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Universal features of polymer shapes
J.A. Aronovitz, D.R. Nelson
To cite this version:
J.A. Aronovitz, D.R. Nelson. Universal features of polymer shapes. Journal de Physique, 1986, 47 (9), pp.1445-1456. �10.1051/jphys:019860047090144500�. �jpa-00210339�
Universal features of polymer shapes
J. A. Aronovitz and D. R. Nelson
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, U.S.A.
(Reçu le 7 mars 1986, accepté le 7 mai 1986)
Résumé. 2014 La forme moyenne adoptée par de longues chaines polymériques en solution diluée dans un bon solvant est étudiée. Deux quantités invariantes par rotation, 0394 et S, sont définies et sont utilisées pour décrire de façon quantitative l’asymétrie moyenne ainsi que le degré d’élongation ou d’aplatissement d’une configuration. L’uni-
versalité de ces rapports d’amplitudes est démontrée et ils sont calculés au premier ordre en 03B5 = 4 - d. Il est montré
que les chaînes ont tendance à adopter en moyenne des configurations asymétriques de forme oblongue.
Abstract 2014 The equilibrium shapes of long chain polymers in a dilute solution in good solvent are investigated Rotationally invariant quantities 0394 and S which characterize the average asymmetry and degree of prolate 2014 or
oblateness of a polymer configuration are defined These amplitude ratios are proved to be universal, and calculated
to first order in an 03B5 = 4 - d expansion. The polymers are shown to be on average quite asymmetric and prolate.
Classification
Physics Abstracts
36.20 - 36.20E - 05.40
1. Introduction
In the last fifteen years the scaling behaviour of long
chain polymer sizes has been rather throughly ana- lysed, following the crucial observation by de Gennes [1] that the problem could be mapped onto the
critical behaviour of a simple magnetic system.
However, questions about the polymers’ shapes have
remained relatively unexplored other than by Monte
Carlo calculations [2-7]. It is clear that averaging over
an isotropic ensemble gives an appearance of spheri-
cal symmetry; however, this does not imply that the typical polymeric realization within the ensemble is itself symmetric. In fact, polymers are surprisingly anisotropic, even if one ignores their self avoidance and simply models them with Gaussian random walks (RWs). This anisotropy is similar to that which has been recently reported for two dimensional cluster
growth problems [8]. As in that case, one expects that the asymmetry has an effect on the flow characteristics of polymeric fluids. For example, it has been proposed [9] that an average polymeric configuration which is prolate, with moderate aspect ratio, could be an important ingredient in a phenomenological theory
of the velocity fluctuation spectrum observed in
experiments on laminar polymeric fluid flow at high
strain rates. These experiments are a step towards
understanding polymeric reduction of turbulent drag,
and so are of technological importance.
In order to study polymer shapes, one must first
decide both how to model the polymer and what is
meant by shape. We will consider polymers long compared to the persistence length, in a solution dilute
enough that a given polymer interacts primarily with
itself We further assume a good solvent. For this case, the monomers which make up the polymer effectively
interact via a weak, relatively short range attractive
potential with a hard repulsive core [10].
As we are interested in large scale properties of shapes in the limit of asymptotically long polymers, we
can coarse grain. We thus study the statistics of conti-
nuous paths r(t), where 0 t T, r(0) = 0, T is proportional to the polymerization index N (see Fig. 1), and a path’s self-interaction is modelled via a
T T
contact
potential A f 0 f 0
dti dt2 ð(r(tl)-r(t2)). Here,0 0
A is a coarse grained average of the solvent mode- rated (thus temperature dependent) monomer-mono- mer potential. It is implicit in this model that the paths r(t) are constrained not to vary on a length scale
shorter then some microscopic coarse graining length
ao = A - ’, where A is the theory’s ultraviolet cutoff.
The discretized version of this model counts self avoid-
ing random walks (SAWs) on a lattice. The above poly-
mer models are expected to share a single universality
class with the physical polymer system [10].
To motivate a definition of polymer shape, we first
consider the standard measures of polymer size. The
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047090144500
Fig. 1. - A typical continuum path r(t). Its end to end distance is r(T), because r(o) = 0.
computationally simplest size measure is the average end to end distance R, where
and z is an average over the ensemble of walks. A
more democratic measure is the mean square radius of gyration RG. To define RG, we first define averages
over a given walk. For X(r) a function of the position r, let
Then the centre of mass of a given walk is r, and
measures, the average excursion of a walk away from its center of mass.
Following Sole [3] we generalize the radius of gyra- tion to a shape tensor Q with components
which measures a given walk’s shape. The spread in Q’s eigenvalues Al >1 A2 >- A3 >1 0 measures the walk’s
asymmetry, and
It is worth noting that the moment of inertia tensor of the walk about its centre of mass is
Various authors have investigated walk asymmetries by computing the normalized average eigenvalues ( Åi )/R/; using Monte Carlo techniques. Their results
are typified by those of Kranbuehl and Verdier [7] who
find that
where the isotropic case would be (1/3, 1/3, 1/3). One
sees that even without self avoidance the typical poly-
mer configuration is quite anisotropic.
Unfortunately it is difficult to compute the quantities z Ai )/R/; analytically, because one must explicitly diagonalize Q for each realization in an ensemble of
polymers for this calculation. To make analytic pro- gress, we have found it convenient to characterize the walks’ shapes by rotationally invariant polynomials
in the components of Q. If we let 1 = tr Q /3 be Q’s average eigenvalue, it is clear that the eigenvalues
ii of the traceless matrix Q = Q - £ 1 characterize the walk’s asymmetry. By definition :
so that tr Q2 gives the variance of Q’s eigenvalues, and
hence is a measure of how much a walk deviates from
spherical symmetry (for which case Ai = 1, so that Q = 0). Because
if the walk is prolate (i.e. A, >> A2 A3), then À1 - I
is positive while A2 - 7 and A3 A are negative, causing their product det Q to be positive. In con-
trast, an oblate walk (Al A2 >> A3) will have a
negative value for det Q. The sign of det Q classifies
a walk as oblate or prolate, while its magnitude is a
measure of how oblate or prolate the walk is.
These parametrizations of polymer shape are illus-
trated for random walks on a cubical lattice in
figures 2(a), (b), and (c). The Q tensors for these
N = 16 384 step walks were diagonalized numerically
and used to determine a set of principal axes. The
walks were then projected onto the three planes ortho- gonal to these principal directions. A prolate walk, an
oblate walk, and a walk which is approximately sphe-
rical are shown. In each case, we have also calculated
and
I-- Fig 2. - Random walks which are (a) prolate, (b) oblate, and (c) almost spherical. The walks have been projected
onto the faces of a cube oriented along the principal axes of Qit The figures result from unfolding three faces of this cube
sharing a common vertex. The rectangles have edges pro-
*
portional to the square roots of the eigenvalues of Qit The
numerical values of the shape parameters d o, and So defined
in the text are also shown.
which are just the quantities (1. 4), and (1.5) norma-
lized by appropriate powers of tr Q (the bounding inequalities are proven in Appendix A). It is evident from figure 2 that the quantities d o and So give good
numerical characterizations of both the asymmetry and shape of these walks.
Because the eigenvectors which diagonalize Q do
not appear explicitly in equations (1.4), it is straight-
forward to average these shape parameters over a thermodynamic ensemble of different walks. To obtain averages which are finite as the polymerization
index tends to infinity, one can normalize by dividing by appropriate powers of R or Rc;, in analogy with (1.6). We have found it convenient to look at the parti-
cular ratios
and
These universal amplitude ratios obey the exact inequalities (see Appendix A),
Note that A, which measures the degree of asymmetry, is bounded above by unity. The sign of S determines
whether walks are predominantly oblate (- 0.25
S 0) or prolate (0 S 2.0). Other amplitude ra- tios, such as tr 0’ >/ tr Q )2, can also be computed using the methods of this paper (see section 4). We have
been unable to find inequalities analogous to (1. 8) for
these cases, however. In all cases, our calculations are
restricted to the ratios of averages of polynomials of Q components. The averages of ratios such as
( tr 4’/(tr Q)2 >, which would also be quite interest- ing, are much harder to compute.
In the remainder of this paper we develop a forma-
lism for computing ratios like those in (1. 7). In sec-
tion 2 we will show how they can be calculated for the RW case, generalize the RW model to include self
avoidance, and develop a generating function which
yields shape ratios in terms of path integrals. In
section 3, we map these path integrals onto Green’s
functions in an 0(n) symmetric spin model in the
n - 0 limit. In section 4, we show that the shape ratios
are determined by asymptotic properties of these Green’s functions at their critical point. This deter-
mination will allow us to use the momentum space renormalization group to compute the ratios A and S to leading order in an expansion in B = 4 - d, and to prove them to be universal. In appendix A, we
derive the inequalities (1. 8). In appendix B we explicitly compute the universal ratio R 2/ RJ as a concrete
example of our method This ratio has been pre-
viously computed by a number of authors [11-13] and
so provides a useful check on the calculation. Finally,
in appendix C, we tabulate a universal amplitude ratio characterizing the shape of two-dimensional random walks.
2. The RW case and the SAW formulation
We first consider the problem of computing the shapes of noninteracting random walks. Choose the ensemble of paths to be that of Brownian motion, so
that a particular realization r(t) satisfies the stochastic differential equation
where Y(t) is a Gaussian random process of mean zero. Equation (2.1) is immediately solved by r(t) =
t
dt’ Y(t’). Averaging r(t) r(t’) yields the basic rela- tion
from which it is apparent that R 2 3
(3 i= i r;(T) x
ri(T) > = 3 DT. Similarly, one can show that Rj =
3 ( Qxx> = DT/2, so that R2jR¿ = 6. Because r is Gaussian, r is too. We can use Wick factorization in
conjunction with (2.2) to compute averages of pro- ducts of many components of r. Thus, we can explicitly
calculate averages of products of Qij’s for the non- interacting (RW) case. It is also worth noticing that
within the RW model different spatial components are
independent, so that some averages factor. For instance
and
We use the results of the RW computations described
above as a check on the combinatorics which enter in the SAW calculation described below. Results for the RW case apply to SAWs whenever the spatial dimen-
sion d is greater than four, which is the upper critical dimension of the theory.
As a next step, we augment the random walks by adding self avoidance. The fact that Y is Gaussian with variance D implies that the probabilistic weight of a given realization r, written P(r), is proportional to
exp -
dt V(t)12 D . Since dr(t)/dt = Y(t), the weight assigned to a particular walk r(t) isIf we now add the contact potential described in the
introduction, we get
where we have chosen to write the coupling constant as A/3 ! to simplify later expressions. If X[r] is a functional of the path r, then the ensemble average of X is given by
In the case of interest, we first sum over all paths of length T which start at the origin and end at R, and then integrate over possible endpoints. R. Symbolically,
where
f Ðr(t) denotes a path integral.
To evaluate the averages in equations (1. 7), we use a generating function. First note that the expression (1.2) for Q; can be rewritten as
Now consider the generating function
It is easy to check by differentiating and comparing with (2.6) that
where means alaqu. We will explicitly show how to compute g with (12 = (13 = 0, which suffices to eva-
luate (2. 8a). The generalizations needed to compute higher moments of Q are straightforward For this case of
(J 1 = Q and q 1 = q nonzero, we shall need the object
where ri denotes r(ti), and
In terms of A R T(q, cr), we have
3. Mapping onto a field theory.
Our strategy for computing Ai will be to exploit the analogy [1] between the polymer problem and 0(n) sym- metric field theory in the n - 0 limit. To implement this mapping, we have generalized techniques developed by Emery [14].
We start by simplifying the double time integrals in equation (2 . 9). Remembering that fq = eq-", we can write
the exponential involving this factor as an integral over a single complex variable 00,
where Xf is a normalization constant, the last line defines a weight factor ’Wf (00), and doo means
(d Re [0.]) (d Im [001). Upon defining h(x) = Jo T dt 6(x - r(t)), the term involving the 6-function can be written 0
We use x to denote a variable which runs over all of three dimensional space, in contrast to r(t), which is restricted to a particular polymer trajectory. After exponentiating this factor, we rewrite it in terms of a functional integral
over an auxiliary field g/(x),
where we have now introduced a normalization constant Xh, and a weight factor ’IDh[ tJ(x)].
We can now rewrite (2.9) in terms of a Feynman path integral
where O(r(t)) = CPo e - iq.r(t).
Applying the standard correspondence between path integrals and quantum mechanics [15] to the last part of (3.4), and using Emery’s analysis [14] we find that
~
where CTe f (T)] is the Laplace transform Jo d Tf(T) exp[- TO], and S(x) is an n component spin field
0
Using the weight functionw(oo) defined in (3. 1), we now integrate out 00. We have that
where S’(q) is the Fourier transform of S2(x). Recalling equations (2. 8-(2.11), and carrying out the final inte- gration over R indicated in (2 .11), we see that
where X > s == f ÐSX e-’/Z is a thermodynamic average using Hamiltonian
Let G(2,m)(Ql, q2 ; ki ... km) denote a connected Green’s function in this theory which has 2 external Sx legs
and mS2/2 insertions, but with the global momentum conserving 6 function 6(Zn qi + Ei kj) factored off. Then
(3.7) can be written more compactly as
where 0 is the operation of taking the limit n - 0 followed by an inverse Laplace transform,
One can similarly show that
and that
The same sort of manipulations lead to a formula for the mean square end-to-end distance of a polymer, namely,
4. Computation and universality of shape ratios.
In order to evaluate expressions like (3. 8), we need to
understand both what the n -+ 0 limit, and the inverse Laplace transform t-’ do to a theory of interacting spins. First, consider the role of n in the perturbative
evaluation of a given Green’s function. We graphically represent the (S2)Z interaction by figure 3a, and an S’(q)/2 operator insertion by figure 3b. Any graph
will pick up a factor of n for each of its closed solid
loops, such as that shown in figure 3c, because a solid loop corresponds to a sum over all n components of S.
Thus, to implement the n -+ 0 limit perturbatively,
one simply adds up all the graphs corresponding to a given Green’s function excluding those containing
solid loops [1]. We explictly work out an example of
this procedure in appendix B.
Fig. 3a. - The graph representing an (82)2 interaction
vertex. The dotted interaction line carries a factor of 1.
Fig. 3b. - The graph representing a momentum insertion.
Fig. 3c. - A graph with a closed solid loop. This loop car-
ries a factor of n and does not contribute in the n -+ 0 limit.
To compute the inverse Laplace transform, we
assume that the Green’s functions of a symmetric (82)2 theory have a simple singularity structure in the complex 0 plane, i.e., that the Green’s functions are
analytic functions of 0 except for a branch cut running
Fig. 4. - The complex 0 plane. The Green’s functions have
a branch cut from - oo to 0,. Accordingly, the C -’ integral
can be deformed from F, to T2.
from 6 = - cc to the critical point 0 = Oc (see Fig, 4).
Using complex analysis, we can deform the contour of integration in the usual Laplace inversion formula
c- 0- 1T G(9) = dge9T G(9) from r1 to r 2. Along r 2 r1
it is clear from the form of the integrand that in the limit of asymptotically large T the leading behaviour ofC;-!T G will be determined by G’s leading behaviour
as 0 -+ 0,,. We know from studies of critical pheno-
mena [16] that G N A(O - Or Accordingly, to
8-+8c
leading order in T
The factor of e8tT will always be cancelled by N, so that we finally can see that computing shape para- meters for long chain polymers reduces to computing
the critical amplitudes and exponents of (S’)’ theory’s
Green’s functions.
These amplitudes and exponents have been tho-
roughly studied using renormalization group tech-
niques [16]. We have found it convenient to use the formalism of renormalized perturbation theory to
compute the amplitudes in an e = 4 - d expansion.
(The exponents are well known). We first rescale the
spins in (3. 7) to give a theory in the standard form with
a propagator - V’ + Jl and Green’s functions
Here G:ntl) is a Green’s function in this standard
theory. We then notice that to leading order in the inverse cutoff ao [16]
where the Zs are nonuniversal renormalization cons-
tants, and GR(",’) is a Green’s function renormalized via minimal subtraction at an inverse length scale K, computed with an infinite cutoff (ao - 0). The scaling properties of GR(n,’) are well known. One finds [16] that
where u* is the fixed point coupling constant, y and v
are critical exponents, X and Y are nonuniversal
quantities and our units have been chosen to set K = 1.
We can now put (4 .1 )-(4 . 4) together with (3 . 7)-(3 .11 )
to get expressions for averages of Q’s. Our results are
tabulated in table I. Notice that f has been cancelled out by the normalization constant N, so that ratios of
Q averages will be universal if and only if the factors of x2mv cancel. This cancelation does occur for the
cases of interest, i.e., for R 21j;tj, tr (()2 >/ (tr Q)2 ) and ( det (0) )j (tr Q)3 ).
Table I gives a given Q moment as a product of a
nonuniversal prefactor which cancels from ratios, a
universal prefactor which only depends on critical exponents, and a Green’s function evaluated at the
matching point u = u*, t = 1. The exponents are known in an E expansion : v = 1/2 + sl 16 and
y = 1 + e/8. Because u* = 3 e/4 + O( e2), the Green’s functions can be similarly expanded To 0(e), we only
need to consider the graphs shown in figure 5. For
Fig. 5. - The graphs which contribute to lim G(2,1) to 0(s).
11-0
To get all contributions, I momentum insertions must be added to the dark lines in all possible ways.
each graph we sum over all possible ways of putting
on the appropriate number of momentum insertions.
When there are more then two insertions the number of independent terms becomes large, as does the algebraic complexity involved in calculating the deri-
vatives with respect to the inserted momenta. To keep
track of all of this we used the symbolic manipulation package SMP [17] to perform the actual computation.
The values of the relevant Green’s functions at the
matching point are listed in table II. In Appendix B
we compute Rg / R.J as an explicit example of how these
calculations work.
From all of the above, we get averages of polyno-
mials of Qij’s in an e expansion. The isotropy of the original theory implies, in three dimensions, that
tr Q > = 3 Q.. >, that tr 02 > = 2( (Q..)2 > +
3 ( (QXy)2 ) - ( Qxx Qyy ». and similar relations for the other averages needed to calculate A and S. We have chosen to proceed by first using these relations
(rather than their 4 - E dimensional counterparts)
to express A and S in terms of averages like ( Qxx ), ( Qxx 6yy X etc. These basic averages are then eva-
luated in an E-expansion. Our results for shape ratios
Table I. - Expressions for Q moments in terms of critical exponents y and v, and Green’s functions evaluated at
the matching point. Here * means evaluated at t = 1 and u = u*l An n - 0 limit is tacitly assumed
