Semidilute region for linear polymers in good solvents

Noda, Ichiro; Higo, Yuji; Ueno, Noboru; Fujimoto, Teruo

doi:10.1021/ma00135a013

Cited by 94 publications

(91 citation statements)

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“…To do so, we need, in addition to the diffusion coefficient D, values for the kinematic viscosity n, the osmotic compressibility ͑≠m͞≠w͒ P,T , and the Soret coefficient. Values for n and ͑≠m͞≠w͒ P,T were obtained from the literature [11,14]. As mentioned earlier, S T of our solutions was measured by Zhang et al [12], yielding values in good agreement with data obtained by Köhler et al [15].…”

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Concentration Fluctuations in a Polymer Solution under a Temperature Gradient

Zhang

Sengers

et al. 1998

Phys. Rev. Lett.

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We have performed small-angle Rayleigh light-scattering measurements in polymer solutions under various externally applied temperature gradients. Our experiments confirm the existence of an enhancement of the concentration fluctuations. The nonequilibrium concentration fluctuations are proportional to ͑=T͒ 2 ͞k 4 , where =T is the applied temperature gradient, and k is the wave number of the fluctuations. The measured strengths of the nonequilibrium fluctuations in the dilute and semidilute concentration regime agree with the strengths calculated from fluctuating hydrodynamics.[ S0031-9007(98) PACS numbers: 61.25. Hq, 05.70.Ln, 66.10.Cb During the past decade, considerable theoretical and experimental effort has been devoted to the study of fluctuations in fluids in nonequilibrium steady states, in particular, when subjected to a constant temperature gradient =T . It is now known that the correlation functions become spatially long ranged [1][2][3], because microscopic detailed balance is violated. These nonequilibrium correlation functions can be observed by small-angle Rayleigh light scattering. Experiments have been carried out with simple fluids, in which case excellent agreement with the theory has been found [4,5].In fluid mixtures, the situation is more complicated because the temperature gradient induces a concentration gradient through the Soret effect. Hence, in addition to nonequilibrium entropy and viscous fluctuations, which are also present in one-component fluids, nonequilibrium concentration fluctuations are present in mixtures [6]. All of these nonequilibrium fluctuations have intensities proportional to ͑=T ͒ 2 ͞k 4 , where k is the wave number of the fluctuations. The existence of these nonequilibrium fluctuations and their dependence on ͑=T ͒ 2 ͞k 4 have been checked experimentally, but the quantitative values of the observed strengths of the nonequilibrium concentration fluctuations were not in agreement with the theory [5,7,8].Dilute polymer solutions appear to be a more suitable system to check the theory for nonequilibrium concentration fluctuations, because the expression for the dynamic correlation function C͑k, t͒ as a function of time is simpler. When the polymer concentration is below the overlap concentration, viscoelastic and entanglement effects are negligible, and ordinary fluctuating hydrodynamics should be applicable to describe the nonequilibrium fluctuations. Furthermore, the collective mass-diffusion coefficient is so small and the Rayleigh ratio so large that, in practice, the concentration fluctuations are dominant. Therefore, for polymer solutions at sufficiently low concentrations, the equations in Refs. [5,6] can be simplified, yielding the following for fluctuations with a wave vector perpendicular to =T :where C 0 is the ratio of the signal to the local oscillator background, which corresponds to the amplitude of the experimental equilibrium correlation function; D is the collective mass-diffusion coefficient, and A c is the nonequilibrium enhancement. A c may be ...

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confidence: 89%

“…From Ref. [11], the radius of gyration and overlap concentration for this polymer in toluene solutions may be estimated as R g ഠ 11 nm and w ‫ء‬ ഠ 3.1%. Five polystyrene/toluene solutions with concentrations below w ‫ء‬ and one slightly above w ‫ء‬ were prepared gravimetrically and were agitated for at least one hour before use.…”

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confidence: 99%

Concentration Fluctuations in a Polymer Solution under a Temperature Gradient

Zhang

Sengers

et al. 1998

Phys. Rev. Lett.

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“…8 When the polymer concentration is beyond the semidilute regime, the osmotic pressure or the osmotic compressibility cannot be described by the renormalization-group theory. 9 Alternatively, the thermodynamic perturbation theory chooses a system of particles interacting by the hard-core potential as a reference system, and treats the long-ranged attractive interaction in a perturbative way. For example, Barker and Henderson proposed a perturbation theory for the system of spherical particles with a square-well potential.…”

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Repulsive and Attractive Interactions between Polystyrene Chains in a Poor Solvent

Oribe

Sato

2004

Polym J

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ABSTRACT:The osmotic compressibility up to high concentrations as well as the second virial coefficient were measured for low molecular weight polystyrenes dissolved in a poor solvent cyclohexane at 35, 25, and 15 C, by sedimentation equilibrium. The results of the osmotic compressibility over wide concentration ranges were favorably compared with a recently developed thermodynamic perturbation theory based on the spherocylinder model bearing a square-well potential, and from the comparison, the hard-core diameter d and the depth " of the attractive square-well potential including in the theory were determined for polystyrene in cyclohexane. Compared with the previous results of d and " for the same polymer in 15 C toluene (a good solvent), it turned out that " increases and d decreases with reducing the solvent quality. [DOI 10.1295/polymj.36.747] KEY WORDS Polystyrene / Osmotic Compressibility / Second Virial Coefficient / Sedimentation Equilibrium / Thermodynamic Perturbation Theory / The distribution function theories for polymer solutions 1-3 express the polymer intermolecular interaction in terms of the potential U 12 ð1; 2Þ of mean force, where the arguments 1 and 2 represent all coordinates specifying the position, orientation, and conformation of polymer chains 1 and 2, respectively. If the polymer chain is divided into N 0 identical (spherical) segments, U 12 ð1; 2Þ may be given bywhere uðR i 1 i 2 Þ is the pair potential of mean force between segments i 1 and i 2 belonging to polymer chains 1 and 2, respectively, which is a function of the distance R i 1 i 2 of the two segments. For neutral polymers, the potential uðR i 1 i 2 Þ consists of short-ranged repulsive and long-ranged attractive interaction parts. The two-parameter theory 1-3 and also the renormalization-group theory 4-7 assume that the intermolecular excluded volume effect on the virial coefficients and the osmotic pressure can be expressed as a function of the binary cluster integral 2 defined by 2 4(k B : the Boltzmann constant; T: the absolute temperature) instead of the full function of uðRÞ. However, this assumption does not necessarily hold. Near the theta condition where 2 vanishes, the ternary cluster integral or the three-segment interaction becomes important in virial coefficients against the two-parameter theory. 8 When the polymer concentration is beyond the semidilute regime, the osmotic pressure or the osmotic compressibility cannot be described by the renormalization-group theory. Alternatively, the thermodynamic perturbation theory chooses a system of particles interacting by the hard-core potential as a reference system, and treats the long-ranged attractive interaction in a perturbative way. For example, Barker and Henderson proposed a perturbation theory for the system of spherical particles with a square-well potential. Their theory includes all perturbation terms using the Padé approximation. Recently, Koyama and Sato 10 extended the Barker-Henderson theory to the wormlike spherocylinder system. In the theory, the i...

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“…The front factor of 2ð3=4Þ 1=3 (¼ $ 1:24) has been introduced in eq 14b in order to ensure the continuity of st at C ¼ C Ã . Utilizing the standard semidilute formula, 22,23 we may express the osmotic pressures Å f and Å b in the forward and backward regimes ( Figure 13) in terms of the total polymer concentration C f and C b in respective regimes as…”

Section: Model For Unimer/f-mer/2f-mer Systemsmentioning

confidence: 99%

Kinetics of Anionic Polymerization of Polybutadienyl Lithium in Benzene: An Osmotic Effect on Propagation Process

Oishi

Watanabe

2007

Polym J

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ABSTRACT:The kinetics of the polymerization (propagation) process of protonated polybutadienyllithium (hPB) in a nonpolar solvent, deuterated benzene (dBz), was examined with 1 H NMR. An oligomeric, deuterated butadienyllithium (oBLi) was utilized as an initiator in order to avoid a contamination of the initiation process to the NMR data. The hPBLi chains mostly formed aggregate with an average aggregation number f ¼ $ 4 through Li at their ends. The residual monomer fraction ðtÞ did not rigorously exhibit the single-exponential decay with time t expected for the conventionally considered propagation through the dissociated chains, ðtÞ ¼ expðÀt=Þ withassociation/dissociation equilibrium constant, k p = propagation rate constant, ½I 0 = molar concentration of initiator at time 0). This deviation from the conventional behavior appeared to be due to competing propagation mechanism through the transiently fused aggregates (suggested from the 7 Li NMR data): This fusion-aided propagation should have been osmotically suppressed on an increase of the hPB concentration (decrease of ) to give the deviation. The propagation was found to be also retarded in the presence of chemically inert deuterated polybutadiene (dPB) chains that just tuned the osmotic environment for the aggregates, lending support to the molecular idea of the osmotic effect on the propagation. Furthermore, the ðtÞ data in the absence/presence of the dPB chains were semi-quantitatively described by a simple model considering the competition of the propagation mechanisms through the fused 2 f -mer aggregates and dissociated chains, with the former mechanism vanishing in the late stage of propagation. These results suggested a non-negligible contribution of the fused aggregates to the polymerization kinetics in particular in the early stage.

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Semidilute region for linear polymers in good solvents

Cited by 94 publications

References 8 publications

Concentration Fluctuations in a Polymer Solution under a Temperature Gradient

Concentration Fluctuations in a Polymer Solution under a Temperature Gradient

Repulsive and Attractive Interactions between Polystyrene Chains in a Poor Solvent

Kinetics of Anionic Polymerization of Polybutadienyl Lithium in Benzene: An Osmotic Effect on Propagation Process

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