Geometrical properties of a Kuhnian polymer chain

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Geometrical properties of a Kuhnian polymer chain

Bertrand Duplantier

To cite this version:

Bertrand Duplantier. Geometrical properties of a Kuhnian polymer chain. Journal de Physique, 1986,

47 (10), pp.1633-1656. �10.1051/jphys:0198600470100163300�. �jpa-00210361�

Geometrical properties ^{of a Kuhnian} polymer ^chain

B. Duplantier

Service de Physique Théorique, ^CEN Saclay, ^{91191 Gif sur Yvette} Cedex, ^France (Requ ^{le 9} janvier 1986, accept6 ^{le 22 mai} 1986)

Résumé.

Nous calculons le facteur de forme H [q] , ^fonction génératrice ^{des moments} du rayon de giration,

et la fonction génératrice G[q] ^{des moments de la} distance bout à bout, par ^un développement ^au premier

ordre en g, ^second coefficient du viriel sans dimension, ^ou en 03B5, où 03B5

4 2014 d, d ^dimension d’espace. ^Ces

fonctions sont calculées dans la limite asymptotique kuhnienne de chaines très longues, ^en bon solvant. On en

déduit des propriétés géométriques universelles de la chaîne kuhnienne, ^comme les rayons de giration moyens d’ordre 2 n, RG[2n], ^{et les} puissances moyennes R [2n], ^{de la} distance bout à bout. Les rapports ^universels

RG[2n] /R ^{[2n] , qui sont calculés, ont un} comportement régulier ^{dans leur} développement ^en g ^ou 03B5, et ce,

également pour n grand. ^{Par une} approche ^{directe, nous} obtenons les coefficients du développement ^en g, ^ou

03B5, ^comme des fonctions de la dimension d. Nous obtenons ainsi ^un pur développement ^{en 03B5, au} premier ordre,

où les coefficients ont leur valeur à d

4. Nous obtenons aussi ^un meilleur développement, ^en g, ^{où les} coefficients ont leur valeur à d

3, ^et où g prend ^{sa valeur g* au} point fixe kuhnien. La distribution de

probabilité de la distance interne entre deux points quelconques de la chaîne kuhnienne est reconsidérée, ^et

nous donnons sa forme universelle normalisée. En particulier, ^nous obtenons des formes très simples ^{dans trois}

cas limites, ^où les deux points 1) ^{forment un} segment fini à l’intérieur d’une très longue chaîne ; 2) ^{forment un} segment ^{fini à} l’extrémité de cette même chaîne ; 3) ^{sont les} extrémités de la chaîne entière. Nous comparons le facteur de forme H[q] ^{aux données} expérimentales ^{obtenues récemment} par Noda et al. ^Dans la région ^de grands q, ^{nous montrons} qu’en effectuant des corrections du polydispersité, ^{on obtient} apparemment ^{un bon} accord entre théorie et expérience.

Abstract.

We calculate the form factor H[q], ^{i.e. the} generating function of the gyration radii, ^{and the} generating function G [q] of the end-to-end radii, ^to first order in the dimensionless second viral coefficient g,

or in ^03B5

4 2014 d, d being the space dimension. These functions ^are calculated in the asymptotic limit of Kuhnian

chains, ^i.e. very long ^{chains in a} good ^solvent. They ^{are used to} determine universal geometrical properties ^of

the Kuhnian chain, like the average gyration ^radii RG[2n]of ^{order 2 n} and average 2 nth powers R [2n] of the end-to-end distance. The universal ratios RG[2n] /R ^{[2n] are} calculated and shown to have a regular g ^{or 03B5-} expansion, ^{even for} large ^{n. In this} approach, ^one obtains the coefficients of the g expansion ^as general

functions of the dimension d. This allows ^us to calculate the numerical values of the geometrical quantities

either by ^a pure 03B5-expansion ^{to first order,} setting d

^{4 in the} coefficients, or, better, ^{in the} g-expansion by using the values of the coefficients at d

3 and the best known value of the Kuhnian fixed point ^{g*. The} probability distribution for the internal distances between any ^two points ^{of the} Kuhnian chain is reconsidered, and its normalized universal form is given. ^In particular ^we get very simple forms for three limiting ^{cases :} 1)

the two points ^{form a finite} segment ^{inside a} very long chain, ^or 2) ^{a finite} segment ^{at the} extremity of the very

long ^{chain, or} 3) ^are the extremities of the whole chain. We finally compare the form factor

H [q] ^{to the} experimental data obtained recently by ^{Noda et al. It is shown that in the} large q region, ^{it is} possible ^{to take into account} polydispersity effects and obtain apparently ^a good agreement ^between theory

and experiments.

1. Introduction.

Very long polymer chains in a good solvent have been the subject ^{of numerous studies} [1-5]. ^The

purpose of this article is ^to study ^more specifically

some geometrical properties ^{of these} long ^chains

with excluded volume, ^{when the} latter is fully developed. ^{Such studies are} naturally important ^for comparison ^with experiments ^{and we shall} actually

perform ^a comparison ^{here. However, it is not the} only motivation of this work. It is well known that very long polymer chains with fully developed ^exclu-

ded volume have universal geometrical properties,

which do not depend of the model employed ^for describing these chains. So, ^{from a} mathematical

point ^of view, these very long chains, ^said Kuhnian,

define a universal system which has very well-defi- ned properties, exactly ^{as the famous Brownian}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100163300

chains define another (but simple) universality ^class.

The latter can be described by ^a continuous model

using the Wiener measure. In the same way,

Kuhnian continuous chains ^are associated with the continuous model of S. F. Edwards [1] ^described by

the probability density

Here r (s) ^{is the} configuration ^{of the chain in d}

dimensional space, s is the absissa (0 -- s -- S) along the chain and S is the Brownian unperturbed

« area » of the chain, ^{such that}

where °R2 is the Brownian end-to-end metln square distance, calculated from the weight (1.1) ^when

b=0.

b is the excluded volume parameter, ^{b -- 0 for the}

model to be defined. For b ⁼ 0, ^{we recover a}

Brownian chain. However for b > 0, in the limit S -o oo , the properties of the Kuhnian chain are

entirely different, ^and belong ^{to a} different universa-

lity ^class.

In principle, all these Kuhnian properties depend only ^on the space dimension d, and all the associated function giving quantities ^{like the form factor, the} probability densities, etc., exist somewhere in the mathematical space ^as functions of dimension d.

They ^{are not} easily accessible, however, ^{and the}

exact theoretical device which we use is the Wilson

E-expansion, ^{where d = 4 - E. Indeed for} d > 4,

one knows that the excluded volume effects are

irrelevant for polymer chains. Then the universality

class of the Kuhnian chains coincide with that of the Brownian chains. For d , 4, ^{on the} contrary, ^the Kuhnian chains ^are entirely ^different objects. ^Instead

of using ^{the «} bare » Brownian area S of the

underlying continuous model (1.1), ^{one considers}

the physical ^{size X} of the Kuhnian chain defined by [4]

in terms of the Kuhnian mean squared ^end-to-end

distance R2, ^when the excluded volume effects are

fully developed. ^{Then all} physical quantities ^{like the} scattering functions, the average distances, ^the geo- metrical probability distributions, ^{are finite functions} depending only ^on their natural variables, ^{on the}

size X, ^{and on the} dimension d. They ^{do not} depend

anymore ^{on the} two-body interaction b of the Edwards model (1.1), ^or any microscopic ^{details and}

it is in this sense that the Kuhnian chains are

universal. It is the purpose of this article to illustrate this interesting ^fact. Following ^a previous study [6],

we give ^various geometrical results written in this universal formalism, ^{i. e. in terms} of X and of d, ^{or E.}

We must also remark that the continuous model

(1.1) requires, ^{to be} defined, ^{the use of a} regulariza-

tion. This is actually ^{true for d -- 2. One can use a}

cut-off so, which is the minimal area between _any

two interaction points along ^a single ^chain. However,

one may also ^use the dimensional regularization

which amounts to continue analytically ^the quantities

calculated for general ^{d. For d} 2, ^the integrals appearing ^{in the} perturbation expansion ^{of the}

model (1.1) ^are convergent. ^{Then one can define}

their analytical continuation in dimension d for d > 2 and toward d ⁼ 4. In this work that is the way

we calculate all partition functions, ^{when a} regulari-

zation is needed.

In reference [6], ^{we obtained} precise information about the swelling of the various parts ^{of a isolated} Kuhnian chain. Elsewhere [7], ^a simple renormalized calculation of the asymptotic form factor of a

Kuhnian chain has been made (see ^also [8]). ^Here

we shall describe in more details the geometrical properties ^{of a} Kuhnian chain. Some partial ^calcula-

tions have already ^{been made} by various authors.

The scattering ^{structure function or form factor was}

calculated by Witten and Schafer [9], who gave the first three terms of its small wave-vector expansion, using field theoretic methods. The same form factor

was also considered by ^{Ohta et al.} [10] ^{and found to}

be very close ^to its Debye approximation. ^{We shall}

here make a more systematic study of this form

factor, calculating ^{all its} moments, i.e. the average radii of gyration RJ2n] ^{of order 2 n. We shall use}

direct renormalization [4] (see ^also [11, 12]). ^We

shall make the calculation as direct and simple ^as possible.

As a result we shall obtain two different approxi-

mations : the first one will be the _pure s-expansion,

with d ⁼ 4 - _E, the second one will be closer to the

physical ^{case d =} 3. We shall compare ^{our results to}

recent experimental ^{data of} Noda’s group. ^We

especially study ^{the case of} large wave-vectors q and show that polydispersity corrections can be impor-

tant. We perform (tentatively) these corrections in detail and find apparently ^a good agreement ^with experiments ^{in the} high q region.

Following ^our geometrical inquiry, ^{we shall also} study systematically ^the generalization of the well known universal ratio [9, 4, 13] N ^{= 6} R 2 IR 2 ^to

general powers, i.e.

We also consider the probability distributions for the distances between _any two points ^{inside a}

Kuhnian chain. We calculate exactly ^{these normali-}

zed probability distributions to first order in 8, in the

simplest way ^{we can} find. Thus, ^we refine the unnormalized preliminary results of Oono and Ohta

[14], and for the probability distributions of the end- to-end distance, we recover a result of reference

[15]. ^{In some} special geometrical ^cases, following

the ideas used in ^our previous ^work [6], ^we find very

simple ^universal probability laws. This apply ^{to a} segment ^{inside a} very long ^{Kuhnian chain, or located}

at the extremity ^{of such a chain.}

The summary is ^as follows :

In section 2, ^we calculate in detail the renormali- zed form factor H [ q] ^to first order in e or 9. In section 3, ^we perform ^the systematic ^series expansion

of this form factor, ^{to all orders} in powers of the

wave vector. This is applied ^{in § 4 to} the calculation of all the moments RJ2n] of the Kuhnian chain.

Section 5 deals with the evaluation of the set of universal ratios N _n. The probability distributions in direct space ^are calculated in section 6. The compari-

son with experimental results for the form factor is

performed in the last section 7. A comparison ^{is also}

made with a result of numerical simulations. Appen-

dix A deals with the direct renormalization of

H[q]. Appendix ^B gives ^the calculation of the

polydispersity corrections required ^{in the} comparison

with experimental ^data.

2. The form factor N[q]

We shall consider the generating ^function [6] :

for an isolated polymer chain. The form factor is then defined by

where f ( q ; S, ^s’, s" ) ^{is the} partition function of a

continuous polymer chain, ^{with two} insertions of

wave vectors q and - q ^at points s’ and s" along ^the

chain. It is dimensionally regularized. ^{The denomi-}

nator of (2.3) ^{is a} convenient representation ^{of the} partition ^{function of a} single ^{chain !E} ( S, b, s ) :

The diagrams contributing, ^at first order in the interaction parameter b, ^to 9 (q; S, s’, s") ^are represented ⁱⁿ figure ^{1. The} resulting contribution to

H[q] (2.2) ^{reads, after some algebra:}

where

The quantity ^z is thus the well known Zimm-Yama- kawa dimensionless interaction parameter. ^{The func-}

tion Ho ( y ) ^{is the} Debye ^function

Fig. 1.

^The diagrams contributing ^{to l2’} ( q ; S, ^s’, s" ) ,

to first order in b.

The functions

correspond respectively ^{to the} diagrams 1, 2, ^{3 of} figure ^1. Incidentally, ^{we note that the} contributions of

diagrams ^{4, 4’} identically cancel in the ratio (2.3).

When the space dimension ^d approachs ^d

^{4, some} diagrams diverge. ^{In the} expansion (2.4) ^of H[q]

it is easy ^{to see} that only 7B diverges corresponding naturally ^{to the} only diagram ^of figure ^{1, which} really

contributes to H and has no insertion inside the interaction loop. The renormalization of expression (2.4) (2.7) ^is extremely simple [6, 7]. ^We eliminate y

q2 S/2 in favour of the physical quantity ^x

where X is the actual size (1.3) of the swollen Kuhnian chain.

One has

where the swelling ^factor f£ 0 ( z) ^reads [6] :

for d = 4 - e.

Eliminating y in favour of x (2.8) ⁱⁿ (2.4) ^and expanding ^to first order in _z, we find the renormalized

quantity

Now this expression is finite when E

4 - d _goes to zero. For seing this, ^{one has to} regroup various ^terms of

(2.10). We do this in Appendix ^B, for any dimension d. Then, using previous ^results [7, 4] ^{it is} possible ^to

write

-

where 9 is the dimensionless second virial coefficient (in scaling form) [4]

where Z ( S, ^{S, b,} e) ^{is the} two-chain connected partition ^{function, and f} (S, b, e) , ^the partition ^function

of a single ^chain (both dimensionally regularized). ^Moreover, in the Kuhnian limit ^{z -} _{oo ,} ^we know that 9 reaches a fixed point ^value

In the following ^{9 will have this} meaning of the Kuhnian fixed point. The result is the renormalized

expression of the form factor H[q] (2.10)

where

I2d, 13d ^{are the same as in the} original expression (2.10). ^{We note that this} renormalized expression,

valid in first order in 9, yields ^an approximate expression ^{of H} for any d. One ^can take the limit d -+ 4 in the factor of 9, ^{but one} may also set d ⁼ 3

directly ⁱⁿ (2.12), taking ^{for 9} the best known

approximation [4]

In this respect, ^our result differs from previous

studies [10] ^where only ^{the pure} E-expansion ^has

been considered. As we shall see later, ^this approach

makes the agreement ^with experiments better. For

large values of x ⁼ q2 X2/2 ^the asymptotic ^behaviour

of H[ q] has been evaluated in reference [7] (see

also [8]). ^{We found}

where v is the usual critical index governing ^{the size}

of the Kuhnian chain

and where hoo ^{reads :}

C being ^{Euler’s constant, C = 0.577...} Naturally,

this behaviour can be obtained from equations (2.12)-(2.16) in the limit d

4. In the same limit d

4, the results of references [9, 10] ^must agree with our expressions (2.12)-(2.16). They ^{are not}

written in terms of the same variables and the

analytical ^{form is not the same.} However, ^{we have}

checked the agreement ^to first orders in the series

expansions, correcting ^{a statement} made in reference

[10] : the results agree also well with the order

x2 expansion ^of H[q] given by Witten and Schafer in reference [9].

3. Series expansions.

The integral expression (2.12) ^of H ( x ) ^{cannot be} given ^{a much more} useful form in terms of known functions. It is however suitable for numerical inte-

gration ^and comparison ^with experimental ^results (see ^Sect. 7). ^{In order to} obtain another piece ^of information, ^{we shall} perform ^{here the} series expan- sion of H ( x ) , and obtain all its moments, ^that is, ^all

the radii of gyration of the Kuhnian polymer ^chain.

Performing the series expansion in powers of x of

H (x ) require careful and systematic calculations.

First of all, ^{we have} trivially ^{for the} Debye ^function Ho (x)

We have found for the I, ^J functions :

Again, ^we want to stress the fact that these functions ^are perfectly finite when d ⁼ 4, ^{as are their series}

expansions.

Inserting these results into the general expression ^of H[q] (2.12), ^{we find, to first order in g :}

where

and

where Ad ( n) has the form

and where Sd ( n ) ^{is defined by}

Actually Sd ( n ) ^{can be} calculated exactly. ^{We find}

For d ⁼ 3, ^this gives, ^{after some} calculations

The limit of (3.10) ^{for d -+ 4,} apparently singular, ^{takes, as it must, a} finite value given by [6] :.

where T (z) - d ln T z ^{is the usual} Eulerian 1/I’-function, ^For d = 4, ^the expression ^of Ad ( n ) simplifies ^{and we find, after some} calculations, ^the simple ^result

where Sd = 4 ( n ) ^{is given by} equation (3.11b).

We have also evaluated the first A’( n ) coefficients for general ^{d and n = 1, 2, 3. We find} successively,

after some arithmetics :

For the specific ^{case d = 4} (which ^{can also be obtained from} equation (3.12)), ^{we find}

For d ⁼ 3, equation (3.13) gives

These figures clearly ^show the difference between an approximation using coefficients taken at d ⁼ 3 and a pure s-expansion, ^{which uses the d = 4} coefficients.

We may compare ^{now our} results to previous calculations. Inserting the values (3.14) ^for

d ⁼ 4 into the series (3.6) ^{of H, and} using ^the Kuhnian fixed point ^value (2.11) ^of 9, ^i.e. 9* = g + ... , ^{0 we find}

the e-expansion ^of H ( x ) ^{near the} origin x ^{= 0 :}

The terms up ^to order x2 have been calculated in different notations by ^{Witten and Schafer} [9] (see ^also [10]),

and agree with ^ours.

°

As already mentioned, ^{our results are more} flexible since we have at our disposal ^a dependence upon the dimension d, which will permit ^{us to obtain} approximate values better than the first order E-expansion.

We are now in position ^for calculating all the radii of gyration.

4. Radii of gyration.

Owing ^{to the} definition (2.2) of the form factor H[q], ^{it is clear that it} yields ^the following expansion ⁱⁿ

powers of q2:

a2n is the angular average, in d dimensional space, of the 2 nth power of the cosine of the azimuthal angle ^{8 :}

where nd represents ^the angular ^{variables in d} dimensions. Using ^{the identities}

dil d = dil d - 1 (sin 8) d - 2 d 8, ^{0-- 0-- iT, and}

where Sd ^{is the area of the unit} sphere ^{in d} dimension = 2 1T d/2 _{, lt ’} is not difficult to obtain the explicit F (d/2) ’

value :

Now we define the radius of gyration Rgn] of order n as

and we have identically

Identifying ^this expansion with the calculated series expansion (3.6), ^{we find} immediately ^{the set} of radii of

gyration :

Using ^{the value} (4.3) of a2n yields ^the explicit expressions of the radii of gyration

This gives ^{the first order} expression in 9 of the universal ratio, relating the 2 nth radius of gyration

RJ2n] ^to the 2 nth power X2n of the size X of the Kuhnian swollen chain.

The universal but trivial result for a Brownian chain is simply ^obtained by setting ^{9 = 0 and X2 = S}

in (4.7). ^{Thus the} analytical coefficient depending ^on (n, d) in front of X2n is Brownian, ^{and must not be} e-expanded.

Let us now consider some particular results. For

n =1, ^we find, owing ^to equations (4.7) (3.13),

which recovers a known result [4]. ^The correspon-

ding e-expansion is, according ^to (2.11) :

a well known result [9]. ^{If we use the} expansion (4.8)

for d

3, and take the best value (2.17) ^{9* = 0.233,}

we find

while the naive e-expansion (4.9) with - = 1 gives

The best known value is given by ^the e-expansion ^to

second order [13] :

which, ^{for E} = 1, yields N 1 ^{== 1 -} 0.0410. Thus ^we

see that the approximation ^{with d = 3 in} (4.8) gives

a slightly better result than the E-expansion ^{in first} order, if result (4.9bis) ^{is taken as a reference.}

However, ^we already ^{see that the second order}

corrections are important. ^{We shall} return to this later. The next moments RJ4], RJ6], ^{are obtained}

from (4.7). They ^{have the} following e-expansions

(see (3.14))

The values obtained by retaining ^{the d}

^{3 values} (3.15) ^{of the A’} coefficients and the value 9* ⁼ 0.233, ^are

which are not very different from the previous ^{ones for E = 1.}

Let us consider now the general pure E-expansion of the radii R G E23. ^{We have} explicitely

where

It is interesting ^to study the behaviour of the E-correction for large ^{values of n. For} large ^{n, one has the} asymptotic ^behaviours

while C ( n j ^remains bounded. Thus ^{we see} that the s-corrections become very large when n increases. The behaviour or n large

is related to the behaviour of the probability distributions P ( r ) ^for large ^{distances r} between any ^two points

of the Kuhnian chain, ^which essentially ^{behave as}

where t, ^{0 --} t -- 1, is the fraction of the chain located between the two points, ^{and 9 is a} coefficient

ØI = 1 + 0 ( e ) . ^These probabilities ^{will be} calculated in the next sections. From the asymptotic ^behaviour (4.16) ^{it is} actually possible ^to reobtain the value (4.11) ^of RJ2n], together ^{with the} asymptotic expression (4.15) ^of A’(n). ^{We shall not} give here the details of this calculation.

5. The universal ratios

The preceding discussion leads us to an interesting point : ^{since, for} large n, large ^{distances are} dominant in

the average values RJ2n], ^{one can} imagine that these RJ2n] ^have essentially ^{the same kind} of behaviour ^as

the average end-to-end distances of order 2 n, ^{defined as}

Until _now, we have measured RJ2n] ^{in terms of](2n =} 2013! ^{and we may} anticipate that the unbounded

growing ^of Ad ( n ) (4.15), ^{as a function of n, is directly} related to this fact. So it is interesting ^{to evaluate the}

universal ratios N n

The factor ( n + 1 ) ( n ⁺ 2 ) is Gaussian and such that ’ N , ^{=1 for Brownian chains,} since for the latter :

The moments R ^[2n] can be found in a previous ^work [6] (see ^also [16]). ^{Let us indicate here the main} steps of a immediate direct calculation, which will be useful later. We consider the generating ^function

which has ^a series expansion ^{similar to} (4.1)

This generating ^{function can} be evaluated exactly ^{in the same} way ^as H[q] in the first sections. We write as

in equation (2.3)

and the diagrams contributing ^{to !E} ( q, S, o ) , ^{at first order in the} interaction b, ^{are shown in} figure 2, ^and

are naturally ^{identical to the} diagrams (0) ^and (1) ^of figure ^{1. We find} immediately

Fig. ^2.

^The diagrams contributing ^to ( q, S, o ) , ^to

first order in b.

where, ^as before, y = q2 S/2, ^{z =} (2 IT )- d/2 bS2 - d/2. The renormalization of G [q] ^{consists just in} replacing ^y by ^{y =} !!’õ 1 (z) ^x (see (2.9)), ^{where x =} q 2X212, ^and by expanding ^{to first order in z = 9 + ....}

The result is extremely simple :

where

Sd(n) ^{is the same finite sum as in} equations (3.8)-(3.11).

Identifying ^this expansion (5.7) ^{with the} expansion (5.4) ^of G [q] , yields ^{the set of radii}

Using (4.3) ^and (5.8), ^{we find} explicitely

in agreement ^with previous ^results [6].

We are now in position ^for calculating ^{the universal} ratios N,,. Using equation (4.6), ^{we find} immediately

Owing ^{to the} expressions (3.7) ^and (5.8) ^off ( n ) , ^[8’ ( n ), ^we find in first order in 9 :

where ABd (n) , Sd ( n ) ^are respectively given by equations (3.8) ^and (3.9). Considering ^now the pure ^e-

expansion ^{of this} ratio n ^{we have to set d = 4 in} Ad ( n ) , Sd ( n ) . ^{Thus we may use the} expression (4.12) ^of

these coefficients to get the universal values for the Kuhnian chain

where 9 -+ ’8 E ^{+ ..., and where}

Now we check that for n = 1, C ( n =1 ) =1/12, ^{as it should} [9]. Moreover, ^{for n} large, C ( n ) ^{is a well-}

behaved function, ^as expected. Indeed, the consideration of the ratio N n (5.10) ^has suppressed ^from And ( n )

the term Sd ( n ) , ^which contained, according ^to equation (4.14), ^{all the} divergences ^of Ad(n) ^for large ^n.

In particular, ^{for n - oo C} ( n ) ^{tends to} the finite limit

which can be evaluated to yield [17]

So we can write the slightly academic, ^but interesting result for ^a Kuhnian chain

°RJ2nJ

while for a Brownian chain the quantity ( n ⁺ 1) ( n ⁺ 2) o R [2n ] ls exactly equal ^{to one, for any n. So we} OR E2n]

have obtained, ^with equations (5.11)-(5.12)-(5.14), ^a complete ^set of universal ratios N n, ^which generalize

the well known universal ratio N _{1 =} 6 RÕ/R2.

6. Kuhnian probability ^densities.

We shall now calculate, in first order in 9, ^the

probability ^densities

of having ^{the two} points s’, s" along ^the chain, 0 , s’, s" , S, ^{in a relative} position ^{r in the d-}

dimensional space. In ^a previous ^work [6], ^{we have}

calculated directly ^{the whole set of moments associa-}

ted with the probability density (6.1). ^{A «} prelimi-

nary » form of the unnormalized probability density

itself has been given by ^{Oono and Ohta} [14], ^but

without any calculation. Moreover, it still contained

a unnecessary dependence ^on microscopic ^parame-

ters. Here we want to calculate briefly ^these probabi- lity ^densities by ^{the same method as} in the first

sections, ^and give ^{them a form as} simple ^and

universal as possible. ^In particular, ^{we shall} give ^the

three universal probability ^laws corresponding ^{to the}

whole chain [15], ^{or to} infinitesimal segments ^located

at the very extremity ^{of the chain, or} inside the chain.

We start with the generating ^function (2.1)

which gives by Fourier transform

h [q, ^s’, s"] ^{can be} easily calculated at first order in z, with the help ^{of the} identity (2.3) h [q, s’, s"] =

Z (q, Defining the useful reduced varia-

St’sit)

*

bles

with

we have [6] :

with

where Il, 12, 13 correspond respectively ^{to the} diagrams 1, 2, 3, ^of figure ^{1. Here} again diagrams 4, ^4’

immediately ^cancel.

The renormalization of h [q, ^s’, s"] ^is quite simple : ^we consider the Kuhnian size Y2 that the fraction of

chain would have, ^{if it were free to swell :}

where z = (2 M ) - dl2 bS2 - d/2 and where ad = du ( 1- u ) ^{0 Ul-d/2 is given by (2.9). We express}

h [q, s’, s"] ^{in terms of the} physical quantity

instead of y’ = q2 tS/2. ^{One has}

Eliminating y’ ⁱⁿ equation (6.4) ^{in favour of} x, ^we find, ^to first order in z, the renormalized expression :

where Ii is the subtracted function

Now all the functions appearing ⁱⁿ (b.11) ^are finite when d -+ 4. We also note that the quantity ^{zt 2 - d/2} appearing ⁱⁿ (b.11) ^is. exactly the Zimm-Yamakawa z-parameter of the fraction t of the chain. In the Kuhnian limit, ^{z -+} oo , it is thus possible ^{to write in} (b.11)

where 9 is the second virial coefficient (in scaling form) and, ^as already ^seen,

in the Kuhnian limit.

We now perform the Fourier transformation of the renormalized expression (6.11), ^{and we} obtain after

some calculations

where 37 is the reduced variable

and where

As already stressed, these functions have ^a finite limit when d -+ 4. We are here essentially interested in the pure E-expansion ^{of the} probability distributions, ^{and thus we} calculate the K functions in the limit d -+ 4. We find after some calculations :

where C is the Euler’s constant C

0.577...,

where

and finally

where

Thus our final result is, up ^to first order in 9 :

where

and where the Kuhnian correction term is given by (6.18) ^and

where K2, K3 ^are given by ^{the set of} equations (6.19)-(6.22).

We notice that the Brownian result is recovered for 9 ⁼ 0, ^{as it must. The} expressions (6.18)-(6.21)

are somewhat similar to expressions ^obtained by

Oono and Ohta [14], but differ on the whole, ^since

these authors calculated only ^the unnormalized

partition ^function.

We shall here exploit ^our results in some particular

cases, where they ^{can be written in a new and} simpler ^universal scaling ^form.

We shall consider three particular ^cases of interest

[6, 18, 19] (See Fig. 3)

a) ^The segment ^is equal ^to the entire chain, ^this corresponds ^{to t} =1, ^{and a’ = a" = 0.}

b) ^The segment is located at one extremity ^{of the} chain, for instance s" ⁼ 0, ^and thus a" ⁼ 0. Moreo- ver, if this segment ^is infinitesimally ^{small, or the}

chain _{is very} long, ^{then t -+ 0 and thus,} according ^to (6.24) : a’ --* ^{+ oo .}

c) ^The segment is inside the chain and very small,

or the chain is very large. ^This corresponds ^to

t--+O and a’--+ _{+ oo ,} a" -+ + 00 .

Let us calculate the functions K2, K3 ^{in these}

limits.

Fig. ^3.

^{The three} geometrical ^cases considered for the

probability distribution : case a, a

0, the .whole chain ;

case b, a ^{= 1 a finite} segment ^{inside at the} extremity ^{of an}

infinite chain ; ^case c, a

3, ^{a finite} segment ^{inside an} infinite chain.

K2 ( f!" , a )

For a --+ 0, ^we have, owing ^to definition (6.16)

For a - + oo , ^we calculate from (6.20)

and therefore

For a’ --* + oo, a" --* + oo , ^we need the expansion

of _o (x ) (6.22) ^{for x small}

which leads for the function K3 (6.21) ^to

a result similar to that for K2 ( f£ , a -+ ⁺ oo ) .

So we have, in the three different cases a, b, ^c described above, for the function IK (f£ , a’ , a")

defined by equation (6.25) : a) ^{The entire chain}

b) ^The extremity ^{of a} very long ^chain

c) The interior of ^a very long ^chain

So we may represent these three cases by ^{an index}

a taking the values 0, 1, ^{3 for cases a,} b, ^c respectively [6] and write in general

We therefore find the expression ^{of the} probability density ^P (6.23) for the three cases a, b, ^{c :}

This formula is a totally expanded expression. ^{It can} be better written in the interesting reexponentiated

form

We have checked that these expressions naturally agree with the ^moments of P calculated directly ^{in a} previous ^work [6]. ^{For a}

0, corresponding ^to the entire chain, equation (6.31) agrees with the result of reference [15], ^{but the} reexponentiated ^form (6.32) ^{is different} from that of reference [15]. (The reexponentiation ^{there is not} minimal). ^For a = 1, 3 ^the probability densities have not been written before.

Here, using the proper scaling ^{variables X =} r2/2 ^{Y2, and 9, we obtain} quite simple universal forms. The

quantity

is a universal function of PI, calculated here in a form involving only 9 ^at first order. We note that expression (6.32) agrees with the expected scaling ^results [18, 19]. ^{At short} distances, ^{we have}

where the exponents

agree with the exponents calculated by different methods in reference [19] ^and [6]. ^{We obtain here the}

universal coefficient to first order in 9, ^or E, which depends ^{on the} position along ^{the chain.}

At large distances, ^the expression (6.32) ^{can be} kept ^as such. It has the expected scaling ^form

since

The indices U a have been obtained in previous ^work [6] by calculating directly ^the high ^{moments of P.}

They ^are identical with the short distance indices

6 a (6.35), ^but this holds true only ^{to the order E. We}

note that the -4 coefficient does not depend ^{on the}

location along the chain. In figure ^{4 we} have repre- sented the three different probability ^functions F (3-, a) = (21Ty2)d/2P(r,a), ^as functions of X ⁼ r2/2 ^Y2, for the three different geometrical

cases a

0, 1, ^{3 of} figure 3. The reference length ^Y

is the Kuhnian size which the fraction of chain under consideration would have, ^{if it were free to swell}

alone in the solvent. Therefore Y does not depend

on the cases a

0, 1, 3 and thus the variable Y has

an absolute meaning. ^{The three functions F} (X, a )

can thus be compared together. ^Then considering

for eXample their maxima fE m,a’ ^one notices that the latter increase with a. This shows an interesting

universal geometrical property ^of the Kuhnian chain : it is more swollen inside ( a

3 ) ^{than at its}

extremities ( a =1 ) , and the local swelling ( a = 3,1 ) is always greater ^{than the} global swelling of ^{the chain} (a = 0). ^We already ^{obtained this} property ⁱⁿ [6] by calculating ^{all the moments of the} probability distributions (see ^also [14]). It is interes-

ting ^{also to} compare these probability ^functions (6.33) (6.36) ^to their Brownian equivalent, ^obtained

for 9

0. It reads simply

and does not depend ^on the location along ^{the chain.}

In figure ^4, the difference between the geometrical properties of the Kuhnian and Brownian chains is thus manifest.

Fig. ^4.

^{The three} different Kuhnian probability ^distribu-

tions F ( f!{ , a ) ^{for a} segment ^located according ^{to the}

three geometrical ^{cases of} figure ^{3a, b, c, with a}

0, 1, ^3.

The variable Y is Y

r 2/2 ^{Y2, Y being the} reference size the Kuhnian segment would have alone in the solvent. For

comparison, ^{the curve} B, FB ( Y ) e- corresponds ^to

a Brownian chain, and is the same for the three cases.

Considering the maxima of the _curves, one clearly ^{sees the} geometrical property of the Kuhnian chain: it is more

swollen inside ( a

3 ) ^{then at} its extremities ( a =1 ) ,

and the local swelling (a = 3,1 ) ^is greater ^{than the}

global ^one ( a = 0 ) .

H[q] given by equations (2.12) (2.16) computed numerically. ^{We set either d}

^{4 in these}

formulae and 9 8 + 1/8 ^{0 in first order, in the}

spirit ^{of the} E-expansion, ^{or we take d}

^{3 in the} integrals (2.13)-(2.16) and the best known value of 9, 9 ^{= 9*}

0.233 in d

3. The first case is presen- ted in figure ^5, the second in figure ^6.

The variable x

q 2 X 2/2 takes values between 0 and 130, ^{in these} figures, ^where H- 1 [ q ] is represen- ted as a function of x (curve R). ^{For both cases and}

for large ^{x, we have also} represented ^the asymptotic

form (2.18)

where v ⁼ 0.588 [21]. ^For ho., the formula (2.20)

taken for E = 1, gives hoo

^{0.89. However, a better}

value can be obtained by interpolating ^{with the} help

of the exact value of hoo for d = 1. Indeed, ^the

Kuhnian limit of polymer ^{chains in one} dimension is that of rigid rods of size X. Thus, the form factor

H[q] (2.2) ^reads exactly :

d=1

Thus we see, owing ^to (2.18), since v =1 for d =1,

that hrL; (d = 1) = o. ^Using then the value

hoo = 2 ( 1 - i (5 - C) ), valid for d = 4 - e,

8

s - 0, ^and hoo ( d =1 ) ^{= 0, we find} by interpola-

tion

The corresponding asymptotic ^curve (As.) ^{is repre-}

sented on both figures 5, ^{6. This curve is calculated}

without any adjustable parameter, ^{and since we use} the best known value of the critical index v ⁼ 0.588,

and the interpolating ^value (7.3), ^{it should} give ^a quite good representation of the real curve for x

large. ^In figures 5, ^{6 are also} represented ^two simple approximations, ^the Debye form factor (curve D)

and the uniform swelling approximation, ^{of Peterlin}

We see that the curves D and P are quite ^different

from the results of renormalization theory, ^{i.e. the}

curves R and As.

Fig. ^5.

^{The inverse} from factor H-’[q] ^{as an universal}

function of q2X2/2. ^D

^{Debye theory ; R :} renormalized

theory ; ^{As :} asymptotic renormalized theory ; ^{P : Peterlin} theory. ^x experimental ^data of Noda et al. without polydis- persity corrections ; · ^{the same} points ^when polydispersity

is taken into account, and induces ^a horizontal translation of the points. d

^{4 means that} the renormalized H [q] ^is

calculated in pure e-expansion, ^with coefficients taken at d = 4.

Fig. ^6.

The inverse form factor as a function of

q2 X2/2, ^{in the d}

³ approximation. ^The only change

with respect ^to figure ^{5 is in the} renormalized curve R which is calculated now in the d

3 approximation, 9 being ^taken equal ^{to 9*}

0.233. The curves D, As, ^P stay unchanged. ^This approximation ^{for R is better, since R is} asymptotically ^{closer to the nearly exact} asymptotic ^curve

As. Curve R is also closer to the experimental points.

We note that the two curves R of figure ^{5 and} figure ^{6 are different for} large ^{x. Since the} asymptotic

curve (7.1) (7.3) ^{should be} good ^{in the} high q region,

it can serve as a reference curve. Thus we see that the curve calculated for d ⁼ 3, 9

^0.233 (Fig. 6) ^is

better than the curve of pure --expansion, ^with

d ⁼ 4 in the prefactors (Fig. 5). ^{The first one is}

indeed closer to the asymptotic expansion. ^{For small}

values of x, ^{the two} predictions ^of renormalization

theory ^{are not} very distinguishable, ^{and even are}

close to the Debye ^curve.

Let us now compare ^our results to the experimen-

tal points of Noda et al. We have taken the ^same

points ^as in reference [20], (Fig. 5), which correspon- ded to three samples ^of polystyrene in toluene.

However, ^{we had to make a} conversion since the abscissa variable of reference [20] ^{was x’ =} q2 RÖ.

Using ^{in three} dimensions N ₁ (see (5.2))

we have x =2= _G 2 R2 N -1 = xl N -1. ^{1 1 We need}

a reasonably ^accurate value of the universal ratio N ₁ which can be found from the second order e-

expansion (4.9bis) [13]. ^Here again ^we ameliorate it

by interpolating ^{to d} = 1, where N I = 1/2 exactly.

Then for d ⁼ 3, ^{we find the} approximate ^value N 1= 0.952. Using then the conversion x = x’ N

we obtained the experimental points ^denoted by simple ^{crosses in the} figures 5 and 6. We then note that these points ^{do not} really ^{fall on} any of the

curves. However, they ^{are closer to the} prediction ^of

renormalization theory (curve R). ^The (too) simple

models of Debye and Peterlin are clearly ^{ruled out.}

In order to obtain a better agreement ^with experi-

mental data, ^{it is} actually necessary ^{to take} polydis- persity ^{into account. For} this, ^{we use a Schultz-}

Zimm distribution of polydispersity 1 + p. It is shown in appendix B that the resulting polydispersity

correction has a very simple form in the asymptotic

limit x 00 . It is

with

Thus the corrected equivalent monodisperse ^{curve is}

shifted to the left. Owing ^to the form of the curve

giving H- 1 [ q], ^{which is} monotonously increasing,

this shift looks like an upward ^{movement of the}

inverse scattering ^curve. This agrees with the polydis- persity corrections made by ^Yamakawa [25] ^{for a} simple Debye from factor. The value x" is, according

to our formalism, the actual value to be given ^{to the} experimental abscissas, ^{since we have chosen to refer}

these data to our theoretical prediction, ^calculated

for a monodisperse system (Appendix B). ^{We have}

done this for the six last experimental points ^{of Noda}

et al. [20]. ^These points correspond ^{to the} sample

called F.2000 in reference [20], ^{i.e. the} sample ^with

the highest ^molecular weight, MW ^{= 21 x 106. Its} polydispersity ^{has been} recently ^studied by ^P. Stepa-

nek et al. in [26]. They ^{found as a} preliminary ^result

1 + p = 1.35, which would be quite ^a large polydis- persity. ^We prefer ^{here to} adopt tentatively ^{a lower}

value. For this, ^we consider the polydispersity ^{of the}

other sample ^{F.850 used in} [20], ^and having ^{the same}

manufacturer [27]. The latter gives ^the polydispersity

for F.850 = 1 + p =1.17. This is already ^an impor-

tant polydispersity, ^{and the} polydispersity ^{of the} sample ^{F.2000, of} greater ^molecular weight, ^could

be greater ^{and close to 1.20} [28]. ^So, using tentatively

the value p ⁼ 0.17, ^{and v =} 0.588, ^{we find for} (7.7) :

The resulting position of the last six experimental points, using (7.6) (7.8), ^{is shown in} figures 5, ⁶ by

full circles. One observes the striking fact that these

points ^{lie now} quite ^{close to the curves} predicted by

renormalization theory ((R) ^and (As», ⁱⁿ figure ^6.

One notices that the agreement ^{is less} good ^with

curve R of figure ⁵ (d ^{= 4} case). ^{All of this is} satisfactory, ^{since the} asymptotic ^{curve, as} already stressed, ^was expected ^{to be} good. ^{The curve R of} figure 6 is close to it, ^{as well as the} experimental points, ^with polydispersity corrections. The Debye

and Peterlin curves are clearly far away from the

experimental data in the large x region.

If we consider now the intermediate values of x in

figures ^{5 and 6 we see} that there is a small branch of the experimental points (up ^{to x =} 60) ^which departs

from the main high ^{x curve} and lies below the latter.

The corresponding points ^{in Noda’s et} al. article [20]

(Fig. 5) ^{are those of the} sample 1-1.002 ^{of lower}

molecular weight ^{than F.2000. It is} important ^{to note}

that the two sets of data for 1-1.002 and F.2000 depart

from each other. In principle, ^one should also

perform polydispersity corrections in this interme- diate regime. However, ^the analytical ^{form of} H[q] being ^not simple in this intermediate x

domain, ^the polydispersity corrections would not take a simple ^{form, but one} expects again, ^{that the} monodisperse equivalent ^{curve will move} upward,

and come closer to the theoretical renormalized

curve R. However, ^{we note that} apparently ^the