J. Phys. Chem. 1993,97, 1480-1487
1480
FEATURE ARTICLE Solvent Dynamics, Local Friction, and tbe Viscoelastic Properties of Polymer Solutions Timothy P. Lodge Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 Received: September 24, 1992; In Final Form: December 1 , 1992
Classical assumptions concerning the contribution of the solvent to the v-stic properties of polymer solutions are shown to be invalid. The presence of polymer affects the mean solvent rotational mobility, in some cases even enhancing it relative to the neat solvent case. The resulting changes in local friction modify both the polymer relaxation times and thesolvent contribution to the solution properties. The effect is quantified through an intrinsic "effective solvent viscosity", [qJ, which can be positive or negative. To a first approximation, [qe] appears as an additive correction to the measured intrinsic viscosity, [ q ] ;this correction can be substantial, particularly for polymers of low ,molecular weight. Furthermore, Ille] largely accounts for the anomalous high-frequency limiting viscosity, designated [q'-], even in those cases where [q'-] is negative. These results remain to be incorporated successfullyinto a kinetic theory framework; however, a phenomenological interpretation is offered. The central, unresolved issue is to predict how the local dynamics of molecules in a mixture are modified relative to the unmixed state; the same issue arises in the dynamics of polymer biends.
Introductioa The conformational dynamics of flexible macromolecules in solution have been studied extensively over the past 40 years. The major features of the chain dynamics are well explained by kinetic theory, for example, through the bead-spring model (BSM) of Rouse' and Zimm.2 Furthermore, the recent incorporation of renormalization group (RG) calculations3of average intrachain distances into the BSM has been shown to provide a satisfactory, albeit approximate, description of the effect of solvent quality on chain dynamics.44 As the equilibrium properties of dilute polymer solutions can also be well understood in the RG framework,' at a certain level it is tempting to view dilute polymer solutions as a "solved problem", although some interesting puzzles r e m a i r ~ . ~ . ~ Despite this success, however, it has become apparent that fundamental assumptions concerning the role of the solvent require reexamination. Classically, the solvent is assumed to influence the polymer dynamics in two ways: directly, as a Newtonian continuum with viscosity, qs, such that all polymer relaxation times scale with qr;and indirectly, through the dependence of the monomer-monomer correlation function on solvent quality. Recently, the 'direct" influence has been demonstrated to be In particular, the presence of the much more polymer chain perturbs the dynamics of the solvent, such that qs is not necessarily the appropriatequantity to represent the solvent contribution to the solution dynamics. Failure to account for this effect can even compromise the interpretation of routine low-frequency or steady-state dynamics measurements, such as the intrinsic viscosity. An illustration of the phenomenon is given in Figure 1.lo The plotted quantity is the mean solvent reorientation time, i s , normalized by its value in neat solvent, 7 O S , versus polymer concentration, c, at two temperatures. The solvent is Aroclor 1248 (A12481, a chlorinated biphenyl extensively employed in polymer solution dynamics studies. The influences of four different polymers, polystyrene (PS), polyisoprene (PI), poly(1,2-butadiene) (1,2-PB), and poly( 1,Cbutadiene (1,4-PB) are compared. Several interesting features of these results should be noted. Most obviously, the solvent dynamics are significantly modified by the addition of polymer. The effect differs in sign: OO22-3654/93/2097- 148OSO4.O0/0
1 .o
h
0 "
$.
0.0
v
-
op
- 1 .o
t 0
10
20
30
g/mL x 100 F i p e 1. N o r m a l i d mean solvent rotational relaxation time for Aroclor 1248 as a function of polymer concentration. Data for four polymers (polystyrene, polyisoprene, poly( 1,2-butadiene). and poly( 1.4-butadiene)) and two temperatures (-1.42 and -1 7.3 "C) are compared. The straight lines represent least-squares fits to each data set. E,
PS acts to retard the solvent rotational mobility, whereas PI and 1,4-PB accelerate it. The magnitude of the effect is surprisingly large: by c = 0.1 g/mL, T~ has changed by up to a factor of 5 . Furthermore, 7 S / ~ 0inS PS and 1,2-PB solutions are, at most, weakly dependent on temperature, T,whereas in PI and 1,4-PB solutions there is a strong sensitivity to T. These results raise several immediate questions: (i) what determines the sign and magnitude of a log(rs/ro,)/~c? (ii) What determines the T dependence of the same quantity? (iii) Which features of these results are universal, and which depend on the particular choices of polymer and solvent? (iv) How are these effects manifested in measurements of polymer solution dynamics? (v) How should kinetic theory by modified in order to account for this behavior? In this article we describe some of the phenomenology of solvent dynamics in polymer solutions and offer a qualitative physical interpretation. Recent attempts to incorporate this effect in the kinetic theory framework are also summarized. For both pedagogical and historical reasons, we begin with a review of the high-frequency viscoelastic properties of polymer solutions, where the importance of polymer-induced changes in the solvent
0 1993 American Chemical Society
Feature Article
The Journal of Physical Chemistry, Vol. 97, No. 8, 1993 1481
dynamics first became apparent. Nevertheless, it should be emphasized that the issues at stake are not confined to high frequencies or even to solutions. Rather, they concern the dynamics of mixturesat the molecular scale and lead to interesting consequences even in steady flow; the relevance to polymer blends as well as to solutions will be underscored.
TABLE I polymer
solvent
polystyrene
Aroclor 1248
t
u*/+*
= 7’ - iq”
=:
I d exp(-i@)
(1)
where u* is the sinusoidally time-varying shear stress, +* is the sinusoidally time-varying shear rate, and i = d(-l).The BSM predicts that the V E properties of a dilute polymer solution subjected to a small-amplitude oscillatory shear at frequency w may be expressed as q*
(cRT/wC{Tk/(l
+ iwrk)) + 7s
(2)
k
where Mis the polymer molecular weight and the (Tk]t=1,2,...,~ are the (model-dependent) relaxation times corresponding to the normal modes of the chain. Four features of eq 2 should be noted. First, the solution dynamic viscosity is expressed as the sum of a polymer contribution and a solvent contribution; no polymer-solvent interaction terms are retained in the solution stress tensor. Second, the solvent contribution is given by qs. Third, qscontrols the polymer contribution through the proportionality r/kBT qslkBT (3) where {is the friction coefficient of a bead (or subchain), which is assumed to follow Stokes’ law. Finally, the polymer term vanishes at high frequencies. Assuming that the measurements are confined to the regime where qs is independent of w , then Tk
lim q* = qs w---
ref
25.0 -4.0 25.0 15.2
14.3 19,24 17 24 13.1 22 14.8 22
25.0
15.7
22
10.0
Aroclor 1254
-8.0 30.0 0.0 30.0 25.0
I5 15 11 -1.0 -2.8 4.3 8 -17 7 22.4
21 21 17 23 23 23 21 36 21 20
poly(methy1-2-n-butyl Aroclor 1254
25.0
13.3
20
25.0 30.0
22.2 18
19 21
Aroclor 1232
Decalin a-chloronaphthalene
hC4P-d The dynamic viscoelastic (VE)properties of interest are the loss and storagecomponents, q’ and q”, respectively,of the complex dynamic viscosity, q*: q*
td-1, T,O C mL/g
dioctyl phthalate toluene
polyisoprene
Aroclor 1248
30.0 30.0 0.0
-4.0 poly( 1,4-butadiene)
Aroclor 1248
polyisobutylene poly(methy1 methacrylate)
paraffin oil
acrylate) poly(a-methylstyrene) Aroclor 1254 Aroclor 1248
negative. This observation first prompted the suggestion that the solvent contributionwas at issue, as the high-frequency solution viscosity was less than that of the neat ~olvent.l0.’~.~~.2~.3’36 The dependences of q’- on polymer, solvent, c, M,and Tare conveniently discussed in terms of an intrinsic quantity, [ q ’ J , defined as [T”l
= lim ((7” - vs)/cqsl A
(5)
Thus, [q’..] measures the initial rate at which the addition of polymer changes 7”. The BSM predicts that [$-.I should be zero and attempts to modify the polymer contribution in eq 2 have thus far generated only positive values of [q’,]. Experimentally, to a good approximation
(4)
7’qs exP{c[v’mll (6) Over the past 30 years the high-frequency regime has been in the dilute regime. Where the effects of M and long-chain explored extensively for a variety of polymer/solvent systems, branching have been studied, [q’-] is independent of both despitesevereexperimentalobstacles. In general, with increasing quantities, confirming that [q’- J represents a local property. frequency, $’tends to zero and q’ tends to a frequency-independent Representatives values for [q’J are given in Table I, including value, designated $-;I5 however, q’- # qsin most This two systems where the values are negative. The majority of the high-frequency “anomaly” has been the focus of a considerable data have been obtained in Aroclor solvents, which exhibit large body of experimental and theoretical work, and a quantitative values of qsand aq,/aT. Both features make Aroclors useful for description remains elusive. Nevertheless, it is now clear that a V E studies on polymer solutions, as the (q]are brought into the substantial fraction of the difference between 7’- and qs can be experimentally accessible frequency range, and because timeattributed to polymer-inducedchanges in the solvent dynamics, temperature superposition may be employed to extend the effective such as those illustrated in Figure 1. frequency range. However, it is clear from Table I that the That q’- differs from qs is not surprising, per se. The BSM phenomenology of [qLJ is not an artifact of using highly viscous aims to describe the lower frequency dynamics of the chain and Aroclor solvents or of solvents in close proximity to a glass adopts a crude representation of a polymer segment, namely, the transition; PS dissolved in simple hydrocarbons such as toluene Stokes-bead-and-Hookean-springunit. This unit represents a and decalin exhibits similar behavior. number of monomers on the order of 10, and dynamics within To appreciate the characteristic magnitudes of [q’4 in Table this subchain are explicitly ignored. The BSM has been modified I, i.e., on the order of 10 mL/g, consider the intrinsic viscosity, in a variety of ways to provide a more relatistic description of [ q ] , which is commonly used to characterize polymer M and local dynamics, and values of q’- > qscan be obtained in several solvent quality. The definition of [qJ is cases. For example, an early modification was the “internal viscosity” approach of Kuhn and K ~ h n , *later ~ developed (7) [VI = 1 2I(s - qs)/msI extensively by Cerf,26 P e t ~ r l i n ?Allegra,Z8 ~ and Williams and c o - ~ o r k e r s ;internal ~~ viscosity can be viewed as a dissipative where q is the measured zero-shear rate solution viscosity, but element in parallel with the Hookean spring. More molecularly it may also be written as specific modificationshave included local chain stiffness, and/or constraints such as side-group orientational c o r r e l a t i o n ~ . ~ ~ J ” - ~ ~ (8) [SI = h’- I l s > / C ~ * l In general, such modifications are inadequate to describe the Thus, if [q‘-] reflects a frequency-independent contribution to magnitude of the experimental values of (q’- - qs), but more q*, it should contribute to the experimental [ q ] . For M = los, importantly, for some polymer/solvent systems (7’- - qs) is
Am-
Lodge
1482 The Journal of Physical Chemistry, Vol. 97, No. 8, 1993
a typical value for [‘I] would be 40 mL/g, and thus [?’-I could contribute on the order of 25% to the measured [VI.
20
Solvent Dynamics in Polymer Solutions In this section recent measurements of solvent rotational dynamics in polymer solutions are described, emphasizing those results that have direct bearing on the solvent contribution to the solution VE properties. The experimental techniques that have been employed most profitably are oscillatory electric birefringence (OEB),lslj depolarized Rayleigh scattering (DRS),37,38 and I3C NMR.I4J9 Aroclor 1248 Solutions. The rotational dynamics of A1248 in solutions containing PS, PI, 1,4-PB, and 1,2-PB have been The birefringence, measured by OEB as functionsof c and T. An, of a liquid under the influence of an oscillatory electric field, EOcos(wr), may be expressed as
(An/E*,X,) = B, + B, cos(wt + J/)= B, + B’+ iB” (9) where X, is the vacuum wavelength of the light beam, B, is the steady Kerr coefficient, and B, is the alternating Kerr coefficient. As the average dipole moment of an A1248 molecule is much greater than those of the monomer structures involved, and as the polymer concentrations employed are low, the frequency dependences of the three OEB functions (B,, B,, and $) reflect the solvent dynamics alone. By superposing plots of these functions for a given solution and T with those for the neat solvent, a horizontal shift factor is obtained which may be equated with ~ , ( c , T ) / s ~T), , ( as plotted semilogarithmically vs c in Figure 1 . The dependence of log(ss(c,T)/~o,(T))on cis approximately linear, and thus exp(4
7,(C,T)1/7OS(T)
(10)
The coefficient A depends on temperature and monomer unit and quantifies the rate at which the addition of polymer alters 7,; for 1,4-PB and PI in A1248, A is negative. Interestingly, for these two polymers [‘I”] is negative at comparable T. We assert that this similarity is not coincidental. Changes in 7, with added polymer reflect changes in local friction, and an “effective local viscosity”, ‘I,, can be defined as
4 4
4
4 00 0
rn‘J
0
.PS
0
n o
m w 4
-20 -20
0
-10
20
10
30
PI
4.0
50
T, OC Figure 2. Intrinsic effective solvent viscosity (ope~isymbols), defined in eq 12, and intrinsic high-frequency limiting viscosity (filled symbols), defined in eq 5 , for Aroclor 1248 solutions of polystyrene, polyisoprene, and poly( 1,4-butadiene), as functions of temperature.
---
10.0
0
0.0
0.0
0
e’ oa
M
3
-P E
h
-10.0
Y
f -30.0
’“.9
PI. OEB 0
.
,
,
. ,;,
1.4-PB. NMR
177,.o;.
,
1,4-PB. DRS
.
,
1
t.A
m.
-5.0
2
-1 0 . 0
-40.0
-50
0
50
100
150
T,OC Figure 3. Intrinsic effective solvent viscosity (discrete points) as a function of temperature, for Aroclor 1248 solutions of polystyrene, polyisoprene, and poly( l,Cbutadiene), obtained by oscillatory electric birefringence (OEB),nuclear magnetic resonance relaxation (NMR), or depolarized Rayleigh scattering (DRS). The smooth curves represent the relative segmental and solvent relaxation times (right-hand axis) from eq 15, following ref 14.
Thus, qc resembles the “microviscosity” sometimes invoked to describe the frictional resistance to local motions in polymer solutions. Although 9, is operationally well-defined, it is not a viscosity in the formal sense; nevertheless, we believe it is a useful construct. By analogy with eqs 5 and 7, an “intrinsic” ve can be obtained as
OEB measurements of T~ at much higher T, other techniques have been used.14,37*38 A summary plot of [‘I,] vs T i s shown in Figure 3, adapted from ref 14; in general, the data from the several experiments agree well. Physicnl picture. The dependence of [‘I,] on T and monomer structure reflects an interplay between local dynamics in mixtures and the proximity to thesolvent glass transition. One expects the relaxation times of the individual components in a mixture to differ from those in the unmixed state. For a polymer solution, the standard assumption is that
In Figure 2 we compare [vel and [‘I”] directly, on the assumption that they both reflect primarily changes in solvent dynamics induced by the polymer, for PS, PI, and 1,4-PB in A1248.I2The similarities in both sign and magnitude for the two quantities are striking, especially in view of the different experiments employed; [?,I is determined by direct measurement of solvent rotation, whereas [r)’,] is obtained from the VE properties of the solution. We take Figure 2 to be prima facie evidence that (i) the addition of polymer can have a strong effect on solvent dynamics, and (ii) this effect can be the major contributor to the high-frequency VE properties of a polymer solution. The ratio T , / T ~ obscures , the strong T dependence of T O , . At -17.3 OC,~O,exceeds1 ms, which isan indication of the proximity to the solvent glass transition ( TB -44 “C), whereas at -1.42 OC, T O , < 1ps. As instrumental limitations haveso far precluded
7,g ‘I, 7 O , (13) where T,,* is the relaxation time for any specified conformational change along the polymer backbone. Thus, eq 13 assumes that the solvent dynamics are unaffected by the polymer, and that the polymer segmental relaxation times scale with 7,. The former assumption is incorrect, as illustrated in Figure 1, but in some cases the second assumption fails as well. A reasonable postulate is that T~~ and 7, will be closer to one another in the mixture than in the unmixedstate, e.g., the presence of the faster relaxing component accelerates the relaxation of the slower component. Thus, we have suggestedm that the sign and relative magnitude of [‘I,] depend on the ratio T ~ ~ / T Typically, ~ , . T~~~1 T O , , and consequently [qe]> 0. However, when T O , > T ‘ ,~, [vel < 0; 1,4-PB and PI in A1248 are two examples. This “inversion” presumably requires a high-viscosity solvent and a locally flexible polymer; the latter condition corresponds roughly to a low polymer TB. A1248 has a TBof 4 4 OC, whereas for the
a,(c,T)
= o,(T)
~,~c,T)/7°,
