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The effect of entanglements on the viscosity of semidilute polymer solutions
Y. Shiwa
To cite this version:
Y. Shiwa. The effect of entanglements on the viscosity of semidilute polymer solutions. Journal de Physique II, EDP Sciences, 1991, 1 (11), pp.1331-1336. �10.1051/jp2:1991142�. �jpa-00247594�
Classification
PhysksAbsnacti
61,25H 05.60
Shomcommunication
The effect of entanglements on the viscosity
of senfidilute polymer solutions
Y. Shiwa
The Physics Laboratories, Kyushu Institute of Technology Iizuka, Fukuoka 820, Japan (Received 29July1991, accepted18September1991)
Abstlract. The explicit crossover behavior of the polymer solution viscosityin the dilute and semidi- lute regimes is presented. The calculation is done incorporating ovo effects: (I) the gradual screening
of both hydrodynamic and excluded-volume interactions, (it) the entanglement constraint. The result shows a continuous change to a reptation-like asymptote.
A semidilute regime of polymer solutions is defined as the region where the polymer volume
fraction is infinitesimally small, but the polymer chains are sufficiently long [I]. In this regime, a specific chain overlaps with a significant number of other chains. While the static properties of
polymer solutions are now fairly well understood, our understanding of the dynamic properties
is still incomplete [2]. Once chains overlap, it is likely that the chains start to entangle even if
the monomer density h infinitesimally small. AS a consequence, in the presence of many polymer
chains, not only the complicated interplay of the excluded-volume and hydrodynamic interactions but also the entanglements have to be taken into account in the theoretical analysis of the polymer
solution dynamics. In particular, the theory of crossover behavior (I.e., the dependence on the
degree of chain overlapping) of dynamical quantities, such as solution viscosity and relaxation rate, has met considerable difficulties.
The important feature of the dynamics of semidilute polymer solutions is that the presence of many polymer chains leads to a screening of the hydrodynamic interaction between the monomers of polymer chains [3,4]. At the same time, the screening of the repulsive excluded-volume inter-
actions between monomers is inplay@. (In this paper I restrict my treatment to the good solvent
case.) Thus the first step toward a quantitative theoretical description for the dynamics of semidi- lute polymer solutions is to establish the overall concentration dependence [6] of these screening
effects. This task has recently been accomplished [4], and the gradual screening of both hydro- dynamic and excluded-volume interactions is revealed. Experimental sedimentation data are ob- served to be consistent [7j with the theoretical prediction [8] which takes this gradual screening
effect into account.
Elucidation of the entanglement has been one of the central themes in polymer physics [9]. A
large number of phenomenological (or intuitive) approaches have been developed, among which
1332 JOURNAL DE PHYSIQUE II N° II
the reptation theory [10-12] is most notable. Attempts to put the entanglement effect on a more
microscopic theoretical basis have also been performed [13-l~. However, all of these attempts
are essentially oriented towards the concentrated solution (or melt) dynamics. Such theories, as they are, are not applicable to semidilute solutions. The concentrated solutions and melts are
characterized by the finite volume fraction of polymers, and accordingly belong to a different
universality class from the semidilute solutions [16,17j.
Perhaps best suited for a microscopic description of the semidilute solution dynamics is the formulation due to Hess [18]. His formulation is only applicable to semidilute solutions due to the use of an Edwards' delta-function Hamiltonian [19] characterized by an excluded-volume pa- rameter; the assumption that the interaction is perfectly localized in space is justifiable only when
the monomer density is infinitesimal [l~. Instead of resorting to such phenomenological postu-
lates as tube, Hess then attributes the entanglement effect to the excluded-volume interaction;
volume exclusion disallows chain crossing, and then the chain connectivity allows the display of the entanglement effect. In general, uncrossability amongst the long and interweaving chains can be realized even if the connected object has zero volume exclusion. Nonetheless, I shall here be
contented with the Hess'view on the molecular nature of entanglements.
Crucial aspects that have been left out in the original theory of Hess (hereafter referred to as
OH> are the gradual screening of both hydrodynamic and excluded-volume interactions. AS is shown elsewhere [20], however, these aspects can be fused with OH. Let us introduce the position
vector c(a,t) for the monomer at the contour position a (0 < a < N) of some specific chain, say jth chain, at time t. Formally following the projection operator technique of OH, I find the
generalized l~angevin equation for c(a,t); in terms of Fourier transformed variables [21],c(q,t =
f dac(a, t) exp(-iqa), etc., it reads
(C(q,t) =
/
dl'(~~(q)fi(t t')7(q)C(q,t') + g(q,tl. (1)Here a momentum (q)-dependent function ((q) represents a frictional force between chain seg-
ments, and 7~~(q) is the Fourier transforrn of the static correlation function d~l (c(a) c (a'>)
d being the spatial dimensionality. The memory function fi(t) is determined via ,
#(1) = 2b(ij f~di'# (i i<) a< (i'j Ii, (2)
where ( + ((0) and A((t)
= d~l (Fev(t) Fey(0))
,
Fev being the excluded-volume force exerted on the test chain. The random force is governed by the autocorrelation
(g(q, tlg (q', t')I
= 1(~~(q)fi (t t') 2'b (q + q')
,
(3)
where I h the unit tensor. I take for ((q) the dressed friction coellicient, which may be obtained
by the mode-coupling method [4] to include the hydrodynamic screening effect. The excluded- volume interaction enters through ((q) and A(. In passing, the following remark h in order at this point. Let us define the mode-relaxation time, rp, using [9a,22] the increment of the solvent
vbcosity in semidilute solutions, bqo, as [21]
bqo = (c/2) ~j rp, (4)
P
where c is the polymer concentration. Then it can be shown [23] that rp isgiven byrp = ((q) /7(q).
In particular, I obtain
I ~~b for 0 < ~b < I, fi(°)
"
~
(l + 2~b)~' for I < ~b.
the so-called entanglement parameter, ~b, has already been calculated [8] with the renormalization- group method as
to O(e), where e = 4 d and ii = exp[e(I + In 2)/4]. In (5) X is the usual overlap parameter [24] which is proportional to cNdv, where v is the excluded-volume exponent, v = (I + e/8) /2; so
that it h proportional to c/c*, c* being the critical concentration of chain overlapping.
I now illustrate the utility of the above refined version of OH to determine the detailed crossover behavior of the solution viscosity. The viscosity, q, is givenby the equilibrium average (in units of the thermal energy kBT = 1)
n = v-i f°'di(J(ijJ(o)1, (61
where J(t is the ~y-component of the momentum flux tensor of the solution, and V is the volume of the system. The flux J(t) consists of two parts, the contribution from the solute polymer (JP)
and that from the solvent (J~). For the moment let us ignore the correlation between JP and J~,
and calculate the polymer vhcosity, bqP, defined bybqP = c f dt (JP(t)JP(0)) JP(t is defined for
one chain,
JP(t) =
/
dac~(a, t)bH(c)/boy(a,t).the free energy functional H(c) associated with a polymer chain of configuration (c) is taken to be of the form of the Edwards'Hamiltonian, HE. In the following, however, I use the linearization
approximation for HE due to Edwards [25]: Hi c) = (1/2) f~7(q)[c(q)]~,f~ e (2w)-1 f dq, in accordance with the definition of7(q). It is not difficult to relate 7(q) to HE in the manner in Ref.
[fj. Then the correlation function (JP(t )JP(0)) can readily be computed from Eqs. (I to (3). lb
carry thin ou~ note that Eq. (I) may be formally solved in the form
C(~>t) " G(~,t)C(~>o) +
/
dl'G (~>t t') g (~>t')the relaxation function G(q,t is found to be exponential at long times; G(q, t)
= exp(-r(q)t),
where
~(~) " fi(°)/TP. (7)
Then bqP = (g/2) f r-I(q), where g e cN is the monomer density, and I have adopted the exponential form fo/G(q,t) since the dominant contributions to the time integration arise from the long time behaviour of G(q,t). Equation (6) reduces to
q qo = (c/2)~o ~ (r[~ + contributions from JP J~couplings) (8)
P
1334 JOURNAL DE PHYSIQUE II N° Ii
In the mode-coupling theory [23], rp and hence r-~(q) are proportional to the solvent vbcosity,
qo, as in the Zimm model in f or good solvent. Therefore, I have put r-~ (q) + qorp~ in (8) for
the reminder of this fact.
As have been already noted in the literature [26], to neglect the correlation between JP and jS h not generally tenable. For instance, the conventional Kirkwood diffusion-type formulation
which utilizes the Oseen tensor to describe the hydrodynamic illteractions h known to be correct
only up to O(e [24]. The Oseen type of interactions can be obtained only if one assumes the
decoupling between the solvent and polymer fields. At present, however, our technical capability
fails to investigate this coupling effect systematically. While awaiting a systematic and rigorous theory, I may attempt a tentative ansa~. Namely, assume that the contribution from the second
term of the rhs of (8) is to renormalize the bare solvent viscosity no as no - lo, where fro is identified as the renormalized solvent viscosity that was studied in our previous mode-coupling theory [4];~o = ~o +Sm. (In Ref. [4]bqo is expressed as fin.) Consequently, it follows that q ~o
=
(c/2)@o £~ rpl 16 complete the calculation, I use the relation (4), find that the specific viscosity,
jp + (q ~o)/no, is written in the form
asp = nl)~/fi(°), (9)
where the specific viscosity in the absence of the entanglement effect is given byq))~ = ((o/no) boo/Qo.
Both the dilute and the asymptotic limits of q~p agree with the results by the standard scaling argument [11]. lb see this, note from (5) that ~b -w X as X
- 0, while in the limit X
- cc, ~k -w
Xvldv-~). Recall also that [4] (o
~ b~o ~ X~ldv-~) as X
- cc. Thus I obtain the crossover from the intrinsic viscosity of the 2imm-good solvent type, [vi --
Ndv-~, to the de Gennes' reptation-
like behavior [27], jp -w N~Q~'I~~~~~, as the concentration is increased. In this connection, it can be shown [20] with the help of the renormalization-group calculation of rp [23] that the lim- iting behavior of the relaxation rate (7) is also in agreement with the reptation>scaling result;
r-~
-w
Ndv and N~Q~I~~~~~'~~~~~~ as X
-
0 and m, respectively.
the asymptotic behavior of the result (9) having been confirmed, figure I illustrates the theoret- ical curves for the reduced viscosity parameter, qR + Qsp/Q[Q) vers us clc* [28]. Also shown are the
experimental data [29,30] that I presume are obtained in the semidilute regime. The data seem to lie more or less consistently with the theoretical curve which incorporates the entanglement
effect. However, we notice that the experimental data from two di]erent groups are widely sepa-
rated from each other to exceed experimental uncertainties, although each data is well described by a distinct universal function of c/c*. It clearly calls for more accurate and systematic exper-
iments to prove or disprove the existence of the universality predicted by (9). It remains open
to clarity the difference between the theoretical asymptote and the experimentally oft-claimed
3.4-power law for the solution viscosity.
Finally it is worth adding the following cautionary remark. As emphasized in the beginning,
the concentrated solutions and melts belong to a different universality class from the semidilute solutions. It is then natural to expect that the entanglements in concentrated solutions require separate treatment from the analysh presented here.
3 cj~
~ ~
d
/ j a
~2 ~
~
~
o
I@~
a
~ o
c
en
~
0
-1 0 2
'08(c/c')
Fig. 1. Reduced polymer viscosity QR veJsas c/c*; QR
" (Q Qo) /QOQ[Q], where c and c* are polymer
concentration and its critical concentration of chain overlapping, respectively, is the monomer concen-
tration, [Q] being the intrinsic viscosity. The lower curve represents the reduced polymer viscosity in the absence of entanglements. In plotting the curves, I have set e
= 4 d = 1. The following symbols are used for experimental data of polystyrene solutions covering wide ranges of molecular weights [JI] in different good solvents. (1) Data in benzene due to Adam and Delsanti (Ref. [29]): M x 10~~
= 20.6(a), 6.77(o), 3.84~o), ~89(e), 1.26(+), 0.422(x ), 0.171(T7). (2) Data due to Takahashi et al (Ref.[30]), (I) in toluene:
M x 10~~
= 4.48( ),0.775( ); (it)in a-chloronaphtalene: 20.6 ), 8.42( ),4.48( ), 1.26( ),0.775( ).
Refemnces
[ii DES CLOIzEAUX, J Phys. France 36 (1975 281.
[2]See, for instance, DES CLOIzEAUX J. and JANNINK G., Polymers in Solution: Their Modelling and Structure (Oxford Univ. Press, Oxford, 199l); FUJITA H., Polymer Solutions (Elsevier, Amsterdam, 1990).
[3] FREED K-E and EDWARDS S-E, J Cfiem Phys. 61 (1974) 3626; FREED K-F, in Progress in Liquid Physics, edited by C-A- Croxton (Wiley, New York, 1978), p. 343, and Macroinoleciiles16 (1983)
1855.
[4] SHIWA Y., OONO Y. and BALDWIN RR., Macro,noleculbs 21 (1988) 208; SHIWA Y. and OONO Y., ibid.
21 (1988) 2892.
[5j OHTA T and NAKANisHi A., J Phys. A: Math. Gen. 16 (1983) 4155; NAKANISHI A. and OHTA I, ibid is (1985) 127.
[6jIn Ihe following, I follow convention and loosely refer Ihe dependence on the degree of chain over-
lapping as concentration dependence. However, we should always keep in mind that the monomer concentration is not a natural variable in semidilute solutions since it is infinitesimally small.
33b JOURNAL DE PHYSIQUE II N°11
[~ NYSTROM B. and RcoTs J., J Polym Sci., Po~ym. Lett. 28 (199l) 101, and J Po~y>n. SCL, Polym Pliys.
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[8] SHIWA Y.,Phys. Rev Lett. 58 (1987) 2102.
[9] GRAE%LEY WW, (a) Adv Po~ym. SCL 16 (1974) 1, and (b) ibid 47 (1982) 67.
[10] DE GENNES PG., I Chem Phys. 55 ( In1) 572; Doi M. and EDWARDS S-E, J Che>n. Sac. Faraday
bans. 2 74 (1978) 1789,1802, 1818.
Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, NY, 1979).
M. and EDWARDS S.E, The Theory of Polymer Dynamics (Oxford Univ. Press, Oxford, 1986).
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VG., Soy Phys. JETP 70(1990) 563.
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S-E, hoc. Phys. Soc. London 85 (1965) 613.
Y., to be published.
a is reallyof finite [engih N, and as such we should have a Fourier series on a, not a trans-
form; by (N/21r) fdq we mean £~, and byq, 21rp/N(p
= 1, 2,...). However, when appropriate,
the upper limit of the contour integration may be extended to infinity as far as we are interested in the asymptotic (N
- cc) regime, where the degree of overlapping can still take an arbitrary
value. Therefore, in my model all chain end effects are consistently ignored, playing no special role in contrast with the reptation model.
[22] SArrO N., Introduction to Polymer Physics (Syokabo, Tokyo, 1971).
[23] SHIWA Y. and BALDWIN P-R-, to be published.
[24] OoNo Y.,Adv Che>n. Phys. 61 (1985) 301.
[25~ EDWARDS S-E, J Pliys. A: Math GeJL 8(1975) 1670.
[26j JAGANNATHAN A., OoNo Y. and SCHAUB B., J Chein Phys. 86 (1987) 2276.
[27j DE GENNES P-G-, Macro>nolbcules 9(1976) 594.
[28] 1have adjusIed the unknown proportional constant between X and clc*. For io, I have used the result obtained in Ref. [4]. As is remarked there, the approdmation utilized for the calculation of io is not strictly reliable in the dilute limit. However, a main interest being in the semidilute regime, I take this approximate result in illustrating the theoretical curves.
[29] ADAM M. and DELSAN7II M.,J Pliys. France 44 (1983) 1185.
[30] TA~SHI Y., NODA I. and NAGASAWA M., Macromoleculbs is 11985) 2220.
