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Submitted on 1 Jan 1989
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Correlations among interpenetrating polymer coils : the probing of a fractal
Brunhilde Kruger, Lothar Schäfer, Artur Baumgärtner
To cite this version:
Brunhilde Kruger, Lothar Schäfer, Artur Baumgärtner. Correlations among interpenetrating polymer coils : the probing of a fractal. Journal de Physique, 1989, 50 (21), pp.3191-3222.
�10.1051/jphys:0198900500210319100�. �jpa-00211137�
Correlations among interpenetrating polymer coils : the probing of a fractal
Brunhilde Kruger (1), Lothar Schäfer (1) and Artur Baumgärtner (2) (1) Fachbereich Physik der Universität Essen, D 4300 Essen, F.R.G.
(2) Institut für Festkörperphysik der Kernforschungsanlage Jülich, D 5170 Jülich, F.R.G.
(Reçu le 24 mars 1989, accepté sous forme définitive le 4 juillet 1989)
Résumé. 2014 Le groupe de renormalisation et le développement en 03B5 sont utilisés pour calculer la distribution de densité d’une chaîne polymère d’indice de polymérisation N2 entremêlée à une
chaîne d’indice de polymérisation N1. La distance entre chaînes est contrôlée en fixant la distance des centres de gravité ou en fixant la position du centre de gravité de la chaîne N2 par rapport au segment central de la chaîne N1. Les énergies libres du système tendent dans les deux cas vers des fonctions universelles finies lorsque N1~ ~, N2/N1 fixé, contrairement à la prédiction de
modèles de champs moyens qui conduisent à un comportement de type « sphère dure ». Les
rayons de giration et la distribution de densité de la chaîne N2 sont également calculés. Lorsque N2 ~ N1 les résultats dépendent fortement de la position relative de la chaîne « sonde »
N2 par rapport à la chaîne N1. Ceci met en évidence d’importantes inhomogénéités dans la densité de la chaîne N1 qui caractérisent un objet fractal. Des simulations numériques de Monte Carlo confirment nos résultats analytiques.
Abstract. 2014 We use renormalization group and 03B5-expansion to calculate the density distribution in a polymer coil of polymerisation index N2, interpenetrating a coil of polymerisation index N1. The distance of the coils is controlled by fixing the center of mass distance or by fixing the
center of mass of coil N2 with respect to the central segment of coil N1. The effective free energies
of the two-coil system in both situations are universal functions which tend to some finite limit as
N1 ~~, N2/N1 fixed. This is in sharp contrast to the prediction of mean field models which lead to hard sphere behaviour. Besides free energies we calculate the radius of gyration and the density
distribution of coil N2. For N2 ~ N1 all our results show a pronounced dependence on the position
of the probe coil N2 relative to coil N1, thus proving the strong inhomogeneity of the density of
coil N1 characteristic for a fractal object. We carry through off-lattice Monte Carlo simulations which nicely confirm our analytical results.
Classification
Physics Abstracts
36.20 - 61.40K - 64.60
1. Introduction.
Long macromolecules in dilute solution are among the most intensively studied fractal objects
found in nature. Due to some special simplification of the renormalization group as applied to polymers we can examine the structure of polymeric coils in much detail. For instance, the
overall shape of a coil is a universal property, and recently some ratios measuring this shape
have been calculated [1]. Correlations among specific segments of the chain can also be reduced to universal functions, which have found much interest [2]. Besides being an
excellent example for the power of the renormalization group, polymeric coils also are
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500210319100
especially adapted to computer experiments, and many theoretical results have been verified
by Monte Carlo simulations [3]. As a rule, the quantitative agreement between renormalized
perturbation theory and simulation experiments is surprisingly good.
In a previous contribution [4] we have analyzed correlations involving the center of mass of
a polymeric coil. We found that such correlations obey scaling and can be calculated by
renormalized perturbation theory, even though the center of mass has no counterpart in normal field theory. Specifically, we presented results for the monomer density distribution within the coil.
In the present paper we aim at more detailed information on the internal structure. The fractal structure of a coil must show up in a complicated instantaneous distribution of regions
of high or low density. In the equilibrium density distribution this structure is completely
washed out. Information can be gained by calculating many-point correlation functions of a
single coil. Another approach, which we use here, consists in analyzing correlations between two interpenetrating coils of polymerization index (« chain length ») Ni or N2, respectively.
For N1 > N2 the small coil N2 can be viewed as probe particle testing the monomer
distribution of the large coil N1.
We then expect that coil N2 can take qualitatively different positions, sitting either in a hole
or on top of a strand of the first coil. Both the effective interaction of the coils and the size of the probe coil will depend on its position.
Specifically, we calculate to the order of one loop two different three-point correlation functions. They give the density distribution in a test coil N2 fixed at different places in coil Nl. In the first case we control the center of mass distance R (2) - R (1) In the second case we
fix R(2 ) cm relative to the central segment r (1) of chain N1. The difference between these two
2 N1
functions is a measure of the inhomogeneity of the density distribution of coil N1. Based on
the results for the three point correlation functions, we give a detailed discussion of the effective free energy of the interacting coils as function of their size and distance, and we
calculate their radius and their segment density distribution for Rcm(1) = Rcm(2) or R(2) = (1
These quantities have not been considered before in the frame of the renormalization group.
Only in ref. [5] there have been given results for an effective free energy as function of the distance of the end points of the chains. This quantity, however, is distorted by the correlation hole surrounding each chain end.
To get independent information on these correlation functions we carry through new
Monte Carlo simulations. The two sources of information give a consistent picture.
The organization of the present paper is as follows. In section 2 we sketch the calculation of the three point correlation functions defined above. The full analytical results are lengthy,
and the one loop corrections are collected in appendices. In section 3 we evaluate the effective free energies, analyzing some limiting cases and comparing our full numerical results both to Monte Carlo simulations available in the literature [6, 7, 8] and to new calculations.
Section 4 is devoted to the radius of gyration and to the density distribution of the test coil.
Also for these quantities new Monte Carlo simulations have been carried through. In
section 5 we summarize the insight into the structure of a polymer coil gained by the present and by previous work. Some formulae relevant for sections 3 and 4 are collected in
appendices A and B. Appendix C gives a Flory-type calculation of the radius of interpenetrat- ing chains.
2. General expressions.
2.1 MODEL AND RENORMALIZATION. - We use a model of self-interacting Gaussian chains.
The configuration of the a th chain is fixed by segment-coordinates r i (a) = 0, ..., N a, the chain
length Na being proportional to the molecular weight. The Hamiltonian of the ath chain is taken as
The microscopic length scale f governs the mean size of a Gaussian segment, and f3 c > 0 is the dimensionless excluded volume parameter. For a two chain system we have to add the interchain interactions
To avoid divergencies due to the 5-function interaction we in the cluster expansion of these products omit all terms in which any segment interacts more than once.
We are interested in correlation functions involving center of mass (c.m.) density operators
and segment density operators
In a previous paper [4] we have analyzed the density distribution of segment j within an
isolated chain
Here
is the single-chain partition function and Q denotes the volume of the system. In the present paper we deal with the cumulants
which give the correction to the density distribution of chain N2 due to the presence of chain
N1. They differ in the way the position of the first chain is fixed.
For N > 1 and length scales r > f the polymer coils are expected to exhibit universal
properties. However, the normal cluster expansion in powers of f3 e in that limit diverges term by term. Renormalization consists in a reorganization of the cluster expansion which absorbs all nonuniversal features into renormalization factors multiplying the quantities of the
unrenormalized theory and leads to a formulation in which the limit of long chains can be
taken. The validity and technical details of the renormalization program have been
extensively discussed in the literature [11, 12, 13]. Specific questions concerning the
renormalization of the center of mass vertex have been addressed in our previous work [4].
We therefore here only recall some of the essential relations.
We replace the unrenormalized quantities /3e, Na by renormalized counterparts g, NaR
where K -1 1 is the length scale of the renormalized theory which plays the role of a free parameter. Below it will be identified with the characteristic scale of the phenomena under
consideration.
The evaluation of perturbation theory simplifies if we replace summations over discrete segments by integration along the chain, thus effectively replacing the discrete chain by a
continuous space curve. This leads to the occurrence in perturbation theory of poles in
e = 4 - d which replace the microstructure dependent contributions in the discrete chain model and have to be absorbed into the renormalization factors. Several equivalent prescriptions for choosing these factors are available. We use the massless renormalization scheme of reference [13] which yields
(In comparison to Ref. [13] we have absorbed a factor of 1/3 into our definition of g.) To
eliminate all e-poles occurring in the evaluation of a physical quantity we in general have to multiply this quantity by appropriate renormalization factors assoziated with operators like
p (a)(r). In the present case these factors are easily deduced from the limit ql = q2 = 0 of
equations (2.10), (2.11)
The second virial coefficient A2(Nl, Nz) is known to renormalize according to
where the dimensionless renormalized virial coefficient A2R is expressed totally in terms of
renormalized variables. Equations (2.15), (2.16) yield
where
and PR is the dimensionless renormalized cumulant in which all e-poles and thus all nonuniversal effects are absent. A factor (4 7r )d/2 has been extracted for convenience. A
corresponding equation holds for Qc.
Equation (2.12) shows that the renormalized coupling constant g for a given unrenor-
malized theory depends on K. This dependence is governed by the renormalization group
equations. We will see below, that the choice of K dependents on Na, and in the limit of interest, Na -. oo, it can be shown that g tends to some finite fixed point value [13]
ZIZ2 via its dependence on g also depends on K, and close to g = g * the renormalization group yields [13]
where
The estimate for d = 3 is based on higher order calculations [14].
Since g * is of order e = 4 - d the renormalized perturbation theory formally transforms
into an expansion in powers of e. We will evaluate all our results for g = g *, i.e. in the excluded volume limit of very long chains, setting E = 1, i.e. d = 3.
2.2 CALCULATION OF PR (q1, q2, N1 R, N2R). Equation (2.10) for P, can be written as
Perturbation theory including a c.m. vertex has been formulated in reference [4], section 2.3.
To order of one loop the diagrams for Ge are given in figure 1. We recall that a c.m. vertex
(heavy dot in Fig. 1) distributes the c.m. momentum q onto all the special points (vertices or
end points) of its chain, in proportion to the length of the subchains between neighbouring special points. If a special point connects two subchains of lengths ne, nr respectively, then the
additional c. m. momentum flowing into this point has the value q (ne + nr )/ (2 N ), where N is
the total length of the polymer considered. Figure la illustrates this rule. Besides being the
source of additional momentum flow, the c.m. vertex also modifies the propagators. Consider
a piece of length n and momentum k between two special points. The corresponding propagator takes the form
Fig. 1. - Diagrams contributing to Gc(q1’ q2, N1,N2). In the tree diagram a) we indicate segment variables and internal momentum flow. Here q = (q2 - ql )/2 N2.
where q again denotes the c.m. momentum of the chain of length N. In the absence of the
c. m. vertex (i. e. for q - 0), this expression reduces to the well known propagator
of polymer theory in the chain length representation.
With these rules the tree graph of figure la yields
where
In the continous chain limit we find
where
Equation (2.23) cannot be simplified further analytically.
With the tree approximation giving rise to nontrivial expressions it is obvious that the one
loop terms are fairly complicated. We therefore do not present the full unrenormalized
expressions. Rather we collect the final renormalized result in appendix Al. Here we only
illustrate the essential steps of calculation, taking diagram 1 d as example.
As in standard field theory the diagrams in four dimension (d = 4) show both quadratic
and logarithmic ultra-violett divergencies which here are due to self-interaction of short pieces
of the chains. The leading divergencies are cancelled by the normalization factor
n 2/ «4 Irf2)d/2 Z(Nl) Z(N2» (Eq. (2.20)), the resulting expressions being finite in
d = 4 - e 4 dimensions. With this subtraction, diagram 1 d yields
the contribution - 1 in equation (2.24) being due to the normalization. For e ---> 0 the power
(x 1 - Xl)- 2 + E /2 leads to a logarithmic singularity which is singled out by an additional
subtraction of the exponential
In the so constructed unrenormalized expansion we now replace l3e, Na by g, NaR according
to equations (2.12)-(2.14), and we reorder according to powers’of g. The tree graph (2.23) yields counterterms, which cancel the 11E poles in the one loop contribution. In the final renormalized expansion diagram 1 d, for instance, contributes
where y E" = 0.577... denotes Euler’s number. J3 stands for Îd evaluated for e = 0 in terms of renormalized variables. The remainder of equation (2.27) is due to the 8-expansion of
Jd - fd (Eq. (2.25)), combined with the counterterm arising from the tree graph.
The resulting expression for PR is not yet in final form. Logarithmic chain length dependence (see the last term in Eq. (2.27)) signals the occurrence of nontrivial power laws.
For standard problems involving only a single chain length N1 = N2 = N these power laws
are easily found by an appropriate choice of K. Fixing K such that
we find from equations (2.13), (2.20)
The condition (2.28) can easily be interpreted physically. It implies that the renormalized
length scale K -1 1 is of the order of the radius of gyration R. - lNv of the chain. With this choice all ln NR terms cancel. Momentum q - qf N JI ’" qRg is measured on the scale of
Rg-1, and Pc(q1, q2, N,N) (Eq. (2.17» contains an overall factor of the volume of a chain :
However, this simple argument is not sufficient for a problem involving two independent
chain lengths. It only establishes the behaviour of the scaling functions under a multiplication
of both chain lengths by a common factor, the scaling functions in addition depending on the
renormalization group invariant N1lN2~ (Rg1/R92 )1/v. No general method is known for
analyzing the scaling functions in the limit N1/N2 = N1R/N2R ->{0} , {0~ } but for the present
problem plausible arguments lead to a reasonable form in which all ln NR-terms cancel. We
first note, that all momenta should be measured on the scale of an appropriate radius of gyration. Restricting ourselves to the fixed point g = g * we therefore replace variables of the
form q . qNaR by q . qN 2" NaRE/8 and we expand the last factor in powers of e. Furthermore
equation (2.15) shows that for ql = q2 = 0, - Pc/N2 reduces to the second virial coefficient
A2. According to a suggestion of De Gennes, A2(N1, N2) for N1/N2 -. 00 should behave as
(see Ref. [5] for details of the argument). This form assumes that the small chain
Nz experiences the large chain Ni as composed of
N1/N2
N blobs of size Rg2. Thus N2and in view of equation (2.29)
follows.
These considerations suggest the ansatz
We find that in PR all terms of form In NR drop out. In tree approximation it takes the form
The one loop corrections are given in appendix Al. This result forms the basis of the further discussion presented in sections 3 and 4.
2.3 CALCULATION OF QR(q1, q2, N1R, N2 R ) . We now consider a situation in which we can
force the first chain to pass right through the second coil.
With the diagrammatic expression for (4 7rf2)d/2 Z(N1) Z (NZ ) f2 - 2 6c given in figure 2, the evaluation closely follows the evaluation of Pc(Ql, Q2, NI, N2) sketched in the previous
section. In particular normalization and renormalization proceed as before. Furthermore
Qc(QI = 0, q2 = 0, N1, N2 ) and P c(QI = 0, q2 = 0, N1, N2 ) coincide, leading us to introduce
QR in analogy to PR (Eq. (2.30)) :
Together with the replacement lj2 N aR ---> q2 N2vaR this again eliminates the explicit contri-
butions In N aRe The tree-approximation reads
The one loop corrections are collected in appendix Bl.
Fig. 2. - Diagrams contributing to Qc(q1, q2, N1, N2).
3. Effective free energies among interpenetrating coils.
3.1 THE INTERMOLECULAR PAIR POTENTIAL, ANALYTICAL RESULTS. - The effective
potential U(R) acting among two polymer coils seperated by the c. m. distance R is defined as
The normalization guarantees the asymptotic behaviour U(R) --> 0 for R --> 00. The pair
correlation function
is related to the second virial coefficient .
A comparison of equations (2.10), (3.1), (3.2) yields
consistent with the sum rule (2.15). With equation (2.30) we find the renormalized expression
It is important to note that A2(R, N1, N2) is an unrenormalized quantity which is directly observable, - at least in computer simulations. Still it can be expressed completely in terms
of renormalized quantities, without any nonuniversal prefactor. Therefore A2(R, N1, N2)
and thus U(R) are universal scaling functions which can depend only on
and on RI Rt2, where R12 is any appropriate combination of the radii of gyration of the two
coils. (For our choice see Eq. (3.16).) Specifically, for Ni = N2 = N the intermolecular pair potential U(R) becomes a finite universal function of RIRg(N).
In the older literature the polymer coils often are visualized as impenetrable spheres. This
idea - going back to Flory - is based on a mean field approximation for U(R) :
Here p (r, N ) denotes the equilibrium segment density in a chain of length N at point r,
measured from the center of mass. Equation (3.8) yields the estimate
where v(...) is the overlap volume of the two chains measured in units of the volume of chain
N2. In the excluded volume limit
U (R ) therefore is predicted to diverge for all v( ... ) >0. the chains behave as impenetrable spheres. This is a clear contradiction to the renormalization group results.
A less severe, but significant, contradiction is found for A2(N1, N2). The hard sphere
model predicts
While the overall scaling ~ N1vd of this expression is correct, the N-dependence is not
consistent with the results of renormalized perturbation theory. Specifically, in the limit N -> 0 the e-expansion yields
which can not be interpreted as 8-expansion of equation (3.10). Rather it is consistent with De Gennes suggestion
As has been pointed out by Grosberg et al. [7] the mean field ansatz (3.8) seriously
overestimates the contact probability of the two coils. Using a contact probability resulting
from scaling arguments they find that U(R) stays finite in the excluded volume limit, in complete agreement with our exact results. Due to the fractal nature, polymer coils thus differ
qualitatively from hard spheres.
We now turn to the results of renormalized perturbation theory. Equations (3.2), (3.5), (2.31) yield in tree approximation
We note in passing that in this approximation the q integral (Eq. (3.5)) indeed reduces to the convolution of segment densities as assumed in the mean field approach, with the important
modification that the densities occur in renormalized form. For R = 0 and in strict e-
expansion (Eq. (3.11)) can be evaluated analytically
Special values :
As expected, U(O) vanishes if the molecular weight of the two chains is of different order of
magnitude. The small chain can slip through the holes of the large chain without feeling any essential interaction.
Pursuing the idea that a small chain N2 visualizes a large chain Ni as composed of blobs of size Rg2, we may suppose that for N - 0 the radial dependence of U(R ) is determined by the density of blobs Rg (N2 ) in the large chain, which in turn is proportional to the segment density itself. This is indeed supported by the tree approximation (3.11), which for N 1 reduces to
The integral is the lowest order approximation to the renormalized segment density in chain Nl. (This can be checked easily from the results of Ref. [4].) An analysis based on the one loop terms yields
Here P cm (RI Rg) is the density distribution in an isolated chain of radius Rg, normalized such that
For an analysis of this function we refer to reference [4]. The prefactor Ñvd -1 is proportional
to the average density of blobs Rg (N2) in chain N1.
We now tum to the numerical evaluation of our full result. We measure R2 on the scale of the weight-averaged radius of gyration of the two chains
thus preserving the symmetry with respect to both chains. In the limits N - 0 or
N->oo this scale reduces to the size of the larger chain, in conformity with (3.14).
R12 being the relevant length scale we then fix K so that KR12‘ 1 to zero loôp order. With
We rewrite
and we expand out the correction ERl in the scaling functions.
For the exponents in the prefactors of equation (3.5) we take the values in three dimensions : v = 0.588, d = 3. There remains the question whether we should carry through
the s-expansion of the scaling function PR in R-space or in q-space. This makes some effect, as
can be seen, for instance, from a calculation of U(R = 0, N =1 ). Evaluating the
d = 3 dimensional Fourier transform of the q-space e-expansion numerically we find
a result, which differs from the strict e-expansion in R-space (3.13) : U(0) = 1.53 E +....
This ambiguity is inherent in the method. If not stated explicitely otherwise, we in the sequel
will use the e-expansion in q-space, carrying through the Fourier transform in three dimensions. This is consistent with our analysis of the prefactors in equations (2.30), (3.5), being based on the behaviour for small q. (Note, that in equation (3.12) we also have expanded out the prefactors.)
In figure 3 we plot U(0 ) as function of Ñ together with the corresponding quantity for coil N2 sitting on top of the central segment of coil N1. A detailed discussion is defered to subsection 3.3. Here we only want to stress that U(0 ) is finite, in contrast to the effective
