Dynamic Molecular Shape Analysis of Configurational Transitions

J. Phys. Chem. B 1997, 101, 4097-4104

4097

Dynamic Molecular Shape Analysis of Configurational Transitions Associated with Melting and Premelting of n-Alkane Chains Gustavo A. Arteca De´ partement de Chimie et Biochimie, Laurentian UniVersity, Ramsey Lake Road, Sudbury, Ontario, Canada P3E 2C6 ReceiVed: July 17, 1996; In Final Form: February 26, 1997X

We present and discuss an improved approach to modeling some aspects of the melting transitions in alkanes, based on the use of single-chain shape descriptors. These descriptors characterize the interplay between molecular shape and chain flexibility as a function of the temperature. In particular, we provide a detailed analysis of the mean (dynamic) molecular shape of single alkane chains at the melting point temperature. By using the known melting points, our aim is to test whether one can estimate the length of the alkane chain exhibiting the first premelting transition. To this end, we have determined the number of carbon atoms required to produce liquidlike conformations whose average shape at the melting point is the same as that of a typical conformer in premelting phase. Our results agree closely with experiments, indicating that the first premelting transition takes place in nonane (n ) 9). We believe that the present approach may provide an avenue to extend the use of single-chain shape descriptors for modeling other phenomena that depend on chain flexibility.

Introduction Techniques based on single-molecule descriptors have been successful at predicting a number of characteristic properties of materials and are routinely employed in computer-assisted molecular modeling.1 Some graph-theoretical techniques provide acceptable correlations between selected bulk properties (e.g., boiling points, molar refractivities) and shape descriptors based on two-dimensional (2D) bond connectivity.2,3 However, these traditional shape descriptors have important shortcomings. For instance, they give a single description for all conformers sharing the same connectivity. Consequently, they fail at modeling properties that depend on three-dimensional (3D) shape and molecular flexibility. This shortcoming is a major drawback for properties such as melting points. Attempts to correlate melting points (and other “hard” properties) of alkanes with single-chain descriptors have been met consistently with failure.4 This testifies to the fact that melting points of alkanes depend strongly on chain flexibility, as well as on intermolecular interactions.5 Any improved modeling of melting phenomena by using single-chain descriptors must account for changes in flexibility due to chain length and temperature. This is the issue we tackle in this work. Our goal is to test whether single-molecule properties can provide useful information on melting phenomena, without resorting to a large-scale simulation of the bulk. Note that the interactions in the bulk are essential for determining the precise melting temperature.5 Since these interactions are not included in this work, our objective is not a correlation between shape descriptors and melting points. We believe that such an approach will likely be fruitless, because there is no thermodynamic basis for expecting that the structure of a single molecule will dictate the melting temperature. However, other important information may, in principle, be extracted from the average three-dimensional shape of a single molecule. For instance, the chain length required for an alkane to undergo a solid-solid phase transition prior to melting. Our working hypothesis is that average shape of a flexible molecule, properly characterized by detailed descriptors, can X

Abstract published in AdVance ACS Abstracts, April 15, 1997.

S1089-5647(96)02153-0 CCC: $14.00

tell us something on the physical nature of the dominant conformations. We reason as follows. Since the average shape of a chain molecule changes dramatically with the temperature, one can expect that some shapes could be characteristic of the crystal phase, the melt, and other intermediate solid phases. One can therefore ask, At a given temperature T, does the mean shape of a single molecule resemble that of the conformers found in the crystal, the melt, or an intermediate phase? If the temperature increases, can an alkane chain reach the typical shape of the conformers in an intermediate solid phase before reaching the average shape in the melt? These are some of the issues motivating this work. An important point should be stressed here. In our present approach, “molecular shape” can be regarded as a quantitative label attached specifically to a conformer. Whereas the energy contribution of a single “fixed” conformer will change with the chosen force field, the value of its molecular shape descriptors will remain always the same. However, the notion of molecular shape can be expressed quantitatively in various ways.6 There is no single descriptor that can be said to account for all features that are relevant to the shape of a molecule. For instance, a single molecular graph cannot capture the three-dimensional structure of a chain. Similarly, the molecular size cannot describe the chain’s spatial entanglements. Common approaches to conformational analysis are normally restricted to calculations of the chain size and moments of inertia. Whereas these descriptors are simple to compute, they may not distinguish between some conformers. Note that “size” is determined by the distribution of atoms in space and not by the way they are connected. The actual three-dimensional path of the chain can be characterized by entanglement descriptors. Recently, it has been shown that a geometrical measure of entanglement can correlate better than the molecular size with some physical properties of closed polymers.7 In this work, we adopt a very general definition of molecular shape. We consider several aspects of the shape of an alkane chain undergoing conformational rearrangements and not only its size. In summary, we study some qualitative features of alkane melting by monitoring a set of flexibility-dependent geometrical © 1997 American Chemical Society

4098 J. Phys. Chem. B, Vol. 101, No. 20, 1997 properties of the chain. Our scheme characterizes molecular shape by averaging several 3D molecular shape features over the accessible conformers. (We refer to this as dynamic molecular shape.) We employ a set of shape descriptors associated with distinct molecular shape features. These are molecular size, compactness, anisometry, and self-entanglement complexity in linear chains. To gain insights on melting phenomena, we contrast the behavior of these conformationally averaged descriptors with the available information on “reference conformers”. These reference structures are taken from the dominant configurations known in the alkane crystal and selected “premelting” phases. The work is organized as follows: In the next section, we review briefly some key details on how n-alkanes melt and discuss how this information is used in our work. The next section presents the shape descriptors used. A following section discusses the “reference” chains with conformational “defects” and gives the details of the configurational sampling of alkane chains. The results section gives our estimations of the chain length required for premelting to take place. The interpretation of the results and their possible impact towards improved molecular modeling of “hard” properties are summarized in the final section. Molecular Shapes of Dominant Chain Configurations Present in Alkane Phases The melting of normal linear alkanes (or paraffin waxes) is a complicated process involving the formation of a number of intermediate polymorphic phases. In some cases, the latter may span a range of approximately 10 °C between the solid and liquid phases.8 The occurrence of disordered (gel-like) phases is sometimes referred to as “premelting.” In most cases, this phenomenon involves first-order solid-solid phase transitions. (The term “premelting” is also used in the literature with a more restricted meaning for continuous transitions, with no recognizable intermediate phases in the bulk. We shall refer to the latter as “continuous premelting.” In this work, the term premelting is used strictly in reference to the occurrence of rotator phases (vide infra).) The number and nature of the intermediate phases depends strongly on n, the number of carbon atoms (or “length”) of the alkane chain.9-12 For n-alkanes with n < 9, no intermediate phases are found between the liquid melt and the regular crystalline solid (the so-called “phase I”). In longer alkanes, several phases can be found depending on the value of n, and on whether n is odd or even. A first intermediate phase (usually called “phase II” or “phase RI”) appears at n ) 9 in odd-n paraffins and it persists until n ) 39 (and possibly n ≈ 43). Phase II is the only intermediate one found until n ) 21. For n g 23, at least three additional intermediate phases are recognized between the solid and phase II. In the case of eVen-n alkanes, an intermediate phase is found between n ) 22 and n ) 38. (Continuous premelting appears to take place only for some chains with n g 50.13) The structure of some of the intermediate phases has been established by combining the information of X-ray, vibrational, and NMR spectroscopies, as well as differential scanning calorimetry and incoherent neutron scattering.9-14 The crystalline solid has a layered structure with an orthorhombic unit cell, where chains in the same layer are packed parallel to each other in all-trans conformations. In contrast, the RI phase presents disordered kinked chains. In odd-n paraffins, the RI phase has a pseudohexagonal face-centered orthorhombic cell. In eVen-n paraffins, the cell is hexagonal. For the purposes of the work, it is important to consider the key conformational differences between single chains in the

Arteca solid, liquid, and RI phases. (These are the phases involved in the actual melting process for n e 39.) Initally, it was assumed that the premelting transitions were associated with the onset of global rotational disorder in individual chains about the main chain axis.15 [Hence their name of “rotator phases”.] However, computer simulations14,16 and experiments11-13,17-20 do not support global chain rotation as a viable mechanism for the “I f II” phase transition. Rather, the mechanism suggested is one in which internal rotations at fixed chain sites play the central role (in addition to translational chain diffusion within a layer). Thus, the RI phase is expected to contain chains with specific conformational defects. [In what follows, for simplicity, we shall refer to a chain with specific conformational defects as a “defective chain”.] In particular, it has been shown that only very few conformational defects are required to account for most of the properties observed.18-20 In the case of phase II, the configurational state is characterized by the presence of chains with distorted endings (single and double end-gauche defects) and “kinks” near the center (central-kinks defects). The distribution of such defects with chain length has also been studied.13 In summary, the melting point transition involves a distinct configurational change depending on the chain length. For chains with n < 9, the phase change “I f liquid” is accompanied by an “all-trans-chain f molten-chain” configurational transition. In contrast, when 9 e n e 43 (odd n), the melting point corresponds to a “II f liquid” phase transition which is characterized instead by a “defectiVe-chain f moltenchain” configurational transition. This is the only background information required for the analysis presented in this work. Several computational studies have dealt with various aspects of bulk alkane behavior.21 Our present goal is more limited. As commented before, we are not interested in approximating quantitative properties, such as melting point temperatures. Single-chain simulations are too simplistic to explain the actual melting point temperatures. However, they may still provide useful information on the chain size and temperature range associated with characteristic configurational transitions. (Qualitatively, a similar situation is found in biomolecular folding. Whereas single-chain studies do not provide glass-transition temperatures, they do produce useful information on the conditions under which chain folding takes place.22) With this objective in mind, the present work tackles the following two questions: (a) Consider the fact that an alkane crystal consists of alltrans chains, the phase II includes defectiVe chains, and the liquid contains average molten chain configurations. Consider now a single chain “molten” at its melting point temperature Tm. We can then ask, Is there a critical chain length, nC, whereby the average shape of the “molten” chain (at Tm) equals the shape of the conformational defects in phase II? (b) Let us assume that such a critical length exists, and nC > 3. Since Tm increases uniformly with n > 3,8 the longer the chain, the larger its average deformation at melting with respect to an all-trans chain. Eventually, when n > nC, the “molten” chains will become more distorted (on average) than the conformational defects in phase II. Consequently, an average shape resembling that of the conformational defects in phase II will not be found at temperatures lower than Tm for chains with n < nC. In this case, the “melting” transition for n < nC would not involve typical premelting rotator phase conformers. Then, one can ask, Is this critical length nC comparable with the experimental chain length at which the rotator phase RI actually appears (i.e., n ≈ 9)? In this work, we address these two questions by performing a detailed shape analysis of the conformations found at the melting point temperatures of a series of n-alkanes. The

Molecular Shape Analysis of Configurational Transitions

J. Phys. Chem. B, Vol. 101, No. 20, 1997 4099

configurationally averaged shape descriptors are evaluated from constant-temperature molecular dynamics trajectories, where the simulation temperature T is set at the melting point Tm for each alkane. The averaged shapes obtained are then compared with those of selected conformational defects. It must be noted that the present approach will address only the issue of the critical length nC at which the first intermediate disordered phase appears before the liquid. Little can be said with certainty about chains longer than the critical length, i.e., n > nC, except that they must exhibit some distinctive conformational disorder prior to melting. Thus, the number and nature of the intermediate phases found after the critical chain length are beyond the scope of this work. Shape Descriptors of Chain Conformations We deal with the shape characterization of n-alkanes in terms of their carbon atom chains, i.e., the hydrogen atoms are excluded from our shape analysis. [During the molecular dynamics simulations, all atoms are taken into account.] Let {ri, i ) 1, 2, ..., n} be the carbon nuclear coordinates (with center-of-mass origin), defining one conformation. A number of shape descriptors are computed from these coordinates and bond sequence. These descriptors convey shape features that, a priori, are distinct and independent, namely, (1) molecular size and compactness, (2) anisometry (i.e., deviation from an isotropic distribution of nuclei), and (3) chain self-entanglements (i.e., a measure of turns and twists of the carbon atom skeleton). These properties are discussed briefly below. It must be noted that these are global descriptors, because they use information of the entire molecule to convey large-scale features. We do not use local descriptors, e.g., the values of dihedral angles.13 (For a discussion on the merits of the various descriptors, see ref 6.) 1. Descriptor of Molecular Size and Compactness. These features are captured by the instantaneous radius of gyration, RG, and its configurational average, 〈RG2〉.23 In conformers with identical atoms and using center-of-mass coordinates (our case), one has n

RG2 ) (1/n)

ri2 ∑ i)1

ri ) ||ri||

(1)

The configurationally averaged radius, 〈RG2〉1/2, is a wellcharacterized property of polymers, conveying their size and chain compactness.23 Nevertheless, for a complete analysis of polymer shape, other properties must also be taken into account. 2. Descriptor of Anisometry. Conformations with similar RG values can still differ in their spatial distribution of nuclei (anisometry). A descriptor of this property is the asphericity Ω, defined in terms of the principal moments of inertia {λi} (computed with center-of-mass coordinates):24 3 1 2 3 (λi - λj)2}{ λi}-2 Ω) { 2 i)1 j)i+1 i)1

∑∑

∑

(2)

In a spherical distribution, Ω ) 0, whereas prolate molecules (e.g., all-trans alkane conformations where λ1 ∼ λ2 . λ3) satisfy Ω ∼ 1/4. The configurational average of the asphericity will be indicated as 〈Ω〉. 3. Descriptors of Entanglement Complexity. Chains with similar size and spatial distribution may still be distinguished by their bond connectivity or “complexity” of their folding pattern. The spatial dependence of bond oVercrossings is a convenient tool to quantify this.25,26 [An oVercrossing is a “double point” where two bonds appear to “cross” over each other, when projected onto a plane perpendicular to a viewing

direction.] In particular, the probability distribution of oVercrossings, {AN(n)}, is a global shape descriptor of folding.26 This distribution gives the probability of observing N overcrossings when projecting rigidly an n-atom chain to an arbitrary plane. [The projections considered are those of the chain to planes tangent to the smallest sphere, centered at the centerof-mass, that encloses the chain completely.26] The overcrossing distribution is normalized as A0(n) + A1(n)+ ... + AmaxN(n) ) 1, where maxN ) (n - 2)(n - 3)/2 for a linear chain. Note that the trivial “chain” with three atoms (n ) 3) cannot have overcrossings (i.e., A0(3) ) 1). The algorithm for the actual computation of the probabilities {AN(n)}, as well as some of its basic properties, has been discussed with some detail in the literature.26 The accuracy of the calculations is determined by m, the number of projections used. For convenience, we follow the computational scheme of averaging several calculations of the distribution {AN(n)} for the same conformer. In this approach, each calculation uses a different number m of randomized projections and a different seed for the randomizer.26 For the present computations, we have averaged six calculations: m ) 10, 15, 20, 25 × 104 points and two different calculations with m ) 30 × 104 random projections each. (In practice, all shape descriptors for each generated conformer are computed with the Fortran program Allxs-md.for, developed by the present author, and executed on an AlphaDEC 255/233 computer, under Digital UNIX 3.2D1.) Two simple descriptors convey some of the essential shape features contained in the overcrossing probability distribution. These are (1) A*, the probability of the most probable number of overcrossings N* (i.e., the maxima of the distribution); and (2) N, the mean number of overcrossings, given by maxN

N)

∑

NAN(n)

(3)

N)0

These geometrical descriptors can be used to characterize the nature of the chain’s self-entanglements.7,26,27 In entangled chains, N take large values and A* is small; in swollen chains, N f 0 and A* ()A0) f 1. [In perfectly planar all-trans alkane conformers, these limits are strictly reached.] The configurational averages of these descriptors are indicated as 〈N〉 and 〈A*〉. The average 〈N〉 exhibits power-law scaling with the number of atoms (or monomers) in the chain: 〈N〉 ∼ nβ. Using random-walk polymer configurations, the estimated value at large n is β ≈ 1.2 ( 0.1.27 This exponent appears to depend little on the nature of the monomer-monomer interaction.27a,b Thus, the overall dependence of 〈N〉 with n, determined with a simple polymer model, can serve as a reference to estimate whether adequate sampling has been reached with more realistic alkane chains. The combined approach of several distinct shape descriptors has proven valuable in monitoring molecular shape stability in hydrocarbons28 and regular conformations in polypeptides.29 In this work, we follow a similar approach to analyze the shape changes that accompany single alkane chains near melting. To this end, we propose to evaluate the mean values of RG, Ω, A*, and N for alkanes at melting point temperatures and compare their values with those for characteristic conformers found in rotator phases. Evaluation of Configurationally Averaged Shape Descriptors for Alkane Chains Three classes of minimum-energy conformers were initially derived for n-alkanes of various lengths, within the context of molecular mechanics. On the one hand, we determined the all-

4100 J. Phys. Chem. B, Vol. 101, No. 20, 1997

Figure 1. Example of conformational defects characteristic of the intermediate rotator phase II. [These are (optimized) minimum energy conformers of tetradecane (n ) 14).]

trans conformational energy (global) minima for selected alkanes in the range 4 e n e 70. These conformers, which mimic the chains in the crystal, are the starting point for a constant-temperature molecular dynamics (MD) sampling. On the other hand, we computed two well-defined classes of conformational defects for each n-alkane considered. These structures, known to be characteristic of the rotator phase II, are the so-called end-gauche defect (EGD) conformers and the central-kink defect (CKD) conformers. In all cases, conformations of single alkane chains have been minimized within the all-atom MM2 force field,30a as implemented in the molecular modeling package HyperChem.30b [Minimizations were perfomed up to a root-mean-square energy gradient in internal coordinates of less than 0.0004 kcal mol-1 Å-1.] The MM2 force field has also been used for the evaluation of MD trajectories. No distance cutoff for nonbonded interactions was introduced. Representative examples of the chain defects are illustrated in Figure 1, which shows the EGD and CKD conformers for tetradecane (n ) 14). With the definition used in this work, the end-gauche and central-chain defects are not distinguishable for n ) 4. For n > 4, their shape descriptions are different. The basic shape features of these conformers are simple: their radii of gyration grow linearly with n and their asphericity approaches 1/4 asymptotically (following a simple law, i.e., 1/ - const × n-2).31 For our present purposes, we are interested 4 only in the shape features characterizing the typical EGD and CKD conformers. The details of how each conformer is achieved dynamically in the bulk rotator phase are not important here. The largest discrimination between the EGD and CKD conformers is provided by the mean number of overcrossings N. Note that the end-chain defect resembles more the alltrans conformer and is less entangled than a conformer with a central kink. The mean overcrossing numbers increase with n in both cases up to the asymptotic values: (N)EGD ) 0.110 ( 0.005 and (N)CKD ) 0.235 ( 0.005 (for n . 1). In contrast, the two conformers are not distinguishable in terms of their maximum probabilities of overcrossings, where we find A* ) 0.956 ( 0.002 for large n, in both cases. [The most probable number of overcrossings is always N* ) 0, for both chain defects.] The above descriptor values correspond to frozen conformations, and they can be compared with conformational averages. We have sampled the range of accessible configurations by using constant-temperature MD trajectories starting from the all-trans global minimum for a chain. Each chain has been studied at

Arteca its corresponding melting point temperature. Nineteen actual cases have been considered: n ) 4-14, in addition to n ) 16, 18, 21, 25, 31, 40, 54, and 70. These alkanes span a range of melting points from Tm ) 134.85 K (n ) 4) to Tm ) 378.5 K (n ) 70).8 The simulations were carried out using a weak coupling to a thermal bath, with a variable coupling constant.32 The approach chosen and the force field are comparable to other standard studies of alkane chains, including a recent simulation of folding in single polyethylene chains.33 The details of our calculations are as follows: (1) We used a small integration step for the equations of motion, ∆t ) 0.5 fs. (2) Starting from the alltrans “crystalline” conformer, an equilibration period of 100 ps was used, with a slow heating to the target temperature Tm. (3) Equilibration was established in all cases within 1 K (standard error) of the corresponding melting point temperature. (4) Equilibration was followed by 200 ps of sampling. Sampling was done every 0.25 ps. (5) As a control, several of the examples were repeated from slighty different (reoptimized) alltrans minima, using different bath relaxation constants and sampling rate (every 0.50 ps). The results of both types of calculations differed only within statistical errors. (6) Shape descriptors were computed for all sampled conformations, by using only the carbon atom coordinates and the chain connectivity, as explained before. In addition to the MD simulations, we have evaluated the averaged shape descriptors in a polymer model with the same number of carbon atoms and with a similar (fixed) carboncarbon distance, l ) 1.53 Å. This model provides a reference to interpret some of the results in actual alkane chains. As the simplest model, we consider random-flight (freely jointed) polymers.23a These are off-lattice chains with constant distance between consecutively bonded “beads” and a negligible excluded volume interaction. Even though off-lattice chains are computationally more demanding than lattice polymers, they are more realistic for our typical chain lengths (n e 70).21a The details of the model have been discussed already in the literature.28 In the present case, we employ a virtually zero radius of excluded volume, the distance within which no nonbonded beads are found (rex ) 0.001 Å). The averaged descriptors have comparable accuracy to those derived by MD trajectories in realistic chains. Calculations have been performed up to n ) 390, in order to reach asymptotic behavior in the mean number of overcrossings.27a,c This polymer model mimics, very roughly, the behavior for hydrocarbon chains in the limit of high temperature. For this reason, they can provide bounds to the values of the shape descriptors in the case of more realistic alkane models at lower temperatures. In the next section, we discuss the results for the configurationally averaged shape descriptors. Their values are compared with those for the conformational defects found in the rotator phase II. Results for the Dynamic Shape Descriptors The MD trajectories computed at Tm provide a distribution of molecular shapes accessible to a single “molten” chain. Figures 2 and 3 show the typical distributions for three distinct descriptors in the case of two example chains (n ) 16 and n ) 54). Note that molecular size and anisometry distribute differently from the mean number of overcrossings N h . Whereas RG and Ω show asymmetrical (χ2-like) distributions, the descriptor N h appears to distribute normally. This distinct behavior suggests that N h is less sensitive to the details of the potential energy function (or the temperature). This is consistent with the observed insensitivity of the scaling law for overcrossing numbers with respect to the monomer-monomer interaction.27a,b


Figure 2. Histograms for the values of shape descriptors found along the MD trajectory of hexadecane (n ) 16) at its melting point temperature (Tm ) 291.2 K). [The ordinates correspond to the frequency of occurrence among the sampled conformations. The abscissas h , and Ω.] correspond to the three distinct descriptors RG, N

We have contrasted the values for 〈N h 〉, 〈Ω〉, 〈A*〉, and 〈N h〉 with the values of the same descriptors for the end-gauche and central-kink configurational defects. The results obtained are displayed in Figures 4-7 and briefly summarized below. Figure 4 shows the variation of the mean molecular size at the melting points. The narrow gray region indicates the range of values for the radii of gyration of the EGD and CKD conformers. Our results show that 〈RG2〉 is indistinguishable from the latter until n ≈ 13. In chains longer than n ) 14, we observe larger configurational fluctuations in size and a marked deviation from the reference conformers. Considering that the simulations are started from the crystalline (all-trans) conformers, the result implies that the longer chains will reach the shapes of the EGD and CKD conformers before the melting point. Whereas the “molten” chains with n < 14 are comparable in size to EGD and CKD conformers, their difference is apparent when analyzing the asphericity (Figure 5). The mean chain

J. Phys. Chem. B, Vol. 101, No. 20, 1997 4101

Figure 3. Histograms for the values of shape descriptors found along the MD trajectory of tetrapentacontane (n ) 54) at its melting point temperature (Tm ) 368.2 K). [The ordinates correspond to the frequency of occurrence among the sampled conformations. The abscissas h , and Ω.] correspond to the three distinct descriptors RG, N

asphericity at melting, 〈Ω〉, exhibits a maximum at n ) 8. In longer chains 〈Ω〉 diminishes, i.e., the chains become more spheroidal. Note that this average shape equals that of the EGD and CKD conformers (gray region) for chains of length n ≈ 10 ( 1. In other words, chains shorter that n ≈ 10 exhibit anisometry values characteristic of EG and CK conformational defects only at temperatures higher than their melting points. Figure 6 shows the results for the maximum probability of overcrossings, A*. (For clarity, the scale uses only the n < 25 data.) In the present case, A* gives a description similar to that of Ω in Figure 5. As commented before, A* is virtually constant for the EGD and CKD conformers (gray region). Figure 6 indicates that 〈A*〉 equals that of the conformational defects at n ) 9. That is, the molten chains with n e 8 are less entangled than the reference EGD and CKD conformers. Again, this suggests that the chain defects present “high-temperature shape features” (T > Tm) for short chains. The mean number of overcrossings (depicted in Figure 7) provides the most details, because it distinguishes between EGD


Figure 4. Variation of mean molecular sizes of single-alkane chains, each one at its corresponding melting point. [Sizes are given in terms of mean radius of gyration, in angstroms. The gray region corresponds to the radii of gyration of the EGD and CKD conformers with the same chain length.]

Figure 5. Variation of mean asphericities of single-alkane chains, each one at its corresponding melting point. [The gray region corresponds to the values of asphericity for the EGD and CKD conformers with the same chain length.]

Figure 6. Variation of mean maximum probability of overcrossings of single-alkane chains, each one at its corresponding melting point. [The gray region corresponds to the values of this shape descriptor for the EGD and CKD conformers with the same chain length. The results are displayed only up to n ) 21, in order to highlight the most important details.]

and CKD conformers. Figure 7 confirms the result in Figure 6 regarding the fact that chains with n e 8 are less entangled than the conformational defects (i.e., 〈N〉 < (N)EGD < (N)CKD). However, note that molten chains equal only the

Arteca

Figure 7. Variation of mean overcrossing number N of single-alkane chains, each one at its corresponding melting point. [Note that this time the shape differences between the EGD and CKD conformers (lines A and B, respectively) are magnified. For a complete comparison, the results of a freely jointed polymer model of an n-alkane is included (line D).]

Figure 8. Conformational “phase” diagram derived from an analysis of self-entanglements in terms of the descriptor N. [For the meaning of the curves identified by the letters A, C, and D, see Figure 7. The region identified as “collapsed chains” corresponds to conformers that could only be found when using a polymer model with attractive potential.]

shape of the EGD conformers at n ) 9 (curve A). The chains need to be slightly longer (n ≈ 11) to reach the shape of the central-kink defects (curve B). Molten chains longer than n ) 11 are always more entangled than the conformers in phase II, which in turn are more entangled than the solid-state conformers (where (N)all-trans ) 0). It must be noted that, even though the complexity of entanglements increases with the temperature, there is a physical upper bound to their value. In the case of n-alkanes, this can be estimated by using a freely jointed chain (where there are no repulsions or attractions between monomers). To this purpose, we have used the random walk with no excluded volume described in the previous section. The results are included in Figure 7 (dashed bars, curve D), and they mark the maximum entanglement complexity that is reasonably accessible to n-alkanes at any temperature. Note that larger values of N could only be achieved in “collapsed chains,” e.g., if the monomers had attractive local interactions (which is not the case for hydrocarbons). The full significance of the results for entanglements can be extracted from Figure 8. The figure (derived from Figure 7) indicates the “configurational phases”, characterized in terms


J. Phys. Chem. B, Vol. 101, No. 20, 1997 4103 changes to some degree when the number of carbon atoms exceeds 10. From the results in Figures 4-7, we can assign this difference to the fact that the shorter chains contain mostly elongated, unentangled, and “unfolded” conformers. In contrast, once the chains reach n ≈ 10, the dominant conformations at melting become more folded and spheroidal than the characteristic conformational defects observed in the intermediate solid phase II. Further Comments and Conclusions

Figure 9. Compared asymptotic behavior of the mean number of overcrossings of a freely jointed polymer chain and the single alkane chains at the melting points. [The line indicates the expected asymptotic scaling exponent β ≈ 1.2.25 The results include chains up to n ) 70.]

of N. The figure provides a single-chain “phase” diagram of melting and premelting transitions. This diagram answers one of the questions we posed at the beginning: There exists indeed a critical chain length, nC, below which the shapes of EG and CK conformational defects cannot be reached before melting. That is, for chains shorter than nC, the solid-liquid transition does not encounter the shape of the latter defects. If n > nC, the shape of the EGD and CKD conformers is found in chain configurations characteristic of temperatures below the melting point temperature. As the results above indicate, the estimated value of nC depends slightly on the aspect of molecular shape considered. Whereas entanglements and anisometry give nC ≈ 8 (determined by EGD conformers), molecular sizes favors a larger value. Our estimate is nC ≈ 10 ( 2, where the shape of end-gauche defects is reached before that of central kinks. The existence of a critical nC value for single alkane chains is also apparent when analyzing the asymptotics of shape descriptors. The most clear case is provided by the mean overcrossing number. As mentioned before, this descriptor follows a law 〈N〉 ∼ nβ, with little dependence on the monomer-monomer interaction (or the temperature of the chain). For polymers of intermediate size (200 < n < 500), the apparent value of the scaling exponent is β ≈ 1.35,27a,c whereas the believed asymptotic value is β ≈ 1.2.27b-d Figure 9 illustrates this behavior for the current n-alkanes in a log-log plot. The “control” set of freely jointed alkane-like chains (white squares) and the actual n-alkanes are depicted at the same n values, up to n ) 70. [The latter is the longer alkane for which we could compute MD trajectories. The calculation of the polymer model is more feasible, and it was continued up to n ) 390.] The dashed line in Figure 9 indicates the asymptotic slope β ) 1.2. A number of observations can be made from this figure: (a) The model polymers appear to be long enough to reach the scaling regime. (b) Qualitatively, the n-alkanes at the melting points parallel the behavior of the model polymers. This suggests that our conformational sampling for alkanes at Tm should be reliable, even for larger n values. (c) The agreement in scaling behavior with the model polymers is most apparent for the alkane chains with n > 10. Even though a log-log plot magnifies the size of the configurational fluctuations in shorter chains (n e 10), their entanglements would appear to follow a slightly different law. In conclusion, Figure 9 is consistent with the emerging picture: the average molecular shape of alkane chains at melting

The results in the previous section suggest that single “molten” chains have a configurational transition relatiVe to EGD and CKD conformers at the critical length of nC ≈ 10 ( 2. This change in the nature of the dominant configuration at nC has two consequences: (a) chains shorter than nC reach liquidlike configurations before attaining the average shape of EGD and CKD confomers; (b) longer isolated chains can only have the average shape of the EG and CK defects at temperatures below the melting point. These features match closely some aspects of the phase transitions that accompany melting in actual bulk n-alkanes. As commented before, experiments show that chains shorter than n ) 9 do not have intermediate rotator phases, whereas longer alkanes exhibit solid-solid transitions before the melting point. The critical length estimated in this work, nC ≈ 10, is close to the point marking the transition from melting to premelting phenomena. Obviously, one cannot explain the details and nature of solid-solid transition by using single-chain simulations. One must take into account interchain interactions to justify why these EGD and CKD conformers play the key role in stabilizing the intermediate rotator phase before melting. Yet, our results suggests that a detailed analysis of molecular shape, relative to selected conformers, can provide a consistent picture on the chains where premelting first occurs. Summarizing, we stress again the intrinsic limitations of the present approach. The results point the likely presence of conformational defects prior to melting for n > 10. However, whether or not these conformational defects can form a stable intermediate “phase” cannot be answered with single-chain calculations and requires further input information. Therefore, the present method cannot predict when a given rotator phase will disappear or whether the conformational defects are characterisitc of an entirely different premelting regime or phases. From a different viewpoint, the results could suggest that chain segments shorter than 10 carbon atoms may be “too hard” to deform, even at the melting point. A local analysis of the shape of some long-alkane conformers gives support to this intuitive explanation. Often we have found that, although longer chains are indeed folded and entangled at melting, they do exhibit short segments which are locally “unfolded” (e.g., quasiall-trans). Figure 10 illustrates this behavior with two snapshots of conformers for n ) 25 and n ) 40. The “unfolded” segments are indicated by letters. There are three such segments in pentacosane, with partial lengths n(A) ) 7, n(B) ) 7, and n(C) ) 8. The four segments in tetracontane have partial lengths n(A′) ) 7, n(B′) ) 8, n(C′) ) 7, and n(D′) ) 8. These features are commonly found throughout the sampled conformers. In a very rough approximation, conformers appear to be formed by folding a number relatively rigid segments with 6-8 carbon atoms (i.e., below the critical value nC). Significant folding at melting can thus appear to take place in hydrocarbons with at least 14 carbon atoms. Previously in the literature, we had noted (in the particular example of a single dodecane chain, n ) 12) that a marked change in shape descriptor fluctuations took place near Tm.28


Figure 10. Snapshots of typical conformers found along the MD trajectories of “long” n-alkanes (n > 16) at their melting points. [Only the carbon chain is displayed. The segments of chain labeled A, B, C (n ) 25) and A′, B′, C′, and D′ (n ) 40) are in almost planar quasiall-trans conformations. These segments contain approximately eight carbon atoms.]

The approach highlighted the potential value of an analysis of molecular shape that includes chain flexibility: it can indicate the range of temperatures where folding into “molten” conformers starts. In the present work, we have extended further the scope of this approach by showing that it is possible to derive information on the occurrence (and placement) of premelting transitions. The method used in this work is semiempirical: it tests single alkane chains against information deduced experimentally for conformers in intermediate phases. Nevertheless, the procedure can be extended to other similar systems where the experimental characterization of solid phases may not be available. Note that the nature of the descriptors used is central to achieve any success. To capture all essential shape features, we have used here a combination of independent threedimensional descriptors that are sensitive to chain flexibility. As experienced before in the literature,4 shape descriptors that do not differentiate between conformers fail when trying to analyze “hard” properties, such as the melting points. We believe that the combined approach presented in this work provides an avenue to extend the usefulness of single-chain descriptors for modeling difficult molecular properties. Acknowledgment. I thank N. Grant (Sudbury) for her useful comments on the manuscript and an anonymous referee for bringing some references to my attention. This work has been supported by an operating grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada. References and Notes (1) (a) Dearden, J. C., Ed. QuantitatiVe Approaches to Drug Design; Elsevier: Amsterdam, 1983. (b) Balbes, L. M.; Mascarella, S. W.; Boyd, D. B. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: New York, 1994; Vol. 5, pp 337-379. (c) Doucet, J.-P.; Weber, J. Computer-Aided Drug Design: Theory and Applications; Academic Press: London, 1996. (2) Randic´, M. J. Am. Chem. Soc. 1975, 97, 6609. (3) (a) Rouvray, D. H.; Pandey, R. G. J. Chem. Phys. 1986, 85, 2286. (b) Hall L. H.; Kier, L. B. In ReViews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B., Eds.; VCH Publishers: New York, 1991; Vol. 2, pp 367-422. (4) (a) Needham, D. E.; Wei, I.-C.; Seybold, P. G. J. Am. Chem. Soc. 1988, 110, 4186. (b) Pogliani, L. J. Phys. Chem. 1995, 99, 925. (5) (a) Flory, P. J.; Vrij, A. J. Am. Chem. Soc. 1963, 85, 3548. (b) Ungar, G.; Stejny, J.; Keller, A.; Bidd, I.; Whiting, M. C. Science 1985, 229, 386. (c) Mandelkern, L.; Prasad, A.; Alamo, R. G.; Stack, G. M. Macromolecules 1990, 23, 3696. (6) Arteca, G. A. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: New York, 1996; Vol. 9, pp 191-253.

Arteca (7) (a) Stasiak, A.; Katritch, V.; Bednar, J.; Michoud, D.; Dubochet, J. Nature 1996, 384, 122. (b) Katritch, V.; Bednar, J.; Michoud, D.; Scharein, R. G.; Dubochet, J.; Stasiak, A. Nature 1996, 384, 142. (8) (a) Broadhurst, M. G. J. Res. Natl. Bur. Stand. Sect. A 1962, 66, 241. (b) Broadhurst, M. G. J. Chem. Phys. 1962, 36, 2578. (9) (a) Doucet, J.; Denicolo`, I.; Craievich, A. F. J. Chem. Phys. 1981, 75, 1523. (b) Doucet, J.; Denicolo`, I.; Craievich, A. F.; Collet, A. J. Chem. Phys. 1981, 75, 5125. (c) Denicolo`, I.; Doucet, J.; Craievich, A. F. J. Chem. Phys. 1983, 78, 1465. (d) Craievich, A. F.; Denicolo`, I.; Doucet, J. Phys. ReV. B 1984, 30, 4782. (e) Doucet, J.; Denicolo`, I.; Craievich, A. F.; Germain, C. J. Chem. Phys. 1984, 80, 1647. (f) Doucet, J.; Dianoux, A. J. J. Chem. Phys. 1984, 81, 5043. (g) Guillaume, F.; Doucet, J.; Sourisseau, C.; Dianoux, A. J. J. Chem. Phys. 1989, 91, 2555. (10) (a) Ungar, G. J. Phys. Chem. 1983, 87, 689. (b) Ungar, G.; Masˇic, N. J. Phys. Chem. 1985, 89, 1036. (11) (a) Sirota, E. B.; King, H. E., Jr.; Singer, D. M.; Shao, H. H. J. Chem. Phys. 1993, 98, 5809. (b) Sirota, E. B.; Singer, D. M. J. Chem. Phys. 1994, 101, 10873. (12) Nozaki, K.; Higashitani, N.; Yamamoto, T.; Hara, T. J. Chem. Phys. 1995, 103, 5762. (13) Kim, Y.; Strauss, H. L.; Snyder, R. G. J. Phys. Chem. 1989, 93, 7520. (14) Ryckaert, J.-P.; Klein, M. L.; McDonald, I. R. Mol. Phys. 1994, 83, 439. (15) Mu¨ller, A. Proc. R. Soc. London Ser. A 1932, 138, 514. (16) (a) Ryckaert, J.-P.; Klein, M. L. J. Chem. Phys. 1986, 85, 1613. (b) Ryckaert, J.-P.; Klein, M. L.; McDonald, I. R. Phys. ReV. Lett. 1987, 58, 698. (c) Ryckaert, J.-P., McDonald, I. R.; Klein, M. L. In Computer Simulation of Polymers; Roe, R. J., Ed.; Prentice Hall: Englewood Cliffs, NJ, 1991; pp 70-78. (17) (a) Snyder, R. G.; Maroncelli, M.; Qi, S. P.; Strauss, H. L. Science 1981, 214, 188. (b) Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G. J. Am. Chem. Soc. 1982, 104, 6237. (c) Maroncelli, M.; Strauss, H. L.; Snyder, R. G. J. Chem. Phys. 1985, 82, 2811. (18) Dorset, D. L.; Moss, B.; Wittmann, J. C.; Lotz, B. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 1913. (19) (a) Stohrer, M.; Noack, F. J. Chem. Phys. 1977, 67, 3729. (b) Zerbi, G.; Magni, R.; Gussoni, M.; Moritz, K. H.; Bigotto, A.; Dirlikov, S. J. Chem. Phys. 1981, 75, 3175. (20) (a) Jarrett, W. L.; Mathias, L. J.; Alamo, R. G.; Mandelkern, L.; Dorset, D. L. Macromolecules 1992, 25, 3468. (b) Stewart, M. J.; Jarrett, W. L.; Mathias, L. J.; Alamo, R. G.; Mandelkern, L. Macromolecules 1996, 29, 4963. (21) (a) Freire, J. J.; Pla, J.; Rey, A.; Prats, R. Macromolecules 1986, 19, 452. (b) Rigby, D.; Roe, R. J. Macromolecules 1989, 22, 2259. (c) Noid, D. W.; Pfeffer, G. A.; Cheng, S. Z. D.; Wunderlich, B. Macromolecules 1988, 21, 3482. (d) Karasawa, N.; Goddard III, W. A. J. Phys. Chem. 1988, 92, 5828. (e) Karaborni, S.; O’Connell, J. P. J. Chem. Phys. 1990, 92, 6190. (f) Doherty, D. C.; Hopfinger, A. In Computer Simulation of Polymers; Roe, R. J., Ed.; Prentice Hall: Englewood Cliffs, NJ, 1991; pp 55-69. (g) Winkler, R. G.; Ludovice, P. J.; Yoon, D. Y.; Morawitz, H. J. Chem. Phys. 1991, 95, 4709. (h) Rey, A.; Kolinski, A.; Skolnick, J.; Levine, Y. K. J. Chem. Phys. 1992, 97, 1240. (i) Boyd, R. H.; Gee, R. H.; Han, J.; Jin Y. J. Chem. Phys. 1994, 101, 788. (j) Mondello, M.; Grest, G. S. J. Chem. Phys. 1995, 103, 7156. (k) Wunderlich, B. Macromol. Symp. 1995, 98, 1069. (22) Karplus M.; Shakhnovich, E. I. In Protein Folding; Creighton, T. E., Ed.; Freeman: New York, 1992; Chapter 4. (23) (a) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. (b) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1985. (24) (a) Rudnick, J.; Gaspari, G. J. Phys. A 1986, 19, L191. (b) Rudnick, J.; Gaspari, G. Science 1987, 237, 384. (c) Diehl, H. W.; Eisenriegler, E. J. Phys. A 1989, 22, L87. (d) Baumga¨rtner, A. J. Chem. Phys. 1993, 98, 7496. (25) (a) Arteca, G. A.; Mezey, P. G. Biopolymers 1992, 32, 1609. (b) Janse van Rensburg, E. J.; Sumners, D. A. W.; Wasserman, E.; Whittington, S. G. J. Phys. A 1992, 25, 6557. (26) Arteca, G. A. Biopolymers 1993, 33, 1829. (27) (a) Arteca, G. A. Phys. ReV. E 1994, 49, 2417. (b) Orlandini, E.; Tesi, M. C.; Whittington, S. G.; Sumners, D. W.; Janse van Rensburg, E. J. J. Phys. A 1994, 27, L333. (c) Arteca, G. A. Phys. ReV. E 1995, 51, 2600. (d) Kholodenko, A. L.; Rolfsen, D. P. J. Phys. A 1996, 29, 5677. (28) Arteca, G. A. J. Phys. Chem. 1993, 97, 13831. (29) Arteca, G. A. Biopolymers 1995, 35, 393. (30) (a) Burkert U.; Allinger N. L. Molecular Mechanics; ACS Monographs: Washington, DC, 1982. (b) HyperChem (Release 4.0 for Windows), Waterloo, Ontario, Canada, 1994. (31) Arteca, G. A. Can. J. Chem. 1995, 73, 241. (32) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Di Nola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (33) Kavassalis, T. A; Sundarajan, P. R. Proc. Can. Chem. Conf. (St. John’s, Nfld.) 1996; Abstract 654.

Dynamic Molecular Shape Analysis of Configurational Transitions

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