A simple model for polymeric fractals in a good solvent and an improved version of the Flory approximation

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A simple model for polymeric fractals in a good solvent and an improved version of the Flory approximation

Daniel Lhuillier

To cite this version:

Daniel Lhuillier. A simple model for polymeric fractals in a good solvent and an im- proved version of the Flory approximation. Journal de Physique, 1988, 49 (5), pp.705-710.

�10.1051/jphys:01988004905070500�. �jpa-00210747�

A simple ^{model for} polymeric ^{fractals in a} good solvent and an improved

version of the Flory approximation

Daniel Lhuillier

Laboratoire de Modélisation en Mécanique, Université P. et M. Curie et C.N.R.S. (UA 229), 4 place ^Jussieu,

75230 Paris Cedex 05, ^France

(Reçu ^{le 7} juillet 1987, ^{révisé le 28 octobre} 1987, accept6 ^{le 13} janvier 1988)

Résumé.

On définit un r-fractal comme une structure self-similaire construite à partir ^{de N} objets élémentaires, ^{et dont le} rayon de giration ^{maximum est de l’ordre de Nr. On} suppose que r ^{est un} exposant superuniversel, indépendant ^{de la} dimension d de l’espace ^{et vérifiant} 1/d~ ^{r ~ 1.} On détermine _pour ces structures la dimension fractale à l’équilibre, ^{la dimension} spectrale, ^{la dimension} d’étalement, ^{etc... en} fonction de r et d. Il apparaît que les r-fractals offrent ^un modèle simple ^pour décrire les polymères, ^{la valeur}

r = 1 correspondant ^aux chaînes, ^{et r}

^{3/4 aux} polymères ^{branchés. On} réinterprète l’approximation ^de Flory

comme donnant une forme ultrasimplifiée de la loi de distribution statistique ^du rayon de giration ^{d’un r-}

fractal. L’expression ^{correcte de cette} quantité ^{conduit à une version} améliorée de ce modèle dont le succès est semble-t-il dû au comportement ^{inattendu d’un} rapport d’exposants qui ^reste obstinément au voisinage ^{de sa}

valeur idéale dans une large gamme de valeurs de r ^et d.

Abstract.

We define a r-fractal as a self-similar structure built from N basic units, ^{and with a maximum} gyration ^radius scaling ^{like Nr. We assume that r is a} superuniversal exponent, independent ^{of the} space dimension d and verifying 1/d~r~1. ^{We determine the} equilibrium ^fractal dimension, ^the spectral ^and spreading dimensions, ^{etc... as a} function of r and d. These r-fractals seem to offer a simple picture ^{of actual} polymers, ^{with r = 1} for chains and r

3/4 for branched polymers. ^We reinterpret ^the Flory approximation ^as giving ^an oversimplified expression for the statistical distribution of the gyration ^radius of r-fractals. The correct expression ^{leads to an} improved version of the Flory ^{model, whose success is} (tentatively) explained by

the rather unexpected ^{behaviour of an} exponent ratio which stays very close ^to its ideal value in a wide _{range of} r and d.

Classification

Physics ^Abstracts

61.40

05.40

05.90

1. Introduction.

The equilibrium gyration ^{radius of a linear} polymer

in a good ^solvent depends ^on the number N of

monomers according ^{to the} scaling ^law

where v depends ^{on the} space dimension d. For a

branched polymer, the relation v (d ) ^{is different.}

Hence, ^{at least one} parameter is necessary ^to describe the internal topology. ^{Possible candidates}

are the spectral ^dimension D, [1] ^{and the} spreading

or connectivity ^dimension Di [2]. ^We would like the

topological parameter ^{to be} independent ^{of d so as}

to make it a real intrinsic (superuniversal) property of the polymeric fractal. The situation is not clear

regarding Ds ^and De since, for branched polymers,

the results are divided into a dependence ^{on d} [3] ^or

no dependence [4]. ^In this paper ^we do not refer to these quantities ^{at the} outset ; ^we prefer ^{to introduce}

a more intuitive exponent, ^{let us call it r, such that} N r is the maximum possible gyration radius of the

polymer. Assuming ^{that r is} independent ^{of d and} using scaling arguments, ^we determine the statistical distribution law for the gyration radius. We then deduce v (r, d ) ^{as well as} the upper and ^lower critical dimensions. Moreover ^we find the length ^{L - Rz of}

the shortest path ^{between two} points separated by ^a

distance of order R. The knowledge ^{of both} v (r, d )

and 3(r, d) ^{leads to} D, (r, d) ^and De(r, d). ^A comparison ^with previous results confirms the value

r = 1 for linear chains and suggests r

^{3/4 for}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905070500

706

branched polymers. Lastly ^we re-interpret ^the Flory

free energy ^{as a} crude approximation for the distri- bution law of the gyration radius. The more correct

expression ^we get ^{for this} quantity may be though ^of

as an improved ^form (i.e. ^with good exponents) ^of

the Flory approximation.

2. The statistical distribution law for the gyration

radius of a polymer ^chain.

Most of the studies concerning the statistical proper- ties of a N mer chain have dealt with XN (-k), ^the

number of different spatial configurations ^{with an}

end-to-end distance x. This distribution law ^can be calculated with great accuracy [5], ^{and its} asymptotic

behaviour for small and large ^{values of i can be}

recovered with scaling arguments [6], ^{see also} fig-

ure 1. But what about the distribution law XN(X)

Fig. ^1.

^The distribution law for the end-to-end distance 1 [6] ; g

(’Y - 1 )/ II [16] ^{and 5 is defined in} (5).

for the gyration radius x ? In principle ^{the later can}

be deduced from the former and the joint probability

that a chain with end-to-end distance 1 has ^a

gyration ^{radius x.} Unfortunately, the related calcu- lations are quite complicated ^{and we want here to} give ^some simple arguments leading ^{to the behaviour}

of XN (X) ^{for small and} large ^{values of x. In fact we} hope ^{that the} knowledge ^{of the} asymptotic ^behaviour

is sufficient to have a good idea of the whole

distribution, ^{as is the case for} AN(,k)- ^{We now}

derive X N (x) for linear (chain) polymers. ^{We first}

use a guess suggested by physical constraints and in a

second step ^we make this guess compatible ^with scaling ^laws.

A polymer chain made of N basic units ^can be found in ^a number X N (x) of different spatial configurations giving ^{it a} gyration radius of order x.

This number ^must strongly decrease in two extreme situations :

1) ^when x : N1/d since the minimum gyration

radius corresponds ^{to a collapsed} polymer ;

2) when x > N since the gyration ^{radius cannot}

exceed the fully ^stretched length.

Since the total number of configurations ^{of a} polymer ^{chain is} J.L N N’Y -1 ¹ [5, 6], ^{if we write}

the probability distribution PN (x ) ^must be such that

J.LN PN(X) strongly ^{decreases for x} N1/d and

x > N. The most simple ^{answer to this} requirement

is to write

where a and 8 are yet ^{unknown but} positive expo- nents. PN(x) gets its maximum for the equilibrium

value xeq

N " with

where k is the (positive) exponent ^ratio

The first point ^to notice is that v (or ^the equilibrium

fractal dimension D = v -1) ^{does not} depend ^{on the}

individual values of « and 5, ^but only ^on their ratio.

The second important point ^{concerns the} exponents

« and 6 which cannot take arbitrary ^{values. Indeed,}

if the polymer ^{chain has a} self-similar structure [7],

one must require ^the probability distribution to appear ^as [6]

Comparing ^the exponents ^{of x in} (1) ^and (3), ^{this can}

occur only ^if

and comparing ^{now the} exponents ^of N, ^{we see that}

« and 5 cannot take but the values

and

where v (d ) ^is given ⁱⁿ (2).

Gathering the above results and discarding ^all

constant factors, the distribution law for the gyration

radius of a polymer ^{chain is} given by

where the exponents ^{a and 8 have the} expressions

and

The only ^{unknown is the} exponent ^ratio k (d ).

The behaviour of PN (x ) ^is reproduced ⁱⁿ figure ²

and it should be compared ^with the related quantity P N (i) which appears in figure ^1. Clearly enough,

the main differences occur for x -- N 1 Id where

PN (x ) ^decreases exponentially ^while P N (i) ^has appreciable values. Note also that the high-x ^be-

haviour of both curves are similar and that the same

Fisher’s exponent ⁵ governs the asymptotic ^be-

haviour. This is not really ^a surprise because for

large x values, ^{most of the} configurations ^{have also} large ^and comparable x values. What is perhaps

more surprising ^{is the} occurrence in the low-x range of the exponent ^a which is in fact the one appearing

in the overlap ^threshold c * -- N - 1/ a and also the

one appearing ^{in the} polymer osmotic pressure which behaves like c1 +" for semi-dilute solutions [8].

In fact this should not be ^a complete surprise (and ^I

am indebted to des Cloizeaux for the argument)

because if we are to derive the free energy from the osmotic pressure p by writting dF = - p dv, ^then

with p ~ c 1 + aand c = n / v ^{we get dF-nc’-ldc}

and consequently F - nC a, ⁱⁿ agreement ^{with the}

term N (N /xd)a ⁱⁿ (6).

Fig. ^2.

^The distribution law for the gyration radius x ; ^a

and 5 are defined in (4) ^and (5) respectively. d ^{is the space}

dimension.

3. The gyration ^radius distribution law for a r-

fractal.

Can ^we extend the above results to branched poly-

mers ? Regarding ^the gyration ^{radius, the main}

difference with chain polymers is the maximum

possible value which is smaller than N. Let us

suppose that this maximum scales like N r with ^an exponent ^r independent of the ^space dimension d and in the range

In other words, ^{let us assume that r is some kind of} superuniversal exponent for all space dimensions

satisfying d , r-1. ^This assumption is obvious for chain polymers (with r

1) but what about branched ones ? Since the ^answer certainly depends ^{on the}

way the polymer ^is built, ^{one cannot} propose really convincing arguments. So it is better to say that in what follows, ^{we consider a} family of fractal objects (let ^us call them r-fractals) ^{which do} satisfy ^the independence ^{of r on} d, ^{and we will see if this} family

shares some common features with what is known

(or admitted) of branched polymers.

Applying ^{the same method as} above, ^{one finds}

that result (4) is still correct, while results (5) ^and (6)

are transformed respectively ^into

and

The exponents ^v (or D) ^{and a are now} functions of both d and r

-i

I-

In results (8) ^to (10), ^the only unknown is the exponent ^ratio k(r, d). ^{Needless to} say that its detailed behaviour cannot be deduced from our

phenomenological approach ^but only ^{from a com-} plete calculation of the distribution law or estimated from accepted values of v (cf. (23)). However, ^there

exists some special ^{cases where a and 5 can be} given specific values. The term N (N/xd)" ⁱⁿ (8) may be

thought ^{of as} the number of contacts between distant basic units, ^{and the term N} (x/N’)/) ^{as the elastic}

energy of the permanent links between adjacent

units. Since the repulsive energy varies ^at least as

N n (if n-body repulsion is dominant or if the n-th visit of a site si forbiden [9], ^{the case n}

² being ^the

most common one) and since the elastic energy varies at least as x2, ^{one deduces}

This prompts ^{us to} define ^the critical dimension

de ^{as one} for ^{which a and 6} get simultaneously ^their

minimum values, ^{i. e.}

708

and

From the expression (9) ^{for a, this occurs for a} single d ^value depending ^{on n and r}

A lower critical dimension d c * ^is sometimes defined

as one for which D

d. According ^to (10) ^this

occurs for

But this result is obvious since a r-fractal has a mass

larger ^{than x1/r and one cannot} immerse it in a space of dimension lower than 1 /r.

If the ideal value of k is known, the way

k (r, d) evolves for d - dc or d::> dc ^{is not known.}

However, ^an increase of d or r means that the fractal

object is less and less compact. ^{One thus} expects ^a

to be a decreasing function of both d and r, and to

keep its minimum value for d > dc. Unfortunately,

the general expression (9) ^{is not} sufficient to assert such a behaviour, ^{unless one makes some} assumption concerning k (r, d ). ^In particular, it is worth noticing

that if k was a constant, ^{or at} the least if the derivatives ak/ar ^and ak/ad ^{were small} enough, ^the exponent a ^{would be a} decreasing function of d and

r indeed.

For future reference, ^{let us} investigate ^{the conse-}

quence of the crude assumption

In this _case, the equilibrium fractal dimension is

for d in the range

where d, is defined in (13). ^The exponents a ^{and 8}

are then given by

and

They ^are decreasing functions of r and d with

a = n -1 and 8 = 2 above dc. ^{As an} example, ^a

chain-like fractal with two-body interactions

(n

^{2, r}

1 ) ^is characterized by ^{« _} (d ⁺ 2)/

2 (d -1 ), 8 = 2 « , ^{D =} (d + 2 )/3 and dc=4. ^It

must be stressed that these results stem from the strong approximation (15), ^{which is} nothing ^{but a} generalization ^{of the} Flory approximation, ^{as we}

now show.

4. Comparison ^{with the} Flory ^model.

The Flory model determines v ^or D from the free energy

where _vo is the so-called ideal value of v. From (18)

one gets

Linear polymers correspond ^to vo

1/2 [10] ^and

branched polymers ^to Vo

1/4 [11]. ^Result (19)

agrees quite ^well (although ^not exactly) ^{with the}

results of exact [12] ^or numerical calculations. But what is the link with r-fractals ?

It appears that the Flory free energy is related ^{to a} distribution law of the gyration radius which ^comes out when in the correct expression (8) ^{one makes the}

two drastic assumptions

and

whatever r and d. Assumption (20) ^is equivalent ^to (15) ^for two-body interactions (cf. ^also (12)). ^Note

that, according ^to (9), assumptions (20) ^and (21) ^are incompatible ^with each other. Nevertheless, with both these assumptions, ^the entropy ^{linked to the} gyration ^radius distribution becomes

which is clearly equivalent ^to (18) provided ^{we let}

The ideal size of ^a polymer ^{in the} Flory ^{scheme is}

thus connected to the maximum gyration ^{radius of a}

r-fractal. Such ^a link is indeed satisfied for linear

polymers ^{and we} find back the obvious value

r

1. but it suggests ^{the more} interesting ^result

Concerning ^the Flory approximation ^{it is} quite

wonderful that a model based on the strong (and incompatible) assumptions (20) ^and (21) gives ^so good results for ^v. In fact, assumption (20) ^{is the} only ^{crucial one because we} already noticed that the exact value of ^a was irrelevant as far as v is concerned. We conclude that the success of Flory’s

model stems from a rather unexpected behaviour of the exponent ^ratio k (r, d ), ^which happens ^to stay ⁱⁿ the neighbourhood ^of its ideal value in ^a large

domain of d dc. ^{In other} words, ^this oversimplified

model works because the exponent ^{5 related to the} elastic properties ^of strongly ^stretched polymers ^is always nearly ^{twice the} exponent ^{a related to the} number of contacts between distant monomers. The

validity ^{of the} Flory approximation (20) ^{can be} appreciated ^{when exact or} numerical values for v are

introduced in the following expression for k deduced from (4) ^and (7)

One finds that for r

1, k stays ^{in the} neighbour-

hood of its ideal value for 2 , d , 4, while for

r

3/4 the few known results are also nearby ^the

value k

2. Note that nothing is known of v for d

1 /r ^{+ e and} consequently, ^{the value} of k for the lower critical dimension dc* is left undetermined.

The point is that in this limit the value of v is r and it is independent ^{of k as} is evident from (10). ^{To sum}

up, for ^{not too low} values of d dc, ^the exponent ratio is _very close to its ideal value, while for the lowest d, ^{the exact value of} k is irrelevant as far as v

is concerned. We think this is the reason why ^the Flory approximation (20) ^{was so} successful ! If ^we take it for granted, ^but want to correct the Flory ^free

energy, ^{we must} replace (18) by

with

in conformity ^with (8) ^and (9). It is thus possible ^to

recover the Flory exponent (19) ^{from a} free energy

compatible ^{with the} requirements ^of scaling ^laws.

5. Squeezed r-fractals.

A straightforward extension of (8) concerns a r-

fractal squeezed ^{in a} tube of diameter f and more

generally ^a r-fractal constrained in all dimensions except ^{one. The} elongation xil in the free direction is

now in the range between N lfd - 1 ^{and N’} provided f N II. The configuration entropy ^{is then}

from which one deduces the equilibrium elongation

and the equilibrium configuration entropy

The second term of the right-hand ^{side is the}

confinement energy of de Gennes [6]. ^{With r}

^{1 we}

recover his results for linear polymers. The results for squeezed ^branched polymers (r

3/4) ^{seem to}

be new.

6. The shortest path exponent ^{and other} properties

of r-fractals.

Generally speaking, ^an approach ^{a la} Flory ^{is an}

easy way ^to find the equilibrium ^{value of a} quantity x depending ^{on a} parameter y ^and limited in the range

The equilibrium ^{value is} _Xeq

y veq ^with

where k * is the ratio of the exponents ^{of the terms}

x / y 11 + ^and y 11- / x. ^{As an example, let us} consider the shortest path ^{L between two} points separated by ^an

euclidian distance R and pertaining ^{to a r-fractal at} equilibrium ^{in a} space of dimension d. This shortest

path ^is always longer ^than R, ^but always smaller than the maximum possible gyration ^{radius, even when R}

is as large ^{as N y.} Hence, ^{in this} problem ^the quantity L depends ^{on the} parameter R ^with

R L R’D. Any Flory approach ^{will lead to an} equilibrium ^{value L - R ’ with}

where k * is undetermined yet. Since for d > dc ^the

shortest path ^{is a} random walk for which z

2, ^the only possible k * value is

and consequently, ^{the best} expression ^{that can be} proposed ^{for z via a} Flory approximation ^is

where D (r, d ) ^{is the} equilibrium fractal dimension

given ⁱⁿ (10). ^In particular, ^{if we trust the value}

r

3/4, ^we get for branched polymers

and with the Flory approximation ^{for D}

The result (27) ^is definitely better than the one

710

proposed ⁱⁿ [14] ^when compared ^{to the numerical}

results got ⁱⁿ [3] ^and [4] ^{for d}

^{2 and d}

^3.

Once i (r, d ) ^and D (r, d ) ^{are known, other} expo- nents can be deduced easily. ^The so-called Einstein relation

allows us to get ^the exponent Dw ^{of a} random-walk

on a r-fractal, while the Alexander-Orbach re-

lation [1]

leads to the expression ^{of the} spectral ^dimension Ds ^{as a} function of r and d. Lastly, ^{if we trust the}

relation

proposed by Havlin et al. [3], ^we get ^the expression

for the chemical or spreading ^dimension De (r, d).

All the main exponents characterizing ^{a r-fractal can}

thus be deduced from (10) ^and (26).

7. Conclusion.

Our first objective in this paper ^{was to} suggest ^{that r-} fractals could offer a simple model for various

polymeric structures. At variance with Flory’s model, ^{we have not} introduced ^an ideal size, ^but instead, ^we proposed the idea of a maximum gy- ration radius of the structure (in fact, ^{both are} related by (22)). ^{This led to our} second aim which

was to give ^{a new} interpretation ^{of the} Flory ^model

and to propose ^an improved version of it, with result (24) ^{as a first} step. ^{It must} be clear that a Flory

model will never supersede ^exact calculations but if it is compatible ^{with them} (as ^{it is} now), ^{it has an}

indeniable pedagogical ^value.

In the same spirit ^{as we} corrected here the Flory

free energy of ^a single polymer, ^{we can also} improve

the Flory-Huggins expression for the free energy of ^a

polymer solution. This goal ^was achieved in [15] ^for

solutions in ^a good ^solvent.

Acknowledgments.

I would like to thank J. des Cloizeaux, ^{M. Daoud} and J. Vannimenus for helpful discussions.

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