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Langmuir 1999, 15, 358-368
Dynamic Structure of Interacting Spherical Polymer Brushes A. N. Semenov,† D. Vlassopoulos,*,‡ G. Fytas,‡ G. Vlachos,‡ G. Fleischer,§ and J. Roovers| Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K., Foundation for Research and TechnologysHellas (FORTH), Institute of Electronic Structure and Laser, P.O. Box 1527, 711 10 Heraklion, Crete, Greece, Fakulta¨ t fu¨ r Physik und Geowissenschaften, Universita¨ t Leipzig, Linnestrasse 5, D-04103 Leipzig, Germany, and Institute for Chemical Process and Environmental Technology, National Research Council, Ottawa, Ontario K1A 0R6, Canada Received July 1, 1998. In Final Form: October 27, 1998 Multiarm star polymers represent a valuable model system for investigating the dynamics of tethered chains, spherical brushes, or grafted colloidal spheres. Because of their topology, the multiarm stars exhibit a nonuniform monomer density distribution leading to a core-shell morphology, which is responsible for their rich dynamic structure. When the stars interpenetrate, they exhibit liquidlike (macrocrystalline) order due to the enhanced osmotic pressure which balances the entropic stretching of the near-core segments and the excluded volume effects. Using dynamic light scattering, we probe three relaxation modes in the semidilute regime: (i) the fast cooperative diffusion, which is characteristic of their polymeric nature (entangled shell arms); (ii) the self-diffusion of the stars (essentially cores), probed because of finite functionality polydispersity, as confirmed by independent pulsed-field gradient NMR measurements; and (iii) the structural mode, which corresponds to rearrangements of the ordered stars. We develop a meanfield scaling theory, which captures all features observed experimentally with good quantitative agreement. The two slow modes, ii and iii, are reminiscent of the behavior of interacting hard colloidal spheres and are governed essentially by the same physics. We propose these model soft spheres as appropriate vehicles for unifying the descriptions of the dynamics of polymers and soft colloidal dispersions.
1. Introduction Colloidal steric stabilization by means of polymers, achieved via attachment of the latter to a particle surface, represents a topic of immense scientific and technological significance.1 The need to obtain a relatively thick polymer layer in order to prevent aggregation due to van der Waals attractions leads to the formation of tethered chain systems which essentially represent spherical brushes. Given the implications of the latter in several fields of colloid and surface science besides stabilization, such as adhesion, lubrication, and surface modification, the understanding of the structure and dynamics of these systems remains a compelling fundamental issue.2 The question of structure, in particular, has been addressed through the use of diblock copolymer micelles, obtained by using the self-assembling nature of diblock copolymers in selective solvents. In this direction, substantial progress has been made in the past years with the use of scattering techniques, which revealed the disorder-to-order transitions in concentrated suspensions of micelles.3 The ordered micelles form, in general, polycrystalline structures, which in turn have a great influence on the suspension rheology.4 †
University of Leeds. Institute of Electronic Structure and Laser. § Universita ¨ t Leipzig. | Institute for Chemical Process and Environmental Technology, National Research Council. ‡
(1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions, Cambridge: New York, 1989 and references therein. (2) Balazs, A. C., Ed. Special issue on theory and simulation of polymers at interfaces. MRS Bull. 1997, 22, 13. (3) McConell, G. A.; Gast, A. P.; Huang, J. S.; Smith, S. D.; Phys. Rev. Lett. 1993, 71, 2102. Gast, A. P. Langmuir 1996, 12, 4060. (4) Lin, E. K.; Gast, A. P. Macromolecules 1996, 29, 390. McConnel, G. A.; Lin, M. Y.; Gast, A. P. Macromolecules 1995, 28, 6754.
However, the dynamics of tethered chains has received much less attention. We recently probed the dynamics of polymer chains anchored on a flat surface with evanescent wave dynamic light scattering,5 but it is important to study the dynamics of polymers on a curved surface (spherical brushes) and, in particular, in the regime where they interact with each other. This problem has not been addressed so far in part because block copolymer micelles are not ideal systems for investigating the dynamic structure factor,3 for instance with dynamic light scattering, which is the most appropriate technique because of its wide range of time response.6 More specifically, they are not truly monodisperse; the cores are not uniform in size and might contain some arms or solvent, and furthermore, there is a chemical heterogeneity between the core and shell.3 The recent synthesis of multiarm star polymers7 has opened the route for the study of the dynamics of interacting chains tethered on curved surfaces, since these materials represent model spherical brushes with soft colloidal character.8 In particular, they are nearly monodisperse homopolymer systems with well-defined cores and without multiple scattering problems at high concentrations (by appropriate solvent selection). Furthermore, because of their nature, they actually represent (5) Fytas, G.; Anastasiadis, S. H.; Segrouschni, R.; Vlassopoulos, D.; Li, J.; Factor, B. J.; Theobaldt, W.; Toprakcioglu, C. Science 1996, 274, 2041. (6) Fytas, G.; Meier, G. In Dynamic Light Scattering. The Method and Some Applications; Brown, W., Ed.; Oxford Clarendon Press: London, 1993. (7) Roovers, J.; Zhou, L.-L.; Toporowski, P. M.; van der Zwan, M.; Iatrou, H.; Hadjichristidis, N. Macromolecules 1993, 25, 4324. (8) Richter, D.; Jucknischke, O.; Willner, L.; Fetters, J. L.; Lin, M.; Huang, J. S.; Roovers, J.; Toporowski, P. M.; Zhou, L.-L. J. Phys. IV 1993, C8, 3.
10.1021/la980794j CCC: $18.00 © 1999 American Chemical Society Published on Web 12/31/1998
Spherical Polymer Brushes
inherently stable suspensions, which exhibit liquidlike ordering at high concentrations,9 and as such, they are ideal model systems for addressing the above-mentioned dynamic issues. Moreover, multiarm star polymers represent a special case of branched polymers; in this respect, the study of their structure and dynamics and the determination of the differences from regular (low functionality) stars10 are also of fundamental importance to the understanding of the behavior of complex soft systems11 and even commercial branched polymers such as metallocene polyolefins.12 In addition, it should be pointed out that the investigation of ordering phenomena (for example, by SANS) based exclusively on structural (static) properties is incomplete, as several dynamic contributions can be involved in the static structure factor of complex fluids.6,13 Rich dynamic response is anticipated from computer simulations14 and dynamic scattering experiments on block copolymer micelles15 and hard-sphere colloidal suspensions.16 Recently we presented a dynamic light scattering study of semidilute solutions of multiarm star polymers.13 It was recognized that these materials exhibit two characteristic length scales, a polymeric scale which is the blob size, and a colloidal scale which is the size of the whole sphere. This unique feature places them in an intermediate regime between polymers and colloids, and therefore, their dynamics reflects these dual characteristics. The observed dynamic processes, i.e., concentration fluctuations (cooperative diffusion) and number density fluctuations (selfdiffusion, structural rearrangements), were also predicted by a mean-field scaling theory which accounted for the monomer density inhomogeneity of a single star and the interactions of interpenetrating stars at high concentrations. In this work, we extend the previous investigation and present a complete theoretical and experimental study of the dynamics of multiarm star polymers. By utilizing a wide range of stars with varying molecular parameters, we show the interplay of the polymeric and colloidal features of these systems. With the support of pulsedfield gradient NMR data, we unambiguously prove that dynamic light scattering can also measure the selfdiffusion in these systems. We show that the departure of their dynamic response from that of linear chains is due to their inherent segment density heterogeneity, which is captured by the theory. Finally, we identify the effects of star functionality, f, and arm molecular weight, Na, on the detected relaxation modes and their relative strengths and dynamics. The paper is organized as follows. Section 2 describes the materials and experimental techniques. Section 3 presents the experimental evidence from the light(9) Willner, L.; Jucknischke, O.; Richter, D., Farago, B.; Fetters, J. L.; Huang, J. S. Europhys. Lett. 1986, 2, 137. Jucknischke, O. Doctoral Thesis, Westfa¨lischen Wilhelms-Universita¨t Mu¨nster, 1995. (10) Fetters, J. L.; Kiss, A. D.; Pearson, D. S.; Quack, G. F.; Vitus, F. J. Macromolecules 1993, 26, 4324. (11) Bishko, G.; Harlen, O. G.; Larson, R. G.; McLeish, T. C. B. Phys. Rev. Lett. 1997, 79, 2452. McLeish, T. C. B., Ed. Theoretical Challenges in the Dynamics of Complex Fluids; NATO ASI Series E, Vol. 339; Kluwer: New York, 1997. (12) Hatzikiriakos, S. G.; Kazatchkov, I. B.; Vlassopoulos, D. J. Rheol. 1997, 41, 1299. (13) Segrouchni, R.; Petekidis, G.; Vlassopoulos, D.; Fytas, G.; Semenov, A. N.; Roovers, J.; Fleischer, G.; Europhys. Lett. 1998, 42, 271. (14) Grest, G. S.; Fetters, L. J.; Huang, J. S.; Richter, D. Adv. Chem. Phys. 1996, 94, 65. Grest, G. S.; Kremer, K.; Milner, S. T.; Witten, T. A. Macromolecules 1989, 22, 1904. (15) Fo¨rster, S.; Wenz, E.; Lindner, P. Phys. Rev. Lett. 1996, 77, 95. (16) Weissman, M. B. J. Chem. Phys. 1980, 72, 231. Segre, P. N.; Meeher, S. P.; Pusey, P. N.; Poon, W. C. K. Phys. Rev. Lett. 1995, 75, 958. Pusey, P. N. J. Phys. (Paris) 1987, 48, 709.
Langmuir, Vol. 15, No. 2, 1999 359 Table 1. Molecular Characteristics of Multiarm Polybutadiene Stars Employed (at 25 °C)
sample
f
Na
Mw
c* × 103 (g/mL)a
PBd 165 6407 12807 12814 12828 12856 12880
2 62 124 125 114 127 122
1570 118 127 240 484 875 1336
170 000 305 000 850 000 1 620 000 2 980 000 6 000 000 8 800 000
5.6 14 17.3 8 5.8 3.3 2.5
RG (Å)b
D0 (cm2/s)c × 10-8
354 100 120 115 216 345 424
26.5 20.4 12.6 8.6 6.3 3.8 3.1
a The determination of the overlap concentration from light scattering is discussed below. b The radius of gyration was determined from light scattering as discussed in the text. c Obtained from light scattering, as discussed in the text.
scattering studies. The theory is developed in section 4, whereas a critical comparison with the experiments is presented in section 5. The appearance of the structural process and its theoretical account are presented in section 6. Finally, section 7 summarizes the main findings and outlines the opened possibilities from and limitations of this investigation. 2. Experimental Section Materials. The synthesis of a series of regular polybutadiene star polymers with high functionality (nominally f ) 64, 128) has been described in detail previously.7 It is based on chlorosilane chemistry, which yields dendritic cores of different generations (third generation for f ) 64 and fourth generation for f ) 128), bearing respective numbers of Si-Cl bonds; the latter are coupled with anionically prepared polybutadiene arms. The important molecular characteristics of the samples used in this work are summarized in Table 1; in this table, Na refers to the number of monomer units per star arm, Mw, the total star weight-average molecular weight, c*, the star overlap concentration, RG, the star radius of gyration, and D0, the translation diffusion coefficient extrapolated at the limit of zero concentration. Dilute solutions of molecularly dispersed stars (no clusters) of optical quality in the good solvent cyclohexane (chosen because of its high refractive index contrast to polybutadiene) were prepared by adding the appropriate amount of polymer with stirring at room temperature and subsequent filtration in dustfree light-scattering cells (o.d. ) 10 mm) through a 0.2 µm Teflon Millipore filter. To avoid degradation problems, an amount of antioxidant, 4-methyl-2,6-di-tert-butylphenol (0.1 wt %), was added to the light-scattering cells. More concentrated solutions were obtained by subsequent controlled slow evaporation of the solvent at room temperature; this procedure took up to 1 week at the highest concentrations reached. The solutions were left to equilibrate before measurements; the corresponding time depended on the concentration and reached up to 2 weeks. The equilibrated samples were checked in the light-scattering setup and exhibited reproducible results and ergodic behavior for the range of concentrations investigated in this work. For comparison purposes, as discussed below, a few samples of 6407 and 12814 stars were also diluted in toluene, another good solvent. It is noted that, at the same polymer concentration, solutions in cyclohexane display about 10 times more scattering intensity than in toluene because of the difference in refractive index increments. All experiments were carried out at 25 °C. Dynamic Light Scattering (PCS). High-quality intermediate scattering functions were obtained using an ALV goniometer with a ALV-5000/fast multi τ digital correlator using 320 channels over a very broad time range, 10-7-103 s. The light source was a Nd:YAG air-cooled laser (Adlas DPY325) with wavelength λ ) 532 nm and single mode intensity 100 mW. In this work, both the incident and scattered light were polarized vertically (V) with respect to the scattering plane. The normalized time autocorrelation function of the polarized light scattering intensity is given in Gvv(q,t) ) 〈Ivv(q,t) Ivv(q,0)〉/〈|Ivv(q,0)|〉2, where q ) (4πn/ λ) sin(ϑ/2) is the scattering wavevector (n is the refractive index
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and θ the scattering angle).6,17, Under homodyne conditions, the Siegert relation holds: G(q,t) ) 1 + f*C2(q,t), where C2(q,t) ) |Rg(q,t)|2, obtained from the experimental G(q,t), is associated with the collective relaxation of the thermal concentration fluctuations of the system; here, g(q,t) ) 〈E(q,t) E(q,0)〉/〈|E(q,0)|2〉 is the normalized field correlation function, where E(q,t) stands for the scattered electric field. The instrument (normalization) factor f* relates to the coherence area and is obtained by means of a standard (a dilute polystyrene solution), whereas R represents the amplitude of the particular relaxation process (it is the fraction of total scattered intensity arising from fluctuations, with correlation times above 0.1 µs); if we denote such a process by “i”, then its intensity Ii is given by Ii ) RiI. In what follows, the three dynamic processes detected are denoted with subscripts “c”, “p”, and “st” to refer to the cooperative, polydispersity, and structural modes, respectively. The absence of a subscript refers to the total scattering intensity I (which is essentially the single translational diffusion process at low concentrations, i.e., isolated stars, as discussed below). Thus, the combined use of static and dynamic light scattering is a very powerful tool for studying multiple relaxation processes. The analysis of C(q,t) proceeded via the inverse Laplace transformation (ILT) assuming a superposition of exponentials:
RC(q,t) )
∫ L(ln τ) exp(-t/τ) d ln τ
(1)
which determines a continuous spectrum of relaxation times L(ln τ). The characteristic relaxation times, τ, correspond to the maximum values of L(ln τ). The ILT, which was realized using the program CONTIN,18 allowed the determination of the relaxation time and intensity of the partitioning modes, as revealed by the distribution relaxation function L(ln τ). Pulsed Field Gradient NMR (PFGNMR). This technique measures, in principle, the incoherent intermediate structure factor of the proton ensemble in the system. A stimulated echo pulse sequence is used, and the spin-echo amplitude is attenuated due to the self-diffusion if two field gradient pulses of magnitude g and duration δ are applied after the first and the third π/2 rf pulse, respectively. The spin-echo attenuation Sinc(q,t) is related to the self-diffusion coefficient by Sinc(q,t) ) exp(-q2tDs), where Ds is the self-diffusion coefficient, t is the diffusion time (the distance between the two field gradient pulses), and q ) γgδ denotes the generalized scattering wavevector, with γ as the gyromagnetic ratio of the proton. The range of q in the present experiments extends up to about 10-2 nm-1. The range of t starts at a few milliseconds and extends to about 1 s, the latter time being determined by the spin-lattice relaxation time T1 of the investigated species. For further details of PFG-NMR, see, for example, ref 19.
3. Experimental Results In the dilute regime, the properties of individual stars were recorded yielding typically a single diffusive relaxation process, which was related to the Brownian diffusion of the star. An example of the normalized C(q,t) for 12807 in cyclohexane (concentration 0.51 wt %) at q ) 0.035 nm-1 is depicted in Figure 1. The inset shows the diffusive character (Γ ∼ q2) of this relaxation rate. The intensity of this single-star motion conforms to the equation7,20 I(q) ) I(0)/(1 + q2RG2/3), where RG is the characteristic size of the star. The so-extracted values of RG are listed in Table 1 and are in good agreement with those obtained by SANS and reported before.9 Furthermore, the extrapolation of the translational diffusion coefficients to zero concentra(17) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (18) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229. (19) Ka¨rger, J.; Pfeifer, H.; Heink, W. Adv. Magn. Res. 1988, 12, 1. Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, U.K., 1992. (20) Fytas, G.; Patkowski, A. In Dynamic Light Scattering. The Method and Some Applications; Brown, W., Ed.; Oxford Clarendon Press: London, 1993.
Figure 1. Normalized scattered field time autocorrelation functions C(q,t) for 12807 at q ) 0.035 nm-1 and two concentrations (dilute, 0 ) 0.51 wt %; semidilute, b ) 5.65 wt %) in cyclohexane, exhibiting one- and two-step decays, respectively. Inset: Respective relaxation rates (0, 9; fast; b, slow) as a function of q2, indicating diffusive processes.
tion yields the single-star translational diffusion, D0 (C(q,t) ) exp(-D0q2t)), which is listed in Table 1 and used below for normalization purposes. As the concentration increases, the character of C(q,t) changes, and a two-step decay is clearly observed. A typical example is shown in Figure 1 for 12807 (5.65 wt % in the semidilute regime22) in cyclohexane at q ) 0.035 nm-1. The fast process is identified with the cooperative diffusion of the stars with c > c*, i.e., the random diffusive motion of the blobs of the stars’ arms which overlap (interpenetrate partially) at this concentration;21,22 this process is the dominant mechanism to relax total concentration fluctuations in semidilute solutions.23 The slow process is also diffusive as shown in the inset of Figure 1, which depicts the relaxation rates as a function of q2, and its origin will be discussed later. It is noted that, for all samples in the semidilute regime, I ∼ q° for both processes (see also Figure 3 below). Figure 2 depicts the concentration dependence of the scattered intensity of the first relaxation process (cooperative diffusion) for all samples investigated; the behavior is universal, as expected. At low concentrations in the dilute limit, where the star translational diffusion is detected, the intensity increases as the concentration of stars increases. At higher concentrations, the cooperative intensity decreases as the concentration increases, because of the negative interference among the interacting blobs, which are also reduced in size.22 The maximum of the intensity profile is identified with the overlapping concentration, c*. The so-determined values of c* are listed in Table 1 and are in reasonable agreement with those reported in previous studies using the intrinsic viscocity.7 It is noted, however, that the present method of c* determination is the most straightforward. The inset of Figure 2 depicts the intermediate scattering function of a semidilute 12807 solution of (2.3c*) versus the normalized decay time, D0q2t, for three different values of q. Such a plot is used routinely in studies of the dynamics of colloidal suspensions,16 and it is shown here for (21) Doi, M.; Edwards, S. F. The theory of polymer dynamics; Oxford: New York, 1986. (22) de Gennes, P. G. Scaling concepts in polymer physics; Cornell University Press: New York, 1979. (23) Jian, T.; Vlassopoulos, D.; Fytas, G.; Pakula, T.; Brown, W. Colloid Polym. Sci. 1996, 274, 1033.
Spherical Polymer Brushes
Figure 2. Concentration dependence of the intensity of the fast relaxation process (cooperative diffusion, or translational diffusion in the dilute single-process limit), indicating a peak at the point of overlap, c*. Data are shown for PBd165 (0), 6407 (O), 12807 (4), 12814 (3), 12828 (]), 12856 (+), and 12880 (×). Lines are drawn to guide the eye. Inset: Representation of the dynamic structure factor of the semidilute solution of 12807 (c ) 2.3c*) as ln C(q,t) versus the normalized decay time D0q2t for three different q’s (0.019 nm-1, 3; 0.024 nm-1, 4; 0.035 nm-1, ]).
Figure 3. Normalized scattered field time autocorrelation functions C(q,t) for 12828 in cyclohexane at q ) 0.035 nm-1 and three concentrations (1.23 wt %, O; 2.19%, 4; 3.4%, 3). Also shown are the distributions of relaxation times obtained from ILT, multiplied by the total scattering intensity I(q), L(ln(τ)) I(q) (solid line, 1.23%; dashed line, 2.19%; dotted line, 3.4%). Inset: Respective intensities of the two relaxation modes (open symbols, cooperative, Ic; solid symbols, slow (polydispersity), Ip) plotted against q.
comparison. Much like in Figure 1, the short- and longtime behaviors of C(q,t) determine the two relaxation processes observed, whereas the fact that the data for three different q’s collapse into a single master curve indicates diffusive processes with q-independent intensities. The evolution of the slow relaxation mode can be observed systematically in Figure 3, which illustrates the normalized C(q,t) at three different concentrations of 12828 in cyclohexane and q ) 0.035 nm-1. As the concentration increases, the departure from the dilutelimit single translation diffusion process becomes apparent, and at high enough concentrations (here up to 3.4 wt %), the two relaxations are clearly separated in time, as revealed by the distribution of relaxation times, which is also shown in this figure. In the inset, the insensitivity of the intensity (normalized to that of neat toluene) to q variations for both modes is demonstrated. Similar results
Langmuir, Vol. 15, No. 2, 1999 361
Figure 4. C(q,t) for 12856 in cyclohexane at q ) 0.035 nm-1 and three concentrations (1.38 wt %, 0; 1.84%, O; 2.89%, 4), along with the respective distributions of relaxation times from ILT, multiplied by the scattering intensity I(q), L(ln(τ)) I(q) (solid, dashed, and dotted lines). Inset: Concentration dependence of the intensities of the cooperative ([) and slow (1) relaxation modes; lines are drawn to guide the eye.
were obtained for all other stars investigated. For example, Figure 4 depicts the normalized correlation functions and respective L(ln τ) distributions (eq 1) for three different concentrations of 12856; the same conclusions as for the case of 12828 above (Figure 3) hold. It is interesting to note the strong concentration dependence of the intensity of the two relaxation processes (inset of Figure 4). For the fast cooperative diffusion, this is understood as already discussed above with respect to Figure 2; however, for the slow diffusive process, this is a new finding which shows stronger dependence than that of the cooperative mode and suggests strong interactions among the stars at high concentrations (star interpenetration). It is also clear from the L(ln τ) representation of Figures 3 and 4 that the cooperative process becomes faster as the concentration increases, as anticipated from the behavior of linear chains,22 whereas the slow mode slows down significantly. The latter, much like the respective intensity behavior, is indicative of strongly interacting systems. At this point it should be mentioned that, as it will be discussed below, multiarm stars form liquidlike structures at concentrations c g c*. This kind of ordering was first predicted by Witten et al.24 and confirmed by SANS measurements showing a peak in S(q) at finite q’s as well as secondary lower peaks at higher q’s.9,25 It is due to the enhanced osmotic pressure around c*, relative to the linear homopolymers, and it relates to the inherent star nonuniform segment density distribution. The presence of structure is consistent with the slowing down of relaxation processes other than cooperative diffusion.26 4. Theoretical Description of the Structure and Dynamics of Multiarm Star Polymers 4.1. Structure of a Single Star. Let us consider a star of functionality f, number of links per arm Na ) N, in a good solvent. The Daoud-Cotton picture27 implies that the size of the star is RG ) R ∼ Nνf(1-ν)/2, where ν = 0.59 (24) Witten, T. A.; Pincus, P. A.; Cates, M. E. Europhys. Lett. 1986, 2, 137. Witten, T. A.; Pincus P. A. Macromolecules 1986, 19, 2509. (25) Ishizu, K.; Ono, T.; Uchida, S. J. Colloid Interface Sci. 1997, 192, 189. (26) Anastasiadis, S. H.; Fytas, G.; Vogt, S.; Fischer, E. W. Phys. Rev. Lett. 1993, 70, 2415. Jian, T.; Anastasiadis, S. H.; Semenov, A. N.; Fytas, G.; Fleischer, G.; Vilesov, A. D. Macromolecules 1995, 28, 2439. Fytas, G.; Vlassopoulos, D.; Meier, G.; Likhtman, A.; Semenov, A. N. Phys. Rev. Lett. 1996, 76, 3586. (27) Daoud, M.; Cotton, J. P. J. Phys. 1982, 43, 531.
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where X1 ) ∫r0 φ(r) d3r ) 4πrc3{1/3 + ν(r1/rc)1/ν} is the total number of monomers in one coat region (where the arms are strongly stretched). Solving eq 4 for φ we get
φ)
φ1
(5)
1 + (φ1V1 - X1)/(Nfυ)
which, together with the expression φ1 ) (rc/r1)3-1/ν, determines implicitly a relation between φ1 and φ, with r1 treated as a parameter. Equations 4 and 5 are valid if cV1 < 1, i.e., X1 < Nfυ. This condition is fulfilled in the semidilute regime, i.e., when φ exceeds the overlap volume fraction,
φ* )
(
1 3 Nfυ Nfυ 4π r 3 4πνr 3 3ν c c
)
-3ν
(6)
4.3. Free Energy. We identify two contributions to the free energy, F, in the semidilute regime,
F ) Fid + Fint
(7)
Figure 5. Schematic representation of multiarm star polymers in good solvent: (a) single star; (b) two interacting stars in semidilute solution.
where Fid ) [φ/(Nfυ)](ln φ) + const is the ideal-gas free energy per unit volume and Fint is due to excluded-volume monomer interactions and arm stretching,
is the Flory exponent. The monomer concentration φ scales with the distance from the center, r, as φ(r) ∼ f[(3ν-1)/(2ν)]r[(1-3ν)/ν]. This equation is valid if φ(r) < 1, i.e., r > rc ∼ f1/2. A region close to the star center, r < rc, is densely packed by monomers. One can consider f arms as being actually grafted to the sphere of radius rc, which can be estimated assuming the minimum area b2 per arm: 4πrc2 ) fb2 so that rc ) b(f/4π)1/2. Note that, in general, we treat the size b as an adjustable parameter. The monomer concentration profile can be thus approximated as
Fint ) (1 - cV1)F(φ1) + cFcoat
φ(r) =
{
(rc/r)3-1/ν, R > r > rc r < rc 1,
(2)
4.2. Semidilute Solution of Stars. The stars are densely packed in the semidilute regime. The concentration is nearly uniform, φ ) φ1, and is increasing according to eq 2 in the vicinity of a star center, if the distance to the center, r, is smaller than another size r1, which is determined by the condition φ(r1) ) φ1, i.e., φ1 ) (rc/r1)3-1/ν, so that the concentration profile is as follows (compare with eq 2):
{
1 φ(r) = (rc/r)3-1/ν φ1,
r < rc rc < r < r1 r > r1
(3)
A star thus consists of a coat of radius r1, where the arms are strongly stretched and where the monomer density profile decreases with distance, and the outside part, where the monomer density φ1 is nearly uniform. Schematically, this situation is sketched in the cartoon of Figure 5. The fraction of total volume occupied by coat regions is p ) cV1, where V ) (4π/3)r13 is the coat volume, c ) φ/(Nfυ) is the number concentration of stars, φ is the mean monomer volume fraction, and υ is the monomer volume; the “uniform” fraction of the total volume is thus 1 - p. Hence,
φ ) φ1(1 - cV1) + cX1
(4)
(8)
In eq 8, (1 - cV1) is the fraction of total volume with uniform monomer concentration, φ1. In this homogeneous region, the arms can be considered as random chains of concentration blobs, like the situation in a semidilute solution of linear chains with the free energy density22 F(φ1) ) (k/b3)φ13ν/(3ν-1), where k is a numerical factor which is determined by the polymer chain rigidity and presumably φ1 , 1. The second term in eq 8 is the free energy of a star ˜ (r) d3 r.28 A scaling argument suggests coat, Fcoat ) ∫rc 0. In the derivation of eq 12 we have assumed N . f 1/2 and rewritten eq 6 as φ* = (3ν)3ν(Vc/Nfυ)3ν-1 ∼ (f1/2/ N)3ν-1. It is noted that the experimentally determined overlap concentrations of Table 1 conform reasonably well to this scaling prediction, especially for the higher Na. Compared to the standard good-solvent scaling,22 φ-0.3, the correction term in eq 12 gives rise to a stronger decrease of Ic with φ. It is well-known that the cooperative mode is diffusive,22 i.e., its thermal decay rate is Γc ) Dcq2, with the diffusion constant, Dc ) sφ(∂µ/∂φ). The sedimentation constant is s ) vm/fm, where vm is the mean polymer velocity induced by a small external force fm acting on each monomer. If a solvent flow is pumped slowly through the polymer matrix by the pressure gradient -∇P, then the mean solvent velocity is vs ) -ζ∇P, where ζ ) s/φ is the permeability of the polymer to the solvent, which, for a semidilute solution of linear chains, scales as the square of the correlation length,22 ξ2 ∝ φ-2ν/(3ν-1). In the case of a semidilute star polymer solution, the local permeability
is not uniform; it is lower near the star center, where the local monomer concentration is higher. An approximate way to find the effective (macroscopic) permeability of the star system is to average the local permeability over the total volume: ζ ) 1/V[∫ζ1(φ(r)) d3r]. Here we assume that the local permeability is determined by the monomer concentration: ζ1(φ) ) ζ0φ-2ν/(3ν-1). Hence, ζ ) (1 - p)ζl(φ1) + cZ1, where Z1 ) ∫r 0. Therefore, this term provides a faster increase of Dc with φ than does the linear chain scaling,22 Dc ∝ φ 0.77. Note, however, that the star architecture effects (correction terms) are more important for Ic (eq 12) than for Dc (eq 14); the ratio of the relative corrections to these quantities, respectively, is (1 + δ)/δ, i.e., it might be large if δ is small. 4.5. Star Self-Diffusion. The presence of inherent functionality polydispersity can cause scattering from polymer density fluctuations related to an exchange between smaller (f1) and larger (f2) stars. Let us consider for simplicity a bidisperse star system with the same arm length N and different functionalities f1 ) f(1 - ) and f2 ) f(1 + ), where is assumed to be small, , 1; in this notation, is a measure of the star molecular weight polydispersity: 2 ∼ Mw/Mn - 1. The treatment of section 4.3 can be easily generalized to the bidisperse system. In the semidilute region (φ* < φ < 1), the total free energy per unit volume encompasses three contributions,
F ∼ Fosm + Fstar + Fid
(15)
where Fosm ) (k/b3)φ3ν/(3ν-1) is the osmotic free energy of the corresponding semidilute system of linear chains (being independent of the polymer architecture), Fstar ) (k/b3)(1/N)β ln(1/φ)(f11/2φ1 + f21/2φ2) is the correction due to the star architecture, and Fid ) [φ1/(Nf1υ)] ln φ1 + [φ2/ (Nf2υ)] ln φ2 is the ideal-gas free energy. Here φ1 and φ2 are the mean volume fractions of f1 and f2 stars, respectively; φ ) φ1 + φ2 is the total concentration; and β ) (b3/υ)(1/(4π)1/2)[ν/(3ν - 1)]. The terms linear in φ1 and φ2 are neglected in eq 15, as these terms do not affect the relevant thermodynamic properties of the system, and the scattering intensity is determined by the free energy expansion in terms of concentration fluctuations δφ1 and δφ2 up to the second order (F(2)). Omitting linear terms,
1 F(2) ) (A11 δφ12 + A22 δφ22 + 2A12 δφ1 δφ2) (16) 2
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where A11 ) A + E + δ1; A22 ) A - E + δ2; A12 ) A = (∂2F)/(∂φ2) = (k/b3)[3ν/(3ν - 1)2]φ3ν/(3ν-1)-2; E = (k˜ /b3)[β(/Nφ)]f1/2; δ1 ) 1/(φ1f1Nυ), and δ2 ) 1/(φ2f2Nυ). Assuming a symmetric mixture of f1- and f2-arm stars (φ1 ) φ2 ) φ/2), δ1 = δ2 = B/2, where B ) δ1 + δ2 = 4/(φfNυ), and introducing new variables x ≡ δφ1 + δφ2 and y ≡ (δφ1 - δφ2)/2, the free energy (eq 16) can be then represented as
1 F(2) ) [Ax2 + By2 + 2Exy] 2
(17)
Thermodynamic stability of the system requires that the quadratic form (eq 17) is positively defined, E2 < AB, i.e., 2 < c2 ≡ const (N/f2)φ1/(3ν-1), where const ) (48π/ν)(υ/ b3)(1/k)(1 + δ)-2 is a numerical constant which depends on the monomer shape parameter, υ/b3. If initially > c, then the system would phase separate in two (at least) phases,29 reducing the degree of polydispersity in each phase below the critical value. Below, we assume for simplicity that , c. In the free energy expansion eq 17, the total monomer density fluctuation x is actually a fast variable that can be relaxed by cooperative diffusion of the polymer matrix moving as a whole with respect to the solvent, so that φ1/φ2 ) const = 1. On the other hand, y is a slow variable that cannot be changed by cooperative diffusion (δφ1 δφ2 ) 0 for φ1/φ2 ) const ) 1), but it can only be changed by mutual diffusion (self-diffusion) of f1 and f2 stars. During the fast cooperative process, the composition fluctuations can be neglected, i.e., we can set y ) 0. Then eq 17 reads F(2) ) (1/2)(Ax2), so that Ic ∼ 〈x2〉 ) 1/A, in agreement with eq 12. While considering a slow relaxation of y, we can assume that the total density (x) is already relaxed by cooperative diffusion. Hence, minimizing F(2) over x for a given y yields F(2) ) (1/2)(B - E2/A)y2 ≈ (1/2)By2, x ) -(E/ A)y. In the semidilute regime φ . φ* and for , 1, the ratio E/A is small, hence |x| , |y|, implying that during the slow process the total density is nearly fixed, while f1 stars and f2 stars exchange their positions in space via mutual diffusion. Because B is related solely to the idealgas free energy, this process is driven by the ideal-gas entropy of mixing of stars of two types, and the mutual diffusion constant is nearly equal to the self-diffusion constant, Ds, of stars. The intensity Is ) 〈x2〉/υ ) E2〈y2〉/ (υA2) ) E2/(υA2B) and the rate Γs of this slow process are given by
f22 (3ν-3)/(3ν-1) φ and Γs = Dsq2 Is ∼ N
(18)
Note that Is ∼ φ-1.6 decreases with concentration much faster than the cooperative intensity, Ic (eq 12), because of the dominance of the entropic coat stretching (term E) in the regime of strong interpenetration. 5. Comparison of Predictions with Experiments It is evident that the theory captures qualitatively the experimental findings observed so far (Figures 1-4). For a quantitative comparison with respect to the scaling predictions, we plot the normalized (per scatterer) experimental cooperative intensities (Ic/cM) and diffusion coefficients (Dc/D0) against the reduced concentration (c/ c*) for all samples investigated in Figures 6 and 7, respectively. First of all, it is indeed confirmed that multiarm stars exhibit a sharp deviation from the semidilute linear chain scaling behavior at c > c* . This deviation is proportional to the parameter δ (eqs 12 and (29) Semenov, A. N.; et al. In preparation.
Figure 6. Normalized scattered cooperative intensity (per concentration and total molecular weight) versus reduced concentration c/c* for all samples investigated: PBd 165 (0); 6407 (O); 12807 (4); 12814 (3); 12828 (]); 12856 (+); 12880 (×). Solid line with slope -1.3 represents the linear chain scaling.
Figure 7. Compiled plot of reduced cooperative diffusion coefficient Dc/D0 versus reduced concentration c/c* for all samples. Symbols are the same as in Figure 6. Solid line with slope 0.77 represents the scaling for linear chains.
14) and it is stronger for larger core size, i.e., for higher f (note that the correction terms in eqs 12 and 14 increase with φ*, and φ* increases with f). This deviation is in agreement with the results of Roovers et al.30 from static light scattering; however, we claim that the current analysis is more accurate, because the extracted cooperative intensity from the dynamics yields correctly the osmotic modulus, whereas the intensity from static measurements bears contributions from the slow mode as well.13 At very high concentrations, it seems that the slope is reduced and tends toward that of the linear chains because of the dominant effect of the osmotic repulsion of the blobs (the osmotic pressure does not increase sharply further24), and this is in harmony with previous observations based on static measurements30 as well as those based on stars with f < 18 with PCS.31 Furthermore, a (30) Roovers, J.; Toporowski, P. M.; Douglas, J. Macromolecules 1995, 28, 7064.
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Figure 8. Compiled plot of the concentration dependence of the diffusion coefficients of the slow (polydispersity) mode, Dp, for all multiarm star samples: 6407 (0); 12807 (O); 12814 (4), 12828 (3); 12856 (]); 12880 (+). Also shown are the self-diffusion coefficients measured by PFG-NMR for 12807 (b); 12814 (2); and 12828 (1). Lines are drawn to guide the eye. Inset: Respective reduced plot of the product Dpη (η being the solution viscosity, taken from ref 32) against c/c* for the various stars.
comparison of Figures 6 and 7 suggests that the effects of the inhomogeneous monomer density distribution on the departure from the linear chain scaling are stronger for the cooperative intensity rather than the cooperative diffusion, in agreement with the theory (eqs 12 and 14; see also discussion in section 4). Figure 8 represents a compiled plot of the diffusion coefficient of the slow mode (Dp) as a function of concentration, indicating a strong slowing down. On the basis of the theory (section 4.5), this is essentially the selfdiffusion of the stars. To prove this experimentally, we carried out independent PFG-NMR measurements on the samples 12807, 12814, and 12828, and the resulting true self-diffusion coefficients (Ds) are also depicted in Figure 8 (solid symbols). The very good agreement with the slow scattering diffusive mode Dp confirms that light scattering, which probes collective motions, can also detect the selfdiffusion through the refractive index contrast induced by the size polydispersity of the stars (essentially it is the size polydispersity of the cores and coat central regions, resulting from the polydispersity in star functionality). It is noted here that with subscript “p” we denote the lightscattering data, whereas with “s”, we denote the PFGNMR data (direct measurement of self-diffusion) and theoretical predictions of self-diffusion, to comply with the essential difference between collective (light-scattering) and self (PFG-NMR) dynamics. These findings, nevertheless, relate Dp with the self-diffusion coefficient of particles with size less than q-1, over the examined concentration range. Excess light scattering at low q’s has been observed in colloidal hard spheres16 and in diblock copolymers,26a,b in which long wavelength order parameter fluctuations are suppressed (correlation hole) in the presence of repulsive interactions. This incoherent scattering is caused by fluctuations in size or by composition polydispersity. In the present case, self-diffusion is measurable because of the concentration (and hence refractive index ∆n) contrast between the core/coat region and the bulk semidilute solution (see also section 4.5), a topological consequence of the starlike architecture. Unlike the situation in diblock copolymer solutions,26a,b Ip should depend on the solvent, since ∆n is actually the polymer (31) Adam, M.; Fetters, L. J.; Graessley, W. W.; Witten, T. A. Macromolecules 1991, 24, 2434.
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Figure 9. Concentration dependence of the intensities of the slow (polydispersity) mode (Ip; symbols are the same as in Figure 8) and the structural mode (Ist; 12856, ×, and 12880, asterisks (*)) for all multiarm stars measured. Dashed and dotted lines are drawn to guide the eye. Solid lines with slopes -1.6 and -2.3 represent the scaling predictions for Ip (eq 18) and Ist (eq 20), respectively.
refractive index relative to the solvent. In fact, Ip ∼ (∆n)2, as verified by the data in two good solvents, toluene and cyclohexane. Polydispersity is actually the reason why self-diffusion coefficients were measured before with dynamic light scattering in different systems, such as hardsphere colloids16 (size polydispersity) and diblock copolymers26 (composition polydispersity). It is further known that the self-diffusion should scale with the solution viscosity1 (rather than the solvent viscosity, as in the case of cooperative diffusion), and this is verified in the inset of Figure 8, which plots the product Dpη against (c/c*) for the range of concentrations studied; η is the solution viscosity, taken from ref 32. This is in clear contrast with the behavior of hard-sphere colloids, where the collective structural relaxation corresponding to the peak of the structure factor, and not the self-diffusion, scales with η-1.16b Concerning the intensity of the slow polydispersity (selfdiffusion) mode (Figure 9), Ip, it is reduced as the concentration increases, especially above c*, as is already mentioned above and discussed in the theoretical section as well. In technical terms, increasing the concentration of stars reduces the refractive index contrast between cores/coats and the “sea” of arm entanglements, which yields a reduced intensity. Given the rather limited number of concentrations investigated for each star system, a comparison with the theoretical predictions should be done carefully in order not to jump to wrong conclusions. Within this limitation, the actual scaling seems to be in reasonable agreement with eq 18, especially for stars with the higher (c/c*), i.e., 12856 and 12880, which are in the regime of Ip reduction. The onset of the latter seems to scale with c*; thus it occurs at much higher concentrations for the stars with shorter arms. The important point to emphasize, however, is the same qualitative trend (decrease) followed by Ip with concentration. Actually, the stronger decrease of Ip with concentration, compared to Ic, as demonstrated in Figures 6 and 9, is captured by the theory as well (eqs 12 and 18). For all samples investigated, at concentrations above the ones reported in Figures 8 and 9, the effects of ordering on the dynamics become more evident through the (32) Roovers, J. Macromolecules 1994, 27, 5359.
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Figure 10. C(q,t) for a semidilute solution of 12880 in cyclohexane with c ) 2.04c* at different q’s ranging from 0.009 nm-1 (0) to 0.035 nm-1 (]). Upper inset: Representation of effective Deff (q). Lower inset: Plot of I(q) profile. Solid lines refer to the dilute single-process limit c ) 0.28 c*.
appearance of nonergodicity and an additional anomalous self-diffusion mechanism (not like the Fickean diffusion of Figure 8) and potential gelation.33 6. Structural Mode The predictions of Witten et al.24 concerning liquidlike ordering of the multiarm stars were confirmed by the SANS measurements of Richter et al.9 The order was found to persist up to the melt state.34 Along these lines, block copolymer micelles were also found to exhibit various macrocrystalline structures.3 To detect, with light scattering, relaxation modes reflecting the presence of order, such as cooperative structural rearrangements, it is necessary to approach qRG ∼ O(1), which becomes feasible with the largest available star, i.e., 12880. Figure 10 depicts the C(q,t) of a cyclohexane solution of 12880 with c/c* ) 2.04, at different q’s ranging from 0.009 to 0.035 nm-1, in a ln-linear representation. It is noted, first of all, that at concentrations below c*, the total intensity I(q) exhibits the expected Debye dependence, and C(q,t) displays a single diffusive relaxation (illustrated by the solid line in Figure 10). The fact that the long-time data for c > c* do not collapse into a master curve versus q2t suggests that the relevant mode (or modes) includes a q-dependent effective diffusion coefficient, Deff. Actually, because of the interstar pair correlations, I(q) reverses its q dependence and C(q,t) deviates from the classical diffusive character (discussed in section 5). The extracted Deff is indeed plotted against q in the upper inset of Figure 10. As in the case of hard-sphere colloids16 and diblock copolymers in the ordering region,26a,b Deff exhibits a slowing down at high q’s due to the liquidlike ordering, implied by the I(q) profile. At the lowest accessible value of q, Deff should approach the value of Dc because of the correlation hole, whereas at the highest value of q, Deff reflects more the effects of structural relaxation (the structural mode); at this concentration (2.04c*), however, the resolution of these two contributions is ambiguous. On the other hand, the total intensity is now q-dependent, and it actually increases with q (lower inset of Figure 10), whereas for both cooperative and polydispersity modes, (33) Vlachos, G.; Vlassopoulos, D.; Fytas, G.; Fleischer, G.; Roovers, J.; Unpublished data. (34) Vlassopoulos, D.; Pakula, T.; Fytas, G.; Roovers, J.; Karatasos, K.; Hadjichristidis, N.; Europhys. Lett. 1997, 39, 617. Pakula, T.; Vlassopoulos, D.; Fytas, G.; Roovers, J. Macromolecules 1998, 31, 8931.
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Figure 11. C(q,t) of 12880 in cyclohexane at c ) 2 wt % and q ) 0.035 nm-1, along with the distribution of relaxation times from ILT, multiplied by the scattering intensity I(q), for c ) 2% and c ) 3 wt % Inset: The corresponding q dependence of the structural intensity, Ist, and structural time per probing area, τstq2 ) 1/Dst for the 2% solution; lines are drawn to guide the eye.
Ic ∼ q°, Ip ∼ q°. This is the strongest evidence of the signature of structure in the range of light-scattering q’s, and the relevant structural mode is responsible for the q dependence in Figure 10. The separation of the two processes (cooperative and self-diffusion) increases with concentration because of the inverse dependence of their relaxation rates with concentration. Hence, C(q,t) develops distinct decays at higher concentrations, but on the other hand, the presence of both polydispersity and structural modes, as well as the fact that qRG e 1, complicates the situation. Therefore, this additional structural mode renders the long-time diffusion analysis (typically used in colloids16) ambiguous, and in order to resolve it, we must use the ILT procedure. The ordered multiarm stars in the liquidlike structure resemble a structure like the representative two-star assembly sketched in Figure 5b; notice the slight polydispersity in core size, which is responsible for the slow mode. One can visualize the cores in this picture as being immersed in a sea of entanglements (blobs) formed by the interacting interpenetrating arms and responsible for the cooperative diffusion, which is of polymeric nature; it relates to the linear-chain blob size. On the other hand, these stars possess an additional large length scale, their radius, as discussed in the Introduction; related to this is not only the star self-diffusion but also the extra mode reflecting collective structural rearrangements of the ordered stars. In Figure 11 the C(q,t) of 12880 in toluene at c ) 2 wt % is depicted along with two distributions of relaxation times from ILT, one at the same concentration, and one at c ) 3 wt %, both at q ) 0.035 nm-1. Interestingly, the ILT reveals three relaxation processes. The fast one is the cooperative diffusion, as it becomes faster with increasing concentration, and the slow one is the polydispersity mode, which slows down significantly with concentration. The clearly resolved intermediate mode is the structural mode, exhibiting distinct characteristics. Its intensity increases with q, as seen in the inset of Figure 11, and it peaks at high q’s outside the range of light scattering. It is important, however, to point out that the inverse effective structural diffusion constant (1/Dst ) τstq2) also increases with q (inset of Figure 11), in harmony with the experimental results from hardsphere colloidal suspensions near their glass transition limit;16 moreover, the theoretical arguments above corroborate this picture. This mode was clearly resolved in the 12856 star as well, at high q’s and concentrations
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Figure 12. Distribution of relaxation times, from ILT, multiplied by the scattering intensity I(q), at three concentrations of 12856 (1.8, 2.9, and 3.1 wt %) and q ) 0.035 nm-1, indicating the evolution of the slow (polydispersity) and intermediate (structural) modes. Inset: Concentration dependence of the ratio of structural to slow (polydispersity) times, τst/τp for 12856 (9) and 12880 (0), at q ) 0.035 nm-1. The line with slope -2/3 represents the theoretical prediction (see text).
above 2.5 wt %, as demonstrated by the ILT representations of Figure 12. Note in this figure the evolution of the slow (polydispersity) and intermediate (structural) modes with concentration35. On the basis of the above observations, a structural mode can be considered from the theoretical standpoint; it is due to local (liquid) ordering of star centers. Its intensity is proportional to the structure factor of the centers, Sc(q):
Ist )
φ (X′)2Sc(q) Nfυ
(19)
if qr1 < 1. Here, X′ ) (X1 - V1φ1) is the excess volume of monomers in a core/coat region. Sc(q) incorporates liquidlike correlations between the star center positions. We define Sc(q) in such a way that it tends to 1 in the limit qR . 1, because in this regime, the structure of the center itself (which is a point by definition) is probed. The structure factor exhibits a maximum (of order unity) at qR ∼ 1 and decays at low q’s as Sc(q) ∼ (qR)2; the latter incorporates the assumption that the system is incompressible during the structural relaxation, i.e., Sc(0) ) 0, which is actually true because a finite compressibility gives rise to the cooperative mode, already taken into account before. Thus, we get, in the regime qR < 1, φ > φ*,
Ist ∼
φ 2 2 f8/3 q R (X′)2 ∼ q2b2 1/3φ(ν-7/3)/(3ν-1) ∝ φ-2.3 (20) Nfυ N
From eq 20, we see that the structural intensity is decreasing with φ with the exponent -2.3, i.e., faster than the intensities of both the cooperative and the slow modes, a trend followed by the experimental data as well (Figure (35) It is further noted that, throughout the wide range of concentrations (extending up to about 50c*, i.e., well into the semidilute regime), and for all different multiarm stars studied in this work, no evidence of ultraslow clusterlike relaxation process was found. This contradicts the recent study of Stellbrink et al. (Stellbrink, J.; Allgaier, J.; Richter, D. Phys. Rev. E 1997, 56, R3772) who reported such an ultraslow mode in polyisoprene stars of much lower functionality (f ) 18) above c*. They associated this mode to a kind of structural glass transition, but on the other hand, they apparently did not find any evidence of self-diffusive or structural relaxation.
9); however, because the theory at this point is not rigorous, the comparison with the experiments is expected to be only qualitative. The intensity of the structural mode is rather small if qR is small. On the other hand, if qR ∼ 1, then the structural intensity actually dominates over both cooperative and slow processes at the overlap concentration φ ∼ φ*; Ist/Ic ∼ f3/2 . 1. Because the structural process involves local rearrangements of stars only, its time is of the order of the self-diffusion time of a star over a distance of order R, the star size, i.e., τst ∼ R2/Ds. The relaxation time is thus predicted to be basically independent of q, although some q dependence of τst in the region qR ∼ 1 might be expected. The ratio of structural to slow diffusion times is thus τst/τs ∼ q2R2 ∼ q2(Nfυ/φ)2/3, i.e., for a fixed q, this ratio is proportional to φ-2/3. Therefore, τst is predicted to display a weaker concentration dependence than the self-diffusion time τs. This prediction is in qualitative agreement with the trends of the experiment results (inset of Figure 12); however, the experimental data (using the slow polydispersity time, τp, as the self-diffusion time) suggest a much weaker concentration dependence of τst at high concentrations; we recall that τp scales with η (inset of Figure 8), in contrast to the corresponding behavior of hard-sphere colloids.16b This discrepancy is consistent with the clear q dependence of τst (inset of Figure 11), not accounted for in the theoretical argumentation of the structural mode. The resolution of this fine issue represents the subject of future work. Nevertheless, it is important to emphasize that the theoretical ideas given here can capture the presence and the qualitative features of the collective structural relaxation. 7. Concluding Remarks Multiarm star homopolymers represent a novel class of topologically complex soft materials possessing properties intermediate between the polymers and colloidal suspensions. The key to their fascinating behavior is the topologically induced inhomogeneous monomer density distribution leading to a core-shell single-star structure. In semidilute solution, the interacting stars form a liquidlike structure due to the enhanced osmotic pressure which balances the entropic stretching of the arms near the core. This situation is reflected in a rich dynamic response, which was investigated thoroughly with dynamic light scattering and PFG-NMR. At concentrations above the overlapping one, three relaxation processes were detected and accounted for theoretically: (i) The fast cooperative diffusion (Ic, Dc) of the entangled arms, a process of polymeric nature, which has an intensity and diffusion coefficient exhibiting stronger concentration dependence compared to the linear chains, the deviation being sharper for the intensity. (ii) The slow self-diffusion mode (Ip, Dp), probed because of the finite functionality polydispersity of the stars, and confirmed by independent PFG-NMR data. This mode slowed significantly with concentration because of the presence of the liquidlike structure, whereas its intensity weakened with concentration, in harmony with predictions, as a result of reduced core-shell refractive index contrast (dominance of entropic stretching of the coat region). (iii) The intermediate structural mode (Ist, τst) due to the local (liquidlike) ordering of the cores, exhibiting an intensity and time increasing with q, in analogy to hardsphere colloids in the glass transition region. This mode was attributed to collective structural rearrangements of the ordered stars. The implications of these results to the understanding of the colloidal steric stabilization and tethered chain
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dynamics, as well as to unifying the physics of these two classes of soft materials, namely polymers and colloids, are evident. Acknowledgment. We are grateful to G. Petekidis and R. Seghrouchni for their assistance with measure-
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ments in the early stages of this work. We would like to acknowledge partial support by the EC (INTAS-39-2505) and the Greek General Secretariat for Research and Technology (PLATON 1997-1998). LA980794J
