J. Phys. Chem. 1995, 99, 816-821
816
Dynamics of Fractional Brownian Walks Parbati Biswas and Binny J. Cherayil*jt Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India Received: February 24, 1994; In Final Form: July 12, 1994@
We investigate the dynamics of polymers whose solution configurations are represented by fractional Brownian walks. The calculation of the two dynamical quantities considered here, the longest relaxation time t, and the intrinsic viscosity [VI, is formulated in terms of Langevin equations and is carried out within the continuum approach developed in an earlier paper. Our results for t, and [v] reproduce known scaling relations and provide reasonable numerical estimates of scaling amplitudes. The possible relevance of the work to the study of globular proteins and other compact polymeric phases is discussed.
1. Introduction Random walks have proved to be extremely useful in representing the configurations of long chain molecules mathematically, and many of the statistical properties of these walks are related to the physical properties of actual polymeric systems.' The correspondence is especially close for solutions of polymers near the 8 point, where interactions between different chain segments are small. Even when such interactions are large, however, as is the case for polymers in good solvents, the behavior of the chains can be represented in terms of perturbations to an underlying random walk process, which can be regarded as a minimal model of the system.* This representation is often quantitatively successful in explaining experimental data on polymer solutions in good solvent^.^ Nevertheless, a considerable amount of effort is needed to achieve these levels of agreement, and in general, it would be far more convenient if excluded volume and other effects could be built into the minimal model itself, without having to treat such effects separately. A model of this kind would, for example, represent polymers in good solvents as self-avoiding walks at the lowest order of approximation, through appropriate choice of statistics, rather than as random walks that experience self-avoidance constraints through the action of an intermolecular potential. It would, therefore, include corrections to random walk behavior in a parameter that "tunes" the configurations of a chain to reflect the properties of the medium. In constructing such models, the hope is, of course, to simplify the mathematical treatment of systems in which there are several sources of non-Gaussian behavior, eliminating, if possible, one or more perturbative terms by absorbing them into the minimal model of the chain (whose properties are presumed to be wellknown) and leaving a smaller number of these terms to be handled by conventional techniques. Several realizations of this program are possible, including descriptions based on Levy flights4 and the Dirac pr~pagator,~ which have been used in the study of stiff chains. More recently, we have found that fractional Brownian walks, a class of stochastic processes that are equivalent to ordinary random walks but allow greater or less degrees of correlation between successive steps in the can be used in related applications. For example, our studies have shown that the path integral representation of these w a k S is useful in examining the question of how the radial dimensions of chains scale with molecular
weight near the critical point.9 In the past, problems involving regions of the phase diagram below the 8 point have generally been addressed in terms of models in which the chains are treated as random walks with attractive segment-segment interactions;1° using fractional Brownian walks, one can instead treat the chains as collapsed polymers with repulsive segmentsegment interactions and so sidestep some of the difficulties associated with the use of attractive potentials. Continuing our efforts to explore the properties of fractional Brownian walks, we consider, in this paper, their application to a few elementary dynamical questions concerning polymers. The aim is less to uncover new results as to extend existing results to a larger class of polymers. To this end, we revisit some old problems in relaxation and transport phenomena; specifically, we consider, in the context of fractional Brownian walks, the decay of the end-to-end distance of long chains, and the intrinsic viscosity of dilute solutions of such chains. Much of what we do is guided by the results and methods of the monograph on polymer dynamics by Doi and Edwards." Future papers will deal with other viscoelastic properties of polymers. In section 2 of this paper, we review the path integral representation of fractional Brownian walks. We next consider, in section 3, their dynamical evolution in terms of a Langevin equation, whose solution can be obtained by a normal mode decomposition of the chain coordinates. This normal mode description is then used to determine the decay of the end-toend distance of the chain and is described in section 4. Section 5 considers the solution of the Langevin equation in the presence of hydrodynamic interactions. The treatment here makes use of preaveraging and other approximations to obtain definite numerical estimates for the longest relaxation time of the chain. A calculation of the intrinsic viscosity is also presented in section 5, and the paper concludes with a brief discussion.
2. Generalized Bead-Spring Model The study of the dynamics of dilute polymer solutions often begins by describing the chains as a collection of friction centers, or beads, connected by harmonic springs. In d = 3 dimensions and in units where the thermal energy ~ B and T the scaled step length Nd are unity, the continuum version of the energy or effective Hamiltonian HOfor this system is3
~~~
Also of The Jawaharlal Nehru Centre for Advanced Studies, Bangalore560012, India. Abstract published in Advance ACS Abstracts, December 1, 1994. +
@
0022-365419512099-0816$09.0010
where ~ ( tis) the spatial location of a bead that is a distance t along the polymer backbone and N is the length of the chain. 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 2, 1995 817
Dynamics of Fractional Brownian Walks
As discussed in ref 8, eq 1 can also be interpreted in terms of the locus of a particle that evolves in “time” z under the action of white noise W’(t), which is the derivative of the Wiener process W(Z).~ A more general kind of motion evidently results if the noise is “colored” and in particular if the particle is governed by the equation
where B‘a(t) is the derivative, along the direction a,of the fractional Brownian motion Ba(s), defined as7
of a Langevin equation in which hydrodynamic interactions between different parts of the chain are completely ignored. Such an approach corresponds to assuming a Rouse model for the dynamics of the chain and is particularly easy to implement. If at time t, then r(z,t) is the distance of a monomer at z from the center of mass of the chain R&), that is, if
then we can write”
(9) with h a number between 0 and 1 and r(. .) the gamma function; then the effective Hamiltonian H , that can be said to give rise to this new motion has been shown to be of the following form8
Clearly, when h = l/2 in this equation, the fractional Brownian walk reduces to the random walk (Le., the Wiener process), and H, reduces to Ho. Moreover, a trivial change of variable allows HOC,also to be expressed as
which indicates that the fractional Brownian walk is in effect a scaled random walk, a result also demonstrated by Maccone7 by a different method. A monomer at the point c1 and t 1 on a chain whose configurations are identified with H , is correlated with a monomer at cz and z2 according to the probability distribution8
where 5 is the monomer friction coefficient and w is a random force that is assumed to satisfy these relations (dictated by the fluctuation-dissipation theorem):
Equation 9 can be solved if a transformation between the coupled coordinates r(z,r) and a set of n modes X,(t) can be found that separates the equation into independent motions of the modes. One such transformation is suggested by the eigenfunctions &(t) derived by Maccone’ as solutions of the integral equation for the autocorrelation of the fractional Brownian motion. In terms of these eigenfunctions
where Jv(. is the Bessel function of order v, with v = 2h/(2h l), yn is the nth zero of the Bessel function of order v - 1, Le., Jv-l(yn) = 0, and A, is the normalization constant
+
Hence, the mean square end-to-end distance scales with chain length N as
(e) of such a chain
C-Nh
(7)
The utility of fractional Brownian walks as representations of polymers is illustrated by this relation; by suitable choice of the parameter h, the average configurational behavior of the chain can be made to correspond to its actual behavior in solvents of different quality, thereby eliminating the need to account in detail for the nature of the intermolecular potential appropriate to the given solvent. For instance, the choice h = 3/5 models polymers in good solvents, while the choice h = l/3 models polymers in compact or collapsed phases. This is not to say that this approach is faithful to the underlying chemical constitution of the chain, even in a coarse-grained sense, only that it is a useful calculational tool. With these preliminary remarks, we now turn to a discussion of how the dynamics of these walks may be described.
0)
A, =
.
..
Nhfl‘zJv(~n>
The 0th normal mode is defined to comespond to the center of mass of the chain, i.e., XO R, = K’Jpd’dtc(t,t). The eigenfunctions of fractional Brownian motion form a complete set and are also orthogonal, in the sense that
so the expression for the normal modes in terms of the chain coordinates is given by
X,(t) = h N d z r(z,t)A,thJ,(Yn(z/~h+ln), n I1 (17) 3. Evolution of Fractional Brownian Walks
One approach to the dynamics of polymers whose configurations are governed by fractional Brownian statistics is in terms
which, together with eq 13 and some elementary identities involving the Bessel function,12 transform eq 9 to
Biswas and Cherayil
818 J. Phys. Chem., Vol. 99, No. 2, 1995
where f is the random variable defined by
and having the same statistics as w. The transformation eq 13 has indeed decoupled the motions of the monomers into the independent motions of the X,(t), and the solution of eq 19 can now be written down at once:
The second line of this equation represents an unproven conjecture* about the exact value of the sum cn=l l/yn2 for general values of h. We have used it here purely for mathematical convenience; the true value of the sum can always be determined to a high level of accuracy numerically. When h = V 2 , eq 28, in original units, reduces to
which may be compared with the Doi-Edwards result for the same normalized correlation function (with ksT= 1 = 1) using the conventional Rouse model:
from which it follows that
1 Here the angular brackets refer to an equilibrium ensemble average, specifically the average over the probability distribution of the chain coordinates P[r(z)], where P[r(r)] = exp( - P ( h
+ 1/2)hNdz l i - ( ~ ) l ~ z ~ - ~ ~ ](23)
or in normal coordinates
Given P[X,], the average in eq 22 is readily calculated:
This average appears in the expression for the decay profile of the end-to-end distance of the chain R, which is treated in the next section.
4. Decay to the End-to-End Distance The connection between eq 25 and R follows from the continuum definition of R:
which, using the eigenfunction expansion eq 13 and standard integral tables,12 is equivalent to
R(t) = NhCX"(Q%J"(Y,)
(27)
=--)
exp{ - 3&2n - 1)2t/N2c}
- 1)2 If the longest relaxation time zr of the end-to-end distance is defined as the reciprocal of the n = 1 mode of the coefficient of t in these two equations, the present model (distinguished by the letters BC) and the Doi-Edwards model (DE)would predict that
The N dependences of these relaxation times are exactly the same, but the scaling amplitudes are quite different. The difference is due to the fact that under 8 conditions (when h = l/2) the Bessel functions, in terms of which the coordinates of the fractional Brownian walk are expanded, reduce to sines and not cosines, the basis set used in the conventional Rouse model. For the purpose of a "normal mode" decomposition, either basis set works, but because the amplitudes depend on the zeros of the eigenfunctions (which are, of course, different for sines and cosines), there are numerical differences between the two models. Comparisons between them are therefore likely to be more informative and meaningful if the predicted values of various quantities are "normalized" against a "fiducial state" for that quantity. We shall have more to say about this later. The Rouse model (hydrodynamic interactions neglected) appears to describe, qualitatively, the dynamics of polymers in the melt," but in dilute solution hydrodynamic interactions are expected to be important. When this is the case, the Langevin approach no longer admits of exact solutions, but it can be usefully implemented approximately, following the methods of Doi and Edwards, as described below. For non-zero hydrodynamic interactions, the Langevin equation reads
n= 1
The time correlation function (R(t).R(O)) = B(t) is thus seen to involve the normal mode average of eq 25, and after substituting this equation into the expression for B(t) and dividing by a factor of B(0) to normalize the expression, one finds, after simplification, that
where H is the hydrodynamic interaction matrix given by
Dynamics of Fractional Brownian Walks
J. Phys. Chem., Vol. 99, No. 2, 1995 819
(original units) (42) with qs the solvent viscosity and P(t1,rz) the unit vector in the direction of r(z1) - r(z2). As a first step in solving this equation, the hydrodynamic matrix is replaced by its average over the equilibrium distribution of the chain Gio)(r,r';rl,r2) (the so-called preaveraging appr~ximation'~).This average is easily calculated, since Gf) is Gaussian (cf. eq 6), and it produces
The longest relaxation time t,is simply the inverse of p?) with n = 1 and can be written as z, = CN3h
This result can be applied not only to the 8 point region (h = '/2) but also, without extra effort, to the good solvent (h = 3/5) and collapsed (h = l/3) regions as well. For these values of h, the longest relaxation times are tr
-(original units) = 1.04N (h = '/J 3s
Introducing the eigenfunction expansion described earlier, along with identities in the Bessel function,12 one can now write the Langevin equation as
where
(43)
(44)
= O.77N3l2(h = 1/2)
(45)
= O.65gl5 (h = 3/5)
(46)
By way of comparison, Doi and Edwards estimate C to be 0.325 for the case h = l/2, while experimentally its value seems to be close to 0.4. The case h = 3/5 corresponds to the good solvent limit, which Doi and Edwards treat by incorporating excluded volume effects into the friction coefficient of a linearlized Langevin equation and assuming a simplified form for the distribution function of the chain segments over which subsequent averages are performed. They do not provide a numerical estimate of C for this limit, apparently because of the severity of some of their approximations. We are not aware of calculated or experimental values of C for collapsed chains (the case h = l13). As remarked earlier, given the differences that arise from the choice of basis set for the normal mode expansion, it is perhaps preferable when comparing the results of different theories or experiments to use "scaled" variables. For example, the longest relaxation times in different solvents can be expressed in terms of the longest relaxation time for the Rouse ( R ) model. In this way, p C / < C ( h = l/2) = 0.18(
