Dilute or semidilute solutions of nonintersecting self-avoiding walk (SAW) polymer chains are mapped onto a fluid of “soft” particles interacting via an effective pair potential between their centers of mass. This mapping is achieved by inverting the pair distribution function of the centers of mass of the original polymer chains, using integral equation techniques from the theory of simple fluids. The resulting effective pair potential is finite at all distances, has a range of the order of the radius of gyration, and turns out to be only moderately concentration-dependent. The dependence of the effective potential on polymer length is analyzed in an effort to extract the scaling limit. The effective potential is used to derive the osmotic equation of state, which is compared to simulation data for the full SAW segment model, and to the predictions of renormalization group calculations. A similar inversion procedure is used to derive an effective wall–polymer potential from the center of mass density pro...

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Accurate effective pair potentials for polymer solutions

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2Δ
4 −γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2Δ
4 −γ. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R

 2
 >/<R

 2
 >=0.14026±0.00007, <R

 2
 >/<R

 2
 >=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponentΔ
1=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy forΔ
1. Earlier Monte Carlo estimates ofv, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R

 2
 >/<R

 2
 >=0.1599±0.0002 and Ψ*=0.2471±0.0003; since Ψ*>0, hyperscaling holds. The approach to Ψ* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2Δ
4 −γ for two-dimensional SAWs.

Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

Dilute or semi-dilute solutions of non-intersecting self-avoiding walk (SAW) polymer chains are mapped onto a fluid of ``soft'' particles interacting via an effective pair potential between their centers of mass. This mapping is achieved by inverting the pair distribution function of the centers of mass of the original polymer chains, using integral equation techniques from the theory of simple fluids. The resulting effective pair potential is finite at all distances, has a range of the order of the radius of gyration, and turns out to be only moderately concentration-dependent. The dependence of the effective potential on polymer length is analyzed in an effort to extract the scaling limit. The effective potential is used to derive the osmotic equation of state, which is compared to simulation data for the full SAW segment model, and to the predictions of renormalization group calculations. A similar inversion procedure is used to derive an effective wall-polymer potential from the center of mass density profiles near the wall, obtained from simulations of the full polymer segment model. The resulting wall-polymer potential turns out to depend strongly on bulk polymer concentration when polymer-polymer correlations are taken into account, leading to a considerable enhancement of the effective repulsion with increasing concentration. The effective polymer-polymer and wall-polymer potentials are combined to calculate the depletion interaction induced by SAW polymers between two walls. The calculated depletion interaction agrees well with the ``exact'' results from much more computer-intensive direct simulation of the full polymer-segment model, and clearly illustrates the inadequacy -- in the semi-dilute regime -- of the standard Asakura-Oosawa approximation based on the assumption of non-interacting polymer coils.

https://arxiv.org/pdf/cond-mat/0009093.pdf

Fabrication of injectable high strength hydrogel based on 4-arm star PEG for cartilage tissue engineering

New developments in the microrheology of complex fluids are considered. Firstly the requirements for a simple modern particle tracking microrheology experiment are introduced, the error analysis methods associated with it and the mathematical techniques required to calculate the linear viscoelasticity. Progress in microrheology instrumentation is then described with respect to detectors, light sources, colloidal probes, magnetic tweezers, optical tweezers, diffusing wave spectroscopy, optical coherence tomography, fluorescence correlation spectroscopy, elastic- and quasi-elastic scattering techniques, 3D tracking, single molecule methods, modern microscopy methods and microfluidics. New theoretical techniques are also reviewed such as Bayesian analysis, oversampling, inversion techniques, alternative statistical tools for tracks (angular correlations, first passage probabilities, the kurtosis, motor protein step segmentation etc), issues in micro/macro rheological agreement and two particle methodologies. Applications where microrheology has begun to make some impact are also considered including semi-flexible polymers, gels, microorganism biofilms, intracellular methods, high frequency viscoelasticity, comb polymers, active motile fluids, blood clots, colloids, granular materials, polymers, liquid crystals and foods. Two large emergent areas of microrheology, non-linear microrheology and surface microrheology are also discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/79/7/074601/pdf

Advances in the microrheology of complex fluids.

Premiere preuve experimentale du fait que, des solutions de polymeres flexibles sont caracterises par une longueur universelle dependant de la concentration qui est aisement mise en loi d'echelle pour etre independante de la masse moleculaire et de la qualite du solvant

Universality for Static Properties of Polystyrenes in Good and Marginal Solvents

An accurate light scattering photometer capable of measuring both the angular distribution of intensity and the intensity fluctuation spectrum of laser light scattered from a wide variety of samples including solutions of polymers and other macromolecules is described. The photometer employs 18 fixed scattering angles ranging from 2.6° to 163° and is sufficiently accurate to measure a molecular radius of gyration as small as 60 A reliably. Stray elastically scattered light normally comprises less than 1% of the light scattered from pure toluene at the smallest scattering angle and is unmeasurable at other angles. The apparatus employs optical fibers to transmit the scattered light to a single photomultiplier tube and functions automatically under microprocessor control, with sample temperature controlled to ±0.1 mK.

Photometer for quasielastic and classical light scattering

We report experimental results for a dynamic length scale ${\ensuremath{\xi}}_{H}$ as a function of polymer concentration $c$ for various molecular weight polystyrenes in the two solvents toluene and methyl ethyl ketone. We find that the quantity $\frac{{\ensuremath{\xi}}_{H}}{{R}_{H}}$ is a function only of the reduced concentration $\mathrm{kc}$, where ${R}_{H}$ is the hydrodynamic radius and $k$ is determined from a virial expansion of the diffusion coefficient. However, the dynamic and static length scales are not proportional.

Hans R. Haller

Papers

Universality for Static Properties of Polystyrenes in Good and Marginal Solvents

Photometer for quasielastic and classical light scattering

Dynamics of Long-Wavelength Concentration Fluctuations in Solutions of Linear Polymers