Internal correlations of a single polymer chain

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Internal correlations of a single polymer chain

L. Schäfer, A. Baumgärtner

To cite this version:

L. Schäfer, A. Baumgärtner. Internal correlations of a single polymer chain. Journal de Physique,

1986, 47 (9), pp.1431-1444. �10.1051/jphys:019860047090143100�. �jpa-00210338�

Internal correlations of a single polymer ^chain

L. Schäfer and A. Baumgärtner (*)

Fachbereich Physik, Universität Essen, ^D-4300 Essen, ^F.R.G.

Institut für Festkörperforschung ^der Kernforschungsanlage Jülich, ^D-5170 Jülich, ^F.R.G.

(Reçu ^{le 28} fgvrier ^1986, accepté ^{le 22 mai} 1986)

Résumé.

Nous calculons à l’ordre d’une boucle les correlations de deux segments, désignés par n1 ^et n1 + n, d’une chaine de polymère ^en interaction avec elle-même de longueur n1 ^{+ n +} n3. Nous analysons ^{en détail le}

second moment et le premier ^moment inverse de la fonction de structure. Nous présentons des résultats de calculs de Monte Carlo pour ^ces quantités ^sous forme de fonctions d’échelle dépendant ^des rapports n1/n ^et n3/n. ^Ces

fonctions deviennent universelles dans la limite du volume exclu. Nous trouvons un bon accord entre les résultats

numériques ^et analytiques. ^{Nos résultats} permettent ^une vérification détaillée de l’hypothèse ^{de «} blob » utilisée communément pour expliquer ^la différence entre le gonflement du rayon hydrodynamique RH ^{et celui} du rayon de

gyration RG. ^{Nous trouvons} que le modèle de ^« blob » n’est pas valable _parce qu’il néglige des effets de bout impor-

tant. Nos calculs de Monte Carlo suggèrent ^{que la} rigidité ^{à courte distance} pourrait ^être responsable ^{de la dif}

férence observée expérimentalement ^entre RH ^et RG.

Abstract.

We calculate to one loop order the correlations of two segments labelled n1 and n1 ^{+ n in a self} interacting polymer ^{chain of} length n1 ^{+ n +} n3. The second ^moment and the first inverse moment of the structure function are analysed in detail. Results of extensive Monte Carlo calculations for these quantities ^are presented

in the form of scaling ^functions depending ^on n1/n, n3/n. These functions become universal in the excluded volume limit. Numerical and analytical ^{results are found to} agree well. Our results allow for ^a detailed check of the blob

hypothesis commonly ^{used to} explain ^the difference in the swelling ^{of the} hydrodynamic ^radius RH ^as compared ^to

the radius of gyration RG. ^{We find} that the blob model is not valid, ^{due to the} neglect ^of important ^{end effects. Our}

Monte Carlo calculations point ^{to short range stiffness as source of the} experimentally observed difference between

RH ^and RG.

Classification Physics ^Abstracts

05.20

05.40

61.25

1. Introduction.

An analysis of the internal structure of a single long

chain molecule in solution is a nontrivial task which has attracted some attention during ^{the last} years.

This problem is of interest not only ^{in its own} right

but also with respect ^{to the} apparent difference in the behaviour of the hydrodynamic ^radius RH ^as compa- red to the radius of gyration RG’ ^Most experiments

find [1] ] that the effective exponent vH governing ^the

chain length dependence of RH ^is systematically ^smaller

than the corresponding exponent VG. This finding -,

being ^not without contradiction [2] -, can be explai-

ned [3] ^{in an} intuitively appealing way by the idea of

« temperature ^{blobs »} [4]. ^{The blob} concept ^assumes that internal parts ^of length n ^{of a chain are less}

swollen than the total chain, ^their swelling ^factor being given approximately by ^{that of a} chain of total

length ^n. The different weighting of internal parts inherent in RH ^as compared ^to RG ^then explains ^the

difference in the effective exponents. ^{To check this} idea one clearly ^{has to} analyse the internal correlations in more detail.

Internal correlations cannot be measured easily.

They have been evaluated numerically by ^exact

enumeration [5] ^or Monte-Carlo [6-10] ^methods.

On the theoretical side normal pertubation expan- sions [11-13] for small excluded volume have been

presented, ^{which via} Flory-type [14] ^or renormaliza- tion [15] arguments ^{have been extended to the} region

of large excluded volume. Some _proper ^renorma- lization _group [R.G.] results exist for the distribution function of the internal distances [16, 17], ^{and the}

2nth moment of this function recently has been dis- cussed in detail [18]. All the different methods _agree in their qualitative ^{results. In} particular ^{it is now well}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047090143100

established ^{that the} swelling crucially depends ^{on the} position along the chain of the internal part ^under consideration. ^A part ^{in the center} of the chain shows

considerably larger swelling ^{than a} part ^{of the same} length ^{but near the end of} the chain. Obviously ^this

effect casts doubt on simple applications of the blob concept [7, 9].

Despite the extensive work published there remain

a number of _open questions.

(i) The second moment r;ltnl +n > of the correla- tion function between segments ni, ni ^{+ n} (compare Fig. 1) has been discussed in detail [ 18]. ^{This moment}

can be used to evaluate RJ. ^{For the} corresponding

evaluation of RH ^{we need a R.G.} calculation of

C I rn l,n 1 + n I - 1 > ^{which has not yet been presented}

Instead, ^the approximation

has been used, ^the validity of which is doubtful.

Fig. ^1.

Notation for internal correlations used in this work.

(ii) ^No comparison has been made between R.G.

results for r;b"1 +" > ^{or (} r"b"1 +" ,-1 > ^{and Monte}

Carlo data. Thus the accuracy of the analytical ^results

is not known.

(iii) ^{No critical} quantitative analysis of the blob-

hypothesis, ^{based on a detailed} understanding ^{of the}

internal correlations, ^{has been} published ^In parti- cular, ^the explanation ^{of the} discrepancy among vG and vH has ^not been checked.

It is the aim of this contribution to clarify ^these

three aspects. Concerning ^{the first} point ^{we here} strongly rely ^{on work of J.} Grigat ^{and one of the} pre- sent authors which has been laid down in reference [19].

To treat the second point ^we carry through ^additional

extensive Monte Carlo calculations. To clarify ^the

third point ^we compare RG, RH ^as calculated directly

via R.G. methods to results arising from various ver-

sions of the blob hypothesis.

Our analysis, combining the results of R.G. and Monte Carlo calculations, ^{leads to} the result that the blob model should not be taken as a valid _represen- tation of internal correlations. It misses important

effects and leads to wrong results for RH.

The structure of our paper is as follows. In section 2

we present one-loop ^results for r;t,n’l +n) ^and

I _{rnt,nl +n} 1-1 ). Section 3 is devoted to our Monte Carlo calculations. Section 4 contains the comparison

between numerical and analytic ^results. RG ^and RH

are calculated in section 5, and section 6 contains the

analysis ^{of the blob model. In section 7 we summarize}

our conclusions.

2. Renormalization _group results for internal corre-

lations.

Z .1 THE CORRELATION FUNCTION IN MOMENTUM RE- PRESENTATION. - We use the standard model of a

selfinteracting ^{chain with N Gaussian} segments ^to calculate the correlation function for two segments

n 1, n 1 ^{+ n} (see Fig. 1). ^{In momentum} representation

this function is defined as

where

and

In (2.1) 3 denotes the partition function of a single

chain so that S is normalized :

1 is the mean size of a Gaussian segment ^and P. > 0

denotes the (dimensionless) ^{excluded volume} para- meter. To first order in P,, equation (2.1) ^is easily evaluated, provided ^{we invoke} the condition N > 1 to replace summations over segment ^indices by integrals. ^{We find}

where

and

The h represent the contributions of the diagrams

of figure ^{2. The other one} loop diagrams drop ^{out. As}

usual we have written the spatial dimensionality ^{of the}

system ^{as d}

^{4 -} E, and we have expanded ^the diagrammatic contributions in _powers of E. In (2.6)

yEu denotes Euler’s ^constant.

,Due to logarithmic divergencies ^{in the} limit9of large

chain length expressions (2.4)-(2.8) ^{are not} directly

useful. The R.G. provides .the additional information needed to set up ^a sensible theory in all the parameter range. It proves and exploits the fact that physical quantities ^{like (T are} invariant under a change ^{of the} microscopic length scale 1 --+ 11A, provided ^{the other}

Fig. ^2. -1-loop diagrams contributing ^to the internal correlation function.

parameters (n, N, n 1, Pe) ^are changed appropriately.

The method has often been explained in the literature and we thus only present ^the equations defining ^our

renormalization scheme. (We ^{use massless renorma-}

lization. For more information see [20] and references

given therein.) The unrenormalized parameters ^are related to renormalized counterparts ^via equations

where (2 .11 ) stands for the renormalization ^{of any}

chain length ^so that the ratios n-1, n3, ^N (Eq. (2. 5ii))

are invariant under the R.G. The renormalization factors Z4/ZZ, ZIZ2 ^are determined as power series in _g by imposing normalization conditions on certain renormalized correlation functions. We here need the results

-

Substituting equations (2.10)-(2.13) ^into equations (2.4)-(2.9) ^we find the renormalized probability

distribution (1)

(1) After this work was finished we received a preprint by

B. Duplantier in which he also calculates this function. His

result is given ^{in terms} of different variables and in a less

explicit form but is equivalent ^{to ours.}

where

Exploiting ^{the structure of the} dilatation _group

together ^{with the} expansions (2.12) (2.13) ^{carried to} higher ^{orders one} derives differential equations gover-

ning ^the A-dependence of the renormalized _parame- ters. Integrating ^these equations ^one finds within the present renormalization scheme

Here _v, OJ denote critical exponents, taking ^{the values}

in three dimensions. f is related to g by

where g*

8/4 ⁺ 0(82) ^{is the} renormalized fixed

point coupling ^{constant. The} excluded volume limit is reached for f

¹ corresponding ^{to £ - 0. The} para-

meter fo, ^which givesf(A

1), ^{is some} unknown, ^but regular, ^function of fie and thus of temperature. ^In the limit fie ^{8 it is} given by equations (2.10), (2.12)

for £ = l,g-+O:

Equations (2.16), (2. 17) ^{are based on a} second order calculation of the renormalization factors.

It is useful to define a generalization ^{of the} z-para- meter of the well known two-parameter theory ^as

Near the 0-temperature ^{we can} assume fo - ^{T - 0,}

in which case the theory ^{reduces to the} two-parameter scheme as has been discussed in detail in reference [20].

Equations (2.14), (2.19) present ^{the basic} equations

to be evaluated in the following. ^Some lengthy ^calcu-

lation [19] shows that the Fourier transform of

s(q, ^n, nl, n3) ^{to order c} is consistent with the result of Oono and Ohta [17] ^and thus also with the limiting

power laws given by ^{Des Cloizeaux} [16].

In the renormalized formalism A is still a free parameter ^{which has to be} choosen in a sensible _way.

The standard procedure ^{is to} identify 11A ^{with some}

correlation length ^{of the} system. ^{Here we will} employ

and _compare two different prescriptions.

A. We fix A by the condition

leading ^to

This corresponds to r,’ , ,,, ,, > - 1’/ A’ ^{and is a}

natural condition as long ^{as we} concentrate on internal correlations.

B. We fix A by ^{the condition}

corresponding to ( r§ ) - ^{[2/;’2, a} natural condition if we are concerned with properties of the whole chain.

2.2 CALCULATION OF r; 1, n 1 +n).

^{The second}

moment of S is defined as

A simple calculation yields

Using equations (2.16); (2.17), (2.19) ^{we can write the} prefactor ^as

where

for T -+ 0 reduces to the internal distance at the 0-

point. ^We thus find the swelling ^factor

Of interest to us is also the ratio

giving ^{the excess} swelling ^{due to the} presence of the

end pieces nl, n3.

Fixing £ by choice A : nR{À)

^{1, we eliminate nR}

from these equations ^{and find} f

f(z), a#

«#(z, fii, n2), ^{PE =} PE{zn, fii, n2). ^In the excluded volume limit _zn ^-+ 00, Le. f -+ 1, PE ^{becomes a universal}

function of n 1, n2 ^with limiting « universal ratios »

All these results coincide with those [18] given by Duplantier, ^{who has} extensively discussed the depen-

dence of the swelling ^on nl, n3, ^{N. We here want to}

add _of only ^a figure showing ^the swelling OC2(Z", nii ^nl)

a piece ^of length n ^{in the middle of a finite} length

chain as function of Zn 1’0; n1/2 [d = 3]. Figure ^{3 shows} clearly ^{that even} outside the excluded volume limit the

swelling takes its maximum value for some pair ^of segments ^well separated from the ends of the chain.

More numerical results are given in section 4.

2. 3 CALCULATION OF I’n I, n I +n I - 1 >.

^{To cal-}

culate the first inverse moment we use the _represen- tation

Fig ^3.

Upper ^{curve :} swelling ^{of a} part ^of length n ^{in the}

centre of an infinite chain. Lower curve : end-end swelling

of a chain of total length ^n. Curves zN = 10 ^or z,

20 :

swelling of the central part (length ^n, n3 ⁼⁼ nl) ^{of finite}

chains characterized by ^total length z-parameters zN

10

or 20, respectively. ^For zn ^-+ 0 the curves coincide with

oe’(z., ^oo, oo). They end for z.

zN (i.e. n ¹

0) ^{on the}

curve aE(zn, 0, 0).

The evaluation of the integral ^is straightforward ^but lengthy. (Compare ^Ref. [15]). ^We present ^{the result} for the hydrodynamic swelling ^factor

We find

In evaluating this result we again ^{fix £} by condition A : _nR

1, ^thus writing (XHE ^as function (XHE(Zn, nt, n3). ^The integral (2.36) ^can be evaluated analytically ^{in the} limiting ^{cases x}

^{0 or x}

^{oo to} yield

(Note ^{that the same} notation a

0,1, 3 ^{for the} limiting ^{cases has been} used in Ref. [18].)

Again ^{we can} define the excess swelling which in the excluded volume limit becomes ^a universal function

of nl, n3 :

The ratio connecting RE and RH ^{also reduces to a} universal function in the excluded volume limit

It takes the limiting ^values

where a is defined as in equation (2.37). ^{With the} help

of an elegant analytical continuation these limiting

cases for RH have also been evaluated in reference [18].

For PEIH it ^{is easily} checked that the results given ^there

coincide with those presented ^here.

The results (2.40) demonstrate that the swelling again depends crucially ^{on the} position along ^the

chain of the piece considered, ^the swelling being

stronger in the middle of the chain. Comparing equation (2.30) ^and equation (2.40) ^{one finds that}

the effect is even stronger ^for RH ^{than for} RE. ^To explain ^{this we refer to} the small distance behaviour of the spatial distribution function. As des Cloizeaux has shown [16] 5(r) ^{behaves as}

where 00 01 03. ^{Thus the} probability ^{of a close}

encounter of the ends of a subchain decreases if the subchain is shifted from the end to the middle of the chain. This effect qualitatively explains ^the position dependence ^{of the excess} swelling pE, PH, and since RH,

as compared ^to RE, weights small distances stronger it is clear that PH should exceed pE.

Numerical evaluation of the complicated ^expres-

sio (2.39) reveals the surprising ^{fact that} pH in all

the _{parameter range} ^is almost proportional ^to pE : PH I"’tJ 1.55 pE. Plots for PH therefore look similar

to those for _pE which have been presented ^{in refe-}

rence [18] (1/2 Fi in that reference coincides with pE).

Further numerical evaluation will be found in section 4 where we compare ^our analytical ^{work to Monte}

Carlo results.

3. Monte Carlo simulations.

The polymer is modelled by ^the « pearl-necklace »

chain consisting ^{of N + 1 hard} spheres of diameter h

0.531 freely jointed together by ^N rigid ^{links of} length ^{I. The} specific size of the hard spheres ^{has been}

chosen in order to avoid as much as possible ^non- asymptotic contribution to the excluded volume effect arising ^from microstructural properties ^inherent

to any polymer ^{model. In fact,} it has been shown by

Monte Carlo renormalization methods [8, 23] applied

to the pearl-necklace model, ^{that at the fixed} point (hll)* -- ^{0.53 the r.m.s.} end-to-end distance exhibit

asymptotic properties (i.e. ^{R oc} NO.59) ^{down to} very short chains of N = 20.

Ensembles of configurations ^{have been} generated by the conventional kink-jump technique [24] : ^a

chain configuration ^is changed locally by trying

rotations of two successive links around the axis

joining ^{their end} points, by ^an angle 0 ^{chosen ran-}

domly from the interval (- AO, AO). ^The parameter

AO is arbitrary. ^{If an end} point ^{of the chain is chosen}

the terminal link is rotated to a new position by specifying ^two randomly ^chosen angles (0, 0) ^{in three} dimensions, ^{with cos 0} being equally distributed in the interval - 1 cos 9 1. If the particular sphere

in its new position intersects with _any other sphere

of the chain, ^{the trial} configuration is discarded and the old configuration ^is retained and counted as

the « new » one.

Simulations have been performed ^for 6 ^{N +} 1 160. Estimates of internal distances rå > ^and

1 rij 1- 1 > ^{with 1 i, j N} have been obtained from

averaging ^{over 3 x 104} (for ^{N + 1} = 160) ^{to 106} (for N + 1 = 6) uncorrelated configurations. ^The

correlation time n separating uncorrelated configura-

tions has been taken in the order of n > N 2’2, ^which

is the well known typical relaxation time of self-

avoiding chains. The statistical error has been esti- mated from the fluctuations of the data; it is in the order of less than 2 %.

4. Comparison of numerical and analytical ^results.

Figure ^{4 shows a} doubly logarithmic plot ^{of several} length scales characteristic of the total chain. Besides

R2(N, 0, 0) ^and Rii l(N, 0, 0) ^we also have plotted

the radius of gyration

and the hydrodynamic ^radius

Over a large range of N all these quantities except ^for the hydrodynamic radius follow a power law which cannot be distinguished from the theoretical law R ’" NV ’" NO.588. This suggests ^{that our numerical} calculations ^are well within the excluded volume limit.

This interpretation ^{is confirmed} by the evaluation of several universal ratios which in table I are com-

pared ^to the results of the e-expansion.

Not too surprising ^the N-dependence ^{of the} hydro- dynamic ^radius RH(N) ^differs markedly ^{from that of}

the other length ^scales. Extracting ^from RH(N) an

effective exponent ^we would find a N-dependent vH,

definitely smaller than v. The non-asymptotic ^beha-

viour is clearly exhibited in figure 5, ^where RH(N)/

RH(N, 0, 0) ^is plotted ^as function of N -’1’. In the excluded volume region this ratio takes a constant value which from figure ⁵ is estimated to be close to 0.8, but the _range of N considered here does not cover the

asymptotic region for that ratio. We will come back to this result below.

Fig ^4.

Doubly logarithmic plot ^of length scales of the total chain as function of N. Squares and circles give ^the

Monte Carlo data

a) R’ ^and R’/;. ^The straight ^lines give the power law N 1.18.

b) RH ’(N) ^and RH I(N, 0, 0). ^The straight ^lines represent the power law N -0.59

Table I.

Universal ratios.

(1) WITTEN, ^T. A., SCHAFER, L., ^J. Phys. ^{A 11} (1978)

1843.

(2) Extrapolated ^from figure ^5.

(3) BENHAMOU, M., MAHoux, G., ^J. Physique ^{Lett. 46}

(1985) ^L-689.

Fig. ^5.

Monte Carlo data for the ratio 4/3 RH(N)IRH(N, 0, 0) plotted ^as function of N-1/2 . The straight ^{line serves to} extrapolate ^{to N}

^oo.

All the other quantities, including RH 1(N, 0, 0),

N > 20, ^{are well described} by ^the asymptotic ^laws.

Therefore it is justified ^to compare the numerically

determined excess swelling pE, PH ^to analytical ^results

evaluated in the excluded volume limit. In figure ⁶

we have plotted

N-n N-n

forni = 0, n-3 =n ^{z =} oo ; or n, =n3 = 2n ’

z

oo, ^as function of n/N. Similarily, ⁱⁿ figure 7,

Fig. ^6. _ p2 00,0, N-n (upper _n part) and I p2 ₃ 00, N-n _2n

N-n

2 n ) ^{(lower part).} Monte Carlo data for different values of N and n are given. The full lines represent ^{the one} loop approximation of section 2.2. The estimated ^error of the data is indicated

Fig ^7.

PH oo, ^0, ---n- ^{(upper part) and 1 3} PH 00, ^{2’n ’}

2ït ^{(lower part).} M onte Carlo data for different values of N and n are given. ^The full lines represent ^{the one} loop approximation of section 2.3. The estimated error of the data is indicated

n - N-N-B ^’ven.

n ^or ( N - n 2 n ^given

In the excluded volume limit all these functions are

universal. This is verified here by ^the scaling ^behaviour.

Within the numerical uncertainty the data for dif- ferent values of n trace out the same curve. Further- more, the numerical data fall close to the theoretical

curves calculated from the results of section 2. This

good agreement suggests ^{that we} indeed have found adequate results for the internal correlations.

To exhibit the deviations from the excluded volume limit which occur if we consider _very short (n 20) parts ^{of the} chains, ⁱⁿ figure ^{8 we have} plotted

as function of ni for several values of N

n + 2 iij.

For n large enough the Monte Carlo data for this

quantity ^{scale as} expected ^{For n 15} scaling ^breaks

down and the data drop ^{down to zero in a} way depend- ing ^{on N. This was to be} expected, ^{since in the limit}

n === 1 the ratio (4.4) ^{has to} vanish. Much less expected, however, ^{is the fact that this} sharp drop ^{cannot be}

understood on the basis of the cross-over equation (2.21) for the effective coupling ^{constant Of course,}

also the effective coupling ^constant decreases with

decreasing ^n, but the well established crossover form

(2.21) ^{leads to a} much slower decrease. A typical ^result,

where the single nonuniversal parameter ^{of the cross-}

Fig. ^8.

Ri(n, Ei, nt)/Ri(n, 0, ² Ei) - ^{1 as} function of nt.

Monte Carlo data for three different chain lengths ^are given. ^{The full curve} gives the theoretical excluded volume limit. The broken curve gives typical theoretical cross-over

behaviour. The data shown do not use the full statistics of

our calculations, ^which explains ^the larger ^{scatter of the} points.

over calculation is fitted close to the plateau value,

is included in figure ^8.

We believe that this marked difference between

analytical and numerical results reflects the difference in the microstructure of the chain models used The Gaussian model has a very soft microstructure,

whereas the Monte Carlo chain is fairly stiff. Thus short parts of the Monte Carlo chain can only weakly

react on the _presence of other parts ^{of the} chain, ^an

effect which cannot be simulated by the Gaussian model. Similar effects of chain stiffness have also been

reported by ^Mattice. (See the discussion connected to Fig.1 in the second paper of Ref. [9].)

We then are led to believe that also the _non-asymp- totic behaviour of RH(N) ^{shown in} figure ^{5 is a conse-}

quence of microstructure rather than indicating ^cross-

over from excluded volume to 0-conditions. Within the Gaussian model we expect ^{a smooth cross-over} of 4/3 RH(N)/RH(N, 0, 0) ^{from -} 0.84 in the excluded volume limit to 1 in the 0-limit. Figure ^{5 shows no} sign ^{of such a} behaviour. Rather the simple propor-

tionality ^{to N - 1/2} exhibited there points ^{to the} importance ^of nonuniversal microstructure effects.

We note that a quite similar result can be found in reference [25], ^where RH(N) ^{has been} calculated by

Monte Carlo methods using ^{chains on a} lattice. Near the 0-point the behaviour Rii 1 N -1/2 (1-Cons>

N - 1/2) ^{is found} Clearly ^{this is} again ^a microstructure effect not contained in the Gaussian model (2).

Comparing figure ^{5a of the first} paper of reference [6]

or figure ^{4 of the} first paper of reference [9] ^to figure ³ given ^{here one} finds that the previous ^{Monte Carlo}

work qualitatively agrees with our results. For a

quantitative comparison ^we have extracted from

figure 3 of reference [7] ^values of p2 , averaged ^over n 1

(2) ^{We thank B.} Duplantier ^for bringing this reference to our attention.

for fixed _n, N. In the excluded volume limit ( cp

0 »)

these data _agree well with the e-expansion. Qualita- tively ^{the recent} work of reference [10] ^also agrees with our results, ^{on the} quantitative ^{level, however,}

there is a marked difference. The data for different chain lengths ^{do not} scale, and the maximum value of the excess swelling ^{is about} 50 % larger ^{than our}

result

5. Correlation lengths of the total chain.

5.1 THE RADIUS OF GYRATION.

We calculate the

swelling ^factor

Replacing summations by integrals ^and using equa- tions (2 , 28) ^and (4 .1) ^{we find}

The n 1-integral ^is easily ^{evaluated to} yield

where

Before evaluating ^the integral (5.5) ^{we need to fix the}

scale parameter ^{Ä. To be} precise, 0152i(nR, x) ^is strictly

invariant under a change of Ä, ^so that the choice of A ^is

irrelevant provided ^{we know} 0152i exactly. ^Given only ^an approximate result, however, the choice of A _may become an important question. Normally ^{one cal-}

culates a’ (N) ^{from the} segment-density correlation function which involves only ^{N as chain} length

variable. Then condition B : NR(A)

^1, is the natural and distinguished choice. With this choice f

f (zN) becomes independent of x, and nR = nR/NR ^x NR

^x.

The integral ^{5.5 can} easily be evaluated analytically

to yield

Choice B

f

f (zN) ^is given by equation (2.23). ^{In the 0- or}

excluded volume-limits equation (5. 7) yields

These results are also found from a direct calculation

[20] ^{of the} segment density correlation function.

(We ^note that in Ref. [20] ^{A was chosen in a different}

but equivalent way. The results are easily ^{shown to be} equivalent ^to 0(s), ^and numerically they ^{can be} brought

into excellent agreement by rescaling zN).

We also can fix A by the choice A : ng(£)

^1.

Then the term - In nR in equation (5. 6) vanishes,

but f ^{becomes a function} of zn = zN Xe/2. The resulting integral ^can be evaluated analytically only ⁱⁿ limiting

cases. We find Choice A

The excluded volume limit happens ^to be insensitive to the choice of A, ⁱⁿ contrast to the limit zN 1.

Numerical evaluation of aG ^{for all zN} shows, however,

that the region where choices A and B differ, ^{is res-}

tricted to very small _zN where a$ = ^{1. In figure 9 we}

Fig. ^9.

aJ(ZN). The upper ^curve gives ^{the result of the one} loop calculation. The curve identified as « Blob(E) » gives

the result of the blob model as explained ^{in section 6.}

have plotted aG(zN) calculated according ^{to choice A,}

and on the scale of that figure, ^no difference between choice A and B is found

5.2 THE HYDRODYNAMIC RADIUS.

The derivation of the previous ^{section can be} repeated ^{for the} hydro- dynamic ^radius.

We again introduce the swelling ^factor

where RH,o has been defined in equation (2.33).

Carrying through ^the n 1-integration ^{we find}

With choice B, aH 1 again ^{can be evaluated} analytically.

Choice B

limiting behaviour :

With choice A we have to evaluate aH 1 numerically.

Choice A

Limiting behaviour :

The cross-over behaviour is plotted ⁱⁿ figure ^10.

Fig. ^10.

^One loop results for aH 1 (zH). ^Curves indicated A

or B are calculated from the full 0 (E) result for aÐA(nR’ nl, n3) fixing ^Z, by ^{choice A or} B, respectively. The insert gives ^these

functions for small values of zN. The ^curves identified as

« Blob » (E) ^or (G) represent ^different versions of the blob

model, ^as explained ^{in section 6.}

As shown in this figure, the choice of A definitely influences our result for aH 1, mainly because of the 10 %

difference in the coefficients of the excluded volume behaviour.

To understand this difference we analyse the excluded volume form of the integral (5.11). With choice A

we find

where G(x) stands for the square bracket in equation (5.12) ^{with the term} 1/2 ln nR ^omitted With choice B the

corresponding expression ^reads

1 03B5

Since v = 1 + T6 ^{+ 0 E2 these two} expressions ^coin-

cide in order s. In the numerical evaluation, however,

the replacement x - v ,X- 0.588 ^-+ x-o.s 1 _ 1 ₁₆ ^In x

makes a big effect in the region ^{of small} x, which

yields ^an important contribution to the integral.

The same analysis ^can be carried through ^for R 2

In the resulting integral additional _powers of x sup- press the region ^x 1, ^and therefore both methods

yield almost the same result.

For (Xu 1 there remains the question which choice

should be preferred ^We have stressed above that

this question ^arises only ⁱⁿ connection with the

approximation ^at hand, ^{thus in} principle ^{it can be}

answered only by calculating higher order corrections.

Inspecting ^our result, ^{we however find} some reason to prefer choice A : as is shown in the insert of figure 10,

aH 1, calculated according ^{to choice B,} changes ^the

curvature for small _zN. Since for zH 1 ^{we do not}

expect cxii 1 ^{to show much} structure, ^we believe that this is an artifact of the method. This points ^{to choice A}

as the better approximation.

Finally ^we analyse ^{the ratio} RG/RH.

Fig. ^11.

^Effective exponents v G ^or v, ^as function of In zN.

VH has been calculated with choosing ^A according ^to pres-

cription ^A.

(From Ref. ^[15] the somewhat different result RG/RH

4/(3Jn)(1 ⁺ 0.07 f) ^can be extracted We do not understand the source of the difference.) According

to equation (5.20) ^the cross-over behaviour of RG

or RH ^is essentially ^{the same.} Specifically, ^{we can}

calculate the difference in the effective exponents

In the last step ^{we have used} equation (2.23). ^The

result yields the estimate vG - vH ^0.001.

Choice A

Equations (5.9),(5.17) yield in the excluded volume limit

In figure ^{11 we have} plotted ^the cross-over behaviour of vG ^or VH’ It is ^seen that _vH VG ^{over a} large range of _zN, the difference _{being quite} small, ^{however :} vG - vH ^0.007.

6. Critical analysis of the blob hypothesis.

Most experiments yield ^a difference vG - vH of the effective exponents ^which considerably ^{exceeds our}

result The experiments ^{have been} explained ^{with the} help of the blob hypothesis [3]. For the sake of our

discussion we decompose ^this hypothesis ^{into three}

statements.

(i) The internal correlations T(q, ^n, n1, n3) ^are go- verned by ^{the same effective} coupling ^{constant as the}

end-to-end correlations of ^a chain of n segments.

(ii) ^For calculating RG ^or Rii 1 ^{we can} replace (Xi(zn’ nl’ n3) ^or aH 1(zn, ni, n3) by (Xi(zn’ ^0, 0) ^or ot 2 (Z., 0, 0)) -1/2, respectively.

(iii) The crude model

is sufficient for qualitative understanding ^{as well as} semiquantitative calculations.

Step (iii) ^has been discussed extensively ^{in the} literature, ^and improved models have been suggested [21, 22]. ^{Here we are not concerned with this} aspect of the blob model.

The first part ^{of the} hypothesis ^{is most delicate.}

We have stressed above, ^{that the choice} of A-, ^and

thus of f

f(A)-, ^is irrelevant provided ^{the exact}

form of T(q) is known. As part ^{of the blob} hypothesis

statement (i) ^{is used to} argue that to a good approxi-

mation ,the nR-dependence ^of cxE(HE)(nR’ nt, n3) ^{can be} represented by cxE(HE)(nR’ ^{0, 0).} Now the relation

with &E(HE)(7il’ n3) independent ^of nR, indeed holds to first order. (Compare Eqs. (2.29) ^or (2.34).) Equa-

tion (6.2), however, ^{cannot hold} rigorously ^{since it} implies ^that &E(HE) ^is independent ^{of the} coupling

constant f :

Thus the nR-dependence, and therefore the crossover

behaviour, ^of 0152E(ÐE)(nR, n1’ n3) ^must di ff er ^{from that}

of 0152E(ÐE)(nR’ ^{0, 0).}

Being ^{valid in first} order, equation (6. 2) ^still suggests that the choice nR(A)

1, being the natural condition for 0152E(ÐE)(nR’ ^{0, 0), is a good} choice for 0152E(HE)(nR, ^ni, n3),

too. This also is supported by ^our discussion of

aH 1 (ZN) ^for zN ;$ ¹ (see Fig. 10). ^We therefore favour choice A : nR(A)

^1, which takes account of part (i) ^of

the blob hypothesis.

Part (ii) is discussed more easily. As is obvious from all the present ^and previous ^{work the} approximation 0152E(HE)(Zn, ^nl, n3) - aE(zn, 0, 0) ^misses part of the swel-

ling. ^{This excess} swelling ^{can be} interpreted ^{as an}

effect of the correlation hole and thus has a well defined physical origin. ^To show its effect in figures ⁹

and 10 we have included the ^« blob-model » results for ocG(zN) ^or 0152M 1 (ZN) calculated according ^to equa- tions (5.5) ^or (5.11), respectively, ^with 0152E(HE)(nR

1, x) replaced by ^{the end to end} swelling ^factor aE(nR = 1, x = 1). The results (o ^Blob(E) ») clearly

demonstrate the great importance ^{of the excess} swelling.

One might try ^{to take the excess} swelling ^into

account by replacing [7] in the blob model the swelling

function aE(nR

^{1, x}

1) by aE(nR

1, x) ; ^i.e. by

the swelling of r;ttnl +n > ^{averaged over the position}

of the piece n

^xN within the chain (compare Eq. (5.6)). By construction this is rigorous ^for R/;.

For RH, however, ^the approximation aH 1(nR

X) 0152F: 1 (nR

1, x) ^{still misses} part ^{of the} swelling, ^{as is}

obvious from figure ¹⁰ (curve ^{« Blob(G)} »). ^{This was}

to be expected, ^{since the excess} swelling ^of 0152HE(Zn, nl, n3) ^is stronger ^{than that of} aE(zn, fill li3).

In summary ^we here have shown that the blob model misses an important physical effect, ^{viz the}

excess swelling ^{due to} the correlation hole. This effect to a large extend cancels the influence of the chain length dependence of the effective coupling

constant considered in the blob model. As a result,

0152O(H)(ZN) ^{is found quite close to} the values calculated without recourse to internal correlations. Incidentally

we note that this discussion leads us to be quite

sceptical concerning ^the interpretation ^{of the numeri-}

cal data of reference [25] ^{with the} help of the blob model.

7. Conclusions.

We have evaluated analytically ^and numerically ^the

second moment and the first inverse moment of the internal distance distribution in ^an isolated polymer

chain. The results agree very well over a wide _range of parameters, ^thus suggesting ^{that our results} provide

a valid basis for further quantitative analysis. ^The only ^serious difference ^occurs for _very short pieces

of the chain. These deviations might ^be interpreted

as indicating ^{a failure of our} analytical ^{work to describe} properly ^the cross-over behaviour from the Gaussian to the self-avoiding ^fixed point. ^We believe that this is not so. On the one hand, ^no reasonable modification of the flow equations (2.16), (2.17) ^can describe the

sharp drop apparent ⁱⁿ figure ^8. On the other hand

our flow equations ^{have been shown to} explain ^almost quantitatively ^the cross-over behaviour of a large variety ^of experimental ^data [20]. Therefore the expla-

nation of the deviations has to be seached outside the

self-interacting Gaussian model. As explained ⁱⁿ

section 4, ^we believe that they reflect the stiffness of the Monte Carlo chain.

Qualitatively ^the difference in the behaviour of

RH(N) ^as compared ^to §(N) ^shown by ^{the Monte}

Carlo data strongly ^{resembles the} experimental ^results.

Our analysis ^therefore suggests that also the experi-

mental results for vH ^are disturbed by microstructure effects which ^are outside the frame of the Gaussian chain model. The blob model cannot consistently explain

the results. It neglects ^the important ^excess swelling

of the interior parts, ^an effect which is of the same order of magnitude ^as the influence of the chain length dependence of the effective coupling ^constant. Impro-

ved versions, being ^{based on some} averaged ^{value of} E ni, n3) still miss about half of the ^excess swelling

of cxHE(n, nl, n3). ^{Therefore we see no} simple ^substitute

for the full calculation of the internal correlations.

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