- Email: [email protected]
Effects of solvent on polymer chain dimensions: a Born–Green–Yvon integral equation study
Fluid Phase Equilibria 150–151 Ž1998. 641–648
Effects of solvent on polymer chain dimensions: a Born–Green–Yvon integral equation study Mark P. Taylor, J.E.G. Lipson
)
Department of Chemistry, Dartmouth College, HanoÕer, NH 03755, USA
Abstract The equilibrium properties of a tangent-hard-sphere polymer chain Žwith site diameter s . in a hard-sphere monomer solvent Žwith monomer diameter D s s . are studied using the Born–Green–Yvon integral equation in conjunction with a two-site solvation potential. The solvation potential is constructed using low-density results for a hard-sphere trimer in a hard sphere solvent. The BGY equation has been solved for polymers of lengths up to 200 for a range of solvent densities. The theory accurately describes the compression of the average polymer dimensions with increasing solvent density. Scaling exponents relating the polymer dimensions to chain length and solvent density are also obtained. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Statistical mechanics; Model; Integral equation; Polymer; Born–Green–Yvon; Solvation potential
1. Introduction The statistical properties of a polymer in solution are strongly dependent on the nature of the solvent w1–3x. In a so-called ‘good’ solvent, the dimensions of the polymer are expanded relative to those of a noninteracting or ideal polymer, while in a ‘poor’ solvent, the polymer dimensions are collapsed relative to the ideal or theta state. A flexible hard-sphere chain with site diameter s in a fluid of hard sphere monomers with diameter D provides a model for a polymer in a solvent. The limiting case of D s 0 Ž i.e., an isolated hard-sphere chain. is commonly used to represent a polymer in a good solvent. For D ) 0, the simple athermal polymer–solvent system exhibits interesting solvent effects w4–7x. In particular, with increasing solvent density, the average dimensions of the hard-sphere polymer are compressed and the magnitude of this compression effect is strongly dependent on the ratio Drs . For Drs - 1, the hard sphere fluid represents a good solvent for the hard-sphere chain at all densities. However, for larger Drs Ž ; 5. , Suen et al. w7x have recently reported evidence that a )
Corresponding author. Tel.: q1-603-646-2390; fax: q1-603-646-3946; e-mail: [email protected]
0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 3 4 3 - 4
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
642
high-density hard-sphere fluid provides a poor solvent for a long hard-sphere chain Ž i.e., the chain undergoes a collapse transition!.. Despite these very interesting size effects, in this brief work, we limit ourselves to the case of a D s s solvent. One approach for studying an n-mer chain in solution is to introduce an n-site solvation potential which accounts for the interactions of the solvent with the polymer w6,8–10x. Approximating the n-site potential as a sum of two-site potentials allows one to map the polymer–solvent problem onto that of an isolated polymer whose segments interact via a density-dependent effective potential. In this work, we introduce a two-site solvation potential and study the effects of solvent density on a hard-sphere chain polymer using a Born–Green–Yvon Ž BGY. integral equation w11x.
2. Theory 2.1. Intramolecular distribution functions In this paper, we consider the equilibrium structure of a single polymer molecule in a monomeric solvent. The polymer is modeled as a flexible chain of n identical hard-sphere monomers with diameter s which are connected by universal joints of bond length s . The solvent consists of N hard spheres of diameter D s s and the solvent density is r s NrV, where V is the total system volume. The configurational properties of the polymer can be expressed in terms of a set of intramolecular site–site distribution functions wi j Ž r; r . which are related to the probability that two sites on the polymer, i and j, are separated by a distance r s <™ r i y™ r j <. The exact expressions for the wi j functions are quite formidable, since they involve integrals over the complete set of the N solvent molecule coordinates. We note that this full problem is amenable to a BGY-type integral equation treatment w11,12x although in here, we take a much simpler tack. A formal simplification in the definition of the wi j functions can be achieved by writing the integrals over the solvent coordinates in terms of an n-site solvation potential w8x. Now, by making the assumption that the n-site solvation potential can be written as a sum of two-site solvation potentials ŽChandler and Pratts’ superposition approximation. , we can map the full problem described above on to that of an isolated chain with a modified site–site potential which implicitly accounts for the effects of the solvent. The resulting density-dependent effective potential between nonbonded polymer sites i and j Ž < i y j < ) 1. is given by hs sol u eff i j Ž r ; r . s ui j Ž r . q ui j Ž r ; r . ,
Ž1.
Ž . where u hs i j r is the original hard sphere potential u hs ij Žr. s
½
r-s r)s
` 0
Ž2.
Ž . and u sol i j r; r is the site–site solvation potential. The site–site distribution function for an n-mer chain whose segments interact via such an effective potential is given by wi j Ž r ; r . s
1
ny2
n
P P exp yb u H . . . H as1 bsaq2 Z Ž r. n
eff ab
ny1
n
a s1
m/i , j
rm , Ž r ; r . P sa , aq1 P d™
Ž3.
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
643
where b s 1rk BT, sŽ r . s d Ž r y s .r4ps 2 is the intramolecular distribution function between bonded sites, and ZnŽ r . is the single chain partition function, which ensures the normalization condition
Hd r 4p r
2
wi j Ž r ; r . s 1.
Ž4.
Due to the constraints of chain connectivity, the wi j Ž r; r . functions are identically zero for r ) < i y j < s . A number of equilibrium configurational properties of the polymer chain may be expressed as averages over these distribution functions. In particular, the mean-square distance between sites i and j is ² ri2j : s d r r 2 4p r 2 wi j Ž r ; r .
Ž5.
H
and the mean-square radius of gyration is ² R g2 : s
1 n2
n
Ý ² ri2j : .
Ž6.
i-j
2.2. BGY integral equation As we have shown previously w11x, an integral equation for the intramolecular site–site distribution functions of an isolated interaction-site polymer can be obtained following the method of Born and Green w13x and Yvon w14x. The BGY equation for the end-to-end distribution function of an n-mer chain is given by ™
™
™
= 1 w 1n s = 1 yb u1effn Ž r ; r . w 1 n q d™ r 2 = 1 ln s12 w 12 n q
H
ny1
™
Ý Hd™rÕ = 1
eff yb u1Õ Ž r ; r . w1Õ n ,
Õs3
Ž7. where w 1Õ n is a three-site distribution function which satisfies the reduction condition w 1n s d™ r Õ w 1Õ n .
H
Ž8.
In order to solve the above BGY equation, we must introduce two approximations. First, the required three-site functions are expressed in terms of two-site functions using a superposition-like approximation and, second, the distribution function between any two sites i and j is approximated by the end-to-end distribution function of a chain of length i q j y 1. With these approximations, Eq. Ž7. can be solved recursively by direct numerical integration for a chain of arbitrary length. For details of the numerical solution the reader is referred to Ref. w11x. This BGY theory has been used to study both isolated hard-sphere and square-well chain and ring polymers w11,15x. The success of the theory in these cases provides validation of the above approximations. In this work, we apply the BGY theory to study a polymer in solution via the
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
644
solvation potential approach described in Section 2.1. The construction of the required solvation potential is described forthwith. 2.3. The solÕation potential Here, we construct a polymer solvation potential which is motivated by low-density results for a hard-sphere 3-mer in a hard-sphere solvent. This task is accomplished by equating a low-density expression for the hard-sphere 3-mer end-to-end distribution function to the analogous distribution function for an isolated 3-mer whose sites interact via an effective potential. To the first order in solvent density, the end-to-end distribution function for a hard-sphere 3-mer is given by w 13 Ž r ; r . s
hs exp yb u13 Žr.
Z3 Ž r .
Ž 0. Ž1. w13 Ž r . q r w13 Žr.
sFrF2s ,
where the zeroth order coefficient is 1 Ž0. w 13 r 2 s12 s23 s sFrF2s , Ž r . s d™ 8ps 2 r and the first-order coefficient can be written as
Ž9.
Ž 10.
H
Ž1. Ž0. w 13 Ž r . s w13 Ž r . d 3 Ž r . q d4 Ž s , s ,r . .
Ž 11 .
The f-bond diagrams comprising this first-order coefficient are defined as p d 3 Ž r 13 . s d™ r4 f 14 f 34 s Ž 16 s 3 y 12 rs 2 q r 3 . s F r F 2 s 12 and
H
d4 Ž r 12 ,r 23 ,r 13 . s d™ r4 f 14 f 24 f 34 ,
Ž 12.
Ž 13.
H
where the solvent–polymer site interaction Mayer f-function is given by y1 r-s Ž 14. 0 r)s . An analytic expression for d4 is provided in Ref. w16x. In Table 1, we compare results for the mean-square end-to-end distance of a hard-sphere 3-mer in a D s s hard-sphere solvent computed fŽr.s
½
Table 1 Mean-square end-to-end distance for a tangent-hard-sphere 3-mer in a hard-sphere solvent Ž Ds s . computed via a first-order density expansion and via Monte Carlo simulation w6x
h
r-Expansion
Monte Carlo
0.0 0.1 0.2 0.3 0.4 0.45
2.50 2.48 2.45 2.43 2.41 2.40
2.50 2.47 2.44 2.40 2.36 2.34
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
645
using the Eq. Ž9. first-order expansion with those from Monte Carlo ŽMC. simulation w6x. The expansion correctly yields the solvent induced compression with increasing solvent volume fraction h s pr D 3r6, although, as expected for a first-order expansion, the magnitude of this effect is underestimated at high density. The end-to-end distribution function of an isolated 3-mer whose sites interact via an effective potential is given exactly w17x by w 13 Ž r ; r . s
eff exp yb u13 Žr; r.
Z3 Ž r .
Ž0. w 13 Žr.
sFrF2s .
Ž 15.
Using the definition of u eff Ž r . given in Eq. Ž 1. and equating the above two expressions for w 13Ž r . ŽEqs. Ž9. and Ž 15.., yields the following solvation potential: Ž1 . Ž0. b u sol Ž r ; r . s yln 1 q r w13 Ž r . rw13 Žr.
sFrF2s .
Ž 16.
In Fig. 1, we plot this site–site solvation potential as a function of solvent volume fraction h. The potential is short-ranged Žvanishing for r ) 31r2s . and purely attractive and its strength increases with increasing solvent density. In the Fig. 1 inset, we show the following reduced second virial coefficient for the Eq. Ž1. effective potential: b 2) s b 2 Ž r . rb 2 Ž 0 . s 1 y 3
2
2
H1 d x x Žexp yb u
sol
Ž x ; r . y 1.
Ž 17.
where x s rrs and b 2 Ž r s 0. is the hard-sphere second virial coefficient. The fact that b 2) is always positive indicates that the effective potential considered here represents a ‘good’ solvent at all densities. In the following, we apply this solvation potential to the problem of a hard-sphere polymer in a monomeric hard-sphere solvent. This potential mimics a very local, solvent induced site–site interaction mediated by a single solvent molecule. The potential does not include any information
Fig. 1. Solvation potential as a function of polymer site–site separation r over a range of solvent volume fractions h as indicated. Inset: reduced second virial coefficient b 2) for the Eq. Ž1. effective potential vs. h.
646
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
about the local solvent structure. Since the interior sites of a long-chain molecule are much more shielded from solvent than the sites of a 3-mer, we expect any 3-mer derived solvation potential to overestimate the compression effects of solvent on a long chain. However, since the above first-order solvation potential underestimates the compression of a 3-mer, we anticipate some cancellation of errors in applying this potential to longer chains.
3. Results Using the single chain BGY equation Ž Eq. Ž7.. and the above polymer solvation potential ŽEq. Ž16.., we have computed the intramolecular distribution functions of hard-sphere chains in a hard-sphere solvent for chains of lengths 4 F n F 200 and solvent volume fractions 0.0 F h F 0.5 Žh s pr D 3r6.. Rather than showing the actual wi j Ž r; r . functions, we present averages over these distribution functions in the form of the polymer mean-square end-to-end distance and radius of gyration defined in Eqs. Ž5. and Ž6., respectively. In Figs. 2 and 3, we show the variation in average polymer size with solvent density. The BGY results for the mean-square end-to-end distance for chains of lengths n s 10, 20 and 30 are shown in Fig. 2 along with corresponding MC data for the polymer–solvent systems w5,6x and, in the case of n s 20, MC data for the analogous polymer melt system w18x. The BGY theory is seen to correctly predict that the polymers are moderately compressed with increasing solvent density. For the n s 10 chain, the BGY theory is quite accurate in comparison with the MC data even out to high solvent density. For n s 20 and 30, the theory gives results in good agreement with the single MC data point at h s 0.20. Despite the limited amount of MC data for the polymer–solvent system, it still seems fair to conclude that the theory works well for short chains. Results for longer chains are given in Fig. 3 where we show the mean-square radius of gyration for chains of length n s 50, 100, 150 and 200. In addition to the BGY results, we also include MC data
Fig. 2. Variation of the mean-square end-to-end distance with solvent density h for a hard-sphere n-mer chain in a hard sphere solvent of diameter Ds s . Solid lines are the results of the BGY theory and the open symbols are MC data from Grayce Ž D, ns10. w6x and from Escobedo and de Pablo ŽI, ns 20; ` ns 30. w5x. The filled symbols are MC data for an ns 20 polymer melt system taken from Yethiraj and Hall w18x.
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
647
Fig. 3. Variation of the mean-square radius of gyration with solvent density h for a hard-sphere n-mer chain in a hard-sphere solvent of diameter Ds s . Solid lines are the results of the BGY theory and the open symbols are MC data for isolated chains w15x. The filled symbols are MC data for the analogous polymer melt systems taken from Yethiraj and Hall Ž', ns 50; B ns100. w18x.
for the analogous n s 50 and 100 polymer melt systems w18x. Again, the theory predicts that the hard-sphere chains are moderately compressed with increasing solvent volume fraction. The rate of compression appears to be comparable to that observed in the polymer melt systems. In fact, looking at the n s 20, h s 0.20 MC data shown in Fig. 2, one might surmise that the behavior of the polymer–solvent system is not that different from the analogous polymer melt system, with the dimensions of a polymer in solution being somewhat larger than those of a polymer in a melt. Our results for the average size of a hard-sphere polymer in a hard-sphere monomer solvent are quite similar to Ž; 10% larger than. those obtained by Grayce et al. w10x, via their Percus–Yevick-type solvation potential, for the average size of a hard-sphere polymer in a polymer melt system. For a long chain in a solvent, one expects a scaling relationship of the form R 2 ; n2 Õ Žwhere R 2 is either the mean-square end-to-end distance or radius of gyration. , where an exponent of 2 Õ s 1 corresponds to a theta solvent while 2 Õ ) 1 and 2 Õ - 1 correspond to good and poor solvents, respectively w1–3x. Our BGY results for ² R g2 : for 100 F n F 200 give exponents of 2 Õ s 1.31, 1.21 and 1.16 for solvent volume fractions of h s 0.0, 0.3 and 0.5, respectively. The corresponding results 2 : for ² r 1n are 2 Õ s 1.34, 1.23 and 1.17. The fact that the h s 0.0 result exceeds the Flory value of 6r5 is due both to the approximations inherent in the BGY approach w11,15x and to the somewhat small chain lengths used in computing this exponent. Nonetheless, this exponent is still a good indicator of solvent condition as we have shown in our BGY study of a square-well chain w11x. Thus, we can conclude that the D s s solvent is a ‘good’ solvent Žfor a hard-sphere chain. at all densities and that increasing solvent density corresponds to making the solvent conditions more ‘theta-like’. These conclusions are entirely consistent with the behavior of the effective potential second virial coefficient shown in Fig. 1. Thus, as suggested above, an athermal 1-mer solvent produces similar effects on the dimensions of a single n-mer chain as an athermal n-mer solvent Žfor which 2 Õ ™ 1 with increasing system density w19x.. Finally, for a long chain in a dense solvent, one also expects the scaling relationship R 2 ; hyg with g f 0.25 for a good solvent w2,20x. The BGY results shown in Fig. 3 for n s 200 give an
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
648
exponent of g f 0.42 for 0.3 F h F 0.5. Although this BGY value exceeds g f 0.25, this latter value of g presumes that one is in the ideal scaling regime Ž i.e., 2 Õ s 1., which is not the case for our BGY results. Thus, it is difficult to draw any conclusions about the accuracy of our theory for long chains at high solvent densities. Additional MC studies would allow us to assess more definitively the range of validity of this theory.
4. Conclusion In this work, we have studied the equilibrium properties of a hard-sphere polymer chain Ž with site diameter s . in a hard-sphere monomer solvent Ž with monomer diameter D s s .. The polymer–solvent system is treated in the context of a polymer site–site solvation potential. Using the BGY integral equation, we have computed the intramolecular distribution functions for polymers over a range of solvent densities. The BGY theory accurately describes the compression of the average polymer dimensions with increasing solvent density. The D s s solvent is found to be a ‘good’ solvent Ž i.e., 2 Õ ) 1. for all densities.
Acknowledgements Financial support from the National Science Foundation Ž grant DMR-9424086. and the Camille and Henry Dreyfus Foundation is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
P.J. Flory, Principles of Polymer Chemistry, Cornell University, Ithaca, 1953. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University, Ithaca, 1979. A.Yu. Grosberg, A.R. Khokhlov, Statistical Physics of Macromolecules, AIP, New York, 1994. J. Komorowski, W. Bruns, J. Chem. Phys. 103 Ž1995. 5756–5761. F.A. Escobedo, J.J. de Pablo, Mol. Phys. 89 Ž1996. 1733–1754. C.J. Grayce, J. Chem. Phys. 106 Ž1997. 5171–5180. J.K.C. Suen, F.A. Escobedo, J.J. de Pablo, J. Chem. Phys. 106 Ž1997. 1288–1290. D. Chandler, L.R. Pratt, J. Chem. Phys. 65 Ž1976. 2925–2940. C.J. Grayce, K.S. Schweizer, J. Chem. Phys. 100 Ž1994. 6846–6856. C.J. Grayce, A. Yethiraj, K.S. Schweizer, J. Chem. Phys. 100 Ž1994. 6857–6872. M.P. Taylor, J.E.G. Lipson, J. Chem. Phys. 104 Ž1996. 4835–4841. M.P. Taylor, J.E.G. Lipson, J. Chem. Phys. 102 Ž1995. 2118–2125. M. Born, H.S. Green, Proc. R. Soc. London, Ser. A 188 Ž1946. 10–18. J. Yvon, Actual. Sci. Ind. 203, 1935. M.P. Taylor, J.L. Mar, J.E.G. Lipson, J. Chem. Phys. 106 Ž1997. 5181–5188. M.J.D. Powell, Mol. Phys. 7 Ž1964. 591–592. M.P. Taylor, Mol. Phys. 86 Ž1995. 73–85. A. Yethiraj, C.K. Hall, J. Chem. Phys. 96 Ž1992. 797–807. P.J. Flory, J. Chem. Phys. 17 Ž1949. 303–310. M.K. Kosmas, K.F. Freed, J. Chem. Phys. 69 Ž1978. 3647–3659.
Effects of solvent on polymer chain dimensions: a Born–Green–Yvon integral equation study Mark P. Taylor, J.E.G. Lipson
)
Department of Chemistry, Dartmouth College, HanoÕer, NH 03755, USA
Abstract The equilibrium properties of a tangent-hard-sphere polymer chain Žwith site diameter s . in a hard-sphere monomer solvent Žwith monomer diameter D s s . are studied using the Born–Green–Yvon integral equation in conjunction with a two-site solvation potential. The solvation potential is constructed using low-density results for a hard-sphere trimer in a hard sphere solvent. The BGY equation has been solved for polymers of lengths up to 200 for a range of solvent densities. The theory accurately describes the compression of the average polymer dimensions with increasing solvent density. Scaling exponents relating the polymer dimensions to chain length and solvent density are also obtained. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Statistical mechanics; Model; Integral equation; Polymer; Born–Green–Yvon; Solvation potential
1. Introduction The statistical properties of a polymer in solution are strongly dependent on the nature of the solvent w1–3x. In a so-called ‘good’ solvent, the dimensions of the polymer are expanded relative to those of a noninteracting or ideal polymer, while in a ‘poor’ solvent, the polymer dimensions are collapsed relative to the ideal or theta state. A flexible hard-sphere chain with site diameter s in a fluid of hard sphere monomers with diameter D provides a model for a polymer in a solvent. The limiting case of D s 0 Ž i.e., an isolated hard-sphere chain. is commonly used to represent a polymer in a good solvent. For D ) 0, the simple athermal polymer–solvent system exhibits interesting solvent effects w4–7x. In particular, with increasing solvent density, the average dimensions of the hard-sphere polymer are compressed and the magnitude of this compression effect is strongly dependent on the ratio Drs . For Drs - 1, the hard sphere fluid represents a good solvent for the hard-sphere chain at all densities. However, for larger Drs Ž ; 5. , Suen et al. w7x have recently reported evidence that a )
Corresponding author. Tel.: q1-603-646-2390; fax: q1-603-646-3946; e-mail: [email protected]
0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 3 4 3 - 4
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
642
high-density hard-sphere fluid provides a poor solvent for a long hard-sphere chain Ž i.e., the chain undergoes a collapse transition!.. Despite these very interesting size effects, in this brief work, we limit ourselves to the case of a D s s solvent. One approach for studying an n-mer chain in solution is to introduce an n-site solvation potential which accounts for the interactions of the solvent with the polymer w6,8–10x. Approximating the n-site potential as a sum of two-site potentials allows one to map the polymer–solvent problem onto that of an isolated polymer whose segments interact via a density-dependent effective potential. In this work, we introduce a two-site solvation potential and study the effects of solvent density on a hard-sphere chain polymer using a Born–Green–Yvon Ž BGY. integral equation w11x.
2. Theory 2.1. Intramolecular distribution functions In this paper, we consider the equilibrium structure of a single polymer molecule in a monomeric solvent. The polymer is modeled as a flexible chain of n identical hard-sphere monomers with diameter s which are connected by universal joints of bond length s . The solvent consists of N hard spheres of diameter D s s and the solvent density is r s NrV, where V is the total system volume. The configurational properties of the polymer can be expressed in terms of a set of intramolecular site–site distribution functions wi j Ž r; r . which are related to the probability that two sites on the polymer, i and j, are separated by a distance r s <™ r i y™ r j <. The exact expressions for the wi j functions are quite formidable, since they involve integrals over the complete set of the N solvent molecule coordinates. We note that this full problem is amenable to a BGY-type integral equation treatment w11,12x although in here, we take a much simpler tack. A formal simplification in the definition of the wi j functions can be achieved by writing the integrals over the solvent coordinates in terms of an n-site solvation potential w8x. Now, by making the assumption that the n-site solvation potential can be written as a sum of two-site solvation potentials ŽChandler and Pratts’ superposition approximation. , we can map the full problem described above on to that of an isolated chain with a modified site–site potential which implicitly accounts for the effects of the solvent. The resulting density-dependent effective potential between nonbonded polymer sites i and j Ž < i y j < ) 1. is given by hs sol u eff i j Ž r ; r . s ui j Ž r . q ui j Ž r ; r . ,
Ž1.
Ž . where u hs i j r is the original hard sphere potential u hs ij Žr. s
½
r-s r)s
` 0
Ž2.
Ž . and u sol i j r; r is the site–site solvation potential. The site–site distribution function for an n-mer chain whose segments interact via such an effective potential is given by wi j Ž r ; r . s
1
ny2
n
P P exp yb u H . . . H as1 bsaq2 Z Ž r. n
eff ab
ny1
n
a s1
m/i , j
rm , Ž r ; r . P sa , aq1 P d™
Ž3.
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
643
where b s 1rk BT, sŽ r . s d Ž r y s .r4ps 2 is the intramolecular distribution function between bonded sites, and ZnŽ r . is the single chain partition function, which ensures the normalization condition
Hd r 4p r
2
wi j Ž r ; r . s 1.
Ž4.
Due to the constraints of chain connectivity, the wi j Ž r; r . functions are identically zero for r ) < i y j < s . A number of equilibrium configurational properties of the polymer chain may be expressed as averages over these distribution functions. In particular, the mean-square distance between sites i and j is ² ri2j : s d r r 2 4p r 2 wi j Ž r ; r .
Ž5.
H
and the mean-square radius of gyration is ² R g2 : s
1 n2
n
Ý ² ri2j : .
Ž6.
i-j
2.2. BGY integral equation As we have shown previously w11x, an integral equation for the intramolecular site–site distribution functions of an isolated interaction-site polymer can be obtained following the method of Born and Green w13x and Yvon w14x. The BGY equation for the end-to-end distribution function of an n-mer chain is given by ™
™
™
= 1 w 1n s = 1 yb u1effn Ž r ; r . w 1 n q d™ r 2 = 1 ln s12 w 12 n q
H
ny1
™
Ý Hd™rÕ = 1
eff yb u1Õ Ž r ; r . w1Õ n ,
Õs3
Ž7. where w 1Õ n is a three-site distribution function which satisfies the reduction condition w 1n s d™ r Õ w 1Õ n .
H
Ž8.
In order to solve the above BGY equation, we must introduce two approximations. First, the required three-site functions are expressed in terms of two-site functions using a superposition-like approximation and, second, the distribution function between any two sites i and j is approximated by the end-to-end distribution function of a chain of length i q j y 1. With these approximations, Eq. Ž7. can be solved recursively by direct numerical integration for a chain of arbitrary length. For details of the numerical solution the reader is referred to Ref. w11x. This BGY theory has been used to study both isolated hard-sphere and square-well chain and ring polymers w11,15x. The success of the theory in these cases provides validation of the above approximations. In this work, we apply the BGY theory to study a polymer in solution via the
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
644
solvation potential approach described in Section 2.1. The construction of the required solvation potential is described forthwith. 2.3. The solÕation potential Here, we construct a polymer solvation potential which is motivated by low-density results for a hard-sphere 3-mer in a hard-sphere solvent. This task is accomplished by equating a low-density expression for the hard-sphere 3-mer end-to-end distribution function to the analogous distribution function for an isolated 3-mer whose sites interact via an effective potential. To the first order in solvent density, the end-to-end distribution function for a hard-sphere 3-mer is given by w 13 Ž r ; r . s
hs exp yb u13 Žr.
Z3 Ž r .
Ž 0. Ž1. w13 Ž r . q r w13 Žr.
sFrF2s ,
where the zeroth order coefficient is 1 Ž0. w 13 r 2 s12 s23 s sFrF2s , Ž r . s d™ 8ps 2 r and the first-order coefficient can be written as
Ž9.
Ž 10.
H
Ž1. Ž0. w 13 Ž r . s w13 Ž r . d 3 Ž r . q d4 Ž s , s ,r . .
Ž 11 .
The f-bond diagrams comprising this first-order coefficient are defined as p d 3 Ž r 13 . s d™ r4 f 14 f 34 s Ž 16 s 3 y 12 rs 2 q r 3 . s F r F 2 s 12 and
H
d4 Ž r 12 ,r 23 ,r 13 . s d™ r4 f 14 f 24 f 34 ,
Ž 12.
Ž 13.
H
where the solvent–polymer site interaction Mayer f-function is given by y1 r-s Ž 14. 0 r)s . An analytic expression for d4 is provided in Ref. w16x. In Table 1, we compare results for the mean-square end-to-end distance of a hard-sphere 3-mer in a D s s hard-sphere solvent computed fŽr.s
½
Table 1 Mean-square end-to-end distance for a tangent-hard-sphere 3-mer in a hard-sphere solvent Ž Ds s . computed via a first-order density expansion and via Monte Carlo simulation w6x
h
r-Expansion
Monte Carlo
0.0 0.1 0.2 0.3 0.4 0.45
2.50 2.48 2.45 2.43 2.41 2.40
2.50 2.47 2.44 2.40 2.36 2.34
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
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using the Eq. Ž9. first-order expansion with those from Monte Carlo ŽMC. simulation w6x. The expansion correctly yields the solvent induced compression with increasing solvent volume fraction h s pr D 3r6, although, as expected for a first-order expansion, the magnitude of this effect is underestimated at high density. The end-to-end distribution function of an isolated 3-mer whose sites interact via an effective potential is given exactly w17x by w 13 Ž r ; r . s
eff exp yb u13 Žr; r.
Z3 Ž r .
Ž0. w 13 Žr.
sFrF2s .
Ž 15.
Using the definition of u eff Ž r . given in Eq. Ž 1. and equating the above two expressions for w 13Ž r . ŽEqs. Ž9. and Ž 15.., yields the following solvation potential: Ž1 . Ž0. b u sol Ž r ; r . s yln 1 q r w13 Ž r . rw13 Žr.
sFrF2s .
Ž 16.
In Fig. 1, we plot this site–site solvation potential as a function of solvent volume fraction h. The potential is short-ranged Žvanishing for r ) 31r2s . and purely attractive and its strength increases with increasing solvent density. In the Fig. 1 inset, we show the following reduced second virial coefficient for the Eq. Ž1. effective potential: b 2) s b 2 Ž r . rb 2 Ž 0 . s 1 y 3
2
2
H1 d x x Žexp yb u
sol
Ž x ; r . y 1.
Ž 17.
where x s rrs and b 2 Ž r s 0. is the hard-sphere second virial coefficient. The fact that b 2) is always positive indicates that the effective potential considered here represents a ‘good’ solvent at all densities. In the following, we apply this solvation potential to the problem of a hard-sphere polymer in a monomeric hard-sphere solvent. This potential mimics a very local, solvent induced site–site interaction mediated by a single solvent molecule. The potential does not include any information
Fig. 1. Solvation potential as a function of polymer site–site separation r over a range of solvent volume fractions h as indicated. Inset: reduced second virial coefficient b 2) for the Eq. Ž1. effective potential vs. h.
646
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
about the local solvent structure. Since the interior sites of a long-chain molecule are much more shielded from solvent than the sites of a 3-mer, we expect any 3-mer derived solvation potential to overestimate the compression effects of solvent on a long chain. However, since the above first-order solvation potential underestimates the compression of a 3-mer, we anticipate some cancellation of errors in applying this potential to longer chains.
3. Results Using the single chain BGY equation Ž Eq. Ž7.. and the above polymer solvation potential ŽEq. Ž16.., we have computed the intramolecular distribution functions of hard-sphere chains in a hard-sphere solvent for chains of lengths 4 F n F 200 and solvent volume fractions 0.0 F h F 0.5 Žh s pr D 3r6.. Rather than showing the actual wi j Ž r; r . functions, we present averages over these distribution functions in the form of the polymer mean-square end-to-end distance and radius of gyration defined in Eqs. Ž5. and Ž6., respectively. In Figs. 2 and 3, we show the variation in average polymer size with solvent density. The BGY results for the mean-square end-to-end distance for chains of lengths n s 10, 20 and 30 are shown in Fig. 2 along with corresponding MC data for the polymer–solvent systems w5,6x and, in the case of n s 20, MC data for the analogous polymer melt system w18x. The BGY theory is seen to correctly predict that the polymers are moderately compressed with increasing solvent density. For the n s 10 chain, the BGY theory is quite accurate in comparison with the MC data even out to high solvent density. For n s 20 and 30, the theory gives results in good agreement with the single MC data point at h s 0.20. Despite the limited amount of MC data for the polymer–solvent system, it still seems fair to conclude that the theory works well for short chains. Results for longer chains are given in Fig. 3 where we show the mean-square radius of gyration for chains of length n s 50, 100, 150 and 200. In addition to the BGY results, we also include MC data
Fig. 2. Variation of the mean-square end-to-end distance with solvent density h for a hard-sphere n-mer chain in a hard sphere solvent of diameter Ds s . Solid lines are the results of the BGY theory and the open symbols are MC data from Grayce Ž D, ns10. w6x and from Escobedo and de Pablo ŽI, ns 20; ` ns 30. w5x. The filled symbols are MC data for an ns 20 polymer melt system taken from Yethiraj and Hall w18x.
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
647
Fig. 3. Variation of the mean-square radius of gyration with solvent density h for a hard-sphere n-mer chain in a hard-sphere solvent of diameter Ds s . Solid lines are the results of the BGY theory and the open symbols are MC data for isolated chains w15x. The filled symbols are MC data for the analogous polymer melt systems taken from Yethiraj and Hall Ž', ns 50; B ns100. w18x.
for the analogous n s 50 and 100 polymer melt systems w18x. Again, the theory predicts that the hard-sphere chains are moderately compressed with increasing solvent volume fraction. The rate of compression appears to be comparable to that observed in the polymer melt systems. In fact, looking at the n s 20, h s 0.20 MC data shown in Fig. 2, one might surmise that the behavior of the polymer–solvent system is not that different from the analogous polymer melt system, with the dimensions of a polymer in solution being somewhat larger than those of a polymer in a melt. Our results for the average size of a hard-sphere polymer in a hard-sphere monomer solvent are quite similar to Ž; 10% larger than. those obtained by Grayce et al. w10x, via their Percus–Yevick-type solvation potential, for the average size of a hard-sphere polymer in a polymer melt system. For a long chain in a solvent, one expects a scaling relationship of the form R 2 ; n2 Õ Žwhere R 2 is either the mean-square end-to-end distance or radius of gyration. , where an exponent of 2 Õ s 1 corresponds to a theta solvent while 2 Õ ) 1 and 2 Õ - 1 correspond to good and poor solvents, respectively w1–3x. Our BGY results for ² R g2 : for 100 F n F 200 give exponents of 2 Õ s 1.31, 1.21 and 1.16 for solvent volume fractions of h s 0.0, 0.3 and 0.5, respectively. The corresponding results 2 : for ² r 1n are 2 Õ s 1.34, 1.23 and 1.17. The fact that the h s 0.0 result exceeds the Flory value of 6r5 is due both to the approximations inherent in the BGY approach w11,15x and to the somewhat small chain lengths used in computing this exponent. Nonetheless, this exponent is still a good indicator of solvent condition as we have shown in our BGY study of a square-well chain w11x. Thus, we can conclude that the D s s solvent is a ‘good’ solvent Žfor a hard-sphere chain. at all densities and that increasing solvent density corresponds to making the solvent conditions more ‘theta-like’. These conclusions are entirely consistent with the behavior of the effective potential second virial coefficient shown in Fig. 1. Thus, as suggested above, an athermal 1-mer solvent produces similar effects on the dimensions of a single n-mer chain as an athermal n-mer solvent Žfor which 2 Õ ™ 1 with increasing system density w19x.. Finally, for a long chain in a dense solvent, one also expects the scaling relationship R 2 ; hyg with g f 0.25 for a good solvent w2,20x. The BGY results shown in Fig. 3 for n s 200 give an
M.P. Taylor, J.E.G. Lipsonr Fluid Phase Equilibria 150–151 (1998) 641–648
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exponent of g f 0.42 for 0.3 F h F 0.5. Although this BGY value exceeds g f 0.25, this latter value of g presumes that one is in the ideal scaling regime Ž i.e., 2 Õ s 1., which is not the case for our BGY results. Thus, it is difficult to draw any conclusions about the accuracy of our theory for long chains at high solvent densities. Additional MC studies would allow us to assess more definitively the range of validity of this theory.
4. Conclusion In this work, we have studied the equilibrium properties of a hard-sphere polymer chain Ž with site diameter s . in a hard-sphere monomer solvent Ž with monomer diameter D s s .. The polymer–solvent system is treated in the context of a polymer site–site solvation potential. Using the BGY integral equation, we have computed the intramolecular distribution functions for polymers over a range of solvent densities. The BGY theory accurately describes the compression of the average polymer dimensions with increasing solvent density. The D s s solvent is found to be a ‘good’ solvent Ž i.e., 2 Õ ) 1. for all densities.
Acknowledgements Financial support from the National Science Foundation Ž grant DMR-9424086. and the Camille and Henry Dreyfus Foundation is gratefully acknowledged.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x
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