J. Phys. Chem. 1984,88, 4599-4608 The application of excited-state intramolecular proton transfer can be considered as a photoinduced chemical laser case, because the tautomer produced is a new chemical species generated in an excited state, emitting to a ground state of zero population. The generic nature of the specific case (-OH proton transfer to O==C-, followed by pyrane ring aromatization to pyrillium) should be recognized, suggesting a whole new class of intramolecular pro-
4599
ton-transfer cases with varied excited state, ASE, and laser behavior.
Acknowledgment. This work was supported by Contract No. DE-AS-05-78EV05855 between the Division of Biomedical and Environmental Research of the U.S. Department of Energy and Florida State University.
FEATURE ARTICLE Toward Better Theories of Polymer Melting John F. Nagle,* Departments of Physics and Biological Sciences, Carnegie- Mellon University, Pittsburgh, Pennsylvania I521 3
P.D. Gujrati, Department of Physics and Institute of Polymer Science, University of Akron, Akron, Ohio 44325
and Martin Goldstein Division of Natural Science and Mathematics, Yeshiva University, New York, New York 10033 (Received: November 18, 1983; In Final Form: March 12, 1984)
Because of the recent disproof of the classical statistical-mechanical theory of polymer melting, there is a need to consider new models as well as to do reliable calculations. This paper focuses upon the incorporation of additional features to those already incorporated in the simplest, but inappropriate (a posteriori), models. Within the context of lattice models the additional feature that appeared a priori to offer the most promise for the modeling of the transition in polyethylene is the use of realistic tetrahedral lattices with further neighbor excluded volume interactions, but our investigations indicate that this feature, by itself, is also inadequate. Realistic lattice models, when constrained to the maximum density consistent with the excluded volume interaction, appear unable to undergo a melting transition at any reasonable temperature, a phenomenon which we term "gridlock". Therefore, it is hypothesized that the observed density decrease at the transition and the concomitant increase in van der Waals energy are essential features that must be incorporated into models and theories of polymer melting.
Introduction The melting/crystallization phase transition that occurs in regular polymers is a phenomenon with several fascinating aspects. One aspect that appears to have received the most theoretical attention in the past decade or so concerns the morphologies of crystal growth and the occurrence of chain-folded and/or reentrant lamellae. These chain-folding phenomena involve nonequilibrium kinetic consideration^.^^ Kinetic considerations are of relatively greater importance in polymers than in low molecular weight crystallization because it requires unusual experimental conditions in order for very long polymer chains to arrange themselves into a true equilibrium crystalline state with the chains aligned parallel and extended to their full length. Nevertheless, extended-chain crystals which consist of relatively few folds have been grown in (1) Debates concerning the ways that the chains are organized between the lamellae, Le. the switchboard model vs. the chain-folded model, are recorded in the Faraday Discuss. Chem. SOC.1979, No. 68. This question will not concern us in this paper. (2) Hoffman, J. D.; Davis, F. T.; Lauritzen, J. I. "Treatise on Solid State Chemistry": Hannay, N. B., Ed.: Plenum Press: New York, 1976: Vol. 3, Chapter 9 . (3) Ishinabe, T. J. Chem. Phys. 1978, 68, 1808. (4) Wunderlich, 8.; Czornyj, G. Macromolecules 1977, 10, 906.
0022-3654/84/2088-4599$01.50/0
linear polyethylene at pressures greater than a few kilobar^;^ the growth is promoted by the intervention of a high-pressure hexagonal phase6 which has low enough density to allow long chains to align and extend themselves within laboratory times. These extended chain crystals are stable when cooled into the usual orthorhombic crystal phase and brought back to atmospheric pressure. The existence of such crystals provides a firm experimental foundation for equilibrium statistical-mechanical studies of the melting transition. Such studies should be more fundamental than the kinetic studies, even if they are not as relevant to ordinary crystallization processes and morphologies. We hope to show in this article that equilibrium theories are also as interesting and challenging. The picture of polymer melting that equilibrium statisticalmechanical theory attempts to mimic is a transition from an extended-chain crystalline state in which the internal degrees of freedom of individual molecules are well ordered, essentially in the conformational ground state, to a fluid phase in which the ( 5 ) Geil, P. H.; Anderson, F. R.: Wunderlich, B.; Arakawa, T. J . Polym. Sci., Parr A 1964, 2, 3707. ( 6 ) Bassett, D. ''Principles of Polymer Morphology"; Cambridge University Press: New York, 1980.
0 1984 American Chemical Society
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1
d2
Figure 1. Square lattice model for a small finite lattice. Each polymer chain has eight monomers. All discussion and formulas apply to the limit of long polymer chains and large lattices: (a) the all-trans ground state; (b) a representative disordered state with conformational energy ngc equal to 21e, where there are 21 "right-angled" polymer sites.
internal conformational degrees of freedom are largely disordered. For plyethylene the extended-chain crystalline state is the all-trans state, and the fluid phase has a mixture of gauche and trans rotations about individual C-C bonds along the chain. Although it would be gratifying if a single universal theory would explain the melting transition of all regular polymers, our studies suggest that a case by case study, with careful consideration given to the particlular properties of each polymer, is the safest approach, so we restrict our discussion in this paper to polyethylene. The melting transition in polymers has been thought to be well understood from a universal statistical-mechanical theory due to Flory,' but this understanding has recently been proven rigorously to be grounded on a calculation which is incorrect, not just in minor quantitative ways but even in its gross qualitative results.8-10 Because there has been some confusion and/or disagreement" concerning the significance of this new result, it is necessary to define some terms and to review the classical Flory theory in terms of its logical constituent parts. Review of Classical Flory Theory The hypothesis that emerges from Flory's original paper' is that melting of polymers is primarily due to (a) internal disordering (rotational isomerism) and (b) excluded volume interactions. A significant aspect of this hypothesis is to regard the large volume increase and the concomitant effect of van der Waals attractive interactions to be minor or nonessential to melting. Flory7 performed a statistical-mechanical calculation, the apparent success of which justified elevating the aforementioned hypothesis to a (7) Flory, P.J. Proc. R. SOC.London, Ser. A 1956, 234, 60. (8) Gujrati, P. D. J. Phys. A : Math. Gen. 1980, 13, L437. (9) Gujrati, P. D.; Goldstein, M. J. Chem. Phys. 1981, 74, 2596. (10)Gujrati, P. D. J . Stat. Phys. 1982, 28, 441. (11) Flory, P.J. Proc. Natl. Acad. Sci. U.S.A. 1982, 79, 4510.
theory. The model upon which the calculation was performed was stated7 to be a lattice model, although no particular lattice was specified. In fact, the calculation applies universally to any lattice.12 For definiteness, we shall begin by considering the two-dimensional square lattice (see Figure 1) and the three-dimensional simple cubic lattice in this section and the three-dimensional diamond, Le. tetrahedral, lattice in later sections. The states of the system consist of configurations of nonbranching chains connecting neighboring lattice sites; each site (except terminal sites on the chains) is connected by one chain to two other lattice sites (see Figure 1). The lattice is completely filledI3 by polymer chains, and these chains have an absolute excluded volume interaction so that no two chains occupy the same lattice site. The energy of the system is simply ngt, where ng is the number of sites at which the two links in a polymer chain are not collinear; each such (gauche) site costs an energy E when compared to (trans) sites with collinear links. This is probably the simplest type of model that incorporates many of the essential features of polymer systems. It is the model of choice for initial study of polymer melting, even though there are difficulties when using it to model real systems such as p01yethylene.l~ Models of this type will be called simple lattice models. Flory performed a kind of mean-field statistical-mechanical calculation for simple lattice model^.^ For the calculation one must imagine the successive placement of polymer links on an initially empty lattice. The essence of the approximation is that the probability that the next site to be occupied is empty is taken to be equal to the fraction of still empty sites. This probability factor is known to be inaccurate because it does not take into account the fact that a site near the chain end, as the chain is laid down, does not have the same probability of being vacant as one a further distance from the chain. A better and equally simple approximation, due essentially to H ~ g g i n s and ' ~ utilized for phase transition studies by Gibbs and DiMarzio,16 takes into account the statistics of nearest neighboring sites in a way which is equivalent to the Bethe a p p r o ~ i m a t i o n . ' ~ J However, ~ for the ~~
(12) Nagle, J. F. Proc. R. SOC.London, Ser. A 1974, 337, 569. (13) Flory7 also considered partially filled lattices but obtained the same qualitative behavior as the limiting model with completely filled chains. (14) The straightforward correspondence between the simple lattice models and polyethylene is that one bond on the lattice corresponds to one C C bond, either with bond length 1.54 A or projected bond length 1.27 A between centers of successivebonds. Then, the high-energy c sites correspond to gauche rotamers and the low-energy collinear sites correspond to trans rotamers. Of course, real polyethylene chains form nearly tetrahedral angles, not 90 or 180° angles. However, the French meaning of the word gauche would seem to encourage its usage to describe 90° turns as well, so despite recent criticism" we will continue with this more general and suggestive usage. In addition to the unphysical geometry of single chains, another unphysical feature of this correspondence is that the interchain s acing is too small; distances between chains should be approximately 4-5 several times the C-C bond length which is also the distance between chains in this model. For this latter reason Flory7." has insisted that a bond on the lattice corresponds to between three and four CH2 units in polyethylene. Unfortunately, any correspondence between states in the model and states in real polyethylene then becomes quite indefinite. Therefore, we prefer the straightforward correspondence which preserves the identification of the degrees of freedom in the model in a oneto-one way with real polymeric degrees of freedom. This preference does not play any role in the major issues discussed in this paper, though considerations of this sort help to lead to more realistic lattice models of polymer melting to be discussed later. (1 5) Huggins, M.L. Ann. N.Y. Acad. Sci. 1942, 43, 1. (16) Gibbs, J. H.; DiMarzio, E. A. J . Chem. Phys. 1958, 28, 373. (17) Bethe, H.A.Proc. R. SOC.London, Ser. A 1935, 150, 552. (18)Details of the differences between the two approximations have been discussed.12 Here we only emphasize that the Huggins-Gibbs-DiMarzio (HGD) calculation gives the correct transition temperature TMfor three exactly solvable polymer-like lattice modelsI2 while the Flory approximation yields a TIMtoo high by more than a factor of 4. Furthermore, if one also uses the straightforward correspondence mentioned in footnote 14, then the TM given by the HGD calculation is 363 k 73 K usin the experimentally accepted This is to be compared to value for polyethylene, c = 0.5 i 0.1 kcal/mol!2 the experimental TM = 415 K for polyethylene and TM = 1650 i 330 K by using the Flory approximation. In contrast, it is not known how to calculate TM from Flory's correspondence7J1 mentioned in footnote 14. The book by Miller (Miller, A. R. "The Theory of Solutions of High Polymers"; Oxford University Press: London, 1948) is an older reference that compares the two approximations and that fully develops the Bethe approximation for polymer solutions but with no rotational isomeric energy and with no phase transitions.
1,
The Journal of Physical Chemistry, Vol. 88. No. 20, 1984 4601
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Figure 2. Calculations for square lattice model. Top: Free energy per lattice site,/; in units of c vs. temperature in units of s / k (notice break in middle scale). The dashed curve labeled H G D is the result of the Huggins-Gibbs-DiMarzio approximation, and the dotted curve labeled F is the result of the Flory approximation. Both curves have a flat portion at low temperature signifying an inactive low-temperature phase. Both have an abrupt change of slope signifying a first-order transition, at k T , = 1 . 4 4 ~and k T F = 6.596, respectively. The solid curve labeled G G is the rigorous upper bound on the free energy and shows that no inactive low-temperature phase exists for this model. T E is the temperature at which the G G curve and the H G D curve cross. Bottom: The density of gauche rotamers, ps, as a function of temperature. Symbols have the same meaning as above.
purposes of this paper the distinction between the Flory approximation and the superior Huggins-Gibbs-DiMarzio (HGD) approximation is not essential. In the language of modern statistical mechanics both are classical result^.'^ Here the term “classical” does not mean nonquantum mechanical, although the calculations and the model are in fact nonquantum mechanical. Rather, it has the dictionary meaning of describing a past era of distinction, in this case an era in statistical mechanics, with its mode of thought and calculations. Accordingly, we will describe the Flory and the H G D approximations collectively as classical calculations for polymer melting. The qualitative result of the classical calculations for polymers is that at equilibrium the simple lattice models remain in their ground states for all temperatures below TM(see Figure 2 ) . We shall describe this behavior as “an inactive low-temperature phase”. At TM, according to these calculations, the system melts abruptly in a first-order To see how the inactive low-temperature phase comes about in these classical calculations, consider the entropy per polymer link, s, which is shown as a function of the density of gauche rotamers, p,, in Figure 3. A striking result of the classical calculations is that there is only a thermodynamically negligible number of states and hence zero entropy when the density of gauche linkages lies below a critical value which is given as pg = 0.23 by the HGD calculation. That is to say, the entropy per polymer link, given by s = ( k / n ) In W(p,) where Wis the number of states with np, gauche rotamers, becomes zero in the limit as m. This does not mean that the number of polymer links n there are no states for such values of p, but only that Wincreases less rapidly than exponentially with the size of the system n. We will call this the entropy gap. In addition, there are two all-trans
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(19) Fisher, M. E. In “Lectures in Theoretical Physics”; University of Colorado Press: Boulder, CO, 1965; Vol. VIIC. (20) Another qualitative result of the classical calculations that will not be dwelt on in this paper is the prediction of complete rotameric disorder in the melt phase. This result is due to the artificial separation of the excluded volume effect from the rotameric disorder that is imposed by the classical approximations. While present theories are not good enough to say how incomplete the rotameric disorder should be, it is clear that complete disorder is not only not demanded by theory but is quite unlikely.i2 Recent Monte Carlo calculationsziindicate very large departures from complete disorder for the square lattice model. (21) Baumgartner, A.; Yoon, D. Y. J. Chem. Phys. 1983, 79, 521.
Figure 3. Entropy/polymer link vs. density of gauche rotamers for the simple square lattice model. The curve labeled F is the Flory approximation. The curve labeled H G D results from the Huggins-Gibbs-DiMarzio classical approximation. The dashed line through the origin and tangent to the HGD curve locates the classical transition parameters, Ap = I/*, and c / k T , is the slope of this line. The tangent point of the d o t d line with slope s / k T locates p8 at higher temperatures. The dash-dot line through the origin and with slope s / k T locates pg = 0 at lower temperatures. The curve labeled G G is a rigorous lower bound to the entropy which shows that none of the above re$ults are correct.
ground states with p = 0, into which the system may condense at low temperatures.92 To locate the most probable value of pg, one minimizes the free energyf = F/n = ep, - kTs(p,) with respect to p,. This is easily done graphically in Figure 3 by taking a straightedge oriented with slope c/kT and sliding it down the paper until it touches the HGD curve or the origin. The point of contact locates the most probable value of pg. At low T, the point of contact is the origin and p, is zero. At high temperatures the point of contact is on the HGD curve and pg > 0. In Figure 3 the dashed straight line, which is tangent to the HGD curve at p, = ‘/z and which also passes through the origin, indicates coexistence between the slope of this line is e/kTMwhich the two values pg = 0 and yields the transition temperature, TM. The result of an inactive low-temperature phase is rather unusual, and so it is worth mentioning that it has a precedent in rigorous statistical mechanics. Such a resuit has been obtained for the Slater KDP six-vertex model, first by Slater’s approximate c a l c u l a t i ~ nthen , ~ ~ by exact calculations for the two-dimensional rn0de1,2~J~ and finally by a proof that this inactive low-temperature phase exists for the generalized model in all dimensions.26 Furthermore, the Slater approximation yields result which are very close to the exact results, and this approximation is mathematically equivalent to the HGD approximation for polyrners.l2 Thus, the prediction of an inactive low-temperature phase by the classical calculations is not an implausible statistical-mechanical result. Finally, it has been noted that the Slater KDP six-vertex model has a one-to-one correspondence with a polymer-like lattice model,lz which lends even further support to the classical result of an inactive low-temperature phase in polymers models.z7 A different criticism of the result of an inactive low-temperature phase is that such a result is implausible because no real physical phase a t finite temperature is so completely dead with a zero specific heat.28 In its naivest forms this criticism is easily rebutted. (22) The second ground state is obtained from Figure la by rotating it by
In the limit of an infinite system the entropy per polymer link is still zero. (23) Slater, J. C. J . Chem. Phys. 1941, 9, 16. (24) Sutherland, B.; Yang, C. N.; Yang, C. P. Phys. Rev. Lezt. 1967, 19, 588. (25) Lieb, E. H.; Wu, F.Y. “Phase Transitions and Critical Phenomena”; Domb, C., Green, M. S., Eds.; Academic Press: London, 1972; Vol. 1, p 332. (26) Nagle, J. F. Commun. Math. Phys. 1969, 13, 62. (27) One difference between the classical calculation and the exact result for the two-dimensional KDP model appears in Figure 3. The KDP model has no entropy gap. Instead, the exact entropy follows the envelope of the dashed curve through the origin which is also tangent to the solid curve.25 Indeed, the entropy gap can only be the result of a calculation that imposes the homogeneity of the states. This is analogous to the results of many classical calculations; for example, the van der Waals equation of state has a metastable loop that must be eliminated by the Maxwell construction. 90’.
4602 The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
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Figure 5. The kinds of states that are counted in the bound to the
(c)
entropy:8-10 (a) These chains are predominantly oriented along the 1 I vertical axis. (b) On a much reduced scale compared to (a), each area
Figure 4. Three kinds of defects in an ordered crystalline state: (a) This dislocation defect cannot occur in simple lattice models. There are two gauche rotamers in the middle chain, but the other four chains involve nonlocal distortions which leave each rotamer basically in the all-trans state. (b) This defect, which involves four gauche rotamers and two vacancies, cannot occur in simple lattice models which are constrained to maximum density. (c) This defect, which involves four gauche rotamers, can occur in simple lattice models. For example, because of its complete emphasis on configurational degrees of freedom, the model in Figure 1 suppresses lattice vibrations and a uniform thermal lattice expansion. To obtain more realistic low-temperature behavior, one simply adds Debydruneisen terms to the free energy given in Figure 2. Further, certain crystal defects, such as the one shown in Figure 4a, are not permitted in lattice models. Their neglect, which results in reduced disorder, is, however,justified if they play only a secondary role in melting. It may also be emphasized that any real crystal will involve many defects, such as dislocations and grain boundaries, whose concentration will be sensitively dependent upon the history of the formation of the crystal. However, the more subtle question is whether there is a nonzero concentration of gauche rotamers in a thoroughly annealed crystal in complete equilibrium when T is greater than absolute zero. Later in this paper we will present an argument for the existence of a nonzero concentration of gauche rotamers, but we emphasize that this answer is by no means obvious. This answer should also not obscure the fact that the crystal phase is conformationally very well ordered. In this regard it may be noted that the more recent experiments on extended-chain crystals4 indicate much less premelting just below TMthan earlier experiment^.^^ We believe that describing the crystal phase as conformationally inactive is quantitatively not far from the truth, and we do not consider this to be a major criticism of the classical theories. With this caveat in mind, the result of the classical calculations for a simple, but reasonable, model agrees so well with experiment that the elevation of the Flory hypothesis to a substantial theory seemed ~ a r r a n t e d . ’ ~ (28) For example, in real KDP the existence of high-energy defects destroys the inactive low-temperature phase. (29) Mandelkern, L. “Crystallization of Polymers”; McGraw-Hill: New York, 1964. (30) It had been recognized by Nagle12and Malakis (Malakis, A. Physica A (Amsterdam) 1976, 84A, 256) that the classical approximations do not reproduce all the features of exactly solvable polymer-like models, but the Flory hypothesis was still accepted as basic.
represents the states in (a), but the orientation is randomly vertical or horizontal from area to area as shown by arrows. Notice that the sizes of the areas are not the same. Such a composite state has no long-range order (and no boundary energies) but can be included in the entropy bound of references.*-l0
Current Status of the Classical Theory and the Behavior of Simple Lattice Models Recently, the harmonious theoretical situation just described has been disrupted by the discovery of a simple and completely rigorous proof that an inactive low-temperature phase does not exist in the square or simple cubic lattice models.*-1° This proof proceeds by explicit construction of a subset of the actual states of the square or cubic lattice models (Figure 5). Even though this subset almost certainly contains only a small fraction of all states with a given fraction of gauche bonds, it suffices to supply an upper bound to the free energy (Figure 2) and a lower bound to the entropy (Figure 3) which exclude the possibility of an inactive phase at any nonzero temperature. This result is devastating to the classical calculations because their method of locating the transition temperature and predicting the order of the transition requires the inactivity of the low-temperature phase (see Figures 2 and 3). Thus, this proof shows that the classical calculations for the phase transition in polymers are not trustworthy. Consequently, the classical theory must be demoted back to the status of an hypothesis unsubstantiated by reliable calculations. It is interesting to compare the status of the classical calculations for polymer melting with the status of classical calculations for critical points. It is well-known that classical calculations give incorrect values for the various critical indices that describe the divergences that occur at critical points,lg but the same calculations usually yield qualitatively reliable phase diagrams and get the general nature of the phases and the phase transitions right.31 In contrast, it appears that classical calculations for polymer melting make the more serious error of predicting an inactive low-temperature phase in cases where none exist. The actual phase behaviors of the square lattice model in Figure 1 or the simple cubic lattice model are completely unknown at the present time, except that there is no inactive low-temperature phase. However, three prominent possibilities have been disc ~ s s e d :(1) ~ no phase transition, (2) higher order transition at an unknown temperature, Tc, and (3) first-order transition at an unknown temperature, T M .Recently, Flory” has advocated the (31) One important exception is the three-state Potts model in two dimensions; see: Baxter, R. J. J . Phys. C 1973, 6, L445. Straley, J. p.; Fisher, M. E. J. Phys. A : Math., Nucl. Gem 1973, 6, 1310.
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Figure 6. A typical disordered state on a finite lattice of the polymer-like analogue of the exactly solvable F-model.12 This state is very similar to the square lattice disordered state in Figure l b except that there is one closed ring and the chain lengths are not all the same; as the lattice gets larger, the latter fault is not so severe, but closed rings still exist. The ground state of this model is identical with the one shown in Figure l a for the square lattice model.
third possibility. The basis of his argument is that the subset of states considered when obtaining the rigorous bounds-10 are “ordered” and, as such, would not be included in his calculation of “randomn states. Unfortunately, no definition is given of ordered and random, and it is unclear that the approximation’ considers only random states or that the states counted in the lower bound to the entropy9 are all ordered.32 Nevertheless, a composite free energy was constructed by patching together the H G D approximate result a t high temperatures with the free energy bound c a l c ~ l a t e d *for ~ ~ low temperatures” (see Figure 2). The ordered/random transition supposedly occurs at TEwhere the two free energies, labeled GG and H G D in Figure 2, are equal, so this method automatically yields a first-order transition, with a A S given by the discontinuity in the slopes a t TE. However, in our view the H G D result and the upper bound to the free energys-10 are just two approximate estimates of the same free energy. The upper bound must be more accurate at low temperatures where it gives a lower free energy than the H G D calculation, and the latter is knowng to be close to the correct value when pg = 2/3. It is not too surprising that the two approximate free energy curves cross somewhere and even less surprising that they cross with unequal slopes, but this fact can hardly be said to provide reliable information concerning the existence of a first-order phase transition. Evidence that simple lattice models need not have a first-order transition is given by the exact result for the F-model. The F-model has a polymeric analogue which is the same as the simple square lattice model except that, like the KDP analogue model, states consisting of finite loops of polymer are allowed (see Figure 6). Of the exactly solvable two-dimensional polymerlike mode l ~the~ F-model ~ , ~is the ~ most realistic. Like the square lattice model, the F-model does not have an inactive low-temperature phase. The F-model does have a phase transition, but it is an infinite-order transition, which means that all derivatives of the free energy are continuous even though the free energy is nonan a l y t i ~at~ Tc. ~ This is the weakest transition imaginable and would be experimentally difficult to distinguish from no phase (32) A suitable order parameter for the square lattice model that one might consider is ph+ = Ph - pv, where Ph is the concentration of horizontal chain links and pv is the concentration of vertical chain links. With this definition, the subset of states, as presented literally>9 is indeed ordered. However, if one first takes a large square lattice of size 2 m X 2 m considers perpendicularly oriented subsets in domains of size 1 m X 2 m, then the order parameter is zero. One may argue that this construction merely consists of two domains, each of which has long-range order. However, one may decompose a square in many different ways to give different subsets (see Figure 5b). It is not at all clear that there is long-range order in such an ensemble of decompositions, where long-range order would be defined as lim (rl - r, m ) (ph:v(rl) ph.&2)) # 0. Our unpublished studies show that the proof of nonexistence of an inactive low-temperature phase goes through even with such irregular decompositions. (33) Malakis, A. J . Phys. A: Math. Gen. 1980, 13, 651.
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transition at all unless one could find a way to measure the order parameter.34 The fraction of gauche rotamers pg at the transition temperature is in this polymer analogue model. In a recent letter21 Monte Carlo calculations for the square lattice model have been interpreted in terms of a first-order transition at kT/c = 0.93 f 0.02, considerably smaller than the value 1.443 obtained from the H G D approximation and much smaller than the value 6.587 obtained from the Flory approximation. These calculations also indicate that pg is about 0.08 in the low-temperature ordered state just before melting. While it is an outstanding and important problem in statistical mechanics to solve the simple square lattice model or any of the simple lattice models exactly or to pbtain reliable approximations, it seems likely that the square or simple cubic lattice models permit too much gradual disordering at low temperatures to be satisfactory models for polymer melting, even if they should turn out to have first-order transitions. In particular, if the transition temperature is at TEin Figure 2, then the computed bound9 yields 7% of the bonds to be gauche in the crystal just below TE. If TM is closer to that given originally by Flory? then the bound yields 32% gauche bonds just below TM.These percentages will increase, probably quite dramatically in the former case, if one had the exact results instead of what are clearly weak lower bounds to the entropy. They would also increase if the model had extra degrees of freedom such as those required to form the defects in Figure 4a. Another argument that further study of simple lattice models may be unrewarding comes from exact results for four two-dimensional polymer-like models. In addition to the polymer-like analogues of the KDP model and the F-model which have already been mentioned, a third polymer-like model is related to the Kasteleyn dimer model (K-model).I2 These three models have the same gross features, such as two high-energy t (gauche) continuations of each chain compared to one low-energy (trans) c o n t i n ~ a t i o n . Differences ~~ in these models are much more subtle, involving rings and spatial anisotropy.I2 While there is a notable universality in that all three models have exactly the same transition temperature, kT, = t/ln 2 = 1.44t, the orders of the phase transitions vary widely: first order for the KDP model, 3/2 order for the K-model, and infinite order for the F-model. A fourth model introduced by M a l a k i ~is ~isomorphic ~ to the Ising model. When confined to the square lattice, it has a logarithmic second-order transition at kTc = 1.136. Also, the KDP model and the K-model have inactive low-temperature phases and the Fmodel and the Ising model do not. These results collectively suggest that real characteristics of polymer chains and the corresponding details of models may play a significant role in determining the nature of, or even the existence of, their phase transitions. Therefore, the phase behavior of the model in Figure 1 may be quite different from the phase behavior of a real polymer. It also seems possible that different polymers might even have different phase transition behavior, so we will concentrate on polyethylene, for which the most is known.
New Directions Because the simple lattice models may be inadequate to describe polymer phase transitions, we believe that it is fruitful to explore new models and new hypotheses. Our procedure is to begin with the simplest models because they are most likely to become amenable to rigorous analysis and to add more complicated (34) The order parameter in the F-model (Nagle, J. F. J . Chem. Phys. 1969, 50, 2813) is similar to the one described in footnote 32. (35) The features mentioned in this sentence in the text completely determine the phase transition behavior according to the classical calculations, which are therefore incapable of distinguishing models of different dimensionality.12 It may also be noted that classical calculations of the Bethe sort probably become qualitatively correct for high enough dimensionality. As an example, recent work of Bhattacharjee et al. (Bhattacharjee, S. M.; Nagle, J. F.; Huse, D. A.; Fisher, M. E. J . Stat. Phys. 1983, 32, 361), which generalizes the K-model of chains to dimensions greater than two, indicates that the Bethe approximation may be correct for dimensions greater than three, but for three dimensions there appear to be logarithmic corrections to the Bethe approximation.
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The Journal of Physical Chemistry, Vol. 88, No. 20, 1984
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Figure 7. Chain states on the diamond lattice: (a) one of the close-packed all-trans ground states of model XOD (The lattice is the diamond (tetrahedral) lattice which is projected onto the plane of the figure. The dashed lines are parallel to the plane of the paper. The vertical, solid lines are the projections of the other two tetrahedral bonds at each lattice site, one bond connecting above the site and the other is the mirror image connecting below. The polymer chains shown in the upper left side of the figure consist of zigzag all-trans chains running along the thick solid lines with average chain direction perpendicular to the plane of the paper. As shown in the lower right-hand side, it is easy to reorient the average direction of all-trans chains by 120 or 240° on any plane without leaving any vacancies. This projection is along the (1 10) crystallographicaxes.); (b) an all-trans ground state of model XlD with 2/3 site occupancy (Occupied sites are shown with heavy circles and occupied bonds by heavy lines.); (c) the all-trans ground state of model (d) an all-trans ground state of model X3aD with p = (Also shown is a protounit cell (doubled) for comparison to the structure X2D with p = of real polyethylene.);(e) the all-trans ground state of model X3bD with p = l/5.
features to the models only as it is demonstrated that they are required. In particular, following the tradition begun by Flory? we will try to find a suitable lattice model.
Spatially Realistic Lattice Models Considering polyethylene, it appears now appropriate to consider three-dimensional tetrahedral (diamond) lattices, upon which the rotational isomerism of chains can be most closely realized.36 Let us begin with a tetrahedral model which is also a simple lattice model in the sense that the excluded volume interaction only excludes conformations with two chain segments passing through the same lattice site. We will refer to this model as XOD, where the X means excluded volume interactions, the D designates the diamond (tetrahedral) lattice, and the meaning of the 0 will become transparent later. Model XOD has a well-ordered, close-packed ground state (no gauche or e sites, all sites filled and all the all-trans chains aligned parallel) as shown in the upper left part of Figure 7a. However, this ground state is not unique; the chains on neighboring planes may be oriented at 120 or 240’ from the well-ordered state, as shown in the lower right part of Figure 7a, so the ground state has a degeneracy of the order of 3”’, where N is the number of lattice sites. This model permits gauche rotamers to be introduced, while maintaining maximum density. It appears, though this is not proved, that this model yields
roughly a factor of 2 in the number of states for each pair of gauche bonds introduced; this would yield a transition at kTM = 2e/ln 2. Furthermore, we have not succeeded in constructing a proof for the nonexistence of an inactive low-temperature phase for this model. Certainly, any such proof will depart considerably from those given for the square or cubic lattice models. It is our present feeling that this simple XOD model may very well have an inactive low-temperature phase followed by a transition similar to that predicted by the classical calculations. Thus, this model is a prime candidate for obeying the Flory hypothesis. However, as Flory has emphasized recently,” this model has some severe failings as a realistic model for polyethylene. The obvious difficulty with the XOD model is that the excluded diameter (about 4.0 A)37of actual polyethylene chains is larger than one C-C bond length (1.54 A), which by definition is the bond length of the diamond lattice. Another indication of the same difficulty is that the density of fully crystalline polyethylene3* is about 1 g/cm3 whereas the density of the ground state of this XOD model is 4.14g/cm3. This suggests that a better model would extend the excluded volume interaction to nearest neighbors and beyond. To help assess how far one should go, some appropriate facts about the diamond lattice are listed in Table I. In the first row of Table I we consider nearest-neighbor sites which perforce are separated by the C-C single bond distance 1.54 A. If one attempts to force carbons from different chains
(36) One additional aspect of real polyeth lene chains which has not
previously played a role in theories of melting7p1rand will also not play a role
in this paper is the inclusion of the higher but finite energy of 8’8- conformations, However, it might also be noted that its inclusion in the simple square or cubic lattice models does not disturb the rigorous proof of the nonexistence of an inactive low-temperature phase.
(37) Pauling, L. “Nature of the Chemical Bond”, 3rd ed.; Cornel1 University Press: Ithaca, NY, 1960. (38) Bum, C. W. Trans. Faraday SOC.1939, 35,482-491. Smith, A. E. J. Chem. Phys. 1953,21, 2229,
Thle Journal of Physical Chemistry, Vol. 88, No. 20, 1984 4605
Feature Article TABLE I: Data for the Diamond Lattice‘ (1) 1 2 3a 3b 4a 4b 4c
(2)
g
t gg gt tt
(3) 1.540 2.515 2.949 3.876 3.556 4.356 5.030
(4) -0.66 0.72 1.88 2.01 2.62 2.94 3.23
(5) XlD X2D X3aD X3bD X4aD X4bD X4cD
(6)
*/, I/,
‘I5
(7) 2.76 1.38 1.04 0.83 0.83
’Column headings: (1) Minimum number, B, of bonds between two carbon sites ( B = 1 is first neighbors, B = 2 is second neighbors etc.). In case there are more than one type of neighbor, a second index, a, b, ..,, is used. (2) Rotamer type of innermost B - 2 bonds. (3) Distances between carbon sites, dcc, in angstroms. (4) Closest distances, dHHin angstroms, between hydrogens attached to the carbon sites in (3). (5) Name of model which disallows all HH contacts in (4) and all closer ones but allows further HH contacts. (6) Maximum packing fraction of all-trans chains in model in (5). ( 7 ) Maximum density of (6) in g/cm3. onto nearest-neighbor lattice sites, then one must force a hydrogen from one chain to lie closer to the carbon atom of an adjacent chain than to the carbon atom to which it is bonded. Disallowing this chemical absurdity yields a new model, to be called X l D , where the X1 refers to the exclusion offirst-neighbor occupancy. The X l D model is no longer a simple lattice model of the kind previously discussed. The X1D model has a unique, maximally packed ground state in which 2/3 of the lattice sites are occupied, but half the chains lie in a direction perpendicular to the direction of the other half, as shown in Figure 7b. We have not succeeded in constructing a proof of the nonexistence of an inactive lowtemperature phase for this model. There is also a unique wellordered ground state (not shown) with all chains parallel but with only ‘I2the sites occupied. Starting from this ground state a proof can be constructed of the nonexistence of an inactive low-temperature phase. Our insistence upon the densest all-trans ground state compatible with the excluded volume interactions follows because attractive van der Waals interactions will make the densest all-trans state more stable than less dense ones; this is a general principle that applies to any of the models considered in this paper. In the second row of Table I we consider next-nearest neighbors, which are allowed in model X1D but which are not allowed in nature because they involve pairs of nonbonded hydrogens that are separated by only 0.72 A. The van der Waals radius for hydrogens given by P a ~ l i n gare ~ ~about 1.2 A, and the closest separation of two hydrogens in crystalline p ~ l y e t h y l e n eis~about ~ 2.49 A. Therefore, we define a model, X2D, that prohibits joint carbon occupancy of next-nearest-neighbor as well as nearestneighbor sites on the diamond lattice. The unique ground state with maximum density of all-trans chains in model X2D consists of a well-ordered parallel array with ‘I3of the sites occupied, as shown in Figure 7c. We have not succeeded in constructing a proof of the nonexistence of an inactive low-temperature phase for model X2D. In rows 3a and 3b in Table I we consider third neighbors which come in two types, depending upon whether the central bond in the triad joining third neighbors is gauche or trans. Both types place two hydrogens closer than Pauling’s 2.4 8, van der Waals diameter, but the difference is now not so great. Indeed, the Occurrence of a gauche rotamer in a single chain also places two hydrogens only about 2 A from each and this costs only about e = 0.5 kcal in energy. Exclusion of only the first type of third neighbor results in model X3aD. This model has a small number of well-defined all-trans ground states in which the chains are all parallel and packed with density ‘Iq, as shown in Figure 7d, as well as some with the all-trans chains not parallel. We have not been able to prove the nonexistence of an inactive low-temperature phase for model X3aD. (39) Rotation and bond angles in single chains are not exactly tetrahedral (Bartell, L. S.;Kohl, D. A. J. Chem. Phys. 1963,39,3097), and this increases the H.-H nonbonded contact distances in a gauche rotamer by about 0.2 A beyond the values given in the text.
Model X3bD, for which all third-neighbor separations of carbons on different chains are prohibited, is the most realistic of the diamond lattice models according to the criterion that it excludes all hydrogen-hydrogen contacts less than Pauling’s van der Waals distance of 2.4 A and permits all other contacts. The densest all-trans ground state for model X3bD that we could find filled sites as shown in Figure 7e. As in the ground consists of state of X l D , half the chains are oriented perpendicular to the other half and there is a degeneracy of 2 in each plane of parallel neighboring chains. (It may be noted that this is also the ground state of model X4aD, because this ground state does not have any g+g+ fourth neighbors.) We have not succeeded in constructing a proof of the nonexistence of an inactive low-temperature phase for model X3bD. The last column in Table I emphasizes a difficulty with models having chains constrained to the diamond lattice. The best model for hydrogen-hydrogen contacts, model X3bD, has too low a density, 0.83 g/cm3, as compared to a density of about 1.01 g/cm3 for real p ~ l y e t h y l e n e . This ~ ~ is not surprising since in reality neighboring chains are not constrained to lie on the same diamond lattice; taking advantage of the continuous translational and rotational degrees of freedom leads to an increase in density in the real crystal. This naturally leads to the question of whether lattice models have any value at all. Following Flory, we believe that they do, but this is an open question that must be considered seriously. In the remainder of this paragraph we address a related but simpler question, namely, which of the tetrahedral models defined is the most realistic one? Hydrogen-hydrogen contacts favor model X3bD, but it has already been noted that hydrogen-hydrogen contacts in the range around 2 A are not absolutely prohibitive. Alternatively, with density as a criterion, model X3aD is clearly best. Yet another criterion for choosing the best model is that an all-parallel all-trans ground state exists. This criterion favors models X3aD and X2D over X3bD and X1D. One refinement of this latter criterion is that all the ground states of a model consist of all-parallel all-trans configurations. This favors model X2D over X3aD because the latter model has many ground states that consist of all-trans chains that are not parallel. However, to break this degeneracy in model X3aD, one could always invoke small anisotropic attractive van der Waals interactions to favor the all-parallel ground state. Another refinement of the ground-state criterion is that the ground state of the model should resemble the actual ground state of polyethylene. This favors model X3aD. The doubled unit cell shown in Figure 7d has a number of similarities to real polyethylene. The greatest difference is that each chain in the middle of the doubled unit cell in Figure 7d is rotated by nearly 90’ around the z axis in real polyethylene. This suggests that one might want to consider a lattice model consisting of two interpenetrating diamond lattices rotated relatively to each other; such a construction also requires small distortions of the diamond lattices away from pure tetrahedral symmetry.40 Returning to the noninterpenetrating lattice models, while we are inclined to think models X3aD and X2D are the best models with X3bD slightly worse, it seems best not to dismiss any of them at present because they have all been useful in helping us to formulate and test our new ideas.
‘Is
Gridlock We have suggested that the XOD model has the following two properties: (i) an entropy gap and an inactive low-temperature phase and (ii) many disordered states even when the model is (40) The physical reason for this rotation of half the chains in real polyethylene is not obvious to us. If one rotates these chains back to the same orientation as the other chains (thereby halving the unit cell), then there are fewer close (
