the interaction between the CM of two polymers of di ering lengths Rg1 and Rg2 suggest that the interaction strength at full overlap should weaken with increasingsize asymmetry, and that the interaction range should approximately scale as R12 = 21 R2g1 + R2g2 [21]. We con rm this behaviour for simulations of a number of di erent length SAW polymers at  = 0 in Fig. 5. 5.3. Phase separation? An interesting prediction is that binary mixtures of Gaussian core particles interacting via the pair potentials V (r) =  exp{−(r=R )2 } will phase-separate over broad √ ranges of the coupling constant ratio |12 |= 11 22 . Semi-quantitatively correct phase diagrams follow already from an analytic mean- eld calculation [17,22], and suggest that polymers in a good solvent will not phase separate at low densities. 260 A.A. Louis et al. / Physica A 306 (2002) 251 – 261 5.4. Relationship with PRISM Each time the present coarse-graining methods are applied to a new type of polymer solution, we need a new set of computer simulations at the monomer level for the parameters of interest. It would be very helpful to nd other, semi-analytic, ways of providing input information that are faster and more exible. One candidate would be PRISM [23], an integral equation method which has been applied to a wide variety of polymeric systems. This requires a way of deriving CM–CM correlations from the monomer-based correlations provided by PRISM. As a rst step in this direction, we have derived an approximate relationship which is much more accurate than earlier, heuristic approaches [24]. This could form the basis for using PRISM or other monomer-based methods as input to our “polymers as soft colloids” approach. 6. Conclusions In summary then, we have shown how to coarse-grain polymers as single “soft colloids”, with just three degrees of freedom each, interacting via density-independent pair, triplet, and higher body potentials. These, however, become rapidly intractable. In parallel, we also derived density-dependent pair potentials which include, in an average way, the e ect of the higher n-body interactions. These e ective pair potentials exactly reproduce the two-body correlations of the underlying polymer solution, and, through the compressibility equation, they reproduce the EOS as well. In a similar way, the e ective one-body potentials which exactly reproduce the one-body density pro les near walls and spheres can also be derived. Because these reproduce the correct adsorptions, the thermodynamics of a polymer solution near a non-adsorbing surfaces are also correctly reproduced by our formulation. We also showed how to extend these methods to derive e ective potentials for polymers in a poor solvent and for mixtures of di erent length polymers. We sketched some ways in which other monomer-based methods such as PRISM could be used as the source of input to derive our potentials. At this point one might ask what has been gained, since at each point direct computer simulations were needed as input to derive the potentials. This question brings us back to the aim stated at the outset: to describe mixtures of many polymers and many colloids. Here our coarse-graining of polymers as soft colloids does result in important simpli cations. For example, we have performed such simulations for mixtures of spheres of radius Rc and polymers with sizes ranging from Rg =Rc ≈ 0:3 to Rg =Rc ≈ 1, and determined the polymer induced phase-separation of the colloids [25]. The e ective polymer density =∗ at the critical point increases with increasing polymer size, but even for the largest polymers it is still in the regime =∗ 6 1, where we expect our formulation to work best. Simulations with a full polymer model would be about two orders of magnitude slower. Without coarse-graining the polymers as soft colloids, such a simulation would have been virtually impossible to perform. A.A. Louis et al. / Physica A 306 (2002) 251 – 261 261 Acknowledgements A.A.L. acknowledges support from the Isaac Newton Trust, Cambridge, P.B. and V.K. acknowledge support from the EPSRC under Grant No. GR=M88839, R.F. acknowledges support from the Oppenheimer Fund, E.J.M. acknowledges support from the Royal Netherlands Academy of Arts and Sciences, as well as from the Stichting Nationale Computerfaciliteiten (NCF) and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for the use of supercomputer facilities. References [1] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1989. [2] H.N.W. Lekkerkerker, W.C.K. Poon, P.N. Pusey, A. Stroobants, P.B. Warren, Europhys. Lett. 20 (1992) 559. [3] E.J. Meijer, D. Frenkel, J. Chem. Phys. 100 (1994) 6873. [4] W.C.K. Poon, et al., Philos. Trans. Roy. Soc. A 359 (2001) 897; H. Lowen, et al., Philos. Trans. Roy. Soc. A 359 (2001) 909; A. Yodh, et al., Philos. Trans. Roy. Soc. 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[16] R.L. Henderson, Phys. Lett. A 49 (1974) 197; J.T. Chayes, L. Chayes, J. Stat. Phys. 36 (1984) 471; L. Reatto, Philos. Mag. A 58 (1986) 37. [17] A.A. Louis, P.G. Bolhuis, J.P. Hansen, Phys. Rev. E 62 (2000) 7961; C.N. Likos, A. Lang, M. Watzlawek, H. Lowen, Phys. Rev. E 63 (2001) 031 206. [18] M.A. van der Hoef, P.A. Madden, J. Chem. Phys. 111 (1999) 1520 and references therein. [19] A.A. Louis, 2001, in preparation. [20] Y. Mao, P. Bladon, H.N.W. Lekkerkerker, M.E. Cates, Mol. Phys. 92 (1997) 151. [21] B. Kruger, L. Schafer, A. Baumgartner, J. Phys. France 50 (1989) 319. [22] R. Finken, J.P. Hansen, A.A. Louis, to appear in Europhys. Lett. (2002); A.J. Archer, R. Evans, Phys. Rev. E 64 (2001) 041 501. [23] K.S. Schweitzer, J. Curro, Adv. Chem. Phys. 98 (1997) 1. [24] V. Krakoviack, J.P. Hansen, A.A. Louis, submitted for publication. [25] P.G. Bolhuis, A.A. Louis, J.P. Hansen, in preparation.