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Long polymer chains in good solvent: beyond the universal limit
Brunhilde Krüger, Lothar Schäfer
To cite this version:
Brunhilde Krüger, Lothar Schäfer. Long polymer chains in good solvent: beyond the universal limit.
Journal de Physique I, EDP Sciences, 1994, 4 (5), pp.757-790. �10.1051/jp1:1994174�. �jpa-00246945�
Classification
Physics Abstracts
36.20E 64.60A
Long polymer chains in good solvent: beyond trie universal Emit
Brunhilde Kriiger and Lothar Schàfer
Fachbereich Physik~ Universitàt Essen, 45117 Essen, Germany
(Received 20 September 1993~ received in final form 12 January 1994~ accepted 17 January1994)
Abstract. We analyze the effects of finite segment size and weak chain stiffness on the
properties of polymer solutions in the dilute limit. We consider bath weak coupling (cross over
from 8- to excluded volume conditions) and strong coupling (very good solvents)~ pointing
eut that the renormalization group in its standard field theoretic implementation allows for two universal branches corresponding to these two different situations. The strong couphng branch implicitly relies on the existence of a finite segment size, a feature which however does net explicitly show up m the renornialized results. Beyond the leading 'corrections to scaling' described by the standard renormalization group flow~ the strong coupling branch necessarily is affected by nonumversal corrections. In field theoretic renormalization we include the first order effects of finite segment size or stiffness
m terms of an expansion of the scaling functions about the universal limit. We compare ta direct renormalization~ where finite segment size or stiffness
are included into the renormahzation group equations. This somewhat magnifies the effects, but still the nonumversal corrections are negligible for chains longer than about 500 freely rotating
umts. Quahtatively the effects in bath renormabzation schemes are the same, consistent with experimental or Monte Carlo results.
1. Introduction.
Long flexible polymers m dilute solutions show universal large scale properties which cau be
explaiued by the reuormalization group il, 2]. This approach exploits invariance properties
under spatial rescalingi which rigorously hold in trie limit of an infinitely dilute solution of infinitely long chairs. (Segment concentration c - 0, polymerization index n - oo.) In
practice trie theory bas been found [1-3] to explain dilute solution data for flexible chains of
lengths n 2 500 and segment concentrations c $ 10 %. Trie theory is restricted to trie region where segments repel each other, i e. to temperatures above trie e-point-
Though the renormalization group approach is supported by a lot of experimental evidence there exist both computer experiments and physical expenments which do not seem to fit into trie universal scheme. This is particularly obvions for trie interpenetration ratio #. This dimen- sionless quantity is constructed from trie second virial coefficient A2 of trie osmotic pressure
and trie radius of gyration RG of an isolated polymer colt.
# = ~
~~
( (l.1)
(2x) G
(Note that we define trie virial expansion in powers of the polymer concentration cp mstead of trie segment concentration c
= cpn.) In trie excluded volume limit of long chains (n - oo) in
a good solventi trie renormalization group predicts [4] that # tends to some universal constant
#~. This result is consistent with both physical and computer expenments, where a value
#~ m 0.245 bas been found [5-7]. Violations of universality are found for trie variation of # with chain length and temperature outside the exduded volume limit. Both standard two-
parameter theory [8] and renormalization group theory in trie way it usually is presented il, 2]
predict # to be a universal function of trie swelling factor a:
~fi = ~fi(a) (1.2)
Here Re(N)
= RG(Ri N) is the radius at trie 8-temperature, where trie effect of trie excluded volume vanishes. Trie renormalization group analysis furthermore suggests that #(a) is a
monotic function. Since #(a) vanishes at trie 8- temperature (a
= 1)~ it must attain its hmiting value iJ~
= #(a - oo) from below. This is in keeping with experiments on moderate solvents where trie 8-temperature can be reached. However, for very good solvents, where all the accessible temperature range is far above the 8-temperature, one often finds that # with
increasing chain length approaches #~ from above [9]~ which violates the scenario sketched here.
Some recent publications are concerned with that problem. Trie work con be grouped into
two -, not disjunct
-, classes. Approaches of trie first dass [6, loi use renormalization theory to
construct equations for # which allow for a behaviour more general than equation (1.2). They
do net introduce new physics into the mortel and indeed they do not go beyond trie standard renormalization group: as bas been pointed out some years ago [3, 19], equation (1.2) only is a
limiting form of the renormalization group result, valid close to the 8-temperature. Inspecting
the general result more dosely, one finds that it allows for two branches #<(T, N), #>(T, N),
bath monotonic, but approaching #~ from below or above, respectively.
A second dass of work [10-12] considers trie effect of chaiu stiffness on the polymer con- figurations. In the standard renormalization group mortel stiffness is neglected smce it has
no nontrivial influence on the limit of long chains. Technically speaking it is an ~irrelevant
perturbation', which in pnnciple can be induded by expanding about the universal limit. No
systematic attempt in that direction has been made. Trie published approaches essentially indude stiffness into trie nonuniversal pararneters of the mortel.
It is the main goal of this article to present systematic renormahzation group calculations, which to leading order mdude trie effects of small chain stiffness and of the finite size of trie
polymer segments. For fairly flexible chains as considered here, both features dosely are re- lated~ since redefining trie effective segment we dearly also must redefine trie effective stiffness
parameter. Indeed we will find bath sources to yield quahtatively similar corrections. (To avoid confusion we should note that trie finite segment size has nothing to do with trie 'finite size effects' discussed in trie theory of critical phenomena. Trie latter are due to trie finite extension of trie sample.) We use two different renormalization approaches. In trie first 'field theo-
retic' method we indude the corrections into the perturbative calculation of the renormalized
scaling functions, which we evaluate directly in three dimensions. The renormalization group flow is not changed. In trie second 'direct renormalization method' we use trie e-expansion
about four dimensions to construct renormalization group flow equations for # or RG> which mclude stiffness and finite segment size. These two methods span a large range of technical
implementations of trie renormalization group idea, and thus a comparison should give some
indication on trie reliability of the results. We however should note that in both methods we treat the irrelevant perturbations on the simplest, essentially unrenormalized, level. This point
is discussed in more detail later.
Before taking up our main theme we recall trie standard renormalization group results for
fully flexible chains. Though these results are not new and ail trie features are imphcit in previous work we feel that a thorough discussion is appropriate, stressing aspects which are not widely appreciated. In particular we point ont that trie most elaborate renormalized
theory, which bas been derived [13] by some very efficient renormalization technique (minimal subtraction) applied to a field theory equivalent to a continuons chain model, in identical form
can be derived from a discrete chain model. In trie present context this is important, since the discrete chain model in contrast to the continuons chair model allows for values of trie
renormalized coupling larger than trie asymptotic exduded volume limit. As a result, values
# > #~, which are allowed by trie renormalization group equations, are unphysical if the
theory is interpreted within trie continuons chain model but are perfectly reasonable if trie discrete chain model is considered as trie underlying theory. Trie renormalized theory itself is
independent of that interpretation.
The orgamzation of this paper is as follows. In section 2 we derme trie model of discrete flexible Gaussian chairs, and ~ve present and discuss trie general renormalization group result.
Trie remainder of trie paper is devoted to trie generalization of the model induding the leading
stiffness corrections. Section 3 presents the unrenormalized results. Section 4 is concerned with field theoretic renormalization. Direct renormahzation is carried ont in section 5. Numerical
evaluation and compantive discussion is contained in section 6. Section 7 summarizes our conclusions. Some technical derivations are contained in appendices. Specifically the relation of our model to a ~wormlike chain' model is presented in appendix A.l. Technical details of
unrenormalized perturbation theory induding stiffness are given in appendix A.2.
2. Standard renormalized theory.
2.1 THE MODEL. We start from a model of discrete Gaussian chairs, composed of n seg-
ments of average size of order f. The configuration of trie chair is fixed by segment coordinates r~, j
= 0,..
,
n, taken as d- dimensional vectors. Trie Hamiltonian for fully flexible chairs is written as
where the exponential part guarantees chair connectedness. fl~ > 0 is the dimensionless ex- duded volume parameter. Trie product extends over all pairs (j, k) of segment indices, with the restriction that in trie duster expansion of the product we omit all terms in which any segment index occurs more than once. This restriction, which is consistent with the neglect of
many body interactions, ehminates divergencies due to the à-function interaction.
2.2 RENORMALIzATION. Trie mortel has to be evaluated perturbatively, the appropnate
tool being trie duster expansion in powers of fl~, which is sketched in appeudix A.2. It is fourra
that the expansion orders in powers of fie n~/~, where e
= 4 d(= 1 in three dimensions). In trie limit of interest, fl~ > 0,n - oo, this expansion parameter diverges and trie expansion
seems useless. This well known problem is solved by renormalization, which allows us to map
trie chair of length n on a shorter chair of length nR. Choosing nR ~ l we cari evaluate trie renormalized theory perturbatively. More precisely, renormalization theory proves trie following theorem: There exists a mapping
Î ~ ÎR> n ~ nR ~))
,
~e ~ i£ ~)) (2.2)
which leaves macroscopic observables invariant. Here éR is an arbitrary length scale which can be chosen to be of trie order of RG so that nR
= 1. Trie renormalized result is independent of trie original segment size é, up to corrections of order (é/RG)~.
As a result we for instance for RG find the relation
Rà(Î,Îl>~e)
" Î~~(~l>~e)
éz
" ÎÎ~R(~R
" 1>i£) (1+ ° (@j (2.3)
G
The function ÉR(1,~) cari be calculated in 'renormalized' perturbation theory in powers of ~.
We should note that the discreteness of trie segments yields Orly corrections
+~ é~/R[ so that
the renormalized results show no exphcit remainder of trie microstructure.
TO make this scheme work we need ta know the mapping (2.2). In renormalization theory
one proves that trie change of the renormahzed parameters under an infinitesimal change oféR obeys simple dilferential equations:
-ÎRù~
= W(~)
(2.4)
-ÎRùillnR = v(~)
The functions W(u), il(~) again con be calculated by renormalized perturbation theory. To a
good approximation we find for the flow of the coupling ~
W(it) = ~~[ ~~ (2.5)
-~ + -(~ ~~)
~ ~
Here the fixed-point coupling ~~ is the nontrivial zero of W(u) and w is trie so-called 'correction to scaling exportent'. In three dimensions w m 0.80. Integrating equation (2.4) for
u with W(u)
given in equatiou (2.5) we find
Îo ~~~ ~Î- ~ ~~~ ~~'~~
We introduced the normahzed couphng f =
~ (2.7)
u~
f~ is an integration constant defined as
fo " ~~~)F ~~ " fief(fie) , (~.8)
where the last equality expresses the expectation that the 'running couphng constant' u(éR) for éR = é is an analytic function of the unrenormalized couphng fl~, clearly vanishing for fl~ = 0.
A result similar to equation (2.6) holds for nRi
nR "
~ a( f~)n (2.9)
Here u is the correlation length exponent~ taking the value v m 0.59 in d
= 3, and a(f~) is
an integration constant, which again is assumed to be an analytic function of f~ and thus of fie. The results (2.6, 2.9) are based on a so-called two loop calculation, improved by inserting
exact values of w, v. They first have been presented in reference [3].
A more accurate form of the renormalization group mapping has been derived by Schloms
and Dohm [13]. Their result is based on Padé-Borel resummation of higher order perturbation theory. Equations (2.6, 2.9) for d
= 3 are found to be a good approximation to these more rigorous results. For the present discussion we therefore ~vill use this approximation, presenting the more rigorous form in section 4.3.
Renormalized perturbation theory in the form of the so-called e-expansion yields the following
results for # or R[:
l~) # =
if il + ~~e + if In
2 +
~
+ O(f~,~)j
(2.10)
3
~~~
8 32 2 48
R[
=
~é( Il ~~e f + (f~,e~)) (2.Il)
3 96
We used the 'minimal subtraction' scheme, consistent with the denvation [13] of the renormal- ization group mapping. These results easily are derived from equations (4.13, 4.15) of section 4.1. Setting nR
= we note from equation (2.9) (or Eq. (4.26) that in the exduded volume hmit n - oo, f~ > 0 the coupling f tends to the fixed point value f~
= 1. Evaluating equation (2.10) for f~ = 1,e = we find a value #~ = 0.259, to be compared to the expenmental result
#~ m 0.245. Improved expressions for #~ RI, based on an evaluation of perturbation theory directly in d
= 3 are given in section 4.3.
2.3 DIscussIoN. Renormahzation group results for # or R[ are abundant in the literature,
and all these results are equivalent in the sense of the e-expansion. We have nothing to add here, except that we expect that the 'optimal' form given in section 4.3 combined with the 'exact~
renormahzation group mapping indeed is optimal for physical dimension d
= 3. Recalling these results we however want to stress some aspects of the general renormalization group scheme
relevant to an understanding of the problems connected to # > #~.
Let us first rewrite the renormalization group mapping (2.6, 2.9) in a more compact form.
We introduce two parameters
_1
~ ~l/2(f f i f ~"" ~l/2
o o o
(2.12) R(
= é~a( fo) fo ~~ ~ n Taking na = 1,e = 1, we can rewrite equations (2.6, 2.9) as
ii
= i- f[~~~-~~àj (2.13)
2
o i,o i s Ç
Fig. 1. é(fo) evaluated according to equation (2.12) with a(fo) + 1. The scale of é is arbitrary.
_~
f(1 f( ~""
= é (2.14)
These results imply that the three parameters 1,n, fl~(T) of the underlying unrenormalized mortel combine into two parameters
é
= v(T)n~/~
(2.15) R~ = 1(T)n~/~
In that sense the renormahzed theory is a two-parameter theory. It, however, is important
to note that 1,É~ in general ailler from the parameters z, RH introduced in the naive two
parameter theory
z = constflen~/~
(2.16) RH fi in~/~
By definition z is a linear measure of the excluded volume strength. In contrast, the proper
scaling variable é is a complicated function of f~. Figure gives é( f~) according to the ap~
proximation (2.12), illustrating that 1( fo) is net even monotonic m fo. It is only by virtue of
analyticity close to the 8~point fl~ « 1, that the relations f~ m fl~, é m z, R~ m Re hold. As
a result, the universal scaling laws take the form (1.2, 1.3) only m the region (T 8) le « 1.
The general renormalization group result reads
ifi " ifi(2)
(2.17) fi2
2 o -2(~)
~ j~2 ~ ~ '
8
as can be seen from equations (2.10, 2. ii, 2.13, 2.14). We thus find a umversal law ~b
= ~b(£i~), the relation among a~ and d~ however involvmg a temperature dependent factor R]/R(
=
f~(T)li~. Trie temperature dependence of1(T) Ii or é for T ~/ B is net universal. This has been pointed eut in references [3, 19].
We furthermore should stress that nonuniversal functions hke /(fl~(T)) (Eq. (2.8)) cannot be calculated reliably, despite the assumed analyticity. These functions m principle ai-e defined
in the limit of an infinitely long chain, so that a proper calculation is equivalent to an exact solution of the problem. Th~s the renormalization gro~p ares net predict the temperat~re dependence more thon implied by analyticity of the non~niversal parameters VIT),1(T).
As is obvious the general results (2.6, 2.9) or (2.13, 2.14) allow for two universal branches (1) fo < f <1
(ii) fo> f>1
,
seperated by a third branch
(iii) fo=f=1
The first branch describes the cross-over from B-conditions to exduded volume conditions. ~b approaches ~b* from below and a~ shows the normal swelling behaviour, starting from a = 1 at é = 0 (compare Eqs. (2.10, 2.Il)). It is this '8-branch' or'weak couphng' branch, which mostly
has been discussed in the literature, for good reasons given below. Typically the assumptions
of naive two-parameter theory é
+w z +w n~/~(T 8) le, Ro m RH are used.
The second branch starts from large exduded volume and may be adressed as ~self-avoiding
walk' branch [6] or 'strong coupling' branch. ~b approaches ~b* from above and the swelling
increases faster than implied by the asymptotic power law. This branch cannot be followed up to1 = 0. From equation (2.14) é ~ 0,f > limplies f ~ cc, so that renormalized perturbation theory breaks down. Furthermore this branch is net connected to the B-point and thus the
definition (1.3) of the swelling torses its direct physical significance. However, the variables é,Éo still are the proper scaling variables.
The separating 'fixed point branch' fo = f
= i yields pure exduded volume behaviour:
~b = ~b*,RG
" const n~. It only is reached if the unrenormalized coupling carefully is adjusted
to guarantee fo
= i. As a result the so-called 'corrections to scaling', which are due to the deviation of f from its fixed point value, vanish identically. It is worth noting that in Monte Carlo simulations sometimes the excluded volume interaction has been adjusted appropriately [14], m order to find excluded volume behaviour for fairly short chains.
In figures 2 or 3 we have plotted the dilferent branches of
~b or a~, using the 'Optimal' results of section 4.3. For the strong coupling branch only the region corresponding to 1.8 > f > i is
shown.
It is very important to note that the branches f < i or f > i are of quite dilferent physical relevance, due to the rote of the nonuniversal corrections neglected so far. These corrections
result bath from the finite segment size and from features net contained in the mortel, like the finite range of the two-body repulsion, three body interactions, chain stilfness, e-t-c-. In the
renormalized formalism they show up as terms of order i~IRI
+w iIn. Strictly speaking they
can be neglected only for n ~ cc. For fo > 0, 1-e- T > 8 fixed, this hmit in view of equations (2.12, 2.13) implies 1 ~ cc, f ~ i, which means that we are restricted to the exduded volume fixed point. A doser inspection shows that the leading corrections due to f # i are proportional
to 2~~~~
+w n~~~
+w n~°.~~, this power dominating the nonuniversal terms
+w
n~~ We thus should condude that of ail renormalization group results only the excluded volume behaviour and trie leading 'corrections to scaling' are relevant physically. This indeed is the situation for general critical phenomena. For the present polymer problem we however can decrease fo
to~vards fo
= 0 by approaching the 8-temperature. We thus con make n large, so that the nonumversal corrections safely con be neglected and at the same time keep 2 constant, f < i, by appropriately taking T ~ 8. Thus the whole weak couphng branch f < i physically is relevant, for long enough chains describing the cross~over from 8-behaviour to the excluded volume hmit. (Rigorously speaking the 8~behaviour m d
= 3 is influenced by three body forces,
a feature which is beyond the present discussion.) This explains [3, 19] why in the evaluation
