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Segment distribution about the center-of-mass in an isolated polymer coil. A renormalization group study
Lothar Schäfer, Brunhilde Krüger
To cite this version:
Lothar Schäfer, Brunhilde Krüger. Segment distribution about the center-of-mass in an isolated polymer coil. A renormalization group study. Journal de Physique, 1988, 49 (5), pp.749-758.
�10.1051/jphys:01988004905074900�. �jpa-00210751�
749
Segment distribution about the center-of-mass in an isolated polymer
coil. A renormalization group study
Lothar Schäfer and Brunhilde Krüger
Fachbereich Physik der Universität Essen, D 4300 Essen, F.R.G.
(Requ le 28 dgcembre 1987, accept6 le 3 fgvrier 1988)
Résumé. 2014 Nous analysons la distribution de segments d’un polymère isolé n’interagissant qu’avec lui-même,
au voisinage du centre de masse. Nous trouvons que la renormalisation de l’opérateur du centre de masse ne
fait pas intervenir de renormalisations nouvelles. Nous calculons la distribution de densité des segments à l’ordre d’une boucle. L’interaction diminue la densité au centre de la chaîne. Cet effet, par ailleurs attendu, apparaît comme la conséquence d’un mécanisme surprenant : l’interaction diminue les fluctuations de segments individuels qui les éloigneraient de leur position moyenne.
Abstract. 2014 We analyse the segment density distribution about the center of mass in an isolated self-
interacting polymer chain. We find that the center-of-mass operator is renormalizable without introducing new
renormalisation factors. The segment density distribution is calculated to one loop order. The interaction reduces the density in the center of the chain. This effect, which was to be expected, is due to a somewhat surprising mechamism : the interaction suppresses fluctuations of individual segments away from their average
position.
J. Phys. France 49 (1988) 749-758 MAI 1988,
Classification
Physics Abstracts
36.20 - 61.40K - 64.60
1. Introduction.
Long polymer molecules in dilute solution form coils which are a well known example of fractal objects. The properties of fractals have found great interest recently, and for polymeric coils the typical shape of the coil [1] or the correlations among
specific segments of the chain [2] have been studied both numerically and by the renormalization group.
In the present paper we are concerned with the segment density distribution within the coil. In a
simplistic approach one might calculate the density
as function of the distance from the segment in the middle of the chain. Indeed, such results could be taken from previous work [2]. However, this distri- bution is distorted by the repulsive interaction between the central monomer and the rest of the chain. To avoid this correlation hole effect we here
calculate the density distribution as a function of the distance from the center of mass. This clearly is the
standard definition of the density profile, but at first sight it gives rise to problems - with the renormali- zation of the theory. The center-of-mass of a polymer
molecule is not equivalent to a renormalizable operator of Landau-Ginzburg-Wilson field theory.
Using the mapping between polymer theory and zero-component LGW theory we however can prove that arbitrary moments of center of mass correlations
are renormalizable without need for new renormali-
zation factors. Therefore center of mass correlations
obey scaling laws similar to the segment density
correlation functions, and the scaling functions can
be calculated within renormalized perturbation theory.
The organization of our paper is as follows. In section 2 we demonstrate the renormalizability, and
we calculate to one loop order the density distri-
bution within an isolated polymer coil. In section 3
we evaluate our results both in momentum space and in real space, first concentrating on the distri- bution of the total segment density. As is to be expected this distribution for the interacting chain is fairly flat in the center and at the surface becomes somewhat steeper. We then discuss the distribution of a specific segment about the center of mass, which
turns out to differ qualitatively from Gaussian statis-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905074900
tics in a surprising way. Section 4 summarizes our
conclusions and gives a short outlook on work
currently under way.
2. Basic formalism.
2.1 THE MODEL. - We consider a single chain
constructed for N Gaussian segments of mean size f.
The Hamiltonian is written as
where f3 c ;?; 0 is the dimensionless excluded volume parameter. We will be concerned with correlation functions involving the segment density operator
and the operator of the center of mass position
Specifically we deal with the probability to find a given segment n at a distance r from the center of
mass. The Fourier transform of the (not normalized)
distribution reads
where a denotes the volume of the system. For all n,
Gem (q == 0, N , n) is proportional to the partition function 3
so that the normalized distribution of the total
density takes the form
2.2 RENORMALIZATION. - Universal properties of
the polymer coils by definition are independent of
the microscopic structure of the chain. However,
normal perturbation theory in powers of Be yields
results which strongly depend on the microstructure,
which for the present model takes the form of discrete Gaussian segments. To extract universal properties we therefore reorganize perturbation theory so as to eliminate the nonuniversal features.
Technically this is carried through by multiplying parameters or operators of the model by renormali-
zation factors which take the form of power series in the renormalized coupling constant g, adjusted such
as to absorb the microstructure effects.
The renormalizability of polymer theory follows
from the observation [3] that a Laplace transform
with respect to chain length maps the polymer model
onto a special Landau-Ginzburg-Wilson (LGW) theory [4] in the formal limit of a zero component spin field. This theory is known to be renormalizable and we thus conclude that polymer theory is renor-
malized using the renormalization factors of zero
component LGW theory. Specifically, the excluded volume constant, which is proportional to the
(S2)2-coupling of LGW theory, is related to the renormalized coupling g by
where K is the momentum scale of the renormalized
theory. The segment density operator is mapped [5]
on the S2 -operator of LGW theory. It therefore is renormalized multiplicatively according to
The chain length obeys
All chain length variables renormalize in the same
way. Thus n/N, where n denotes an interiour
segment of the chain, is invariant under renormali-
zation.
The renormalization factors Z, Z2, Zn are chosen
as power series in g in such a way that they absorb all
microstructure effects. Different prescriptions are available, but for the present problem we must use a
scheme where the Z-factors are independent of the
mass m 0 2 of field theory. In mapping the polymer
model on LGW theory mo is introduced as Laplace
variable conjugate to N. A mass-dependent renor-
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malization scheme therefore mixes up renormali- zation with the inverse Laplace transform which
gives back the renormalized form of polymer theory [6]. We therefore use the massless renormalization of reference [4]. (Note, that our definition of g
differs from that given in Sect. IX of Ref. [4] by a
trivial factor of 3.)
The operator locating the center of mass
has no immediate counterpart in LGW theory.
Formally writing
we observe that the factor 1/N cannot be mapped
on a local operator of LGW theory. (N is related to an expection value rather than to a local operator.) However, since p (r ) is multiplicatively renormaliz- able any power of Rcm is renormalized according to
1 r B k
(Compare Eqs. (2.8), (2.9)). All renormalization factors drop out. This implies that all moments of
center of mass correlations are renormalizable.
p m (r) does not introduce any renormalization factor at all.
This discussion yields simple relations between unrenormalized and renormalized density distribu-
tions
Note that the renormalization factor of the segment density operator is cancelled by the normalization.
In deriving equation (2.14) we have used the ap-
proximation
valid for large N.
Nonlinear scaling laws follow by standard argu- ments [4]. In the renormalized theory the momentum
scale K is a free parameter, and a change of
À = K f can be compensated for by appropriate changes of NR and g. The functions N R (À ), g (A ) are calculated from differential equations ex- pressing the dilatation symmetry of the system. In a
two loop approximation one finds
Here v and w denote critical exponents.
9 * = E+0 (E2) is 4 the fixed point coupling constant
and go is some unknown, but regular, function of temperature and chemistry of the solution. It is
independent of N. We impose the condition
which fixes g = g (A ) or A as functions of go and N.
Condition (2.17) eliminates NR, leaving us with the scaling laws
The scaling functions pcm. Pc. depend on the scaling
variables n/N, g = g(A (N, go)), and
pcm and Pcm can be calculated perturbatively.
2.3 UNRENORMALIZED PERTURBATION THEORY.
We expand the correlation function G cm (q, N, n ) (Eq. (2.4)) in powers of f3e. Individual terms of the
expansion can be represented by diagrams (Fig. 1) in
Fig. 1. - Zero loop (a) and one loop (b-d) contributions to Gem (q, N, n ).
which the polymer is represented by a full line and
interacting segments are connected by a dotted line.
A wiggly arrow represents the segment density operator. The center of mass density operator is represented by an arrow attached to a heavy dot,
which distributes the center of mass momentum to all the special points (interacting segments, chain ends, or density insertions) of the diagram. To avoid
short range divergencies we impose the constraint that any part of the chain between two special points
consists out of at least one segment.
We now consider a part of the chain between two special points n and n’. The integration over the
coordinates rj, n j n’, yields
where
Fig. 2. - (a) Part of a diagram, relevant for the interac- tion among segments n, n’ ; (b) Momentum flow in the
zero loop diagram.
We use the notation f k = ddk (2 _X)-d . The propa-
gator Go k, n, q is symmetric under the oper- ation k - - k which corresponds to a reflection of the chain. In the exponent we have omitted terms of relative order lln. Equation (2.23) differs from the
propagator normally encountered in polymer theory by the presence of the term - nq2 N which is due
to the momentum nqln flowing into the considered subchain in order to localize the center of mass of the total chain.
We next consider a vertex connecting segments n and n’ (Fig. 2a) :)
Collecting all factors containing ’n we find
Here equation (2.21) contributes both the factor
involving q/N and the factor (4 7Tf2 )d/2. The terms
involving k> or k result from the Fourier represen- tation (2.22). The result (2.25) shows that the additional momentum q (n> - n, )/ (2 N ) flows into
the interacting segment in order to localize the center of mass. Momentum conservation at the
density insertion is expressed by the same 5-function (2.25) with k’ replaced by the external momentum q.
Integration over the endpoints of the chain fixes the momentum of the first or the last propagator to
k = nq/(2 N) or k = - (N - n ) q/ (2 N ), respect- ively. The contributions q/2 N for all special points sum up to the total center of mass momentum.
All prefactors (4 7Tf2 )d/2 cancell.
With these rules the zero loop contribution
(Fig. la) to Gcm(q,N,n) yields
Figure 2b shows the momentum flow and demon- strates that momentum conservation is obeyed. The
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result (2.26), which is correct for a noninteracting chain, is of course well known [7].
To show the structure of the one loop calculation
we evaluate diagram lb. We find the contribution
This result de ends strongly on the microstructure, i.e. on the discreteness of the chain. For a continous
chain, E --+ dnl dn2, the factor (n2 - nl )- dl2 for d >_ 2 yields a divergent result. The microstructure
ni n2
dependence is reduced by normalization of the density distributions. We find, using an obvious notation
where
Equation 2.27 yields
where tj = njIN ; t = n/N. In this expression we for all d 4 can take the continuum limit
The other diagrams yield the contributions
It is easily checked that Jc (Q, t ) is finite for all d 6.
2.4 RENORMALIZED PERTURBATION THEORY. -
We now use equations (2.7), (2.9) to replace f3 e and N by renormalized quantities. Expressions
for the renormalization factors adequate for our
calculation are [4]
We use the standard notation E = 4 - d.
Equations (2.9), (2.33) yield
where the scaling variable q = q/ K (Eq. 2.20)
occurs naturally. Substituting these results into equa- tion (2.28) and expanding consistently to first order
in g we find
We have used equation (2.17) : NR = 1. The 1 / £-poles cancell against contributions of Jb or Ja. To show this we expand the integrals in powers of E, a procedure consistent with the g-expansion since g -- g * -- E. This yields
where le (q2, t) = Ie (q2, t), evaluated for d = 4 and
Here yEu = 0.5772... denotes Euler’s constant. The
integrals cannot be simplified further. Equations (2.37), (2.38), (2.32) constitute our basic result.
3. Discussion.
3.1 DISTRIBUTION OF THE TOTAL SEGMENT DEN- SITY. - Equations (2.14) and (2.37) yield
where we used the fact that diagrams Ib or Id yield
the same contribution to P em (q, N, f3 e). For q - 0
this result can be evaluated analytically :
The coefficient of q2 yields the well known result for the radius of gyration.
From A and R we can build the critical ratio
It measures the tendency of the system to create a well defined surface. This is seen by comparing the
result for a Gaussian coil
to the result for a sharply bounded sphere
The sharpness of the surface shows up in a smaller value of i’l. In the present case we find from equations (3.3), (3.4)
where we replaced g by its fixed point value : 9 = e /4 and kept the full dimensionality d in the prefactor. iq is decreased compared to a swollen
Gaussian coil, which shows that the region of space
occupied by a selfinteracting coil is better defined than for a noninteracting chain.
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Evaluating equation (3.7) for c = 1 we find
31 - 0.85, a result very close to a == 0.82 extracted from Monte Carlo data [8]. This agreement, how-
ever, is somewhat fortuitous since both R and A show large first order corrections. Indeed, evaluat- ing ’ji directly for d = 3 with R, A taken from
equations (3.3), (3.4) evaluated for E = 1 we find R ~ 0.27 which shows that a first order calculation
only gives an order of magnitude estimate of this ratio. For a comparison we note the values for d = 3 for a Gaussian coil or a sphere, respectively : AGauss = 1 , Rsphere = 0.19. So the essential statement is that A for an interacting coil is found somewhere between the Gaussian coil and the uniform sphere
values.
For large moments q - o0
where a = 1/12 in the zero-loop approximation. For
the one-loop contribution the leading behaviour is of the same form (3.8), but with a = 1/48. This shows that perturbation theory is likely to break down in the limit of large q, the first order term dominating
the zero order. However, the absence of a leading
term ~ In q at least shows that for small distances r- 0 there is no power law behaviour Pc. (r) - ra, 0 a = 0 (ê), characteristic of the correlation hole. Rather P em (r), r -+ 0, smoothely approaches
some non-zero limit.
To calculate the spatial distribution of the segment density we have inverted the Fourier transform in three dimensions
taking E =1 in P,.,, (q, N ). Figure 3 compares the result in the excluded volume limit g = g * to the segment density of a noninteracting three-dimen- sional chain (g == 0). Since r is plotted on scale Rg, the curve g = 0 in fact represents the swollen Gaussian approximation, which has been widely
discussed in the literature. As was to be expected we
find that the interaction diminishes the density in the
center of the coil, shifting it to r ~ Rg, where a slight
increase of the segment density is observed. In the absence of a correlation hole, the density stays finite for r --+ 0, however. For rlrg > 2 the density for the
interacting coil decreases faster than expected from
the swollen Gaussian approximation. This is shown very clearly in figure 4, where we plotted In Pcm as
function of (r / Rg)2. In this figure we also included Monte Carlo data taken from the work of Olaj and
Zifferer [8]. In the logarithmic plot the difference between our calculation and the swollen Gaussian
approximation is not very large for (r / Rg)2 $ 4, still
our result gives a significantly better fit to the Monte
Fig. 3. - Pcm = (4 7T )"2 Rg Pcm (r, N ) plotted as function
of r/Rg, The full line is the excluded volume result. The broken line represents the swollen Gaussian approxi-
mation.
Fig. 4. - Logarithmic plot of the total segment density.
The circles denote Monte Carlo data of Olaj and Zifferer
[8]. Full an broken lines as in figure 3.
Carlo data. Note, that in the excluded volume limit
P em (r) is a universal function of rlrg, not involving
any fit parameter. Also for larger r our curve is closer to the data, though In P em still is somewhat
large. At least a part of this discrepancy is due to the
fact that the Monte Carlo chain is fairly short (50
steps on a cubic lattice), so that distances
(r/Rg )2 > 4 certainly feel the finiteness of the chain.
3.2 DENSITY DISTRIBUTION FOR A SPECIFIED SEG- MENT. - Our results allow for a more detailed
analysis. Equation (2.37) gives the distribution func- tion for a given segment n. Let us again consider the first moments of this distribution
The mean squared distance of segment n from the
center of mass is to be calculated as
In figure 5 we have plotted
as function of t = n/N, which is again a universal
function. It is somewhat surprising to see that the
first order correction is numerically very small. The discussion of the previous section suggests that in particular the inner segments, n/N ’" 1/2 are pushed
out compared to the swollen Gaussian approxi- mation, so that the chain can decrease its central segment density. As figure 5 shows, this is not what actually happens. Contrary to our expectation, the
central segments I n - 1 -5 N 1/2 0.2 are found closer to
the center of mass !
Fig. 5. - R’IR 2 as function of n/N. Full or broken lines- denote the excluded volume or swollen Gaussian be- haviour, respectively.
How does the chain decrease its central density ?
A first answer is found by an analysis of the critical ratio
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This ratio measures the relative fluctuations of
(rn - Rem)2 about its mean value. For a Gaussian chain these fluctuations are large and independent of
n :
2
They are suppressed considerably by the interaction,
as is seen from figure 6. This suggests that the
Fig. 6. - 3tn as function of n/N. The horizontal line at
3tn = 2/3 is the Gaussian chain result.
density in the center of the interacting chain is
decreased since the segments stay closer to their average position (rn - Rem)2 ’" R2 and large fluctu-
ations towards rn - Rem are inhibited. Indeed, this
idea is verified by a calculation of the r-space distribution for a given segment n. Figure 7 shows
Fig. 7. - Pcm = (4 7r )3/2 Re Pcm (r, N, n ) as function of
r/Rg. The values of n/N are indicated. The insert compares the results for n/N = 0.5 or 0.2 to the swollen Gaussian approximation (broken lines).
Fig. 8. - ln (Pcm(r, N, t ) Rg3 ) as function of r2/Re. Para-
meter values : a) t = n/N = 0.5 ; b) t = 0.25 ; c) t = 0.
The fat or thin lines give the excluded volume result or the
swollen Gaussian approximation, respectively. Circles give data of Olaj and Zifferer.
P cm (r, N, n ) as function of r/Rg for several values of
n/N, evaluated for g = g *. We note that P cm (r, N, n ) for a Gaussian chain is a pure Gaussian for all n. Thus in the swollen Gaussian approxi-
mation any given segment with highest probability is
found in the center of the chain. It therefore must
