J. Phys. Chem. 1984, 88, 6618-6633
6618
Screening of Hydrodynamic Interactions in Dense Polymer Solutions: A Phenomenological Theory and Neutron-Scattering Investigations D. Richter, Institut f u r Festkorperforschung, Kernforschungsanlage Julich, D-5170 Jiilich, Federal Republic of Germany
K. Binder,* Institut fiir Festkorperforschung, Kernforschungsanlage Julich. D-5170 Jiilich, Federal Republic of Germany, and Institut f u r Physik. Universitat Mainz, 0-6500 Mainz, Federal Republic of Germany'
B. Ewen,f and B. Stiihns Institut f u r Physikalische Chemie, Universitat Mainz, 0-6500 Mainz, Federal Republic of Germany (Received: June 8,1984; In Final Form: September 14, 1984) We present a study on single-chain relaxation of linear polymers covering the full concentration range from dilute solution to the melt. Assuming incomplete screening of the hydrodynamic interaction between different segments of a chain, we introduce a phenomenological theory of chain dynamics in semidilute and concentrated solutions and calculate explicitly the intermediate coherent scattering function SWh(q,t).As a new dynamical regime, we predict a second Zimm regime due to incomplete screening and give explicit crossover conditions. Experimentally, we studied the single-chain relaxation on labeled PDMS/C6D5Cl solutions applying neutron spin-echo spectroscopy and neutron backscattering. In the dilute regime the Zimm model is a good description of the experimental spectra for 0 as well as for good solvents. In the semidilute solution the experiments allowed a first direct observation of hydrodynamic screening and led to an evaluation of the hydrodynamic screening length SH for several concentrations. Other than suggested theoretically, iH nearly coincides with the excluded volume screening length 4,. The details of the experimental line shape and the overall quality of the data description support strongly the model of incomplete screening as opposed to total screening. In concentrated solutions and the melt Rouse-like relaxation dominates. A microscopic determination of the only parameter of the Rouse model, namely the segmental friction coefficient over the segment length squared, agrees well with that from macroscopic shear viscosity measurements, demonstrating the basic correctness of the molecular foundations of the Rouse model. We also show that attempts to interpret our data in terms of the reptation model do not work. but we cannot exclude that reptative motions become relevant on larger length and time scales.
I. Introduction Long flexible polymers in dilute and concentrated solutions have been a subject of longstanding interest, both from an experimental and theoretical point of view.'-3 While the static properties of such systems are now fairly well understood, dynamic properties clearly are much less under~tood;~ explicit systematic theories mostly concern the dilute or semidilute solution regime, while the concentrated solution regime requires drastic approximation^."^' Thus, a large part of the theoretical discussion is based on simplified phenomenological models, such as the Rouse the Zimm model~o~2' and the reptation m ~ d e I . ~ ~However, -~' even if one disregards the more complicated regime of the dynamics on shorter length and time scales, where effects due to the local structure and stiffness of the chains and direct van der Waals interactions between monomers come into play,32-34the precise regimes of applicability of these simple phenomenological models are not so clear. In particular, the crossover regimes from one model to the other need a t t e n t i ~ n . ~ ~ , ~ ~ ? ~ ~ With regard to the experimental situation, there clearly exists a wealth of viscoelastic data37-39and also some information on the diffusion of chains as a whole,40*41 but very little has been known on the dynamics of internal modes of the chain, a type of motion which is accessible best via inelastic neutron-scattering techniques such as the neutron spin-echo technique.42 Previous work has been devoted to a study of dilute solutions43and melts44 (again we are not concerned with work probing the nonuniversal behavior on short scales mentioned above33,34*45).The present work tries to fill in this gap by presenting a neutron inelastic scattering investigation of the dynamics of internal motions of polymers in solution over the whole range of polymer concentrations c, from the dilute solution to the melt.46 The analysis Present and permanent address. 'Present address: Max-Planck Institut fiir Polymerforschung, c/o Institut fiir Physikalische Chemie, Universitat Mainz, D-6500 Mainz, Postfach 3980, Federal Republic of Germany. Present address: Chemische Werke Marl-Hiils, D-4370 Marl, Federal Republic of Germany.
0022-3654/84/2088-6618$01.50/0
of such data requires the use of phenomenological theoretical predictions for the coherent scattering function S(q,t),with q being (1) Flory, P. J. 'Principles of Polymer Chemistry"; Cornell University Press: Ithaca, NY, 1971. (2) Yamakawa, H. "Modern Theory of Polymer Solutions"; Harper and Row: New York, 1971. (3) de Gennes, P. G. "Scaling Concepts in Polymer Physics"; Cornell University Press: Ithaca, NY, 1979. (4) Edwards, S. F.; Freed, K. F. J . Chem. Phys. 1974, 61, 1189. (5) Freed, K. F.; Edwards, S. F. J. Chem. Phys. 1974, 61, 3626. (6) Freed, K. F.; Edwards, S. F. J. Chem. SOC.,Faraday Trans. 1, 1975, 71,2025; J. Chem. Phys. 1975,62,4032. Freed, K. F. In "Progress in Liquid Physics"; Croxton, C. A., Ed.; Wiley: London, 1978; p 343. (7) Freed, K. F.; Metiu, H. J . Chem. Phys. 1978, 68 4604. (8) Muthukumar, M.; Freed, K. F. Macromolecules 1977.10, 899; 1978, 11, 843. (9) Freed, K. F. Ferroelectrics 1980, 30, 277. (IO) Perico, A.; Cuniberti, C.; J . Polym. Sci., Polym. Phys. Ed. 1983, 17, 1983. (11) Freed, K. F.; Perico, A. Macromolecules 1981, 14, 1290. (12) Muthukumar, M.; Edwards, S.F. Polymer 1982, 23, 345. (13) Doi, M. J. Chem. Phys. 1983, 79, 5080. (14) Oono, Y.; Kohmoto, M. J. Chem. Phys. 1983, 78, 520. Shiwa, Y.; Kawasaki, K. J . Phys. C 1982, 15, 5345. (15) Perico, A.; Freed, K. F. J. Chem. Phys. 1983, 78, 2059. Freed, K. F. J . Chem. Phys. 1983, 78, 2051. Freed, K. F. Macromolecules 1983, 16, 1855. (16) Freed, K. F.; Perico, A. Faraday Symp. Chem. SOC.1983, 18, 29. Perico, A.; Freed, K. F., to be submitted for publication. (17) Freed, K. F.; Perico, A. Faraday Symp. Chem. SOC.1983, 18, 221. (18) Rouse, P. E. J . Chem. Phys. 1953, 21, 1272. (19) de Gennes, P. G. Physics (Long Island City, N.Y.) 1967, 3, 37. (20) Zimm, B. H. J . Chem. Phys. 1956, 24, 269. (21) Dubois-Violette, E.; de Gennes, P. G. Physics 1967, 3, 181. (22) de Gennes, P. G. J . Chem. Phys. 1971, 55, 572. (23) Edwards, S. F.; Grant, J. M. V. J . Phys. A: Math., Nucl. Gem 1973, 6, 1169, 1186. (24) Doi, M.; Edwards, S. F. J . Chem. SOC.,Faraday Trans. 2 1978, 74, 1789, 1802, 1818; 1979, 75, 38. (25) de Gennes, P. G. J . Chem. Phys. 1980, 72, 4756; J. Phys. (Paris) 1981, 42, 735 (26) Curtis, C. F.; Bird, R. B. J . Chem. Phys. 1981, 74, 2016, 2026. (27) Graessley, W. W J Polym. Sci., Polym. Phys. Ed. 1980, 18, 27. (28) Doi, M. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 1005; 1983, 21, 667.
f 3 1984 American Chemical Societv
Single-Chain Relaxation of Linear Polymers the momentum transfer in the scattering and t the time. These phenomenological theories must contain parameters such as concentration-dependent friction coefficients, hydrodynamic screening lengths &(c), etc. which then can be obtained explicitly by fitting the data to the theoretical expressions. The aim of such an analysis hence is twofold: (i) to check that the framework provided by the phenomenological theory is adequate and (ii) to obtain experimental information on quantities such as EH(c),which otherwise are not directly measurable but for which exist some predictions due to more microscopic theories (e.g. ref 5 and 12). In the course of this analysis, we quickly had to realize that the standard a s ~ u m p t i o n that ~ ~ ~ hydrodynamic * interactions are screened out completely if one considers distances exceeding EH(c) is not adequate, at least for the system considered (poly(dimethylsiloxane) (PDMS) dissolved in deuterated chlorobenzene). Rather, what happens is that due to the residual viscosity q ( c ) of the solution, there remains some residual hydrodynamic interaction over arbitrarily large distances (we term this phenomenon “incomplete hydrodynamic screening”). As a consequence, additional regimes of Zimm-like relaxation20,21are predicted and found, which have not been identified p r e v i o ~ s l ybut , ~ ~we like to mention that there is recent support for this idea also from more fundamental microscopic theories.]’ Thus, in section I1 we shall expose an extension of the phenomenological theory of the neutron-scattering function S(q,t) for concentrated solutions, where we generalize previous work on Zimm-Rouse model^^^^^^ to account for this incomplete hydrodynamic screening. Since great care is needed in extracting any model parameters from corresponding neutron data, it is necessary to present a theoretical analysis of the resulting S(q,t) in some detail, starting from the limiting cases of the dilute solution (described by the Zimm model) and the melt (described by the Rouse model). As will also be discussed, there is no evidence that entanglement effects leads to any significant deviations from Rouse behavior, for the accessible range of q and t, even in melts;@hence, there is no need to complicate the present analysis of concentrated solutions further by considering reptation effects. Section I11 then summarizes the experimental procedures. Section IV accounts for the “simple” limiting cases, i.e. the dilute solution and the melt, while the complicated intermediate concentration regime, where two different crossovers are observed, is analyzed in section V. In all cases detailed comparisons with the theoretical predictions are performed, and it is shown that in fact meaningful parameters (29) Giaessley, W. W. Adv. Polym. Sci. 1982, 47, 68. (30) Leger, L.; de Gennes, P. G . Annu. Rev. Phys. Chem. 1982, 33, 49. (31) Graessley, W. W. Faraday Symp. Chem. SOC.1983, 18, 7. (32) Allegra, G. J . Chem. Phys. 1974, 61, 4910. Allegra, G.;Ganazzoli, F. Macromolecules 1981, 14, 1110. (33) Allegra, G.;Higgins, J. S.; Ganazzoli, F.; Luchelli, E.; Bruckner, S. Macromolecules, in press. (34) Stuhn, B. Dissertation, Universitat Mainz, 1983. (35) de Gennes, P. G . Macromolecules 1976, 9, 587, 594. (36) Baumgartner, A,; Kremer, K.; Binder, K. Faraday Symp. Chem. SOC. 1983, 18, 37. (37) Ferry, J. D. “Viscoelastic Properties of Polymers”; Wiley: New York, 1980. (38) Martel, C. J. T.; Lodge, T. P.; Dibbs, M. G.: Stokich, T. M.; Samrnler, R. L.; Carriere, C. J.; Schrag, J. L. Faraday Symp. Chem. Soc. 1983, 18, 173 and references therein. (39) Osaka, K.; Kurata, M. Macromolecules 1980, 13,671 and references therein. Adam, M.; Delsanti, M. J. Phys. (Paris) 1982, 43, 549 and references therein. (40) Klein, J. Nature (London) 1978, 271, 143; Philos. Mag. [Part] A 1981, A43, 771. (41) Hervet, H.; Leger, L.; Rondelez, F. Phys. Rev. Lett. 1979, 42, 1681. (42) Mezei, F.; Ed. “Neutron Spin Echo”; Springer-Verlag: West Berlin, 1980. (43) Richter, D.; Hayter, J. B.; Mezei, F.; Ewen, B. Phys. Rev. Lett. 1978, 42, 1681. (44) Richter, D.; Baumgartner, A.; Binder, K.; Ewen, B.; Hayter, J. B. Phys. Rev. Lett. 1981.47, 109; 1982,48, 1695; to be submitted for publication. (45) Allen, G.;Higgins, J. S.; Maconnachie, A,; Gosh, R. E. J Chem. SOC., Faraday Trans. 2 1982, 78, 21 17. Nicholson, L. K.; Higgins, J. S.; Hayter, J. B. Macromolecules 1981, 14, 836. (46) A very brief account of part of our results has been given by: Ewen, B.; Stiihn, B.; Binder, K.; Richter, D.; Hayter, J. B. Polym. Commun. 1984, 25, 133. See also ref 36.
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6619 can be extracted from this comparison. In the regime of very concentrated solutions (section V.3) the data are found to fall entirely within the Rouse regime. A comparison with zero shear viscosity measurements reveals a good agreement between microscopic and macroscopic data. Section VI finally contains our conclusions. 11. Theoretical Considerations In this section we give a phenomenological discussion of the coherent scattering due to one polymer chain in a solution, extending the treatment of de G e n n e ~ . ’ ~ ~So * ’ far, the scattering function has been derived for the very dilute solution limit, assuming either a Rouse19or a Zimm model.21 Here we generalize this treatment to more concentrated solutions, introducing the phenomenological concept of “incomplete screening of hydrodynamic interactions”. For introducing the notation and for the sake of comparison, we first briefly summarize the treatment for the Rouse and Zimm model^'^^^^ (subsection 11.1). In subsection 11.2 we calculate the eigenmodes for concentrated solutions, while the coherent scattering function is discussed in subsection 11.3. Entanglement effects are discussed in subsection 11.4, and scaling considerations are presented in subsection 11.5. 1. Quasi-Elastic Scattering by Dilute, Ideal Polymer Solutions. Let us consider a long flexible chain in a 8 solvent, such that internal distances between subunits of the chain obey classical random walk statistics: dividing the chain into N subunits marked by points ?,, ..., FN, with intervals Zn = ?n+l - Fn, we require that there are no orientational correlations between different “effective links” or segments (;At) z m ( t ) ) = u2Bnrn
(2.1)
being the segment length. Then the probability distribution p,,(R) that two points rn and r, are a distance R apart is a simple Gaussian u
with
((yn - ?rn)2)
= 3unm2= u21n - m(
(2.3)
and the static coherent scattering function is given by the wellknown Debye function4’ where RG2= Nu2/6 is the gyration radius of the coil. We shall restrict attention to the regime RG-’ 1 eq 2.25 simplifies to
where the function f(uzlrn - nl/tH2) is defined as Equation 2.26 reveals the existence of a characteristic wavenumber pc, defined by 70 EH = -(12pc)’/*
1 1 6
For distances small in comparison to tHwe have u21m - nl/(HZ pc, it is the second term on the right-hand side of eq 2.26 which dominates the eigenmode spectrum which has a Rouse-like form (7p-10: p 2 ) , but the prefactor is enhanced by a factor of order t H / o in comparison with the usual case. For p < pc, on the other hand, the first term dominates and we have a Zimm-like spectrum, 7;’ a p3IZ. The prefactor in this law is reduced by a factor a o / q in comparison with eq 2.1 1, however. We hence encounter two crossovers when p decreases: for p 65HT/u2the spectrum changes from the “unscreened Zimm relaxation” to the “enhanced Rouse relaxation”, and at p = pc (defined in eq 2.27) it changes again to a “screened Zimm relaxation”. Of course, this last regime becomes physically meaningful only if pc exceeds the lower cutoff provided to the spectrum by the finite linear dimensions of the polymer coil, pcutoff = 1/N. This question will be discussed further in subsection 11.5. In the case where pc exceeds pcutoff the double crossover identified in the eigenmode spectrum also shows up in the coherent scattering function, as discussed below. 3. The Coherent Scattering Function for the Model of Incompletely Screened Hydrodynamic Interactions. The coherent scattering function can be expressed in terms of the eigenmode spectrum 7 p as follows
(2.28) Here we consider the continuum limit of eq 2.12 which is appropriate for small wavevector 4‘. Rescaling variables as q2uzs/6 = x
If we would put
to/? equal
to zero, eq 2.25 would become
q2u2y/6 = p
(2.29)
(49) Cf. eq 4.10 of ref 15 but note that the sign of the term proportional to p1I2should be a minus instead of a plus sign.
6622 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
> qtH we find (42tH2YT)3/2x 9 1 90
7
and hence the screened Zimm relaxation follows, for t eq 2.35
-
(2.37)
from
a,
(2.31)
rewriting eq 2.30 as
In the inverse case, qo/q > 1 eq 2.34 can be simplified
(42tH2Yr)3/2=
71
(2.41)
and then eq 2.35 leads to the unscreened Zimm relaxation, for 1-m
Scoh(q,t)
1
swh(q,o)
e ~ p [ - & q ~ o ’ ( ( 2 a ) l / ’ z W t ) ~ /(2.42) ~
As a consequence, we find that the asymptotic decay of the normalized scattering function is of the form Scoh(q,r)
exp[-constant $a2( Wr).]
(2.43)
scoh(q,o)
(2.33)
Defining a quantity yTas solution of the equation where the argument of the exponential is unity
-+ (~2tHZyT)2
7
(2.34)
where the exponent n changes from n = 2 / 3 (for q& 1 the hydrodynamic interaction is not screened, and hence F(x>>l,y) = F, (2.51) with F , being a constant independent of y because the characteristic frequency for unscreened Zimm relaxation is independent of chain length. In the inverse case, x = qt c* but c
