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Polymer crystallization: density functional theory and application to polyethylene
Volume 179, number 4
CHEMICAL PHYSICS LETTERS
26 April 1991
Polymer crystallization: density functional theory and application to polyethylene John D. McCoy, Kevin G. Honnell, Kenneth S. Schweizer and John G. Curt-0 Sandia NationalLaboratories,Albuquerque,NM 87185, USA Received 23 January I991
A general microscopic approach for the crystallization of flexible polymer melts is formulated and applied to polyethylene.
Site-site polyatomic density functional (DF) theory has contributed to a number of interesting innovations [ l-41. The polyatomic DF theory of freezing, however, is considerably more difficult to apply than its monatomic predecessor [5,6]. This increase in technical difficulty is due to both the enhanced structural complexity of molecular crystals and the increased mathematical complexity of the DF theory itself. Disregarding the delicate effect of the form of the atomic potential, one can think of monatomic freezing as depending only upon a single length scale (i.e. the atomic diameter), while polymer freezing is sensitive to a large number of length scales corresponding to bond lengths, angles, and backbone flexibility in addition to a site diameter. A change in this distribution of length scales (arising, for instance, from a change in the gauche-trans rotational conformer energy) affects the freezing transition in ways which cannot be accounted for by the simple scaling procedure [7] appropriate for the monatomic case. The present Letter describes a tractable scheme to evaluate the polyatomic DF theory for the difficult case of polymers, and here we sketch our major results #‘. Density functional theory is not the only alternative in the study of the freezing transition. In principle, computer simulations provide an unambiguous approach. Unfortunately, continuous-space simulations of polymer crystallization are prohibitively computer intensive. However, the situation, $a A more detailed discussion is to be found in ref. [ 81. 374
both in terms of theory and simulation, is greatly simplified if the polymers are confined to a lattice. A number of years ago, F’lory developed an incompressible lattice model using a very simple mean-field theory [ 91, but the results of this treatment are suspect for two distinct reasons. First, the lattice model alters the original system in physically unreasonable ways. For instance, an incompressible lattice requires that there not be a density change upon freezing (unrealistic, since polyethylene exhibits a roughly 30% density change); the multiple length scales associated with intra- and inter-chain correlations are lost on a lattice; and it is not clear what a bead on a lattice physically represents. Second, the basic statistical approximation of the Flory mean-field theory ignores essentially all intra- and inter-chain correlations. Nagle et al. [lo] argue that these inherent approximations fatally distort the physics. While we are not prepared to make a definite statement about the adequacy of Flory’s approach, we present here a methodology which circumvents the weaknesses of the lattice mean-field treatment. Although simple coarse-grained models of polymer chains are useful for qualitatively addressing certain long wavelength questions about the structure and dynamics of polymeric melts, it is not clear if such models are adequate for describing strongly first-order phase transitions. Indeed, we find that a chemically realistic model of the polymer chain is required, and in our opinion the present work employs the minimum amount of chemical detail needed to adequately describe the crystallization of polyeth-
0009-2614/91/$ 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
CHEMICAL PHYSICS LETTERS
Volume 179, number 4
ylene, (-CH,-), where N is the degree of polymerization. Specifically, we have used the rotational isomeric state (RIS) model of Flory [ 111 where each CH, group is described by a single Lennard-Jones sphere. Density functional theory requires liquid-state pair correlation functions as input. While a separate problem from DF theory per se, calculation of these correlation functions is essential to the application of DF theory and has only recently become possible for polymers [ 12,13 1. As seen in fig. 1, we have been able to use polymer-REM (reference interaction site model) integral-equation theory [ 131 to calculate the pair correlation functions of polyethylene (as exemplified by the static structure factor) to a high degree of accuracy [ 121. In general, RISM [ 151 #2,as well as polymer-REM, is a non-penurbative, off-lattice method to calculate intermolecular site-site correlation functions, i.e. the pair correlation function g(u) and the direct correlation function C(T) (where chain-end effects have been averaged over [ 13] ) . The intramolecular structure factor must be supplied as input into RISM theory. For a flexible molecule, this is generally a difficult task since the intramolecular distribution is a non-linear functional of the intermolecular distributions, and vice versa [ 13,151. However, for the special case of long chain melts a 1)2There is no relation between the RKM theory and the RIS model of ref. [ 111.
1’5 K 1.0 rl
0.5
2
0.0 -0.5
5
k (kl)
to
Fig. 1. The structure factor for polyethylene is calculated [ 121 and compared to scattering measurements [ 141. Systemvariables are: T=413 K, p,=O.781 g/cm’; N=6429. (-_) PolymerRISM; ( 0 ) scattering [ 141.
26 April 1991
great simplification occurs since it is well established that the intramolecular distribution function is nearly the same as that of an ideal chain without long-range excluded volume #3. Polyatomic DF theory is based on a site-site description of molecular interactions. It was first derived in ref. [ 1 ] ; however, we more closely follow the derivation given in ref. [ 2 ] which examines flexible molecules of identical sites. In brief, the excess Helmoltz free-energy functional of the solid is expanded about the liquid of density pL. The primary result of either derivation [ I ,2] is an expression for the grand-potential functional difference, A W, between the solid and liquid (both at the same temperature and chemical potential) which is obtained from a Legendre transform of the Helmholtz free-energy expansion. This functional becomes the thermodynamic grand-potential difference when it is evaluated for the true solid p(r). On the other hand, if p(r) is not the true inhomogeneous density, AW is larger than the thermodynamic grand-potential difference, by a second-law argument. Hence, the minimum of AW can be used to predict the equilibrium solid density. Moreover, if the liquid of density pL were the coexistence liquid at that temperature, the minimum of A W would be zero (i.e. mechanical equilibrium). The phase transition is located (at a particular temperature) by stepping through trial liquid densities until A Wmin=O. The quantities of interest calculated in this manner are the densities at coexistence of both the liquid and the solid, as well as the crystal lattice parameters. Of great importance is the selection of an “ideal” system which defines the excess free energy. This importance is the direct result of the truncation of the functional Taylor series. Naturally, if the series were to be carried to all orders, an ideal system would be unnecessary. It has been claimed [ 21 that the ideal system should be selected so that the ideal liquid and the true liquid have the same structure: the so-called “equal-structure” criterion. While this is an important requirement, we suggest that the inherent interactions in the ideal system should be the same as in the true system: the “equal-interactions” crite1(3That is, the melt is a &solvent for itself. The intramolecular distribution function depends only upon temperature and not upon density.
375
Volume 179, number 4
rion. For rigid molecules the two criteria are identical; however, for flexible molecules this is not the case. The approach we advocate is to use the “equalinteractions” ideal system and to enforce the “equalstructure” criterion by altering the density functional formalism. This ideal system differs from a system of real chains only in that the sites on different chains do not interact (i.e. an ideal gas). The ideal interactions, then, are entirely intramolecular and are of two varieties: local and non-local. The local interactions are of the RIS type, while the nonlocal interactions are of the excluded-volume-type. In order to enforce the “equal-structure” criterion, we add thermodynamic variables which permit the ideal chain interactions to be renormalized in the liquid. To describe the liquid structure, the local interactions must take on slightly different values and the global consequences of non-local interactions must be completely screened out [ 111. Hence, we use as our ideal system a “bare” RIS system which has the gauche-trans energy difference equal to the gas phase value of n-butane. An additional thermodynamic variable in the formalism couples to the gauche-trans energy and renormalizes it in the melt. We find by a simple first-order perturbation calculation in the energy difference AE that (in units of kT where k is the Boltzmann constant and T, the temperature ) A wbare
=
A w-
Nf,
bE,
wheref, is the fraction of gauche bonds in the liquid. The quantity RTAE=Eg -Ei (where R is the gas constant) is the difference between the bare gauchetrans energy Eg and the renormalized energy Ez required in order to reproduce the correct intramolecular structure in the melt. The standard Ei used in the RIS model of polyethylene is 500 cal/mol [ I 11; the best experimental values of EE are 890 cal/mol [ 161 and 970 cal/mol [ 171. Thus, we use RTAE= 430 cal/mol as a reasonable value. Combining the above results with the previous work [ 1,2] yields a final expression #4for the grand * In order to make the non-linear inversion problem discussed in ref. [I ] tractable, the final expression is derived under the assumption that the solid density is constant in certain volumes and zero in others. The grand potential found with this restriction is an upper bound.
376
26 April 199I
CHEMICAL PHYSICS LETTERS
potential per site (A W,,,/N)
in the long-chain limit
AW*= E In(d) - 1 2P, v drdr’c(r-r’)Ap(r)Ap(r’)
X
-f,dE,
V
where V is the volume; ps, the bulk solid density; Ap( r) =p(r) -pL; c(r), the direct correlation function found through polymer-RISM calculations; and L, the Nth root of the single-chain partition function for a RIS chain (with pentane effect) [ 111. The value of lnl is positive and increases with temperature thereby destabilizing the crystal phase. The explicit chain parameters which we use are: carbon-carbon bond length= 1.54 A; bond angle = 112”; gauche torsional angles= f 120” [ Ill; and the inter-chain Lennard-Jones parameters: flu= 3.923 A; and cu/k =72 K [ 181. In order to numerically implement the polymerRISM approach, a hard-sphere diameter must be selected. The usual method of doing this is by matching the hard-core and Lennard-Jones liquid structures, which we do by employing the highly accurate WCA (Weeks-Chandler-Andersen) [ 191 (see also ref. [ 15 ] ) perturbation prescription formulated in its polymeric context [ 13 1. For the specific problem of crystallization, it is also reasonable to alternatively select the effective hard-sphere diameter by requiring that the hard-sphere system crystallizes at the same liquid density that the true system crystallizes at. Unfortunately, it is not known where the hardsphere RIS system crystallizes; however, if a monatomic hard-sphere system is selected to reproduce the crystallization behavior of a Lcnnard-Jones system, the resulting hard-sphere diameter is about 5% smaller than that determined by the WCA procedure [ 71. For a temperature of 4 13 K, the WCA criterion yields a polyethylene hard-site diameter of qi, = 3.70 A. On the other hand, by matching the liquid crystallization densities by the methods of this Letter, we find a diameter of acry= 3.90 A. The two values again differ by about 5%, giving us confidence in our procedure. For either value of a, we can make use of the hardsphere nature of the c(r) along with the delta function nature of the solid singlet density (due to ne-
Volume 179,number 4
CHEMICALPHYSICSLETTERS
glect of torsional oscillations)
to rewrite the free en-
ergy as AW*=
h PL
In&?(O)
Table I Crystallization at 4 13K
(ps2;pL)2 5 theory u=3.70A
where E(0) is the zero-wavevector Fourier transform of c( r) and the sum over R is restricted to those sites separated by less than the hard-sphere diameter. Notice that there is a compressibility contribution (the term containing (p,-pL)*) which favors the solid having the same density as the liquid, but there is also a competing contribution (the p&O) term) which favors driving the solid to its maximum packing density. Upon free energy minimization, the p,t( 0) term is dominant, resulting in a large density change upon freezing. Another interesting aspect of the above equation is that the p,?(O) term steadily decreases with a, while the C c( ]R] ) term decreases in a disjoint manner. It makes a sudden jump when u becomes larger than another site-site separation in the solid. Hence, stabilization of the solid is predicted for systems where Q is slightly less than one of the site-site distances in the all-trans chain. Polyethylene appears to be a good example of this effect. Here the all-trans site-site separations are 1.54,
2.54, and 3.91 A while qiq is 3.7 8, and Q, is 3.90 A. Thus, polyethylene would be predicted to be easily crystallizable, which is consistent with experiment. We have concentrated on the phase transition at 4 13 K - the experimental freezing temperature at atmospheric pressure. Because the solid (for fixed 0) will maximize its density, the solid density will be the same for all trial liquid densities. When we restrict the solid to have the experimental orthorhombit crystal structure [20] and u is set equal to Si,=3.70 A, we find p,d= 2.74. Here we have minimized the free energy with respect to the lattice constants “a” and “b” of the unit cell perpendicular to the chain backbone. The lattice constant “c” parallel to the backbone is fixed at the experimental value of 2.54 A. We find II/U= 1.95 and b/g= 1.09. The coexistence liquid density is calculated to be pLrr3= 1.93, giving the fractional density change II=@,pL)/pL=420h compared with the experimental value [ 211 of ~~27%. On the other hand, when we set cr
26 April 1991
u=3.90A experiment ref. [21] ref. [20]
Liquid density
Solid
a
b
density
(A)
(A)
k/cm’]
(g/cm’)
0.880 0.783
1.25 1.13
7.21 7.60
4.03 4.25
0.7834 -
0.9962 _
7.706
4.936
equal to a,,=3.90 A, we find p,a3=2.88 and pt_a3=2.00, giving ~=43Oh. These results are summarized and compared to experiment in table 1. Remarkably, the level of agreement is comparable to that of the DF results for the simpler molecular and atomic systems [ l-3,6]. The overestimation of the solid density and fractional volume change is largely due to a well-known weakness of the RIS model: torsional disorder is ignored #5 thereby permitting the solid to be completely localized. In other words, the p8 found by minimizing A W* is the highest possible density which does not require overlap of the effective hard-sphere sites. In reality, torsion would blur the density peaks and reduce both the solid coexistence density and volume contraction. Another interesting finding of this work, which has general implications for macromolecular phase transitions where site localization is of paramount importance (e.g. protein folding and homopolymer collapse), is that many commonly used coarse-grained polymer models (i.e. Gaussian and freely jointed chains) are completely inadequate for the study of crystallization. We find that these models fail to freeze because they possess an unrealistically large amount of entropy in the liquid phase. Most of this single chain disorder must be lost upon crystallization, and the packing contribution to the free energy cannot compensate for the loss. Immediate extensions of this work will address issues such as the role of attractive interactions, finite chain length, variable backbone stiffness, and other crystal structures (e.g. monoclinic) [ 81. Generali” For a complementary work which effectively retains torsion but neglectsspecificsite structure, see ref. [ 22 1. 377
Volume 179, number 4
CHEMICAL PHYSICS LETTERS
zation to treat other inhomogenous polymeric problems, such as interfacial profiles in immiscible blends and micro-phase-separation in diblock copolymers, is also possible. Finally, we wish to emphasize that there are no adjustable parameters in this theory. (Our use of a,, was for illustrative purposes only.) That is, once the chain parameters and intermolecular potentials are selected from common literature values, the freezing transition is uniquely determined. Thus, the polymer-RISM liquid state theory combined with the flexible polyatomic DF approach provides the first off-lattice, molecular-level theory of polymer crystallization. JDM is grateful to David Chandler for advice and guidance as well as for financial support through the early stages of this project. This work was supported by the US Department of Energy under Contract No. DE-AC-04076DP00789.
References [ I] D. Chandler, J.D. McCoy and S.J. Singer, J. Chem. Phys. 85 (1986) 5971,5977; I.D. McCoy, S.J. Singer and D. Chandler, J. Chem. Phys. 87 (1987) 4953. [2] J.D. McCoy, S.W. Rick andA.D.J. Haymet, J. Chem. Phys. 90 (1989) 4622; 92 ( 1990) 3034; SW. Rick, J.D. McCoy and A.D.J. Haymet, J. Chem. Phys. 92 (1990) 3040. [3 ] K. Ding, D. Chandler, S.J. Smithline and A.D.J. Haymet, Phys. Rev. Letters 59 (1987) 1698. [4] W.E. McMullen and K.F. Freed, J. Chem. Phys. 92 ( 1990) 1413.
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26April 1991
[5] T.V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19 (1979) 2775. [ 61 A.D.J. Haymet and D.W. Oxtoby, J. Chem. Phys. 74 ( 1981) 2559; B.B. Laird, J.D. McCoy andA.D.J. Haymet, J. Chem. Phys. 87 (1987) 5451. [7] L. Verlet and J.-J. Weis, Phys. Rev. A 5 ( 1972) 939. [ 8 1J.D. McCoy, K.G. Honnell, KS Schweizer and J.G. Curro, in preparation. [9] P.J. Flory, Proc. Roy. Sot. A 234 (1956) 60. [ 101J.F. Nagle, P.D. Gujrati and M. Goldstein, J. Phys. Chem. 88 ( 1984) 4599. [ 1I ] P.J. Flory, Statistical mechanics of chain molecules (Wiley, New York, 1969). [ 12 ] KG. Honnell, J.D. McCoy, J.G. Curro and KS. Schweizer, to be published. [ 131 KS. Schweizer and J.G. Curro, Phys. Rev. Letters 58 ( 1987) 246; K.G. Honnell, J.G. Curro and K.S. Schweizer, Macromolecules 23 (1990) 3496. [14] A.H. Narten, J. Chem. Phys. 90 (1989) 5857. [ 151 D. Chandler and H.C. Andersen, J. Chem. Phys. 57 ( 1972) 1930; D. Chandler, in: Studies in statistical mechanics, Vol. 8, eds. E. Montroll and J. Lebowitz (North-Holland, Amsterdam, 1982) p. 274. [ 161 D.A. Compton, S. Montero and W.F. Murphy, J. Phys. Chem. 84 (1980) 3587. [ 171 A. Verma, W. Murphy and H.J. Bernstein, J. Chem. Phys. 60 (1974) 1540. [ 181 J.-P. Ryckaert and A. Bellemans, Faraday Discussions Chem. Sot. 66 (1978) 95. [ 19 ] J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. [20] P.R. Swan, J. Polym. Sci. 56 (1962) 403. [ 21 ] B. Wunderlich and G. Czorni, Macromolecules IO ( 1977 ) 906. [ 22] J.V. Selinger and R.F. Bruinsma, to be published.
CHEMICAL PHYSICS LETTERS
26 April 1991
Polymer crystallization: density functional theory and application to polyethylene John D. McCoy, Kevin G. Honnell, Kenneth S. Schweizer and John G. Curt-0 Sandia NationalLaboratories,Albuquerque,NM 87185, USA Received 23 January I991
A general microscopic approach for the crystallization of flexible polymer melts is formulated and applied to polyethylene.
Site-site polyatomic density functional (DF) theory has contributed to a number of interesting innovations [ l-41. The polyatomic DF theory of freezing, however, is considerably more difficult to apply than its monatomic predecessor [5,6]. This increase in technical difficulty is due to both the enhanced structural complexity of molecular crystals and the increased mathematical complexity of the DF theory itself. Disregarding the delicate effect of the form of the atomic potential, one can think of monatomic freezing as depending only upon a single length scale (i.e. the atomic diameter), while polymer freezing is sensitive to a large number of length scales corresponding to bond lengths, angles, and backbone flexibility in addition to a site diameter. A change in this distribution of length scales (arising, for instance, from a change in the gauche-trans rotational conformer energy) affects the freezing transition in ways which cannot be accounted for by the simple scaling procedure [7] appropriate for the monatomic case. The present Letter describes a tractable scheme to evaluate the polyatomic DF theory for the difficult case of polymers, and here we sketch our major results #‘. Density functional theory is not the only alternative in the study of the freezing transition. In principle, computer simulations provide an unambiguous approach. Unfortunately, continuous-space simulations of polymer crystallization are prohibitively computer intensive. However, the situation, $a A more detailed discussion is to be found in ref. [ 81. 374
both in terms of theory and simulation, is greatly simplified if the polymers are confined to a lattice. A number of years ago, F’lory developed an incompressible lattice model using a very simple mean-field theory [ 91, but the results of this treatment are suspect for two distinct reasons. First, the lattice model alters the original system in physically unreasonable ways. For instance, an incompressible lattice requires that there not be a density change upon freezing (unrealistic, since polyethylene exhibits a roughly 30% density change); the multiple length scales associated with intra- and inter-chain correlations are lost on a lattice; and it is not clear what a bead on a lattice physically represents. Second, the basic statistical approximation of the Flory mean-field theory ignores essentially all intra- and inter-chain correlations. Nagle et al. [lo] argue that these inherent approximations fatally distort the physics. While we are not prepared to make a definite statement about the adequacy of Flory’s approach, we present here a methodology which circumvents the weaknesses of the lattice mean-field treatment. Although simple coarse-grained models of polymer chains are useful for qualitatively addressing certain long wavelength questions about the structure and dynamics of polymeric melts, it is not clear if such models are adequate for describing strongly first-order phase transitions. Indeed, we find that a chemically realistic model of the polymer chain is required, and in our opinion the present work employs the minimum amount of chemical detail needed to adequately describe the crystallization of polyeth-
0009-2614/91/$ 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
CHEMICAL PHYSICS LETTERS
Volume 179, number 4
ylene, (-CH,-), where N is the degree of polymerization. Specifically, we have used the rotational isomeric state (RIS) model of Flory [ 111 where each CH, group is described by a single Lennard-Jones sphere. Density functional theory requires liquid-state pair correlation functions as input. While a separate problem from DF theory per se, calculation of these correlation functions is essential to the application of DF theory and has only recently become possible for polymers [ 12,13 1. As seen in fig. 1, we have been able to use polymer-REM (reference interaction site model) integral-equation theory [ 131 to calculate the pair correlation functions of polyethylene (as exemplified by the static structure factor) to a high degree of accuracy [ 121. In general, RISM [ 151 #2,as well as polymer-REM, is a non-penurbative, off-lattice method to calculate intermolecular site-site correlation functions, i.e. the pair correlation function g(u) and the direct correlation function C(T) (where chain-end effects have been averaged over [ 13] ) . The intramolecular structure factor must be supplied as input into RISM theory. For a flexible molecule, this is generally a difficult task since the intramolecular distribution is a non-linear functional of the intermolecular distributions, and vice versa [ 13,151. However, for the special case of long chain melts a 1)2There is no relation between the RKM theory and the RIS model of ref. [ 111.
1’5 K 1.0 rl
0.5
2
0.0 -0.5
5
k (kl)
to
Fig. 1. The structure factor for polyethylene is calculated [ 121 and compared to scattering measurements [ 141. Systemvariables are: T=413 K, p,=O.781 g/cm’; N=6429. (-_) PolymerRISM; ( 0 ) scattering [ 141.
26 April 1991
great simplification occurs since it is well established that the intramolecular distribution function is nearly the same as that of an ideal chain without long-range excluded volume #3. Polyatomic DF theory is based on a site-site description of molecular interactions. It was first derived in ref. [ 1 ] ; however, we more closely follow the derivation given in ref. [ 2 ] which examines flexible molecules of identical sites. In brief, the excess Helmoltz free-energy functional of the solid is expanded about the liquid of density pL. The primary result of either derivation [ I ,2] is an expression for the grand-potential functional difference, A W, between the solid and liquid (both at the same temperature and chemical potential) which is obtained from a Legendre transform of the Helmholtz free-energy expansion. This functional becomes the thermodynamic grand-potential difference when it is evaluated for the true solid p(r). On the other hand, if p(r) is not the true inhomogeneous density, AW is larger than the thermodynamic grand-potential difference, by a second-law argument. Hence, the minimum of AW can be used to predict the equilibrium solid density. Moreover, if the liquid of density pL were the coexistence liquid at that temperature, the minimum of A W would be zero (i.e. mechanical equilibrium). The phase transition is located (at a particular temperature) by stepping through trial liquid densities until A Wmin=O. The quantities of interest calculated in this manner are the densities at coexistence of both the liquid and the solid, as well as the crystal lattice parameters. Of great importance is the selection of an “ideal” system which defines the excess free energy. This importance is the direct result of the truncation of the functional Taylor series. Naturally, if the series were to be carried to all orders, an ideal system would be unnecessary. It has been claimed [ 21 that the ideal system should be selected so that the ideal liquid and the true liquid have the same structure: the so-called “equal-structure” criterion. While this is an important requirement, we suggest that the inherent interactions in the ideal system should be the same as in the true system: the “equal-interactions” crite1(3That is, the melt is a &solvent for itself. The intramolecular distribution function depends only upon temperature and not upon density.
375
Volume 179, number 4
rion. For rigid molecules the two criteria are identical; however, for flexible molecules this is not the case. The approach we advocate is to use the “equalinteractions” ideal system and to enforce the “equalstructure” criterion by altering the density functional formalism. This ideal system differs from a system of real chains only in that the sites on different chains do not interact (i.e. an ideal gas). The ideal interactions, then, are entirely intramolecular and are of two varieties: local and non-local. The local interactions are of the RIS type, while the nonlocal interactions are of the excluded-volume-type. In order to enforce the “equal-structure” criterion, we add thermodynamic variables which permit the ideal chain interactions to be renormalized in the liquid. To describe the liquid structure, the local interactions must take on slightly different values and the global consequences of non-local interactions must be completely screened out [ 111. Hence, we use as our ideal system a “bare” RIS system which has the gauche-trans energy difference equal to the gas phase value of n-butane. An additional thermodynamic variable in the formalism couples to the gauche-trans energy and renormalizes it in the melt. We find by a simple first-order perturbation calculation in the energy difference AE that (in units of kT where k is the Boltzmann constant and T, the temperature ) A wbare
=
A w-
Nf,
bE,
wheref, is the fraction of gauche bonds in the liquid. The quantity RTAE=Eg -Ei (where R is the gas constant) is the difference between the bare gauchetrans energy Eg and the renormalized energy Ez required in order to reproduce the correct intramolecular structure in the melt. The standard Ei used in the RIS model of polyethylene is 500 cal/mol [ I 11; the best experimental values of EE are 890 cal/mol [ 161 and 970 cal/mol [ 171. Thus, we use RTAE= 430 cal/mol as a reasonable value. Combining the above results with the previous work [ 1,2] yields a final expression #4for the grand * In order to make the non-linear inversion problem discussed in ref. [I ] tractable, the final expression is derived under the assumption that the solid density is constant in certain volumes and zero in others. The grand potential found with this restriction is an upper bound.
376
26 April 199I
CHEMICAL PHYSICS LETTERS
potential per site (A W,,,/N)
in the long-chain limit
AW*= E In(d) - 1 2P, v drdr’c(r-r’)Ap(r)Ap(r’)
X
-f,dE,
V
where V is the volume; ps, the bulk solid density; Ap( r) =p(r) -pL; c(r), the direct correlation function found through polymer-RISM calculations; and L, the Nth root of the single-chain partition function for a RIS chain (with pentane effect) [ 111. The value of lnl is positive and increases with temperature thereby destabilizing the crystal phase. The explicit chain parameters which we use are: carbon-carbon bond length= 1.54 A; bond angle = 112”; gauche torsional angles= f 120” [ Ill; and the inter-chain Lennard-Jones parameters: flu= 3.923 A; and cu/k =72 K [ 181. In order to numerically implement the polymerRISM approach, a hard-sphere diameter must be selected. The usual method of doing this is by matching the hard-core and Lennard-Jones liquid structures, which we do by employing the highly accurate WCA (Weeks-Chandler-Andersen) [ 191 (see also ref. [ 15 ] ) perturbation prescription formulated in its polymeric context [ 13 1. For the specific problem of crystallization, it is also reasonable to alternatively select the effective hard-sphere diameter by requiring that the hard-sphere system crystallizes at the same liquid density that the true system crystallizes at. Unfortunately, it is not known where the hardsphere RIS system crystallizes; however, if a monatomic hard-sphere system is selected to reproduce the crystallization behavior of a Lcnnard-Jones system, the resulting hard-sphere diameter is about 5% smaller than that determined by the WCA procedure [ 71. For a temperature of 4 13 K, the WCA criterion yields a polyethylene hard-site diameter of qi, = 3.70 A. On the other hand, by matching the liquid crystallization densities by the methods of this Letter, we find a diameter of acry= 3.90 A. The two values again differ by about 5%, giving us confidence in our procedure. For either value of a, we can make use of the hardsphere nature of the c(r) along with the delta function nature of the solid singlet density (due to ne-
Volume 179,number 4
CHEMICALPHYSICSLETTERS
glect of torsional oscillations)
to rewrite the free en-
ergy as AW*=
h PL
In&?(O)
Table I Crystallization at 4 13K
(ps2;pL)2 5 theory u=3.70A
where E(0) is the zero-wavevector Fourier transform of c( r) and the sum over R is restricted to those sites separated by less than the hard-sphere diameter. Notice that there is a compressibility contribution (the term containing (p,-pL)*) which favors the solid having the same density as the liquid, but there is also a competing contribution (the p&O) term) which favors driving the solid to its maximum packing density. Upon free energy minimization, the p,t( 0) term is dominant, resulting in a large density change upon freezing. Another interesting aspect of the above equation is that the p,?(O) term steadily decreases with a, while the C c( ]R] ) term decreases in a disjoint manner. It makes a sudden jump when u becomes larger than another site-site separation in the solid. Hence, stabilization of the solid is predicted for systems where Q is slightly less than one of the site-site distances in the all-trans chain. Polyethylene appears to be a good example of this effect. Here the all-trans site-site separations are 1.54,
2.54, and 3.91 A while qiq is 3.7 8, and Q, is 3.90 A. Thus, polyethylene would be predicted to be easily crystallizable, which is consistent with experiment. We have concentrated on the phase transition at 4 13 K - the experimental freezing temperature at atmospheric pressure. Because the solid (for fixed 0) will maximize its density, the solid density will be the same for all trial liquid densities. When we restrict the solid to have the experimental orthorhombit crystal structure [20] and u is set equal to Si,=3.70 A, we find p,d= 2.74. Here we have minimized the free energy with respect to the lattice constants “a” and “b” of the unit cell perpendicular to the chain backbone. The lattice constant “c” parallel to the backbone is fixed at the experimental value of 2.54 A. We find II/U= 1.95 and b/g= 1.09. The coexistence liquid density is calculated to be pLrr3= 1.93, giving the fractional density change II=@,pL)/pL=420h compared with the experimental value [ 211 of ~~27%. On the other hand, when we set cr
26 April 1991
u=3.90A experiment ref. [21] ref. [20]
Liquid density
Solid
a
b
density
(A)
(A)
k/cm’]
(g/cm’)
0.880 0.783
1.25 1.13
7.21 7.60
4.03 4.25
0.7834 -
0.9962 _
7.706
4.936
equal to a,,=3.90 A, we find p,a3=2.88 and pt_a3=2.00, giving ~=43Oh. These results are summarized and compared to experiment in table 1. Remarkably, the level of agreement is comparable to that of the DF results for the simpler molecular and atomic systems [ l-3,6]. The overestimation of the solid density and fractional volume change is largely due to a well-known weakness of the RIS model: torsional disorder is ignored #5 thereby permitting the solid to be completely localized. In other words, the p8 found by minimizing A W* is the highest possible density which does not require overlap of the effective hard-sphere sites. In reality, torsion would blur the density peaks and reduce both the solid coexistence density and volume contraction. Another interesting finding of this work, which has general implications for macromolecular phase transitions where site localization is of paramount importance (e.g. protein folding and homopolymer collapse), is that many commonly used coarse-grained polymer models (i.e. Gaussian and freely jointed chains) are completely inadequate for the study of crystallization. We find that these models fail to freeze because they possess an unrealistically large amount of entropy in the liquid phase. Most of this single chain disorder must be lost upon crystallization, and the packing contribution to the free energy cannot compensate for the loss. Immediate extensions of this work will address issues such as the role of attractive interactions, finite chain length, variable backbone stiffness, and other crystal structures (e.g. monoclinic) [ 81. Generali” For a complementary work which effectively retains torsion but neglectsspecificsite structure, see ref. [ 22 1. 377
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CHEMICAL PHYSICS LETTERS
zation to treat other inhomogenous polymeric problems, such as interfacial profiles in immiscible blends and micro-phase-separation in diblock copolymers, is also possible. Finally, we wish to emphasize that there are no adjustable parameters in this theory. (Our use of a,, was for illustrative purposes only.) That is, once the chain parameters and intermolecular potentials are selected from common literature values, the freezing transition is uniquely determined. Thus, the polymer-RISM liquid state theory combined with the flexible polyatomic DF approach provides the first off-lattice, molecular-level theory of polymer crystallization. JDM is grateful to David Chandler for advice and guidance as well as for financial support through the early stages of this project. This work was supported by the US Department of Energy under Contract No. DE-AC-04076DP00789.
References [ I] D. Chandler, J.D. McCoy and S.J. Singer, J. Chem. Phys. 85 (1986) 5971,5977; I.D. McCoy, S.J. Singer and D. Chandler, J. Chem. Phys. 87 (1987) 4953. [2] J.D. McCoy, S.W. Rick andA.D.J. Haymet, J. Chem. Phys. 90 (1989) 4622; 92 ( 1990) 3034; SW. Rick, J.D. McCoy and A.D.J. Haymet, J. Chem. Phys. 92 (1990) 3040. [3 ] K. Ding, D. Chandler, S.J. Smithline and A.D.J. Haymet, Phys. Rev. Letters 59 (1987) 1698. [4] W.E. McMullen and K.F. Freed, J. Chem. Phys. 92 ( 1990) 1413.
378
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[5] T.V. Ramakrishnan and M. Yussouff, Phys. Rev. B 19 (1979) 2775. [ 61 A.D.J. Haymet and D.W. Oxtoby, J. Chem. Phys. 74 ( 1981) 2559; B.B. Laird, J.D. McCoy andA.D.J. Haymet, J. Chem. Phys. 87 (1987) 5451. [7] L. Verlet and J.-J. Weis, Phys. Rev. A 5 ( 1972) 939. [ 8 1J.D. McCoy, K.G. Honnell, KS Schweizer and J.G. Curro, in preparation. [9] P.J. Flory, Proc. Roy. Sot. A 234 (1956) 60. [ 101J.F. Nagle, P.D. Gujrati and M. Goldstein, J. Phys. Chem. 88 ( 1984) 4599. [ 1I ] P.J. Flory, Statistical mechanics of chain molecules (Wiley, New York, 1969). [ 12 ] KG. Honnell, J.D. McCoy, J.G. Curro and KS. Schweizer, to be published. [ 131 KS. Schweizer and J.G. Curro, Phys. Rev. Letters 58 ( 1987) 246; K.G. Honnell, J.G. Curro and K.S. Schweizer, Macromolecules 23 (1990) 3496. [14] A.H. Narten, J. Chem. Phys. 90 (1989) 5857. [ 151 D. Chandler and H.C. Andersen, J. Chem. Phys. 57 ( 1972) 1930; D. Chandler, in: Studies in statistical mechanics, Vol. 8, eds. E. Montroll and J. Lebowitz (North-Holland, Amsterdam, 1982) p. 274. [ 161 D.A. Compton, S. Montero and W.F. Murphy, J. Phys. Chem. 84 (1980) 3587. [ 171 A. Verma, W. Murphy and H.J. Bernstein, J. Chem. Phys. 60 (1974) 1540. [ 181 J.-P. Ryckaert and A. Bellemans, Faraday Discussions Chem. Sot. 66 (1978) 95. [ 19 ] J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54 (1971) 5237. [20] P.R. Swan, J. Polym. Sci. 56 (1962) 403. [ 21 ] B. Wunderlich and G. Czorni, Macromolecules IO ( 1977 ) 906. [ 22] J.V. Selinger and R.F. Bruinsma, to be published.