Draining effect in dilute polymer solutions

Physica A 174 (1991) 223-234 North-Holland

DRAINING EFFECT IN DILUTE POLYMER SOLUTIONS

Y. SHIWA The Physics Laboratories, Kyushu Institute of Technology, lizuka, Fukuoka 820, Japan

Y. OONO Department of Physics, Materials Research Laboratory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received 14 August 1990 Revised manuscript received 14 January 1991

The emphasis is laid upon the presence of a dynamic parameter in the kinetic model of the dynamics of dilute polymer solutions. The parameter represents the strength of dynamical polymer-solvent interactions. The draining effect on transport properties is considered with due attention paid to this physically significant parameter. It is shown that most of the conventional dimensionless ratios such as Flory's P-factor, intrinsic viscosity factor, etc.. arc nonuniversal even in the scaling limit.

1. Introduction

The hydrodynamic interaction among monomers has a drastic effect on the polymer chain dynamics, and consequently on the dynamical properties of polymer solutions. If the strength of the hydrodynamic interaction is altered, significant changes in transport properties result. The study of the draining effect, i.e., the study of the effect due to weaker~ing of the hydrodynamic interaction, has a long history. Probably the most systematic conventional theory [1] is the theory based on the Kirkwood-Riseman equation [2]

F i = r . , ( l ~ . -- u ° ~ • _ o , , , - , ) - 3". L (1) i~j

where F i is the force exerted on the solvent by the ith chain unit and a]~ is a translational friction coefficient of the chain unit; v, is the velocity of the ith unit, and u i is the original solvent velocity evaluated at the position of the ith chain unit in its absence, ~j being the Oseen tensor. The intrinsic viscosity, ['0], is calculated from this (with the preavcraging approximation) as

0378-437!/9!/$(13.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

224 Y. Shiwa, Y. Oono I Draining effect in dilute polymer solutions

NA '°12 N2F(8oN"2), (2) [hi = 36Mr/

g:o (3) ~° = 6Vr6-'~3 ,r/,~l '

F(x) = ~'rr = n2(1 + xlx/-~) " (4)

where N is the degree of polymerization, and ~s is the solvent viscosity; N A is Avogadro's constant, and M is the molecular weight of a polymer, I being the monomer unit size. The parameter 8 0 is called the draining parameter.

There is, however, a very strange thing .about this accepted structure. If we want to change the draining parameter with a given polymer and solvent (i.e., with N, l and 7/S fixed), we must change the friction coefficient. Hence, in the limit 8o---->0, there must be no friction, so that the intrinsic viscosity vanishes. Nothing meaningful remains if one simply takes the limit 8 0---> 0. Hence in the conventional theory, the so-called free-draining limit is not defined by the limit 8 o --0 (as it should be if 8 0 is a control parameter representing the draining effect) ~1, but by the limit that the friction coefficient is small and yet nonvan- ishing. Alternatively, one may say that in the conventional approach the friction coefficient in hydrodynamic interaction terms and that in the un- perturbed term are treated distinctively, the latter being always kept finite in the limit 8o-->0. Since there is only one parameter (friction coefficient) in the original model, this ad hoc procedure is unavoidable.

The conventional model could not properly treat the draining limit in the mathematically legitimate sense of the word. We trust, however, the intuitive physical picture behind the conventional theory. Namely, the draining effect and the translational friction constant can be changed independently. To incorporate this picture, we need an extra parameter. The purpose of the present paper is to point out that the required extra parameter is the local solvent viscosity, or A o (see (10) below), which depends on the microscopic details of the dynamical solvent-polymer interactions. The existence of the parameter is realized only when one starts from a kinetic equation with which one describes behavior occurring on scales ~apttttal / . . . . : - ' anu-'~ . . . . . . . tg ; l i lpLl l t l l I lX : i l i tg ; l . . . . . . lllgLll~t~'-l:--4"~

between the fully microscopic and the fully macroscopic scales; the Kirkwood- Riseman equation is valid for the latter scales. As we will sec in this paper, the parameter appears explicitly in various conventional so-called "universal"

"~ One might as well choose h = 6,,N l'z as control parameter, so that the draining limit is defined as h- -*0 , N - - , ~ such that [~/] remains fixed. In this paper, however, we shall ~.dhere to the less s~h i s t i c a t ed interpretation, 6,, = (,.

Y. Shiwa, Y. Oono I Draining effect in dilute polymer solutions 225

ratios. We can classify these ratios into two classes. That is, a truly universal ratio independent of )t 0, and one dependent on A 0 even in the scaling limit (e.g., in the good solvent and nondraining limit). Most of the conventional ratios containing dynamical quantities, such as the intrinsic viscosity factor and Flory's P-factor, etc., belong to the latter class. Namely, since A 0 is a nonuniversal constant, almost all the purportedly universal ratios lose their universality.

In section 2 the kinetic model for dilute solution dynamics is summarized. In doing so, we define the coupling parameter which allows us to take the draining limit correctly. In section 3 the consequences of this model are discussed in detail. Explicit formulas for "universal ratios" as well as for the effective dynamic exponents are given to describe the crossover behavior due to the excluded-volume and/or the draining effects. Section 4 is devoted to closing remarks. A detailed application of the theory to the initial decay rate of the dynamic structure factor will be presented elsewhere [3], in which the theoretical prediction is compared successfully with experimental results.

2. Kinetic model equation of polymer solution dynamics

The use of a kinetic equation such as the time-dependent Ginzburg-Landau (TDGL) equation has led to the major success of modern theories of critical dynamics [4]. The correlation length of the order-parameter fluctuation and the time scale associated with the critical slowing down justify the existence of the kinetic regime with intermediate spatial and temporal scales between the fully microscopic and the fully macroscopic (hydrodynamic) scales. The large num- ber of the degree of polymerization suggests the existence of the same sort of kinetic regime for the polymer dynamics. This idea has guided us to propose a set of kinetic equations which describes the time evolution of polymer chains and solvent. It reads [5] in appropriate units

Oc(~,t) all{c} f at - ~°'$c(~-,t) + dru(r, t ) 6 ( r -c ( r , t ) )+ f¢ (~ , t ) , (5)

all{c} Ou(r,at t) _ ~_. (,tleV2u(r, t) - f dT $c(r, t) 6 ( r - c(~', t)) + fu) . (6)

Here c(~-, t) is the position vector of the monomer unit at the contour position (0 <~ I" ~< No, N o being the bare polymerization degree) at time t. The solvent velocity field u(r, t) at a space-time point (r, t) i~; assumed to satisfy the incompressibility condition, V . u = 0 ; accordingly, the tensor operator


selects the transverse part of the vector field it is applied to. The coefficients ~'0 and ~e are the bare friction coefficient per segment of the chain, and the solvent viscosity, respectively. The Gaussian stochastic noises are assumed to be governed by the following covariances:

(L(r , t) f~(or, s)) = 2 ~ o I ${$('r - - 0") 6 ( t - s) , (7)

( £ ( r , t) fu(r', s)) = -27/eVZ13(r- r ' ) 6 ( t - s ) , (8)

where I is the d-dimensional unit tensor. The dynamical model thus defined ensures that the system relaxes to an equilibrium state with the probability distribution ~ exp(-H{c} - ~ j" dru2), where the free energy functional, /-/{c}, associated with polymer chains of configuration {e} is taken to be of the type of an Edwards Hamiltonian [6],

No No No

1 v° f d'r Id t r6 (c ( ' r ) - c (o ' ) ) / + g 0 0 0

(9)

Here v o is the bare excluded-volume parameter. An important remark is in place at this point. The parameter r/e, which

appears in our kinetic equation (6), is the local solvent viscosity in the neighborhood of a polymer chain, not the neat solvent viscosity r/s as used in the conventional theories. In the kinetic (semi-microscopic) level of description the bare Onsager coefficients such as 7/e are assumed to be determined by the processes which occur on a very short time scale. Therefore r/e itself depends on the microscopic details of the dynamical solvent-polymer interaction, and is not necessarily equal to the macroscopic (bulk) viscosity, r/s, as stressed by Oono and Kohmoto [7]. This observation leads us to define a dimensionless parameter,

a~ =r/~/r/¢, (10)

as a measure of the strength of the dynamical solvent-polymer coupling. As will be shown below, this parameter makes it possible to define the free- draining limit correctly.

The above set of kinetic equations can be solved perturbatively, the expansion parameters being v,, and G/r/~. it turns out that when treated by the renormalization-group technique [8], the expansion is done in terms of the variable e = 4 - d, where d is the spatial dimensionalitv. For example, we can

Y. Shiwa, Y. Oono / Draining effect in dilute polymer solutions

calculate the intrinsic viscosity to order e from our kinetic model as [7]

227

NA [7/]=n-~ 1-2 1

o L t/2

(2~)2 A 0 ~ 6 + ~ (11)

Here the quantity L is a phenomenological reference contour length scale. The parameters v and ~ are renormalized counterparts of o o and ~'o, respectively.

Notice that the intrinsic viscosity (11) vanishes in the limit in which the conventional draining parameter, ~/7/S (cf. (3)), vanishes. This is of course physically correct; since there is no friction between solvent molecules and monomers in this limit, the solvent properties cannot be modified. Particularly, there should not be any increase of the solution viscosity due to the presence of the chain. However, in the conventional theory [1] the free-draining limit is taken in a more "sophisticated" way. Namely, one takes the limit of vanishing ~/7/s only in the correction terms, keeping intact the zeroth order term (which is also proportional to ~/~s). This is an extremely artificial scheme, and yet unavoidable in the conventional theory. As we observed before, the conventional theory assumes that A 0 = 1; as a result there is no other control parameter than ~/rt s to describe the draining effect.

At this point it shouid be clear from (11) that the free-draining "limit" in the exact sense of the word is to be defined by the limit in which A 0 vanishes. Equivalently and )et more properly, we should define the free-draining limit by the vanishing of a dimensionless parameter s ¢ given by

2 ^

'~o~" (12) ~__-- L -~/2 7/s

Now that the well-controlled free-draining limit has been defined, we can discuss the dynamic crossover from the free draining to the nondraining limit (or vice versa) of transport properties. This is the subject of the next section.

3. Draining crossover

We now know that, when applied to the kinetic model (5)-(0), the renormalized Kirkwood-Riseman (RKR) formalism and the full dynamical theory with the aid of the Green-Kubo (GK) formalism give identical results

228 ¥. Shiwa, Y. Oono ! Draining effect in dilute polymer solutions

for the intrinsic viscosity and the diffusion constant to order e [9-12] #2 Beyond order e, there is no justification for the RKR formalism, which utilizes the Markovian approximation to eliminate the solvent velocity field [8]. Since the order e 2 calculation of transport coefficients via the GK formalism is at present prohibitively difficult, no reliable way to o b t a i n ~(e 2) results in an internally consistent way has been found. Hence, in the following we confine ourselves to the order e consequences of our kinetic equation model.

Based on the kinetic model, various transport properties have been calculated in refs. [7, 13, 14]. Note, however, that although the parameter A 0 was retained in the final results in ref. [7], it was eventually equated to unity on purely empirical grounds. The grave significance (as we will see shortly) of the parameter was not fully recognized there. Along the same lines, the sub- sequent references assumed A 0 = 1 at the very start of their calculation. Keeping this point in mind, we can obtain correct crossover formulas of the transport properties by transcribing the results given in those references. Namely, the renormalization-group equations for the transport properties in terms of u and ~ are intact if we interpret the dimensionless parameter ~ as in eq. (12). Also the crossover formulas of various observable quantities are exactly the same as given in ref. [13] except for multip'kative powers of A 0. For example, the intrinsic viscosity, [7/], and the translational friction coefficient, f, are given as

M ,E'Tl "2

NAN(1 + w) - ' " ] a'2 [nl= 6(2.trf '2 x°2{1 + (511 + ~'(1 + ~.),,81-3,4}-,

~'(1 + ~')"8 (43 x l! + ~'(1 + exp ~-~ ~-

f MN(I + w)-'"l

~" 7 ~" a(1 + ,r) -3/4 ) 1 + ~" 12 e 1 +-----~ 1 + 6(1 + ~)-3 ,4 ,

2,1r 2 = (2 .0 .2 ag'{1 + 311 + + ¢) .8]-3 ,4}- ,

(13)

X ~'(1 + ~.),,s ( 1

l 1 + ((1 + ~ r ~ 3 , 4 * , , s exp ]-~ 1 + ~ 1 + 3 ( ! + ~ ' ) - ~ / 4 • (14)

,2 In ref. [11] it is shown that even with hydrodynamic interactions, the RKR scheme is as reliable as the GK formalism, but ~e hastily concluded that this was not the case with self-avoiding interaction~,;. In ref. [12], however, it is analytically shown that even with both hydrodynamic and ~lf-avoiding interactions the RKR result for the intrinsic viscosity is identical to that obtained by the GK formalism to order e. Y.O. apologizes for erroneous conclusions explicitly stated in refs. 19-111.

Y. Shiwa, Y. Oono / Draining effect in dilute polymer solutions 229

The parameters sr and ~ are defined as

;~ = ( 2 " t r N / L f / 2 w ( 1 + w) -~/8 , w = u / ( u * - u) , (15)

6 = (1 + w)3'4(.~.-] - 1), P . " - u * ~ l u ~ * , (16)

where u* = , r r 2 e / 2 and ~* = 2"rr2e are the values of the nontrivial fixed points of u and ~. Eqs. (13) and (14) correspond, respectively, to (5.8) and (4.28) in ref. [13].

Notice that [7/] or f itself is not only a function of variables ~ and 6, but also given in terms of phenomenological parameters N, w and A0, which are not directly observable. If one can get rid of N, w and /to by taking some combination (ratio) of the directly observable quantities, one calls such a ratio universal; as will be shown below, when the ratio is expressed in terms of only ~" and 6, its asymptotic value in the scaling limit becomes constant independent of the microscopic details of the system. The following dynamical ratios were often claimed to be universal in the literature:

M[,] A 2 M U~/n =- N A R 3 , U,a/n = [71 '

f (M[nl/NA) ''3

vI'R=- ,7 Ro ' v"'r- fins

(17)

where R G is the radius of gyration, and A: is the osmotic second virial coefficient. Using the corrected results (13) and (14) as well as the known results [8] for the static quantities R C and A 2, we can get the explicit formulas for the above ratios.

To streamline the resulting expressions, let us introduce the following parameters:

~'(1 + ~).8 ~-- 1 + ~'(1 + ~. ) , ,8 , 0 - 1 + 6 [ 1 + ~(1 + ~ . f / s ] - 3 / q • ( 1 8 )

In terms of three parameters, 0, 0 and/t 0, the ratios defined by (17) then read as follows (setting e = 1):

Un/R = (3'n')3/2 0 e x p ( l + ~ b + 7 0 ) 24 /to z

(19)

UA/r~ = 3 / t 0 exp[(~6 + ½ 1 n 2 ) $ - 7 0 1 , (20)

230 Y. Shiwa, Y. Oono I Draining effect in dilute po l ymer sohaions

Ur~R = V~.rr3,2 0 exp(~ + ~-~2' + ~20) , (21)

•4/3 0 O -2'3 exp( - ~ ¢ + PsO). Uno-= (24),,3.a. (22)

From , h o e , , a v n l i e l t f n r T n n | ~ ~ n a t t t e n n c l n d e t h a t n n n e o f t h e c o n v e n t i o n a l - - L l t l t ~ t . t t . l ' ~ ¢ V ~ l . 3 a z v z ' t . ~ o ' ~ a = a ~ . ~ . . ~ = . v . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ratios of (17) (including Flory's P-factor, which is propo~ional to b)/R, the Flory-Scheraga-Mandelkem parameter proportional to Un/y, and the intrinsic viscosity factor @ proportional to U,m) is universal. These is no theoretical reason to believe universality of these conventional ratios.

However, we can construct a truly universal ratio by looking at the formulas just obtained. For instance we find that

rI~[71]MD (23) - l V , , , .

is universal, and is given by

U.o,R = ~ exp( ] + ~ 0 + ~ 0 ) . (24)

Here D - kaTI f is a translational diffusion constant, kBT being the tempera- l r t r r d - 2 ture in the units of energy. This ratio is identical to ,.,n/R/u r/n, and was

suggested by Weill and des Cloizeaux [15] long ago. As might be guessed now, effective exponents describing the crossover

behavior are universal in the same sense that they are functions only of the two parameters ~b and O. The longest relaxation time, -r, scales as N ~z asymptotically [16], where v is the excluded-volume exponent and z is the dynamical critical exponent. The effective exponent associated with r is thus defined via

Oln~- (vz)c,, = a In N " (25)

From the explicit result for ~" which has already been obtained [17] #3 we get, to

. . . . . I A "~ In eq. (5.19) of this paper the iv.~egrands in the :×pressions for 1 and J sh . . . . read [ i - (t - sin t ) /Irp] ( i - cos t ) / t ~ - 1/2t and [cos t - (t cos t + sin t ) /~rpl / t - I It, respectively. Ac- cordingly, eqs. (5.21) are to be replaced by

I ( 3 211 - ( - 1 y ] ! 3 2 ) 5 ( p ) = ~ 2 + (~p)" 2p c in ( , ' r p ) - ~pp si('np) + ~pp si(2~rp) ,

1 si0rp) $~{ p) = -¢ inOrp ) - ~p - ,rp

where - o n ( x ) = ci(x,~ - a,~ - In x and si(x) = - J ' " dt sin(t)/ t .


C(e),

( v z )~ f r = d v - e ( 5 qJ - 3 0 ) , (26)

with v = (1 + e/8)/2. The initial decay rate, ,O(k), of the coherent scattering function scales as k z for large k [16], k being the magnitude of the scattering vector. Then we may define the effective dynamic exponent, zeff, as

01nO(k) z~ff = lim (27)

k--.~ 0In k

From the known expression [18] for g2(k) we obtain

z~f,= d + e ( 1 - -~q,- 3 0 ) . (28)

In order to demonstrate that the crossover formulas (24), (26) and (28) correctly describe various scaling limits, let us study the effective dynamical exonent zcf f, (28). First we consider the good-solvent (self-avoiding) limit, u---> u*. In this limit we find from (15) and (18) that ~b---> 1. In the nondraining limit (~----> ~*) we see from (16) and (18) that 0--', 1. In the free-draining limit (Ao---> 0), on the other hand~ 0~ bei_ng asymptotically proportional to A 2, will vanish. Thus we find that in the good-solvent and free-draining limit zeff---->4-e/4, which agrees with the scaling prediction [16], z - - 2 + 1/u , to ~(e). The good-solvent and nondraining limit yields zeff---> d as required again. The Gaussian case (u-->0, or ~'--->0) can be studied similarly. In this case, q,----> 0. Since 0----> ) (see (30) below) in the nondraining limit, we get zeff----> d, while in the free-draining limit O vanishes, so that zeff-->4; they are in agreement with the scaling results, too.

The above illustration, in turn, supports our previous statement that a quantity is universal when it can be expressed solely in terms of the two parameters q, and O. In the scaling limit where the coupling constants u and ~: take fixed-point values, both ~ and 0 take on corresponding constant values irrespective of microscopic details. Hence follows the universality.

4. Concluding remarks

True universal quantities can be expressed solely in terms of two parameters, q, and O. The parameter qJ describes the excluded-volume effect, whereas the draining crossover is depicted by the parameter 0. A partial draining contained in the parameter O can be most easily revealed if we take the Gaussian limit of


this paranmter. In the limit ~---~ 0, we have

6-"> - 1 + u l u 3 u (29) ~/~* 4 u* '

so that the parameter 0 reduces to

19 = Zn (30) ] ( Z . - +

in the Gaussian limit. Here in (30) we have defined the counterpart of the

conventional draining parameter:

[ 2 ~ N \ E:2

z.- T ) (31)

Through O of the form (30) the partial draining effect (even in the Gaussian limit) is contained in any universal quantity.

Incidentally, (29) and (30) correct a result obtained in refs. [8, 9, 13], where it was argued that no partial draining is implied when N---~o0 in the Gaussian chain solutions. However, it should be clear by now that this conclusion was inevitable within the framework of the conventioal theory where A o = 1. The reason is as follows. It is noticed that in our model, which is written down under the assumption of overdamped dynamics, the bare friction constant ~0 cannot be infinitesimally small. If so, we must take into account the inertial effect, which is completely ignored in our model. This implies that the renormalized friction constant ~ cannot be vanishingly small, either. Hence, if the parameter )t o were missing from (12), we would always have Z n >> 1 in the long-chain limit (i.e., in the scaling limit). The ensuing conclusion would be that only the situation with O = 4 (the nondraining-limit value) is physically legitimate in the asymptotic Gaussian limit. After all, if a model is to capture a partial draining in the asymptotic region, it requires the parameter A 0.

Concerning the above crossover parameters, we should emphasize that within any model so far used, no interrelation between the excluded-volume effect and the draining effect can be derived theoretically. If we could fix the draining parameter ~ independently of the excluded-volume parameter u, then the excluded-volume effect would favor the draining effect [19] as can be seen from (16). However, there is no theoretical justification for such an independent change of the parameters.

Finally, we comment on the universal ratio. Many ratios that were conven- tionally called universal have been shown to be in fact nonuniversal; they explicitly involve the dynamic parameter A 0, which depends on microscopic


details of the system #4. There are, however, true universal ratios such as U.o/n - r/s[n]MD/N A k B T R 2 and UAflvlR ~- A 2Mf/ns[rl]RG. In particular, U~o/n is directly related to the ratio of familiar expansion factors as a~/

2 = Uno/R/(U~o/R)e, where (UnoiR)e designates the O-solvent value of o~ S0tH

UnOIR; the quantities a,7, a s and t~ H denote the expansion factors for intrinsic viscosity, root-mean-square radius of gyration, and hydrodynamic radius of

~s • polymers, respectively. Although our concern in .i~s paper is not the interpretation of a specific experimental result (which wql bc presented elsewhere [3]), it is worth adding that our formula (24) predicts thz, t

3 2 a, /a st~H = 0.889 (32)

in the asymptotic limit. The experimental estimates have been compiled recently [21] #5 to yield 3 2 an/t~sa H~0.9, in satisfactory agreement with the prediction (32).

Acknowledgements

Y.O. was supported, in part, by the National Science Foundation through Grants No. NSF-DMR-87-01393 and No. NSF-DMR-90-15791.

References

[1] H. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971). M. Bixon, Annu. Rev. Phys. Chem. 27 (1976) 65.

[2] J .G. Kirkwood and J. Riseman, J. Chem. Phys. 16 (1948) 565. [3] Y. Shiwa, to be published. [4] i'.C. Hohenherg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [5] Y. Oono and K.F. Freed, J. Chem. Phys. 75 (1981) 1009.

Y. Shiwa, Phys. Rev. Lett. 58 (1987) 2102. [6] $.F. Edwards, Proc. Phys. Soc. 85 (1965) 613.

• 4 In this connection, we should mention Fixman's findings [20]. Fixman's extensive dynamical simulations show that either internal rigidity (with constrained bond length or bond angle) or internal friction (or internal viscosity as it is often called) generates a coupling between local motions and fluctuating hydrodynamic interactions, which affects the global translational diffusion constant. This effect seems to persist to arbitrarily large chain lengths to yield nonuniversal values of Flory's P-factor.

• 5 It should be pointed out that in this reference the term "universality" is defined differently from ours. There the ratio is called nonuniversal when the quantity is sensitive to the degree of partial draining. Conversely, if the quantity is a function of the excluded-volume parameter only it is referred to as universal.

234 Y. Shiwa, Y. Oono / Draining effect in dilute polymer solutions

[7] Y. Oono and M. Kohmoto, J. Chem. Phys. 78 (1983) 520. Y. Oono and M. Kohmoto, Phys. Rev. Lett. 49 (1982) 1397.

[8] Y. Oono, Adv. Chem. Phys. 61 (1985) 301. [9] Y. Oono, AIP Conf. Proc. 137 (1985) 187.

[10] A. Jagannathan, B. Schaub and Y. Oono, Phys. Lett. A 113 (1985) 341. [11] A. Jagannathan, Y. Oono and B. Schaub, J. Chem. Phys. 86 (1987) 2276. [12] B. Schaub, D.B. Creamer and H. Johannesson, J. Phys. A 21 (1988) 1431. [13] Y. Oono, J, Chem. Phys. 79 (1983) 4629. [14] Y. Oono and M. Kohmoto, J. Chem. Phys. 79 (1983) 2478. [15] G. Weill and J. des Cloizeaux, J. Phys. (Paris) 40 (1979) 99. [16] P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1980). [17] Y. Shiwa and K. Kawasaki, J. Phys. C 15 (1982) 5345. [18] A. Lee, P.R. Baldwin and Y. Oono, Phys. Rev. A 30 (1984) 968. [19] S.-Q. Wang, J,F. Douglas and K.F. Freed, J. Chem. Phys. 87 (1987) 1346. [20] M. Fixman, J. Chem. Phys. 89 (1988) 2442, and references cited therein. [21] K.F. Freed, S.-Q. Wang, J. Roovers and J.F. Douglas, Macromolecules 21 (1988) 2219.

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Draining effect in dilute polymer solutions