Physica B 296 (2001) 184}189
Spherical polymeric brushes viewed as soft colloidal particles: zero-shear viscosity Dimitris Vlassopoulos *, George Fytas , Stergios Pispas , Nikos Hadjichristidis Foundation for Research and Technology } Hellas (FO.R.T.H.), Institute of Electronic Structure & Laser, Heraklion 71110, Crete, Greece Department of Chemistry, University of Athens, 15771 Zografou, Athens, Greece
Abstract We examine the concentration dependence of the zero-shear viscosity of two chemically distinct classes of spherical brushes, multiarm star polymers and giant diblock copolymer micelles. We show that the e!ective volume fraction based on the hydrodynamic, rather than the static, size of these objects is the appropriate scaling parameter for obtaining a generic description of the relative viscosity for systems of varying size and softness. We demonstrate that it is possible to span the whole range from polymeric behavior to that of hard colloidal spheres by tuning the sphere softness at the molecular level (arm size and functionality). 2001 Elsevier Science B.V. All rights reserved. Keywords: Micelles; Star polymers; Colloids; Viscosity
1. Introduction In recent years the need to bridge the gap between polymers and colloids has emerged as an important topic in soft condensed matted physics [1,2]. The reason stems from the desire to explore the behavior of polymerically stabilized colloidal spheres with increased softness, where the grafted layer dynamics is a signi"cant contributing factor to the overall behavior of the particle. A number of complex #uids such as block copolymer micelles [3], star polymers [4], polyelectrolyte microgels [5] and hard spheres with long grafted layers [6],
* Corresponding author. Fax: #30-81-391-305. E-mail address:
[email protected] (D. Vlassopoulos).
exhibit structural and dynamic properties with features attributed to both polymeric and colloidal character. It is thus evident that the use of model systems will play a crucial role in understanding how to tune the properties of such complex #uids by controlling their polymeric and colloidal contributions. On the practical side, combining the distinct colloidal and polymeric hallmarks in a single mesoscopic system may lead the way for the molecular design of new materials. The former primarily relate to long-range order obtained at relatively low-volume fractions and to Brownian stresses, whereas the latter to short-range order at high concentrations and entropic stresses. Recently, we exploited the dual character of multiarm star polymers [4,7]. Due to their nonuniform monomer density distribution, arising from
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 7 9 8 - 5
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their synthesis, these model macromolecular objects represent soft spheres composed of a central core and a grafted corona, the polymeric layer [8]. Depending on the functionality and arm molecular weight, the core/corona size ratio can vary, remaining much smaller than 1 and yielding particles of di!erent but high softness. As predicted by the theory, in addition to the core, there is a &coat' region of stretched nearly ideal arms, which could be considered as part of the core, i.e., contributing to the colloidal nature of the stars; however, this core}coat region in these homopolymers is not so clearly de"ned in that is it rather soft and deformable, and in addition to arms number ( f ) and degree of polymerization (N ), its size also depends on the volume fraction. When dispersed in concentrated solutions, the stars exhibit a weak liquid-like order as a consequence of the enhanced osmotic pressure which outbalances the entropic stretching of the corona [4,9]. This order is manifested as a peak in the scattering intensity which corresponds to the most probable core}core distance. The dynamics re#ects the corona contribution (polymeric character: cooperative di!usion, arm relaxation) as well as the core contribution (colloidal: self-di!usion, structural rearrangements). Likos et al. [10] have proposed an expression for the interaction potential of these stars, including a logarithmic repulsion at short distances and a Yukawa form at large distances; this potential was proven to work well for stars with f"18. On the other hand, block copolymer micelles, prepared by dispersing a linear block copolymer in a selective solvent for one block, also exhibit a nonuniform monomer density pro"le similar to that of the stars, as demonstrated by SANS measurements of the form factor [11]. Such spherical objects with typical hydrodynamic radius R of about 40 nm and a hard core of about 20 nm, are found to order into crystalline lattices at high concentrations [3,12]. In order to explore the dynamics at the most probable core distances, very large such soft particles are needed, which are amenable to dynamic light scattering studies. This task was successfully accomplished with the synthesis of very high molecular weight asymmetric diblock copolymers of styrene and isoprene dispersed in decane, a selective solvent for the isoprene block. The resulting nearly
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Table 1 Characteristic sizes of stars and micelles used Sample
f
N
12 880 128 1336 12 807 128 127 6407 64 118 3280 32 1648 SI-2M 1470 20676 SI-600 1710 5632
R (nm)
c* (mg/ml) Static
c* (mg/ml) Hydrodynamic
68 15 12 51 400 185
2.5 17.3 14 1.9 } }
18.7 103 94 11.5 30.8 16
spherical micelles had dimensions about one order of magnitude larger than the stars and a very high functionality [13] (see Table 1). A remarkable result is that neither class of these soft particles, namely hyperstars and giant micelles was found to exhibit crystalline order for the concentrations studied. This seems at odds with experimental observations on several diblock copolymer micelles [12,14,15], which, however, exhibit much smaller size and functionality but similar polydispersity, as well as with recent computer simulations [16]. Apparently, the concentration window for crystallization is very small for both, and seems reduced as the softness of the particles is enhanced. It is actually the osmotic pressure (&f ) that drives these systems to crystallize, and its further increase at high concentrations that suppresses the crystallization process [9]. Given the interplay of polymeric and colloidal properties in stars and micelles, it is tempting to try obtaining a generic description of the dynamics of such mesoscopic systems covering the whole range from hard spheres to polymers. This is exactly addressed in this work, which focuses on the zeroshear viscosity of stars, micelles and hard spheres.
2. Experimental The multiarm star 1,4-polybutadienes employed in this comparative study were described in numerous publications [4,7,17,18]. We discuss in this paper four such stars with varying functionality and arm size. Their characteristics of these stars are shown in Table 1. Note that there are two determinations of the overlap concentration, as discussed below; note also that there is a very small
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polydispersity in functionality which remains below 4%. We also used two types of poly(styreneb-isoprene) micelles dispersed in decane, forming polystyrene cores. The "rst one was based on a diblock copolymer with "0.26 and total molecular .1 weight M "1 900 000 g/mol [13], whereas the sec ond was obtained from a diblock copolymer with
"0.32 and M "566 000 g/mol. The polymers .1 were anionically synthesized yielding polydispersities M /M (1.1; their sizes are listed in Table 1. The zero-shear viscosities were measured using a Rheometric Scienti"c ARES-HR 100FRTN1 sensitive rheometer, equipped with a #uids bath temperature control and Couette or parallel plates (diameter 25 mm) geometry, whereas some of the star (3280, 12 807, 6407) [18] and the hard sphere data were obtained from the literature [19,20]. The sizes were determined from dynamic light scattering measurements.
3. Results and discussion Fig. 1 depicts the relative viscosity ( / , with the zero shear viscosity and the solvent vis
cosity) as function of concentration for the various stars and micelles used in this work. It can be observed that the high functionality stars, corresponding to less soft spheres, exhibit a very strong concentration dependence at high concentrations, reminescent of the behavior of hard spheres. This can be appreciated by comparing the behavior of the two stars 6407 and 12 807; clearly the latter has a larger core (r &f [4,8]), and as such the vis cosity increase is sharper at high concentrations, whereas at low concentrations the behavior is indistinguishable. On the other hand, the lowest functionality star used, with a large arm molecular weight (3280), behaves essentially as a polymer solution, reaching the semidilute solution scaling of linear polymers, &c [21]. The di!erent con centrations at which the viscosity shuts up are due to the di!erent overlap concentrations of the stars (Table 1). Similar remarks hold for the two micelles investigated. Actually, the inset of Fig. 1 shows some typical rheograms of SI-600, from which the low-shear limiting viscosity (identi"ed as ) can be determined. The data of Fig. 1 suggest the need to obtain a generic master plot for all systems. In this respect,
Fig. 1. Concentration dependence of the relative viscosity of stars and micelles. Stars: 3280 (*), 6407 (䉭). 12 807 (䉫) and 12 880 (䊐). Micelles: SI-600 (䢇), SI-2M (䊏). Inset: typical rheograms indicating the determination of the zero-shear limiting viscosity for the micelle SI-600 at di!erent concentrations: 0.66wt% (䢇), 0.81 wt% (䉭), 0.94 wt% (£), 1.3 wt% (䊐), 1.6 wt% (*), 2 wt% (;), 2.32 wt% (䉫).
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Fig. 2. (a) Relative viscosity of stars as function of an e!ective volume fraction c/c*, with c* the overlapping concentration determined from static light scattering (topological e!ects): 12 880 (䊐), 12 807 (䉫), 6407 (䉭) , and 3280 (*). Inset: schematic illustration of a multiarm star polymer. The circle represents the e!ective core formed by the touching monomers of neighboring arms. (b) Relative viscosity against e!ective volume fraction,
, based on the hydrodynamic radius, for the stars and micelles (SI-2M: 䊏, SI-600: 䢇). Hard sphere data are also included in this plot for comparison (PMMA in decalin, 640 nm [19]: 夽, 602 nm [20]: X). Inset: schematic illustration of a diblock copolymer micelle. The central part is the core made of the insoluble polymer.
the star data are reduced in Fig. 2a, where the normalized concentration is based on the c* (deter mined from static light scattering measurements [4]; see Table 1). This c* re#ects the polymer be havior as it is the crossover concentration where segments from di!erent polymers touch each other.
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This reduced quantity is practically an e!ective volume fraction. For low concentrations c/c*(1 all star data collapse in the same curve. However, as the light scattering data imply, at c/c*'1 the strong viscosity enhancement re#ects the topological e!ects in these systems. Furthermore, a master curve can no longer be obtained, as in the regime of star}star interaction the viscosity behavior signi"es the softness of these spherical objects, and consequently their interaction potential: 12 807 with the largest functionality and shortest arms is the closest to hard sphere behavior, whereas 3280 with the lowest functionality and largest arms size is the closest to conventional polymeric behavior. An intermediate behavior is observed between the two limiting cases. It is thus the functionality which primarily controls the softness (or hardness) of these nearly spherical particles; f a!ects signi"cantly the core size, as well as the interaction potential. The arm size N is also important as it controls the overall size of the particle (R&f N [4,8]), but secondary, and this is demonstrated in Fig. 2a. Given the fact that the zero-shear viscosity of hard spheres diverges at volume fractions of about 0.6 [22], it is evident from Fig. 2a that the stars would diverge at much higher volume fractions, suggesting either very soft interactions even for 12 807, something that contradicts their interaction potential [10], or the chosen reduction is not appropriate for representing all particles in one plot, irrespectively of softness. This remark leads to another reduction, based on the particles hydrodynamic size. Indeed, Fig. 2b illustrates the relative viscosity as function of the e!ective volume fraction ( ) for hyperstars and giant micelles. The latter quantity is the volume fraction of an e!ective sphere with dimension the overall hydrodynamic radius of the soft particle (R ); this quantity is practically the normalized concentration c/c*, where c* is now determined from the translational di!usion coe$cient, measured via dynamic light scattering [4] or viscosimetry [18] (Table 1). The same approach was actually used in order to describe the relative viscosity of di!erent micelles of much smaller size (up to a factor of 10) and functionality and their departure from hard sphere behavior [23,24]. In this "gure, some typical hard sphere data are also
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included for comparison; they are data on PMMA particles with a tiny grafted layer of poly(12-hydroxy stearic acids), PHSA, suspended in decalin, having two sizes, 602 and 640 nm, obtained, respectively, from Refs. [19,20]. The data collapse into a master curve for low-volume fractions (0.1, whereas deviations as observed at higher values of
. This plot is really meaningful as the deviations from the hard sphere limit are gradual and consistent with the above discussion. More speci"cally, as one increases the softness of the particles, one reaches higher critical volume fraction corresponding to the divergence of the viscosity. Actually, 12 807 and 12 880 are virtually collapsed, con"rming the dominant e!ect of functionality, as already discussed. This is reminiscent of the collapse the selfdi!usion data of stars with the same functionality and di!erent arm size into a master curve, when plotted against the same &hydrodynamic' reduced concentration [25], con"rming the validity of the generalized Stokes}Einstein relation. The micelle data are also consistent as SI-600 exhibits much higher functionality than SI-2M (Table 1). Note, however, that based on the above arguments, one would expect the two micelles to be &harder' than the 128-arm stars on the basis of their functionality. We believe that this is not the case because at the same time the arm of the micelles are extremely long, and thus the softness of these systems, de"ned as S"(R !r )/r is high; it assumes the values of 5.7 and 3.1 for SI-2M and SI-600, respectively, based on the estimation of r using the form factor obtained from light scattering data. These values are similar to or slightly larger than those of Refs. [12}15], where various crystalline phases were observed (typically FCC order was found for harder and BCC for softer micelles). As crystallization was not observed in this work, this may provide a hint about the role of functionality, as already indicated. A comparison with the stars possessing di!erent internal structure cannot be made, as their core is deformable (in contrast to the glass, thus rigid micellar core) and it is thus impossible to quantify the changing core-like high monomer density region of the squeezed interacting stars at high concentrations. As expected, the Krieger}Dougherty empirical / "[1! / ]\ ( [26], proposed to de
scribe the viscosity}concentration curves of hard spheres, is not appropriate in this case (here 2.5 is the Einstein value for hard spheres). It can be used in principle to determine the e!ective volume fraction at which the viscosity diverges, by consid ering the stars and micelles as e!ective hard spheres. In such a case all softer than hard sphere particles would exhibit '0.6, as observed even
by eye in Fig. 2b; such a deviation relates to particle interpenetration and &squeezing'. However, this approach fails for systems like to polymer-like 3280 star or the SI-600 micelle with an extended smooth rise in viscosity followed by divergence in a narrow concentration range. Alternative expressions used in hard sphere colloids, such as the Quemada equation [27,20] produce similar results. It is interesting that even for very small softness (S"0.02}0.2), sharp deviations from Brownian hard spheres are found, and attributed to the layer deformability [28,29]. Perhaps the most interesting result implied from Fig. 2b is the extreme sensitivity of the zero-shear viscosity data on the hydrodynamic size. Deviations from the solvent viscosity and even divergence are noted at &hydrodynamic' volume fractions below 1, whereas the respective &static' volume fractions are above 1 (Fig. 2a). Clearly, hydrodynamic interactions are important [13,30}33] and should be taken into account for obtaining the right scaling. Moreover, this approach leads to a better and more meaningful (compared to the static case) reduced plot (Fig. 2b). Roughly, even at low shear the e!ective hydrodynamic size is reduced, leading to apparent &harder' particles. The partial success of this crude incorporation of the hydrodynamic e!ects into the volume fraction, yielding the scaling plot of Fig. 2b, is rather surprising as the softness changes with concentration (in addition to shear rate and volume fraction, which are not considered here). As the deviation from the hard sphere master curve relates to the compressibility of the layer, this plot serves indeed as an indicator of the particle softness [13,28,29,34]. Finally, all particles of Fig. 2b were found to obey the generalized Stokes}Einstein relation, / +D /D , where D is the translational di!u * sion and D the long-time di!usion. It is noted here *
D. Vlassopoulos et al. / Physica B 296 (2001) 184}189
that the softness plays virtually no role here, other than allowing reaching higher volume fractions in the #uid (pre-gel) state due to particle interpenetration and compression/deformability. This is o!ered as an observation not fully understood yet. An explanation of this e!ect was recently o!ered for hard spheres based on mode coupling theory [35], but this approach would not work for soft particles.
4. Conclusions We investigated the concentration dependence of the zero-shear viscosity of multiarm star polymers and giant diblock copolymer micelles. The overlapping concentration based on the hydrodynamic radius was shown to be the appropriate reduction parameter for the concentration, allowing for a generic description of the relative viscosity and comparison of particles with varying softness, from hard spheres to polymers. Whereas micelles and stars exhibit distinct di!erences in their core structure, which renders a direct quantitative comparison ambiguous, both provide ideal systems for tuning the interaction potential via control of the functionality and arm size.
Acknowledgements We are grateful to J. Roovers and W.B. Russel for providing some of the star and the hard sphere data, respectively, and for useful discussions.
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