Extended rotational isomeric model for describing

Extended rotational isomeric model for describing the long time dynamics of polymers Marina Guenza and Karl F. Freed Citation: J. Chem. Phys. 105, 3823 (1996); doi: 10.1063/1.472203 View online: http://dx.doi.org/10.1063/1.472203 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v105/i9 Published by the American Institute of Physics.

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Extended rotational isomeric model for describing the long time dynamics of polymers Marina Guenzaa) Istituto di Studi Chimico-Fisici di Macromolecole Sintetiche e Naturali, National Research Council, Genova, Italy and The James Franck Institute and Department of Chemistry, The University of Chicago, Chicago, Illinois 60637

Karl F. Freed The James Franck Institute and Department of Chemistry, The University of Chicago, Chicago, Illinois 60637

~Received 27 February 1996; accepted 29 May 1996! An extended rotational isomeric states ~RIS! model is used in conjunction with the matrix expansion method for describing the long time dynamics of flexible polymers in solution. The extended RIS model is derived directly from the potential functions that contain hindered torsional potentials, nonbonded interactions, etc. The matrix expansion method for describing the long time dynamics contains equilibrium conformational averages which are evaluated here from the extended RIS model. The theory effectively assumes that the torsional barriers provide the dominant mechanism for the decay of orientational correlations in the polymer chains. The theory is applied to united atom alkane chain dynamics where previous Brownian dynamics simulations with the same potentials are available for an unambiguous, no-parameter test of the theory. The present computation of equilibrium averages with the extended RIS model represents a significant advancement over the prior treatments that evaluate the equilibrium averages using Brownian dynamics simulations. The comparison with the previous approach indicates the degree to which bond angle fluctuations affect the orientational time correlation functions. © 1996 American Institute of Physics. @S0021-9606~96!51933-1#

I. INTRODUCTION

Natural and synthetic polymers display rich and diverse properties.1 Part of this behavior is a consequence of the close relationship between viscoelastic characteristics and molecular structure.2 The detailed connection between viscoelastic properties and chemical structure can only be understood through a description of the complex molecular motions.3,4 In particular, the study of the local polymer dynamics5–7 can provide useful information concerning important dynamical processes that affect the macroscopic mechanical and viscoelastic properties. Unfortunately, polymers of biological and technological interest are mostly of very high molecular weights, thereby providing serious computational impediments to the study of their local motions by molecular dynamics ~MD! simulations. An increase in the molecular weight slows down both the global and local motions and renders the simulations almost prohibitive for the interesting and important longer time motions.8,9 Brownian dynamics simulations enable calculations for longer time scales, but only at the expense of using a simplified description of polymer–solvent interactions. Clearly, we require theoretical guidance in extending the accessible shorter time simulations to describe the longer time motions. Recent theoretical advances10–12 have provided a statistical mechanical theory ~with realistic interaction potentials! for the dynamics of flexible polymers in solution. The tema!

Present address: Departments of Materials Sciences and Chemistry, University of Illinois, Urbana, Illinois 61801.

J. Chem. Phys. 105 (9), 1 September 1996

poral evolution of the system is described through a generalized Langevin or Smoluchowski equation for the coordinates of the friction points ~beads! in the chain.13 This equation is derived following the standard projection operator techniques14 that are presumed to apply to the description of the polymer motion in solution: the large difference between the polymer and solvent molecular weights implies that the polymer and solvent motions occur on very different time scales. This time scale separation is usually taken as justifying the use of projection operator methods for extracting the dynamics of the slow polymer motions. The projection operator procedure yields an approximate diffusion equation15–17 in which there are, in principle, harmonic forces between all beads on the polymer and in which the solvent mediated hydrodynamic interactions are preaveraged. The harmonic forces describe the chain connectivity through a generalization of the nearest-neighbor Rouse matrix to account for local chain stiffness. The equation also contains a set of memory function matrices that describe the corrections to the generalized Rouse-like3 model, to the preaveraging of the hydrodynamic interactions, etc. The memory function matrices are customarily neglected since they are difficult to compute and since little is known to enable their adequate modeling. Thus, the solvent is described as a source of friction and random forces, influencing the slower polymer motions. The bead friction coefficient then emerges as the only required input to the theory for describing the solvent. A fundamental step for providing a good description of the polymer dynamics resides in the definition of the slow

0021-9606/96/105(9)/3823/15/$10.00

© 1996 American Institute of Physics

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M. Guenza and K. F. Freed: Extended rotational isomeric model

variables and the friction coefficients. The theory may be developed by choosing as slow variables either the positions of individual atoms, residues, or groups of atoms. Different choices modify the required friction coefficients and the computed local persistence length in the molecule, and there is obviously a trade-off between using the most microscopic atomistic description and the constraints on computer time.18 Introducing a more detailed description of the preaveraged hydrodynamic interactions and of the generalized Rouse matrix serves to minimize the corrections from the neglected memory terms, but extensive comparisons between theory and simulations are required to test these approximations and, in particular, to determine when it is permissible to neglect the memory terms ~an approximation that produces the optimized Rouse–Zimm theory!. Several recent works have been devoted to investigate new theoretical methods for computing the memory functions. An exact evaluation of the memory function is possible only for one-dimensional systems without hydrodynamic interactions,19,20 while more realistic three-dimensional systems require approximate treatments.10,21 The most promising of the new approaches is a matrix method10,19 which in certain circumstances can be related to the projection operator methods with an extended set of relevant variables. This matrix method has thus far been applied to systems without hydrodynamic interactions, i.e., to systems whose exact dynamics are presumed to be governed by a diffusion or Smoluchowski equation. The time correlation functions of interest are expressed in terms of the eigenvalues and eigenvectors of the diffusion operator, and the eigenvectors are expanded in a set of basis functions. The long time dynamics are then described using as input information only equilibrium statistical averages and the friction coefficients. The computed correlation functions are tested against the same computer simulations that provide the numerically exact solution, and identical potential functions are used in both the theory and the simulations. Since it is obviously impossible to expand the eigenfunctions of the diffusion operator in a complete set of basis functions, the accuracy of the matrix method relies on the ability of choosing a truncated set of basis functions that is adequate for describing the long time dynamics, which are, in general, inaccessible to computer simulations for large complex polymer systems. Various basis sets have been studied, and the approximations inherent in the theory have been tested against Brownian dynamics simulations.10–12,21 Due to the rapid increase in the dimensions of the matrices required when the molecular weight of the molecule increases, the initial studies have been limited to smaller molecules ~octane, pentadecane, pentacosane!. However, remarkable progress has been made by taking advantage of intrinsic symmetries and the use of a truncated modecoupling type approximation, for which the required dimensions of the matrices are drastically reduced.11 The previous works obtain the requisite equilibrium input information from the same Brownian dynamics simulations that are used to provide the numerically exact time correlation functions for testing the theory. On one hand,

such a procedure suffices to ascertain the appropriate approximations for describing the influence of the memory terms on the long time dynamics. Moreover, the construction of the equilibrium input information from the simulations generates the time correlation functions from the theory using over an order of magnitude less computer time than the generation of the time correlation functions directly from the simulations. However, on the other hand, even these simulations can become prohibitive as the system becomes larger ~higher molecular weights! and as the time scales become longer. Thus, it is necessary to ascertain the possibility of computing the input equilibrium information for the matrix method directly from the potential functions using equilibrium statistical mechanics. The present work derives an extended rotational isomeric state ~RIS!22 model directly from the potential functions, uses the extended RIS theory to evaluate the input equilibrium information for the matrix method, and then calculates the time correlation functions from the matrix method. Applications are provided for united atom models of alkane chain Brownian dynamics where the previous simulations10 are available for testing the accuracy of the RIS approximations involved. The popular RIS model, reviewed in Flory’s classical book,22 correctly reproduces experimental data for the global equilibrium properties of polymers in ideal solutions, including such properties as the characteristic ratio, chain dipoles, etc. The theory explicitly focuses on the torsional potentials and the interdependent torsional interactions.23,24 These torsional interactions govern the dynamics of the conformational transitions across the torsional barriers which drastically influence the local dynamics.25 Thus, a natural first approximation to the full complicated potential function is to represent it solely in terms of the set of torsional potential at a set of discrete angles in order to describe contributions to the dynamics from small fluctuations in torsional angles. This approximation corresponds to an extended RIS model which is straightforward to handle because available computer resources permit retaining more than the three traditional ~RIS model! values for each of the torsional coordinates. The full potential energy is approximated in terms of individual torsional potentials and a potential involving the interaction of successive torsional angles. This model is then used to compute the required input equilibrium information for the calculation of the time correlation functions with the matrix method. Thus, we test here whether the RIS model, which is so successful for describing equilibrium properties of polymers, also can accurately describe the long time dynamics. One traditional approach to study local dynamics in polymer systems involves a consideration of the elemental conformational transitions which in the alkanes are associated with trans↔gauche transitions about individual dihedral angles. The dynamics on time scales longer than the characteristic times for these barrier crossings must therefore represent the cumulative influence of many such conformational transitions along with the other librations in each of the conformational wells and the overall chain rotation. This longer time dynamics may, therefore, be followed by postulating a

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master equation to describe the individual conformational transitions,26 often using a rotational isomeric states model ~RIS! to describe the polymer conformations.27,28 These RIS dynamics models represent an extension to continuous space of the pioneering lattice model treatment of polymer dynamics by Verdier and Stockmayer which introduces an elemental set of conformational transitions on the lattice.29–31 In one sense, this traditional approach is an approximate treatment based on using long individual trajectories with a subsequent time average ~and perhaps an average over trajectories with different initial conditions! to obtain a statistical mechanical description of the systems’ dynamics. We follow a rather different, but equally valid statistical mechanical approach based on the solution of the appropriate diffusion equation governing the dynamics. The general principles of statistical mechanics dictate that the average dynamical properties of the system may be obtained alternatively from solutions of the Liouville equation ~which for our case is the diffusion equation!. Thus, our approach is to obtain approximate solutions for the eigenfunctions and the eigenvalues of the Liouville operator ~the adjoint of the diffusion operator!. In principle, the complete dynamics can be represented in terms of a complete set of these eigenfunctions and eigenvalues. When time correlation functions are expressed ~exactly! in terms of the eigenfunctions and the eigenvalues of the Liouville operator, the correlation functions contain a set of equilibrium averages and exponentials involving the eigenvalues, which are the collective relaxation rates for the system as a whole. It is impossible to extract individual conformational transitions from the individual equilibrium averages or relaxation rates since both represent the cumulative, cooperative motions of the whole system. This impossibility parallels the impossibility of extracting an individual trajectory from a solution of the diffusion ~or Liouville! equation corresponding to the system initially in an equilibrium state. Thus, the individual barrier crossing transitions are not absent in our approach; they are present in the relaxation rates and in the equilibrium averages, but only in a complicated many-body sense. This general statistical mechanical principle has its well known manifestation in the fact that the long time limit of the Verdier–Stockmayer model exactly reproduces Rouse model dynamics in which individual conformational transitions appear to be absent but, in reality, have their cumulative influences present. The computation of a complete set of eigenfunctions and eigenvalues of the diffusion equation would appear, at first glance, to be an impossible task. However, important simplifications are induced by our goal of viewing only the long time dynamics, i.e., the portion of the dynamics that are most difficult to follow using trajectories. This long time dynamics turns out to be simplest to obtain using the distribution function method for the following reasons: the longest relaxation times ~apart from the overall center of mass diffusion of the chain! are associated with the chain rotation and other motions that are qualitatively similar to the longest wavelength Rouse modes. These slowest modes of motions therefore correspond to the smallest eigenvalues of the Liouville operator. Variational methods are well developed for obtaining

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accurate lowest eigenvalues and eigenvectors, but the question is whether these easier to obtain lowest eigenvectors are adequate for describing long time polymer dynamics. Support from the expected adequacy comes first from the long time Rouse model limit of the Verdier–Stockmayer model. More support is from the optimized Rouse–Zimm model computations by Perico and Guenza5,6 which exactly reproduce and extend to include influences of chain stiffness earlier dynamical conformational transition models of Helfand, Monnerie and their co-workers.30–31 Our previous noparameter comparison of the optimized Rouse ~OR! model with simulations indicate that the OR approach is insufficient to describe the ‘‘internal friction’’ contributions to the long time dynamics of the alkanes.10 Improved approximations are necessary to both the eigenfunctions and eigenvalues of the Liouville operator, and a mode coupling method for these contributions provides excellent agreement between theory and simulations.11 While extremely successful, the Liouville operator approach does not indicate which motions are most important in affecting the long time dynamics. The present approach addresses this interesting physical question by using analytical methods based on the assumption that dynamics is governed primarily by motions involving the dihedral angles. The Brownian dynamics simulations not only provide the numerically exact time correlation functions for comparison with the theory, but the use of simulation data for comparisons with the previous matrix method computations also enables us to understand the importance of including the fluctuations in bond angles for computing the time correlation functions. Contributions from the bond angle fluctuations, etc., are implicitly contained in the previous matrix method treatments but are absent in the present RIS approach, which effectively assumes that the relevant dynamics are associated only with the motion of the torsional degrees of freedom. To the extent that the new theory departs from previous matrix treatments using the trajectories to provide the equilibrium inputs, the bond angle fluctuations are relevant to the dynamics. If the bond angle fluctuations are found to contribute significantly, it is, in principle, possible to discretize the bond angles and further generalize the extended RIS-type models to include these angles,32,33 but the complexity of the computations would grow enormously. Section II presents the general theory for the diffusion dynamics of a molecule in a random ‘‘white noise’’ solvent. An outline is provided for the use of the matrix method to compute what is equivalent to the information contained in the memory function matrix. Section III briefly describes the united atom alkane potentials used for both the Brownian dynamics simulations and for the RIS approach. The standard RIS model is extended in Sec. IV to enable computation of the equilibrium averages required for the matrix method. The theory is applied in Sec. V to describe the united atom model dynamics of the octane molecule. Octane provides a significant test of the theory because it has enough flexibility to pose the interesting dynamical problems of describing the dynamical manifestations of the flexibility while still containing enough local stiffness to make a simple Rouse model


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very poor for describing the dynamics.10 An analysis of the individual Brownian dynamics trajectories for octane demonstrate the presence of several trans↔gauche transitions in a typical 30 ps slices. Thus, the system is undergoing many conformational transitions during the correlation times for the correlation functions displayed in the figures. Consequently, the long time dynamics reflects the cumulative influence of many conformational transitions as well as the librational motions.10–12 Computed dihedral angle correlation functions ^cos[ w i (t)2 w i (0)]& decay rather quickly and are, therefore, too rapid to be well represented by the matrix expansion method.46 The situation for octane in this regards is quite similar to that found for longer chain alkanes. II. GENERAL THEORY

The diffusive dynamics of a molecule in dilute solution is conventionally described through a Fokker–Planck, or Smoluchowski equation for the coordinate distribution function C~r,t! of the molecule.13 The molecule is represented as a chain of n segments ~beads! immersed in a hydrodynamic continuum. Each segment acts as a point source of friction with a friction coefficient z related to the diffusion coefficient D through the Einstein relation3,4 D5

k BT , z

~1!

with k B Boltzmann’s constant and T the temperature. The distribution function C~r,t! designates the probability of finding the system with configuration $r% at time t, where $r5ri , i51,...,n% is the collection of all individual beads coordinates. The probability distribution function evolves in time according to the Smoluchowski diffusion equation

] C ~ r,t ! 5DC ~ r,t ! , ]t

~2!

with the diffusion operator D defined below. The equilibrium solution of Eq. ~2! is the Boltzmann distribution C ~ r! 5Z

21

exp~ 2 b U ! ,

~3!

where the partition function is Z5 * dr exp(2 b U), b 5(k B T) 21 , and U5U~r! is the potential energy for the interacting beads of the chain. Since the system is not far from equilibrium, it is possible to use the techniques of linear response theory,14 in particular of projection operator methods. The point of departure involves defining the slowly varying dynamical variables. These slow variables define the subspace onto which we project the dynamics of the system. A first natural choice of the slow variables follows from the popular Rouse4 model of polymer dynamics as the collection of beads or group positions $r%. The time evolution of the bead coordinates is determined by L, the adjoint of the operator D,

] r~ t ! 5Lr~ t ! , ]t where L is given by

~4!

n

L5D† 5D

( $ ¹ r 1 b @ ¹ r U ~ r,t ! –¹ r % . i

i51

i

~5!

i

Following the standard Mori–Zwanzig projection operator techniques the projection operator onto the subspace of the bead variables is defined as P 0 5ur&^r–r&21^ru where the bracket notation ^ A u B & ~or ^ AB & ! designates the scalar product of A and B,

^AuB&5

† * 1` 2` exp@ 2 b U ~ r !# A ~ r ! B ~ r ! dr

* 1` 2` exp@ 2 b U ~ r !# dr

,

~6!

and Q 0 5(12 P 0 ) is the projection operator orthogonal to P 0 . Applying the projection operator P 0 on the left to both sides of the diffusion equation yields the generalized Langevin equation for the bead variables10,17 as

] r~ t ! 5L 0 r~ t ! 1fr~ t ! 2 ]t

E

t

0

d t F ~ t ! r~ t2 t ! ,

~7!

with the frequency matrix L0 given by L 0 5 ^ ru Lu r&^ ru r& 21 .

~8!

Ignoring hydrodynamic interaction and using the same diffusion coefficient for all beads, reduces Eq. ~8! to L 0 52D ^ ru r& 21 .

~9!

The random forces are given by fr(t)5exp~Q 0 Lt)Q 0 Lr, and the memory function matrix is F ~ t ! 52 ^ fr~ 0 ! u fr~ t ! &^ ru r& 21 52 ^ ru LQ 0 exp@ Q 0 Lt # Q 0 Lu r&^ ru r& 21 .

~10!

Memory functions have generally been studied for cases of a single relevant variable. Very little is known concerning the memory function matrices for many variable, flexible chain dynamics. Ignoring the memory function matrix reduces the diffusion equation to the more general optimized Rouse equation for the bead variables,17

] r~ t ! 1DAr~ t ! 5v* ~ t ! , ]t

~11!

where A5^rur&21 generalizes the Rouse A matrix and describes both the connectivity and the local stiffness of the chain, while v*(t) is a column vector of the Gaussian random velocity fluctuations which satisfy

^ v*i ~ t ! –v*j ~ t 8 ! & 52D d i j d ~ t2t 8 ! .

~12!

Inclusion of the memory term imparts to the velocity autocorrelation function of Eq. ~12! a non-Markovian behavior. The matrix A is calculated in terms of the static bond correlation matrix U as A5MT

F G 0

0

0

U

M,

~13!

where the matrix U is obtained from the equilibrium averages 2 U21 i j 5 ^ li –l j & / ^ l & ,

~14!



involving the bond vectors li 5ri11 2ri with i51,...,n21. The matrix M is the bead-to-bond vector transformation matrix ~l5Mr!, which is a function of the polymer architecture. For a linear polymer this matrix has the elements M1i 5n 21 , Mii 51,

i51,...,n, ~15!

i52,...,n,

Mi11i 521,

i51,...,n21.

The bond correlation matrix U21 describes the thermally averaged correlations between the bond vectors in the chain. The classic Rouse model reduces U21 to the simple form ^li –l j & 5l 2 d i j in which the molecule is taken as having no local stiffness. The more general optimized Rouse model produces an A matrix with more than just the nearestneighbor couplings in order to describe the dynamical influences of local chain stiffness. This optimized Rouse equation can be solved exactly in terms of the eigenvalues and eigenvectors of the matrix ^rur&21. Recent applications of the optimized Rouse model and its extension to include hydrodynamic interactions ~the optimized Rouse–Zimm model! have been provided to systems of increasing complexity, such as star polymers34,35 or polypeptides.18,36 However, comparisons with exact solutions for one-dimensional systems and with Brownian dynamics simulations for realistic alkane chain models demonstrate that the memory terms can not generally be ignored.10 Several methods have been developed for approximately solving the original complete diffusion equation. The Mori continued fraction formalism for computing memory functions37 produces highly nonlinear recursion relations that quickly render the problem totally intractable.19,20 More success has been achieved with the matrix expansion method in which the distribution function C~r,t!/C~r! is expanded in a complete set of orthonormal eigenfunctions $Ca % of the dynamical operator L. In this way, an arbitrary correlation function may be written in terms of the eigenvalues $l% and eigenfunctions $C% of L as

^ g ~ t ! u h ~ 0 ! & 5 ( exp~ 2l a t ! ^ g u C a &^ C a u h & ,

~16!

LC a 52l a C a .

~17!

a

The computation is then re-expressed in terms of the solutions of the eigenvalue problem. Equation ~17! is treated approximately through an expansion of the eigenvectors in a set of suitably chosen basis function $ F i % , C a5

(i C ia F i .

~18!

The basis set expansion in Eq. ~18! converts the eigenvalue Eq. ~17! for L into a generalized matrix eigenvalue problem, FC5SCL,

~19!

3827

with S i j 5 ^ F i u F j & the metric matrix, L the diagonal matrix of eigenvalues of the diffusion operator L, and n

F i j 52 ^ F i u Lu F j & 5D

( k51

K U L ]Fi ]F j ] rk ] rk

~20!

the equilibrium force constant matrix. The matrix C is normalized using the condition CT SC51. The correlation function of Eq. ~11! can now be written in terms of the basis functions as

^ g ~ t ! u h ~ 0 ! & 5 ( exp~ 2l a t ! a

3

S( j

S( D a

C ja ^ F j u h & ,

C ia ^ g u F i &

D ~21!

where it is important to notice that Eq. ~21! only requires equilibrium averages as input information for calculating the dynamical properties. Even the collective relaxation rates $ l a % are computed from the equilibrium matrices F and S in Eq. ~19!. The required equilibrium averages have been calculated previously from Brownian dynamics or molecular dynamics simulations, and various different basis sets have been tested for their accuracy in describing the long time dynamics. It is straightforward to show10 that the first order theory with basis set $li % reproduces the optimized Rouse theory for the dipole time correlation functions of bond vectors.43 When the functions g and h may be represented as linear combinations of basis functions, it may be shown12 that Eq. ~21! also emerges from a projection operator treatment with the basis $ F i % chosen as the extended set of relevant variables and with the residual memory terms neglected. The matrix method with a mode coupling basis ~see below! has been shown to reproduce well the long time chain dynamics for alkane chains of increasing molecular weight.10,12 The realistic potentials produce many transitions between different chain conformations on the time scale of the bond orientational correlation functions.21 More complicated problems may arise for oligopeptides where the high complexity of the potential landscape may produce conformational transitions on a vast array of time scales. More detailed descriptions may be required for treating those slower conformational transitions that have inverse transition rates comparable to time scales being probed.38 We follow the previous applications of the matrix method by choosing an expansion in the bond vectors and mode coupling terms to yield a basis of the form $li ,li ~l j –lk !,...%, where (i, j,k51,...,n21). The previous calculations with the equilibrium averages calculated from Brownian dynamics simulations begin by obtaining the optimized Rouse eigenfunctions $ j i ,i51,m•••n21 % as the first order approximation to the eigenfunctions of L generated using only the bond vectors $li % as a basis. Then the ~second order! mode coupling basis functions of the form (1) (1) j (1) i ( j j j k ) for i, j,k51,...,Q,n21 are appended to the


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F

cos q i

T5 sin q i cos w i sin q i sin w i

FIG. 1. Cartesian reference frames for the rotational isomeric states model, for the all-trans configuration. The z axes are not shown but are perpendicular to the plane of the figure, with directions alternating up and down from one reference frame to the next. f i is the diheral angle for bonds li21 , li , and li11 , while u i is the bond angle between bonds li and li11 .

basis with Q incremented to produce convergence which generally appears for Q!n21. This procedure reduces the computational problem of Eq. ~19! and provides an accurate representation of the slow dynamics.11 In this work, the input matrices are calculated directly from the potential U~r!, rather than from Brownian dynamics simulations. This calculation proceeds by first computing Ramachandran maps for successive torsional motions and then by converting these maps into a generalized rotational isomeric states model ~RIS! as described in Sec. IV. This assumption provides a better understanding of the importance of the torsional and other motions upon the chain relaxation since the RIS computation proceeds with ~almost! fixed bond lengths and bond angles. Thus, the statistical averages for the S and F matrices in the mode coupling basis set are calculated with the extended rotational isomeric states model that uses the identical realistic united atom alkane chain potential previously utilized in the Brownian dynamics simulations of the same alkane systems. The prior Brownian dynamics simulations employ twelve trajectories, each of duration 5.6 ns.

III. MODEL

The RIS model22 describes the configurational statistics of a chain whose conformation is expressed in terms of a set of bond vectors $li % (i51,...,n). The computations are conveniently performed by defining local coordinate sets on each of the bond vectors. The x i axis of the coordinate system affixed to bond i is taken to lie in the direction of bond i; the y i axis is in the plane defined by the bonds i21 and i, with its positive direction chosen to render its projection on x i21 positive; while the z i axis is chosen to produce a righthanded Cartesian coordinate system ~see Fig. 1!. The local coordinate systems on successive bond vectors are related by the matrix T that transforms a vector in the reference frame of bond i11 to its representation in reference frame i,

sin q i

0

2cos q i cos w

sin w i

2cos q i sin w i

2cos w i

G

,

~22!

where w i is the dihedral angle between the planes defined by the respective bond pairs (i21,i) and (i,i11), q i is the bond angle between bonds vectors i and i11, and the bond vectors connect two bonded united atoms groups ~which may be CH2 and/or CH3 groups for an alkane chain!. An important feature of the generalized RIS model is the constraint that successive bond angles are fixed. Thus, no consideration is given to the small fluctuations in these bond angles during the dynamical motions. The traditional RIS model assumes that the potential energy depends only on the set of torsional angles $ w i % with a single torsional energy E( w i ) for each torsional angle and interdependent torsional potentials E( w i , w i11 ) involving several nearest-neighbor ‘‘beads’’ along the chain. Recent treatments have even included interactions between the next nearest torsional angles,33 but we retain only the nearestneighbor terms. The partition function and various statistical averages are calculated by discretizing the torsional angles to enable the use of convenient matrix methods as summarized in Flory’s book.22 The traditional approach includes, for instance, only the trans and gauche conformations as the discretized set, but improved computer power enables us to use a larger set of torsional angles, leading to computations that effectively evaluate the averages over torsional degrees of freedom by a discrete quadrature rather than an integral. We briefly describe the full united atom alkane potential energy functions used previously in the Brownian dynamics simulations and used here to produce the Ramachandran conformational maps. By employing the same potentials for both the theory and simulations, a comparison of the new theoretical results with the simulated correlation functions provides the most stringent test of both the model and its implicit approximations. These full potential energy functions depend on all bond angle and bond lengths in addition to the torsional angles. After describing the full potentials, we indicate how they are converted into Ramachandran maps for successive torsional angles. The RIS model torsional potential E( w i ) and the interdependent torsional potentials E( w i , w i11 ) are then computed from these Ramachandran maps as described below. The potential is composed of a sum of four terms,40 U TOT5U bond1U angle1U dihedral1U Lennard-Jones ,

~23!

where the bond length potential is defined as

U bond5

(a @ V adc~ l a ! 1V rdc~ l a !# ,

~24!

where V adc(l a ) arises from attractive distance restraints and V rdc(l a ) is from repulsive distance restraints,41


V adc~ l a ! 5

and

V rdc~ l a ! 5

H H


0, 1 2

if 0,l a ,r 0

k ~ l a 2r 0 ! 2 ,

if r 0 ,l a ,r 0 1Dr,

k ~ l a 2r 0 2 Dr ! Dr,

2k ~ l a 2r 0 1 21 Dr ! Dr, k ~ l a 2r 0 ! 2 ,

0,

if 0,l a ,r 0 2Dr

if r 0 2Dr,l a ,r 0 ,

if r 0 ,l a ,

1

(i 2 K q~ q i 2 q 0 ! 2 ,

~27!

i51,...,N22,

with the constant K q 5460 kJ mol21 rad22 and the equilibrium bond angle q05111°. The dihedral potential42 is U dihedral5

(g (i C i cosi~ w g ! ,

g 51,...,N23,

~28!

where wg is the dihedral angle and the $ C i ,i50,..,5% are constants given in Table I. The nonbonded interactions are represented by a Lennard-Jones 6-12 potential U Lennard-Jones5

( 4e i, j, j2i.3

FS D S D G s r i, j

12

2

s r i, j

6

, ~29!

i, j51,...,N,

where r i, j is the distance between the two united atom units i and j and where e and s are shown in Table II. The values of e and s for the Lennard-Jones 6-12 potential between united atoms CH2 and CH3 are calculated as geometric averages of the parameters in Table II. The solvent enters the dynamics only through the constant diffusion coefficient D, which depends on the friction coefficient z by Eq. ~1!. The friction coefficient for all united atom groups is calculated using the van der Waals radius R of 1.8 Å, a solvent viscosity h of 0.01 poise, and Stokes law with stick boundary conditions at the beads’ surface,

z 56 p h R.

~30!

As noted above, the RIS approach generally assumes that the bond lengths and bond angles are fixed. This approximation is customarily justified by arguing that small deviations from the equilibrium values produce small contri-

TABLE I. Constants in the dihedral potential function of Eq. ~28! ~from Ref. 40!. i C

0 9.2789

1 12.1557

~26!

a51,...,N21

where r 0 51.53 Å, Dr50.23 Å, and k512.96 eV/Å2. The bending potential40 is a function of the bond angle qi , U angle5

~25!

a51,...,N21

if r 0 1Dr,l a

1 2

1 2

3829

2 213.1201

3 23.0597

4 26.2403

5 231.4950

butions to equilibrium properties. The accuracy of this approximation, however, may be quite different for equilibrium and dynamical properties, a point that the present study partially addresses. We introduce the leading corrections due to small bond angle and bond length fluctuations by calculating the conformational maps both with and without a minimization of the total potential energy with respect to all the bond lengths, all the bond angles, and the remaining dihedral angles not directly involved in the maps. Thus, we find the minimum energy for the system with a fixed value for a pair of successive dihedral angles w i and w i11 . The resulting optimized ~i.e., minimum energy! molecular geometry is very slightly modified from the unoptimized geometry for the torsional angles near their equilibrium conformations, but it becomes heavily distorted for high energy conformations. Just as the equilibrium conformations are the ones mainly affecting the partition function, no significant improvements in describing the dynamics are obtained from the minimization procedure, thereby validating this aspect of the RIS ansatz. The next main approximation of the rotational isomeric states approach is the use of discrete conformational states for the individual torsional angles. In the traditional RIS model, these angles are chosen to coincide with the potential minima. This approach corresponds to replacing the continuous distribution of conformations by a discrete distribution over several rotational isomeric states. The original RIS description of the alkane chains takes the states as corresponding to the three minima for the butane molecule, the trans ~0°!, gauche ~120°! and -gauche ~2120°! dihedral angles. As a test on this approximation, we have extended the calculations to consider a wider range of dihedral angles that are equally spaced with intervals 120°, 60°, 30°, 20°, and 10°. The 120° interval case generates the traditional three-state RIS model, while the others include more angles in addition to these three. Both the minimized and nonminimized Ramachandran maps have been calculated for the five different sets of dihedral angle intervals. Inserting these dis-

TABLE II. Constants in the Lennard-Jones potential of Eq. ~29!.

CH2 , CH2 CH3 , CH3

e ~K!

s ~nm!

70.4 90.7

0.3965 0.3786


3830


IV. THE MATRIX METHOD WITH THE EXTENDED ROTATIONAL ISOMERIC STATES MODEL

FIG. 2. Ramachandran map for the two first consecutive dihedral angles in the octane molecule. We present both the three-dimensional potential surface and a contour map ~on the w1 – w2 plane!.

cretized energy functions E( w i ) and E( w i , w i11 ) into the standard RIS method enables computation of the correlation functions necessary for input to the matrix method. Once again no significant improvements in the final time correlation functions are obtained from the energy minimization approach. Of course, this result may not generally apply for molecules with low energy barriers between minima or with very shallow potential minima for the fluctuations in bond angles. As an example, Fig. 2 displays the Ramachandran conformational map for the first pair of dihedral angles in the octane molecule. This energy map is obtained allowing the molecule to relax to the most energetically favorable molecular geometry by minimization of the total potential energy, for fixed values of the pair of dihedral angles. Higher values of the energy maxima emerge when the energy is not minimized, while the potential minima are not appreciably changed. The same qualitative behavior appears in the conformational maps for the other pairs of dihedral angles. Figure 2 is presented to demonstrate the relative smoothness of the energy surface in support of our discretization of the angles down to a spacing of 10°. The matrix expansion theory represents the dynamics of the system through equilibrium statistical averages involving scalar products of increasing order in the bond vectors. Our calculations require knowledge of statistical averages that are second ^li –l j &, fourth ^~li –l j !~lk –lm !&, and sixth ^~li –l j ! ~lk –lm !~lp –lq !& order in the bond vectors. To accomplish this, we extend the RIS method to compute the sixth order terms ~see the Appendix!, while the formulas for evaluated both second and fourth order statistical averages are known22 and are summarized in the next section in order to provide the necessary definitions for our treatment of the sixth order terms. Some minor differences from the conventional RIS treatment arise from our use of statistical matrices that differ for each pair of dihedral angles and of an arbitrary discretization of the dihedral angles.

Following the successes of the matrix expansion method in describing the long time dynamics of the united atom alkane system,10,11 we again choose the basis set as involving all the bond vectors $li % in the molecule along with mode coupling terms constructed from linear combinations of the trilinear products $li ~l j –lk !% of bond vectors. The first order basis set is designated as F15$li % with i51,...,n. As described below in Eq. ~35!, the first order generalized Rouse modes $ji % are obtained as a linear combination of the $li %. These modes are then used to construct the trilinear basis functions F35$ji ~j j –jk !% with i, j,k51,...,n. The second order basis then contains the functions in the sets F1 and F3 . We can then generalize to pentalinear functions F5 , etc. The metric matrix S of Eq. ~19! has the symmetric block form

S5

F

s11

s13 •••

s31

s33

•••

•

•

•

•••

•

s~ 2i21 ! ,1

•

•••

s~ 2i21 ! , ~ 2i21 !

s1,~ 2i21 !

G

,

~31!

with sab the matrix of the scalar products between the basis functions in the sets Fa and Fb . The first order treatment has the basis F1 which generates the matrix ~s11!i j 5^li –l j &. The matrix s11 is just the bond correlation matrix U of Eq. ~14!. The second order treatment contains the matrices s11 , s13 , s31 , and s33 , where the matrix elements are defined by ~s11!i j 5^li –l j &, ~s13!i, jkm 5^~li –j j !~jk –jm !&, and ~s33!isp, jkm 5^~ji –j j !~jk –jm !~js –jp !&, etc. The equilibrium force constant matrix F has the similar block matrix structure

F5

F

f11

f13 •••

f1,~ 2i21 !

f31

f33 •••

•

•

•

•••

•

f~ 2i21 ! ,1

•

•••

f~ 2i21 ! , ~ 2i21 !

G

,

~32!

where fi j is given by fi j 5D

(k

K U L ]Fi ]F j ] rk ] rk

in agreement with Eq. ~20!. The first order treatment reduces the F matrix to f11 , f115DaaT ,

~33!

where the matrix a is the rectangular matrix obtained by removing the first row from the matrix M of Eq. ~15!. The construction of a basis set only in terms of bond vectors corresponds to taking the center of the coordinate system as diffusing with the center of mass ~equivalent here to the center of friction! of the alkane molecule. This procedure sufficies for calculating the internal modes of motion and has the further advantage of reducing the matrix dimension by a half when use is made of the molecule’s head-totail symmetry.



The first order theory employs a basis set expansion in terms of the bond vectors. When applied to dipole correlation functions, the first order treatment is equivalent to the use of the generalized Langevin equation that is produced with the bond vectors as slow variables and with the memory functions neglected. This generalized Langevin equation is equivalent to the optimized Rouse equation for the bond vectors,

] l~ t ! 1DaaT Ul~ t ! 5av* ~ t ! , ]t

~34!

provided the memory functions are ignored.43 Expressing the optimized Rouse equation in terms of the bond vectors as in Eq. ~34! simplifies the calculation of various dipole and orientational time correlation functions that are represented in terms of the bond vectors.36,43 On the other hand, the optimized Rouse equation in a bead position representation @see Eq. ~11!# is more convenient for computing time correlation functions for properties, such as the static and dynamic structure factors,35 that are naturally expressed in terms of bead coordinates. Equation ~34! can be solved in the standard manner by transforming to normal coordinates ja , n21

li 5

(

a51

Qia ja ,

~35!

with Q the matrix of the eigenvectors of the product matrix DaaT U in Eq. ~34!. The first order approximation with the bond vector basis set F1 produces Q equal to the matrix C in Eq. ~19!. Passing now to the second order approximation, the full second order treatment requires the diagonalization of the product of the matrices S21 and F @see Eq. ~19!# whose dimensions grow as the cube of the number of bonds. Recent calculations demonstrate that enormous computational savings are possible without any loss of accuracy by expressing the trilinear basis function in terms of the optimized Rouse eigenfunctions $ j i % and by choosing the matrix dimension such that the approximation retains the minimum number of the slowest, most relevant trilinear modes of motion for the molecule.10,11 Thus, we append to the first order basis set $l% the mode coupling basis functions n21 n21 n21

ja ~ jb –jc ! 5

(i (j (k

Qai Qb j Qck li ~ l j –lk ! ,

~36!

for a,b.c51,...,q,n21. The matrix expansion method has been found to converge by truncating this basis for q!n 21. This convergence is obtained since the slowest optimized Rouse modes contribute most to the long time behavior of the time correlation functions. For example, the slowest mode a51 describes the rotation of the molecule about its center of mass. The observed convergence of time correlation functions for q!n21 produces a very significant reduction in required computation times because the dimensions of the matrices s13 , s33 , f13 , and f33 are enormously diminished.

3831

In general, the nth order approximation contains all scalar products symbolically written as ^~l–l!2i21 & for i51,...,n. The required scalar products are of the form of conformational averages. They therefore are calculated here by extending techniques previously developed for the rotational isomeric states model.22 The distribution function entering in the statistical averages of Eq. ~6! is computed here from the same potential of Eq. ~23! used for the Brownian dynamics simulations. The rotational isomeric states approximation assumes that the molecule is in one of n discrete conformational states corresponding to one of the potential minima, where n53 in the usual RIS approach. However, we take n53, 6, 12, 18, or 36 states with equally spaced dihedral angles in our extended calculations which may therefore be considered discretized quadrature approximations to a continuous RIS theory with integrations over the dihedral angles. Following the RIS approach for interdependent rotational potentials, the energy of a given molecular configuration $ F i % is expressed as a sum of energies for nearest-neighbor pairs of dihedral angles, E F 5E ~ w 1 ! 1E ~ w 1 , w 2 ! 1E ~ w 2 , w 3 ! 1•••1E ~ w n24 , w n23 ! ,

~37!

where the energies E( w i , w i11 ) are computed with the other dihedral angles ~jÞi or i11! taken as fixed in the all-trans conformation and where the first energy term is a function only of the first dihedral angle. The conformational energies E( w i21 ) for a single dihedral angle and E( w i21 , w i ) for a pair of fixed dihedral angles are obtained as described in Sec. III. The zero of energy is defined as that for the all-trans dihedral angle conformation. Thus, the potential for a single dihedral angle w i21 is obtained from the conformational map by fixing all the other dihedral angles in the trans conformation as represented symbolically by E ~ w i21 ! 5E ~ t 1 ,t 2 ,...,t i22 , w i21 ,t i ,...,t n23 ! .

~38!

The potentials for a pair of neighboring dihedral angles w i21 and w i are simply expressed in this notation as E ~ w i21 , w i ! 5E ~ t 1 ,t 2 ,...,t i22 , w i21 , w i ,...,t n23 ! 2E ~ t 1 ,t 2 ,...,t i22 , w i21 ,t i ,...,t n23 ! . ~39! Thus, the pair energies E( w i21 , w i ) contain the influences of all nonbonded interactions, which are the only sources of interdependent torsional angle interactions with the potentials of Eq. ~23!. Our numerical calculations show that the energy rapidly approaches a constant value for dihedral angles not near the chain ends. For example the difference E( w 3 , w 4 ) 2E( w 2 , w 3 ) is less than 1% as are the subsequent differences. When treating higher molecular weight alkanes, this suggests the possibility for simplyfing the calculations by assuming that the E( w i21 , w i ) becomes constant beyond i53 ~from each chain end!. We briefly review some of the basic RIS theory to describe our computations more explicitly and to introduce


3832

M. Guenza and K. F. Freed: Extended rotational isomeric model TABLE III. Possible combinations of the bond indices for the six bond statistical averages. i5 j5k5m5s5 p i5k5s, j5m5 p i5s, j,k,m, p i, j,k5s,m5 p i5 j,k5m,s5p i,k,m,s, j,p i, j,k,s,m, p i, j,k,s, p,m i,k,s, p,m, j i,k,s, p, j,m

i5k5s, j,m5 p i5s,k, j,m, p i5s,k, p, j,m i, j,k5s,m, p i,k5s, j,m, p i5 j,k,s,m, p i,k, j,m,s, p i,k, j,s,m, p i,k, j,s, p,m

necessary notation for explanation of the extension to compute the six order bond vectors averages that have not previously been evaluated. Following the RIS approach for interdependent rotational potentials, the statistical weight for one particular chain configuration is given by n21

V $F%5

)

i52

~40!

ui ,

where ui is the generalized RIS statistical weight matrix for the ith dihedral angle. For example, if the ith dihedral angle is in the generalized RIS state w i 5 h ~a discrete value for the dihedral angle w i ! and the dihedral angle i21 is in state w i21 5 z , these two neighboring dihedral angles contribute to the total statistical weight matrix in Eq. ~40! through the matrix element u z h ,i 5exp~ 2E z h ,i /k B T ! .

~41!

The partition function Z is obtained as

F) G n21

Z5J*

h52

~42!

uh J,

with the vectors J* and J given by J*5@1 0•••0# and JT 5@1 1•••1#. The quantities J* and J appear as in the traditional RIS method22 to extract the correct summation of terms from the product of u matrices. The average of the scalar product of bond vectors li and l j over all chain configurations is then given by the standard RIS expression22

^ li –l j & 5l 2 Z 21 J* ui21 ~ En ^ mT !@~ u ^ E3 ! 3 ~ En ^ T!# j2i21 ~ En ^ m! um2 j21 J

~43!

for a molecule containing m5n21 bonds each of average length l. The matrix T is defined in Eq. ~22!; the matrix En is a @n3n# identity matrix; and the use of the direct product ^ follows the standard RIS method as does the use of the matrix m, mT 5 @ 1 0 0 # ,

~44!

to extract the required terms from the product of matrices.22 The conformational maps for different pairs of bonds along the chain are, in general, not equivalent. In fact, while the molecule has a twofold mirror symmetry between both ends, between both bonds next to the ends, etc., the chain ends have different static and dynamic properties from the

i5k5s,m5 p, j i5s,k,m, j, p i5s, j,k, p,m i5 j5k5m,s5 p i,k5s,m, p, j i5 j,k,s, p,m i,k,m, j,s, p i,k,s, j,m, p i,k,s, j, p,m

i5k5s,m, p, j i5s, p, j,k,m i,k5s, j,m5 p i,k5s,m5 p, j i5 j,k,m,s, p i, j,k,m,s, p i,k,s, p,m, j i,k,m,s, p, j i,k,s,m, j, p

interior beads along the chain. In general, the Eqs. ~42! and ~43! must replace ui21 by the product of the different bonddependent ui matrices calculated from the Ramachandran maps through Eqs. ~38!, ~39!, and ~41!. Therefore, the product in Eq. ~42! must be interpreted as ui21 5u1 u2 u3 •••ui21 ,

~45!

with ui usually different from u j for iÞ j. If the molecule has a high enough molecular weight ~e.g., pentadecane!, the ui matrices in the interior of the chain become equivalent, and it is possible to simplify the calculations by treating differently the u matrices for the terminal and interior portions of the molecule. This also implies that the same set of computed ui matrices may be used for united atom alkane chains of arbitrary molecular weights. The four-bond scalar products ^~li –l j !~lk –lm !& are calculated in analogy with the procedure explained in Chap. IV, Sec. 8 of Flory’s book.22 The method is extended here ~see the Appendix! to treat the six bond scalar products that appear, for instance, in the matrix s33 . The standard procedure for calculating the four bond scalar products ^~li –l j !~lk –lm !& requires treating seven different possibilities for the bond indices, namely i5 j5k5m, i5 j,k5m, i5 j,k,m, i, j ,k,m, i5k, j5m, i,k,m, j, and i,k, j,m. The number of possibilities dramatically increases for the six bond statistical averages, but the use of chain symmetry enables reducing the number of the necessary calculations to 37 different products of matrices ~see Table III!. The statistical averages for each of these combinations is calculated in a different way, and an example of the method is described in the Appendix. The RIS averages for the scalar products of the bond vectors are used to calculate the S and F matrices that enter Eq. ~21! defining the time correlation functions. Thus, these time correlation functions are computed with the same potentials used in the Brownian dynamics simulation. The particular calculated quantities are the dipolar positional time autocorrelation functions for the interbead distance rab , C ab ~ t ! 5 ^ rab ~ t ! •rab ~ 0 ! & / ^ rab ~ 0 ! •rab ~ 0 ! & ,

~46!

where rab5ra2rb , and ra is, in general, the position vector for bead a ( a , b 51,...,n). The time correlation functions are then compared with the numerical results from the Brownian dynamics simulations. This comparison provides a deeper insight into the relative importance of the different



3833

contributions from the potential function to the long time dynamics of the flexible alkane chain molecules. V. RESULTS

Several different time correlation functions have been calculated with the extended RIS approach. All the correlation functions are checked against the corresponding ‘‘exact’’ results from Brownian dynamics simulations.10–12 The correlation functions [C 12(t)], [C 14(t)], and [C 18(t)] provide information on motions involving different parts of the molecule and, as a consequence, are related to different physical properties. The comparison of the RIS correlation functions with the previous theory provides an understanding for the influence of the torsional angle barrier crossings on the chain dynamics. The torsional potentials ~and the associated conformational transitions! are taken to represent the main terms in the intramolecular potential for the rotational isomeric states model, as all the other degrees of freedom ~bond stretching and angle bending! are totally frozen. Although our model permits the bond angles to relax to the most energetically favorable chain conformation for a fixed neighboring pair of dihedral angles, no significant differences arise in the calculations from allowing the bond angles to relax. The theory is illustrated for the dipole time correlation C 12(t), C 14(t), and C 18(t) of the octane molecule using a united atom model and realistic potential functions. These examples include the fastest and the slowest decaying dipole correlation functions. The correlation function C 12(t) investigates the local motion of the terminal bond and is the fastest of the bond relaxations. A single bond vector is known to lose its orientational correlation more rapidly than the vectors connecting more distant nonbonded beads of the chain. The motion of the vector connecting the middle and the end of the molecule is described by the correlation function C 14(t). This vector relaxes on an intermediate time scale. The slowest relaxation process is provided by the correlation function C 18(t) that describes the global motion of the molecule ~the end-to-end distance!. This motion influences the bulk properties of the material, such as the intrinsic viscosity or shear modulus in the melt state. ~The overall chain diffusive motion is not considered as an internal process.! The differences in time scales are greatly enhanced for molecules of increasing molecular weight. The correlation functions have been calculated from the diffusion equation for the bond vectors with and without contributions from the memory terms. These calculations correspond to expanding the eigenfunctions of the diffusion operator in the first or the second order basis sets, respectively, as described in Sec. III. The first order treatment with the extended RIS model produces the optimized Rouse correlation functions which are compared with the first order ‘‘simulated’’ ones in Fig. 3. The first order simulated correlation functions are equivalent to those calculated from the generalized Langevin equation @Eq. ~34!# without the memory terms, when the equilibrium averages of the two bond scalar products are evaluated directly from the simula-

FIG. 3. Comparison between the first order dipole bond time correlation functions C 12(t), C 14(t), and C 18(t), as calculated with the optimized Rouse approximation using as input data the statistical averaged two-bond scalar products ^li –l j & from the Brownian dynamics simulations ~solid lines! and from the extended rotational isomeric states model ~dashed lines!.

tions. The first order RIS model correlation functions appear to overestimate the correlations between neighboring bonds, describing the chain as stiffer than in the simulated cases. The same qualitative effect persists in the higher order treatment. The greater stiffness evidenced by the RIS approach probably emerges because the theory does not include the influence on the dynamics of bond angles fluctuations. Both the first order simulated and the first order extended RIS correlation functions only qualitatively describe the ‘‘exact’’ octane molecule relaxation processes given by the Brownian dynamics simulations. The comparisons with the latter numerically exact correlation functions in Figs. 4–6 exhibit the importance of including contributions from the memory terms.

FIG. 4. Comparison between the dipole bond time correlation function C 12(t) as calculated directly from Brownian dynamics simulations ~solid line! and by the matrix method. Dotted line gives C 12(t) from the first order basis set with input from the Brownian dynamics; short dashed line is from the first order extended RIS model; dot–dashed line is from the second order basis set with input from the Brownian dynamics simulations; and long dashed line is from the second order basis set from the extended RIS model.


3834



The use of higher order basis functions introduces more detailed dynamical information into the theoretical correlation functions. The leading second order treatment for Q51 includes only trilinear contributions from the slowest internal mode ~the rotational mode!. When this Q51 correction is included, the extended RIS correlation functions display excellent agreement with the Brownian dynamics simulations for the bond correlation functions C 14(t) and C 18(t) ~see Fig. 7!. Since the theory is designed to describe the long time dynamics, our inadequate treatment of the memory functions is evident in the discrepancy at short time in the Q51 correlation function C 12(t) in Fig. 4.


FIG. 7. Comparison between the bond time correlation functions C 12(t), C 14(t), C 18(t), as calculated directly from Brownian dynamics simulations ~solid lines! and by the matrix method with the second order basis set from the extended RIS model ~dashed lines!.

Increasing the order Q of the trilinear combinations of modes included in the calculation does not introduce any significant improvement in the calculated time correlation functions through Q54 ~see Fig. 8!. The changes are less than 1%. Thus, the Q52 slowest mode approximation in second order appears to be converged and quite successful in producing excellent agreement with the simulated data. In contrast the second order theory, with the input equilibrium averages evaluated from Brownian dynamics simulations, requires retention of trilinear modes through order Q53. The origins of this slightly slower convergence are presently unclear but probably arise from the angle bends as discussed below. Figures 4–6 compare the first and the second order RIS time correlation functions C 12(t), C 14(t), and C 18(t) with the numerically exact simulated correlation functions.

FIG. 8. Comparison between the dipole time correlation functions C 12(t), C 14(t), C 18(t), as calculated by the matrix method with the second order basis set from the extended RIS model with the trilinear combination of the first modes. Q51 ~solid lines! and with trilinear combinations of modes through Q52 ~dashed lines!. ~Calculations through Q54 are virtually identical to those shown for Q52.!



The differences between the two sets of second order curves are very close to the corresponding differences in the first order curves. Calculations of the dipole time correlation functions have also been performed with increasing numbers of rotational isomeric states as explained in Sec. III. No relevant differences or improvements appear. This finding implies that the relative energies ~statistical weights! of the three potential minima ~trans, gauche, and –gauche! provide the main contributions to the statistical averages. This result presumably reflects the fact that all the correlation functions in the figures have characteristics times longer than the inverse conformational transition rates, so these C i j (t) reflect the average of many such transitions. The above results are consistent with the recent analysis of molecular dynamics simulations that is designed to investigate the importance of various librational motions on the time evolution of the bond orientational correlation functions.44,45 A filtering of the high frequency motions from the simulation data produces an overestimation of the bond orientational correlation times, in qualitative agreement with our findings. On the other hand, we extend the usual RIS model and allow the molecule to relax to the most energetically favorable conformation for fixed values of any pair of successive interdependent dihedral angles and by using a large number of rotational isomeric states ~i.e., a large number of points in the Ramachandran map!. These extensions partially overcome the simplicity of the traditional RIS model, introducing to some extent the contributions from torsional angle librations, but not from librations in the bond angles. Some calculations have been performed for molecules with higher molecular weights. The generalized RIS statistical weight matrices u for pairs of dihedral angles in the interior of the molecule become almost identical, and the solution of the matrix problem @Eq. ~19!# becomes illconditioned because of linear dependencies. This is a consequence of the fact that internal bonds of flexible homopolymer chains relax with very similar correlation times. The first order treatment is, however, soluble, and the dipole correlation functions exhibit the same qualitative behavior as for octane. Therefore, we have not attempted to overcome the numerical difficulties using previously developed methods, mentioned in Sec. IV. Previous computations using simulations to produce the equilibrium inputs have occasionally encountered illconditioned S matrices, generally because of working with too short a trajectory. The S matrix is formally a nonnegative metric matrix, but numerical approximations ~such as using a short trajectory! can yield unphysical negative eigenvalues of the S matrix. Longer trajectories generally repair the damage and produce positive definite S matrices as required. The success of the extended RIS approach confirms the utility of this very simple model as correctly describing not only the equilibrium properties of flexible chain molecules but also the more complex dynamical processes. As a consequence, it is reasonable to suppose that the multiple barrier crossings of the conformational dihedral angles provide the

3835

main contributions to the molecular relaxation processes. The local angular motions appear to produce some corrections which we may estimate as the difference between the first ~or second! order computations with the RIS model and those with simulated equilibrium input averages. VI. CONCLUSIONS

The calculation of the memory function term in the generalized Langevin equation, or equivalently in the Smoluchowski equation, is necessary to describe the long time dynamics of molecules in solution. This calculation represent an extremely challenging problem in statistical mechanics, and the complexity is enhanced when, instead of simple liquids, we focus on the dynamics of flexible protein or polymer chains. To date, several advanced techniques have been used to analytically and/or computationally attack this complicated problem. The matrix method, combined with a mode coupling approximation, appears to be the most promising approach for treating the influence of the memory terms on the long time dynamics of flexible chain molecules in solution. This paper presents the first computation of the memory terms directly from the potential functions using an extended rotational isomeric states ~RIS! model to describe the statistical properties of the molecule. This direct computation contrasts with the prior use of Brownian dynamics or molecular dynamics simulations for generating the equilibrium input information required for the dynamical theory. While the prior use of simulations for the equilibrium averages provides a very stringent test of the theory and has aided in developing an accurate mode coupling approximation for the long time dynamics, the present treatment circumvents the need for the simulations. The use of an extended RIS model has enabled us to investigate the relative importance of torsional and bond angle bending portions of the intramolecular potential on the memory functions describing the long time influence of the internal chain friction. The excellent agreement between the exact bond correlation functions from the Brownian dynamics simulations and the correlation functions calculated from the matrix method, with statistical averages evaluated using the RIS model, would appear to demonstrate that the torsional motions and their multiple conformational transitions provide the dominant long time relaxation mechanism for united atom alkane chain. However, this excellent agreement emerges partially through a cancellation of errors. The small but nonnegligible difference between the theoretical correlation functions, obtained with RIS or simulated equilibrium averages, arises mainly from the dynamical influence of bond angle fluctuations. The matrix method tends to describe the molecule as slightly too flexible, while the RIS treatment makes the molecule somewhat too stiff because of the neglect of the angle bending dynamics. It is, in principle, possible to develop a more sophisticated RIS-matrix method with averages over several discrete values for the bond angles. Such a treatment should make the RIS theory correlation functions coincide with those obtained using simulated


3836


equilibrium averages. Nevertheless, the present approximation is quite adequate for the united atom alkane systems, a feature that accords well with the observed predominance of torsional angle conformational transitions as the relaxation mechanism in the Brownian dynamics simulations ~as well as in molecular dynamics simulations47!. While the present computations are limited to describing octane dynamics, the previous computations exhibit identical trends for octane, pentadecane, and pentacosane. Hence, octane is long enough to be generic. The situation may be more complicated in molecular dynamics simulations for the protein fragment ACTH~5–10! where the correlation time for backbone conformational transitions greatly exceeds those for the individual bond dipole correlation functions. Hence, the decay of the dipole correlation functions in the ACTH~5–10! fragment is driven by a combination of torsional and bond angle fluctuations within a single conformational well. It, therefore, remains to be seen whether a similar torsional RIS treatment sufficies for the protein dynamics and whether the bond angle bending fluctuations become more important. While it is possible to generalize the RIS approach further and to develop a treatment with interactions between three ~or more! successive dihedral angles,41 the RIS formalism is only capable of describing short range correlations along the chain. Thus, as is well known, the RIS method fails to describe the long range correlations induced by excluded volume interactions, correlations that produce a ‘‘nonclassical’’ dependence of chain dimensions on the polymerization index. One possible method for circumventing this difficulty involves applying the RIS approach to describe the short range correlations and then appending renormalization group descriptions48 of excluded volume for the interbead mean square separations ^~ri 2r j !2&.49 This type of ansatz would incorporate the ‘‘nonclassical’’ equilibrium dimensions into an improved matrix method representation of the dynamics. On the other hand, when the equilibrium averages are taken from simulations, all the nonclassical equilibrium dimensions are included in the matrix method. However, it is still not possible to provide a complete theory for both short range chain stiffness ~with realistic intermolecular potentials! and long range excluded volume. The rotational isomeric states model is known to reproduce fairly well the static properties of the polymer chains ~e.g., the characteristic ratio

and the end-to-end distance!. The use of the RIS model in conjunction with the mode coupling matrix method is found to be quite accurate at also describing the long time dynamic properties of united atom alkane chains. Of particular interest is the fact that a three-states RIS model serves to treat the cumulative effects on the dynamics of the multiple conformational transitions. ACKNOWLEDGMENTS

This work is supported, in part, by NSF grant DMR9530403 and by ACS PRF grant 29067-AC7. We thank Xiao-Yan Chang for providing the data on Brownian dynamics simulations and for useful discussions. APPENDIX

The six-bond scalar products are calculated from an extension of the procedure for calculating the four-bond scalar products as described in Flory’s book,22 Chap. IV, Sec. 8. It is convenient to introduced a short hand notation that simplifies the formulas. The direct products of the identity matrices En with the transformation matrix T @Eq. ~22!# and the matrix m @Eq. ~45!# are expressed in this compact notation as @~ u ^ E3 !~ En ^ T!# 5 ~ uT! , @~ u ^ E3 ^ E3 !~ En ^ T ^ T!# 5 ~ uTT! , @~ u ^ E3 ^ E3 ^ E3 !~ En ^ T ^ T ^ T!# 5 ~ uTTT! , ~ En ^ m! 5 ~ m! ,

~ En ^ E3 ^ m! 5 ~ E3 m! ,

~ En ^ m ^ E3 ! 5 ~ mE3 ! , ~ En ^ E9 ^ m! 5 ~ E9 m! , ~ En ^ m ^ E9 ! 5 ~ mE9 ! , ~ En ^ E3 ^ m ^ E3 ! 5 ~ E3 mE3 ! .

The four-bond scalar product of Eq. ~44! for a molecule containing m5n21 bonds becomes in this notation,

^ li –l j & 5l 2 Z 21 J* ui21 ~ mT !~ uT! j2i21 ~ m! um2 j21 J, with Z and u defined in Eqs. ~42! and ~41!. As an example of the six-bond averages, we write the 11 possible different scalar products ^~ls –lp !~li –l j !~lk –ll !& obtained when the indices s and i apply to the same bond vector. Using the above short hand notation these cases yield

s5i5k,p5 j5l:

l 6 Z 21 J* ui21 ~ mT mT mT !~ uTTT! j2i21 ~ mmm! um2 j21 J,

s5i5k, p, j5l:

l 6 Z 21 J* ui21 ~ mT mT mT !~ uTTT! p2i21 ~ E9 m!~ uTT! j2 p21 ~ E3 m!~ m! um2 j21 J,

s5i5k, j5l, p:

l 6 Z 21 J* ui21 ~ mT mT mT !~ uTTT! j2i21 ~ E9 m!~ E3 m!~ uT! p2 j ~ m! um2p21 J,

s5i5k,p, j,l:

l 6 Z 21 J* ui21 ~ mT mT mT !~ uTTT! p2i21 ~ E9 m!~ uTT! j2 p21 ~ E3 m!~ uT! l2 j21 ~ m! um2l21 J,

s5i,p,k,l, j:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! p2i21 ~ E3 m!~ uT! k2p21 ~ mT !~ uTT! l2k21 ~ E3 m!~ uT! j2l ~ m! um2 j21 J,

s5i,k, p,l, j:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! k2i21 ~ mT !~ uTTT! p2k21 ~ mE9 !~ uTT! l2p21 ~ E3 m!~ uT! j2l21 ~ m! um2 j21 J,

s5i,k,l,p, j:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! k2i21 ~ mT !~ uTTT! l2k21 ~ E9 m!~ uTT! p2l21 ~ E3 m!~ uT! j2l21 ~ m! um2 j21 J, J. Chem. Phys., Vol. 105, No. 9, 1 September 1996



3837

s5i,p,k, j,l:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! p2i21 ~ E3 m!~ uT! k2p21 ~ mT !~ uTT! j2k21 ~ mE3 !~ uT! l2 j21 ~ m! um2l21 J,

s5i,k, p, j,l:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! k2i21 ~ mT !~ uTTT! p2k21 ~ mE9 !~ uTT! j2 p21 ~ mE3 !~ uT! l2 j21 ~ m! um2l21 J,

s5i,p, j,k,l:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! p2i21 ~ E3 m!~ uT! j2 p21 ~ m! uk2 j21 ~ mT !~ uT! l2k21 ~ m! um2l21 J,

s5i, j,k,p,l:

l 6 Z 21 J* ui21 ~ mT mT !~ uTT! j2i21 ~ E3 m!~ uT! k2 j21 ~ mT !~ uTT! p2k21 ~ mE3 !~ uT! l2p21 ~ m! um2l21 J.

As already outlined in Sec. IV, the u matrices are generally not equivalent. As a consequence, all the above products of matrices must be interpreted analogously to Eq. ~45! as, for example, ~ uT! j2i21 5 ~ ui T!~ ui11 T!~ ui12 T! ••• ~ u j2i21 T! .

When the molecular geometry is optimized by an energy minimization, the T matrices are also inequivalent. In fact, different optimized bond angles emerge both as the dihedral angles vary and for different pairs of dihedral angles. However, only small corrections from these geometry variations appear, and a constant T matrix approximation suffices. It is likewise possible to express the other six-bond scalar products that arise when the indices s and i are different, but the formulas are more complex. The calculations of the sixth-order scalar products involve matrices of dimension @n327,n327# with n the number of discrete conformational states used to calculate the statistical averages. Thus, there is a significant growth of computational time with an increase in n, and this also leads to a rapid increase with the extension of the basis set to second order. Hence, alternatives to the usual RIS procedure may prove useful in further applications, but our finding that n53 suffices has helped to keep the matrices from becoming unmanageable. J. D. Ferry, Viscoelastic Properties of Polymers ~Wiley, New York, 1970!. R. T. Bailey, A. M. North, and R. A. Pethrick, Molecular Motion in High Polymers ~Clarendon, Oxford, 1981!. 3 H. Yamakawa, Modern Theory of Polymer Solutions ~Harper and Row, New York, 1971!. 4 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics ~Clarendon, Oxford, 1986!. 5 A. Perico and M. Guenza, J. Chem. Phys. 83, 3103 ~1985!. 6 A. Perico and M. Guenza, J. Chem. Phys. 84, 510 ~1986!. 7 A. Perico, J. Chem. Phys. 88, 3996 ~1998!. 8 G. D. Smith, D. Y. Yoon, and R. L. Jaffe, Macromolecules 28, 5897 ~1995!. 9 M. D. Ediger and D. B. Adolf, Adv. Polym. Sci. 116, 73 ~1994!. 10 X. Y. Chang and K. F. Freed, J. Chem. Phys. 99, 8016 ~1993!. 11 W. H. Tang, X. Y. Chang, and K. F. Freed, J. Chem. Phys. 103, 9492 ~1995!. 12 X. Y. Chang and K. F. Freed, J. Chem. Phys. 104, 3092 ~1996!. 13 H. Risken, The Fokker–Planck Equation: Methods of Solution and Applications ~Springer, New York, 1989!. 14 R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II ~Springer, Berlin, 1991!. 15 M. Bixon, J. Chem. Phys. 58, 1459 ~1973!. 1 2

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