# Review: Kirkwood–Riseman Model in Non-Dilute Polymeric Fluids

## Abstract

**:**

## 1. Introduction

#### 1.1. The Hydrodynamic Scaling Model

**First**, the hydrodynamic scaling model presumes that the dominant interactions between neutral polymers in solution are the solvent-mediated hydrodynamic forces. Chain crossing constraints are taken to provide at most secondary corrections. How is this possible? Because hydrodynamic forces are strong, the nearby segments of different polymer molecules move in unison with each other, so the effects of chain crossing constraints are greatly reduced. When two chains are close to each other, each chain drags the other along, rather than each chain acting as a stationary obstacle to block the other chain’s movements.

**Second**, following the Kirkwood–Riseman [5] model, each polymer chain is treated as a line of frictional centers (“beads”) separated by a series of frictionless links (“springs”). The hydrodynamic interactions between beads on different chains are taken to be described by the Oseen tensor [4] and its modern short-range extensions [48].

**Third**, the above assumptions are used to obtain a pseudovirial expansion for the concentration dependence of each transport coefficient, as a power series in concentration.

**Fourth**, to extend the model to elevated concentrations, we have recourse to self-similarity [23] or to renormalization group methods [41]. The renormalization group method of choice is the Altenberger–Dahler positive function renormalization group [49,50,51,52,53]. Altenberger and Dahler developed this group from Shirkov’s general treatment of renormalization analysis, based on functional self-similarity [54,55,56]. While renormalization group methods are indirect, they allow one to extrapolate lower-order pseudovirial expansions to elevated concentrations.

**Fifth**, the quantitative success of the hydrodynamic scaling model is in part based on polymer statics. In particular, it has been theoretically predicted [57] and experimentally demonstrated [57,58] that, in solution, polymer coils contract as the polymer concentration is increased. This fairly modest degree of chain contraction has a substantial effect on the predicted concentration dependencies of the polymer transport coefficients.

#### 1.2. Reptation/Scaling and Hydrodynamic Scaling Models Compared

#### 1.3. Historical Matters Aside

#### 1.4. Precis of the Work

## 2. Single-Chain Behavior

#### 2.1. The Models

#### 2.2. Kirkwood–Riseman Model

## 3. Extended Kirkwood–Riseman Model

#### 3.1. Bead–Bead Hydrodynamic Interactions

#### 3.2. Chain–Chain Hydrodynamic Interactions

## 4. Extended Kirkwood–Riseman Model: Self-Diffusion

**∇**being the variable with respect to which the derivatives are taken. Taking the spherical averages, one finally reaches [41]:

#### Short-Range Hydrodynamic Effects

## 5. Extended Kirkwood–Riseman Model for the Viscosity

#### 5.1. Flow Fields from Scattering of a Shear Field

#### 5.2. Power Dissipated by Chains in a Shear Field

#### 5.3. The Total Shear Field

#### 5.4. Linear and Quadratic Terms—The Huggins Coefficient

## 6. From Pseudovirial Series to Higher Concentrations

#### 6.1. Self-Similarity Approach

- (a)
- For large polymer chains, $\nu =0.5$, except perhaps at very low concentrations.
- (b)
- For short polymer chains at all concentrations, $\nu =1.0$.
- (c)
- For the probe diffusion coefficient ${D}_{p}$, the radius ${R}_{h1}$ of the probe does not depend on the concentration, so $\nu =1-3x/2\approx 5/8$.

#### 6.2. Positive-Function Renormalization Group

## 7. From Renormalization Group to Universal Scaling

## 8. Polymer Solution Viscoelasticity from Two-Parameter Temporal Scaling

#### Two-Parameter Temporal Scaling: Fundamental Approaches

## 9. Brief Description of Individual Historical Papers

## 10. Phenomenological Evidence

#### 10.1. Measurements of the Hydrodynamic Interaction Tensor

#### 10.2. Concentration and Molecular Weight Dependencies

#### 10.3. The Bead Diameter a

#### 10.4. Dielectric Relaxation Spectroscopy

#### 10.5. Viscosity and Solvent Quality

#### 10.6. Transition to the Melt

## 11. Summary and Directions for Future Development

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Phillies, G.D.J. The Kirkwood–Riseman Model of Polymer Solution Dynamics Is Qualitatively Correct. Polymers
**2023**, 15, 1995. [Google Scholar] [CrossRef] [PubMed] - Phillies, G.D.J. Simulational Tests of the Rouse Model. Polymers
**2023**, 15, 2615. [Google Scholar] [CrossRef] - Rouse, P.E. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. J. Chem. Phys.
**1953**, 21, 1272–1280. [Google Scholar] [CrossRef] - Zimm, B.H. Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss. J. Chem. Phys.
**1956**, 24, 269–278. [Google Scholar] [CrossRef] - Kirkwood, J.G.; Riseman, J. The Intrinsic Viscosities and Diffusion Constants of Flexible Macromolecules in Solution. J. Chem. Phys.
**1948**, 16, 565–573. [Google Scholar] [CrossRef] - Phillies, G.D.J. Self-Consistency of Hydrodynamic Models for the Low-Shear Viscosity and the Self-Diffusion Coefficient. Macromolecules
**2002**, 35, 7414–7418. [Google Scholar] [CrossRef] - Brinkman, H.C. The Viscosity of Concentrated Suspensions and Solutions. J. Chem. Phys.
**1952**, 20, 571–581. [Google Scholar] [CrossRef] - Riseman, J.; Ullman, R. The Concentration Dependence of the Viscosity of Solutions of Macromolecules. J. Chem. Phys.
**1951**, 19, 578–584. [Google Scholar] [CrossRef] - Saito, N. Concentration Dependence of the Viscosity of High Polymer Solutions. I. J. Phys. Soc. Jpn.
**1949**, 5, 4–8. [Google Scholar] [CrossRef] - Saito, N. A Remark on the Hydrodynamical Theory of the Viscosity of Solutions of Macromolecules. J. Phys. Soc. Jpn.
**1952**, 7, 447–450. [Google Scholar] [CrossRef] - Yamakawa, H. Concentration Dependence of Polymer Chain Configurations in Solution. J. Chem. Phys.
**1960**, 34, 1360–1364. [Google Scholar] [CrossRef] - Edwards, S.F.; Freed, K.F. Theory of the Dynamical Viscosity of Polymer Solutions. J. Chem. Phys.
**1974**, 61, 1189–1202. [Google Scholar] [CrossRef] - Freed, K.F.; Edwards, S.F. Polymer Viscosity in Concentrated Solutions. J. Chem. Phys.
**1974**, 61, 3626–3633. [Google Scholar] [CrossRef] - Freed, K.F.; Edwards, S.F. Huggins Coefficient for the Viscosity of Polymer Solutions. J. Chem. Phys.
**1975**, 62, 4032–4035. [Google Scholar] [CrossRef] - Freed, K.F.; Perico, A. Considerations on the Multiple Scattering Representation of the Concentration Dependence of the Viscoelastic Properties of Polymer Systems. Macromolecules
**1981**, 14, 1290–1298. [Google Scholar] [CrossRef] - Altenberger, A.R.; Dahler, J.S.; Tirrell, M. On the Theory of Dynamic Screening in Macroparticle Solutions. Macromolecules
**1988**, 21, 464–469. [Google Scholar] [CrossRef] - Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, UK, 1986. [Google Scholar]
- Lodge, T.P.; Rotstein, N.; Prager, S. Dynamics of Entangled Polymer Liquids: Do Linear Chains Reptate? Adv. Chem. Phys.
**1990**, 79, 1–132. [Google Scholar] - Skolnick, J.; Kolinski, A. Dynamics of Dense Polymer Systems: Computer Simulations and Analytic Theories. Adv. Chem. Phys.
**1990**, 78, 223–278. [Google Scholar] - Phillies, G.D.J. Universal Scaling Equation for Self-Diffusion by Macromolecules in Solution. Macromolecules
**1986**, 19, 2367–2376. [Google Scholar] [CrossRef] - Phillies, G.D.J. Phenomenology of Polymer Solution Dynamics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Phillies, G.D.J.; Ullmann, G.S.; Ullmann, K.; Lin, T.-H. Phenomenological Scaling Laws for “Semidilute” Macromolecule Solutions from Light Scattering by Optical Probe Particles. J. Chem. Phys.
**1985**, 82, 5242–5246. [Google Scholar] [CrossRef] - Phillies, G.D.J. Dynamics of Polymers in Concentrated Solution: The Universal Scaling Equation Derived. Macromolecules
**1987**, 20, 558–564. [Google Scholar] [CrossRef] - Phillies, G.D.J. The Universal Scaling Equation for Macromolecule Self-Diffusion. Polym. Prepr.
**1987**, 28, 356–357. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Peczak, P. The Ubiquity of Stretched-Exponential Forms in Polymer Dynamics. Macromolecules
**1988**, 21, 214–220. [Google Scholar] [CrossRef] - Phillies, G.D.J. The Hydrodynamic Scaling Model for Polymer Dynamics. In Proceedings of the Nuclear Physics B, Third University of California Conference on Statistical Mechanics, Davis, CA, USA, 27–29 March 1988; Volume 5A, pp. 214–219. [Google Scholar]
- Phillies, G.D.J. Quantitative Prediction of α in the Scaling Law for Self-Diffusion. Macromolecules
**1988**, 21, 3101–3106. [Google Scholar] [CrossRef] - Phillies, G.D.J. The Hydrodynamic Scaling Model for Polymer Self-Diffusion. J. Phys. Chem.
**1989**, 93, 5029–5039. [Google Scholar] [CrossRef] - Phillies, G.D.J. Chain Architecture in the Hydrodynamic Scaling Picture for Polymer Dynamics. Macromolecules
**1990**, 23, 2742–2748. [Google Scholar] [CrossRef] - Phillies, G.D.J. The Hydrodynamic Scaling Model for Polymer Dynamics. J. Non-Cryst. Solids
**1991**, 131–133, 612–619. [Google Scholar] [CrossRef] - Yu, H.; Binder, K.; Schweizer, K.; Mckenna, G.; Muthukumar, M.; Sillescu, H.; Donth, E.J.; Struik, L.; Lodge, T.P.; Phillies, G.; et al. Polymer Diffusion, Dynamics Furthermore, Viscoelasticity—Discussion. J. Non-Cryst. Solids
**1991**, 131–133, 742–754. [Google Scholar] - Nelson, K.; Spiess, H.; Montrose, C.J.; Roessler, E.; Angell, C.A.; Richter, D.; Meier, G.; Binder, K.; Ediger, M.; Sillescu, H.; et al. Viscous-Liquids Furthermore, Glass Transitions—The Linear Response Regime. J. Non-Cryst. Solids
**1991**, 131–133, 378–394. [Google Scholar] - Phillies, G.D.J. Range of Validity of the Hydrodynamic Scaling Model. J. Phys. Chem.
**1992**, 96, 10061–10066. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Clomenil, D. Probe Diffusion in Polymer Solutions under Theta and Good Conditions. Macromolecules
**1993**, 26, 167–170. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Kirkitelos, P.C. Higher-Order Hydrodynamic Interactions in the Calculation of Polymer Transport Properties. J. Poly. Sci. B Polym. Phys.
**1993**, 31, 1785–1797. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Quinlan, C.A. Analytic Structure of the Solutionlike-Meltlike Transition in Polymer Solution Dynamics. Macromolecules
**1995**, 28, 160–164. [Google Scholar] [CrossRef] - Phillies, G.D.J. Hydrodynamic Scaling of Viscosity and Viscoelasticity of Polymer Solutions, Including Chain Architecture and Solvent Quality Effects. Macromolecules
**1995**, 28, 8198–8208. [Google Scholar] [CrossRef] - Ngai, K.L.; Phillies, G.D.J. Coupling Model Analysis of Polymer Dynamics in Solution: Probe Diffusion and Viscosity. J. Chem. Phys.
**1996**, 105, 8385–8397. [Google Scholar] [CrossRef] - Phillies, G.D.J. Quantitative Experimental Confirmation of the Chain Contraction Assumption of the Hydrodynamic Scaling Model. J. Phys. Chem. B
**1997**, 101, 4226–4231. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Lacroix, M.; Yambert, J. Probe Diffusion in Sodium Polystyrene Sulfonate—Water: Experimental Determination of Sphere-Chain Binary Hydrodynamic Interactions. J. Phys. Chem. B
**1997**, 101, 5124–5130. [Google Scholar] [CrossRef] - Phillies, G.D.J. Derivation of the Universal Scaling Equation of the Hydrodynamic Scaling Model via Renormalization Group Analysis. Macromolecules
**1998**, 31, 2317–2327. [Google Scholar] [CrossRef] - Phillies, G.D.J. Polymer Solution Viscoelasticity from Two-Parameter Temporal Scaling. J. Chem. Phys.
**1999**, 110, 5989–5992. [Google Scholar] [CrossRef] - Phillies, G.D.J. Temporal Scaling Analysis: Viscoelastic Properties of Star Polymers. J. Chem. Phys.
**1999**, 111, 8144–8150. [Google Scholar] [CrossRef] - Phillies, G.D.J. Temporal Scaling Analysis: Linear and Crosslinked Polymers. J. Polym. Sci. B Polym. Phys.
**2002**, 40, 375–386. [Google Scholar] [CrossRef] - Phillies, G.D.J. Low-Shear Viscosity of Non-Dilute Polymer Solutions from a Generalized Kirkwood-Riseman Model. J. Chem. Phys.
**2002**, 116, 5857–5866. [Google Scholar] [CrossRef] - Phillies, G.D.J. Viscosity of Hard Sphere Suspensions. J. Coll. Interf. Sci.
**2002**, 248, 528–529. [Google Scholar] [CrossRef] - Merriam, S.C.; Phillies, G.D.J. Fourth-Order Hydrodynamic Contribution to the Polymer Self-Diffusion Coefficient. J. Polym. Sci. B Polym. Phys.
**2004**, 42, 1663–1670. [Google Scholar] [CrossRef] - Kynch, G.K. The Slow Motion of Two or More Spheres through a Viscous Fluid. J. Fluid Mech.
**1959**, 5, 193–208. [Google Scholar] [CrossRef] - Altenberger, A.R.; Dahler, J.S. A Renormalization Group Calculation of the Viscosity of a Hard-Sphere Suspension. J. Coll. Inter. Sci.
**1997**, 189, 379–381. [Google Scholar] [CrossRef] - Altenberger, A.R.; Dahler, J.S. Application of a New Renormalization Group to the Equation of State of a Hard-Sphere Fluid. Phys. Rev. E
**1997**, 54, 6242–6252. [Google Scholar] [CrossRef] - Altenberger, A.R.; Dahler, J.S. Self Similarity, Scaling and Renormalization Group Theory Used to Generate Equations of State for Hard-Particle Fluids. Polish J. Chem.
**2001**, 75, 601–616. [Google Scholar] - Altenberger, A.R.; Siepmann, J.I.; Dahler, J.S. Functional Self-Similarity, Scaling and a Renormalization Group Calculation of the Partition Function for a Non-Ideal Chain. Phys. A
**2001**, 289, 107–136. [Google Scholar] [CrossRef] - Altenberger, A.R.; Dahler, J.S. The Role of Self-Similarity in Renormalization Group Theory. Adv. Chem. Phys.
**2002**, 123, 267–354. [Google Scholar] - Shirkov, D.V. Renormalization Group and Functional Selfsimilarity in Different Branches of Physics. Theor. Math. Phys.
**1984**, 60, 778–782. [Google Scholar] [CrossRef] - Bogoliubov, N.N.; Shirkov, D.V. Quantum Fields; Pontecorvo, D.B., Ed.; Benjamin Cummings Publishing: Reading, MA, USA, 1983; Appendix IX. [Google Scholar]
- Shirkov, D.V. Renormalization Group in Modern Physics. Int. J. Mod. Phys.
**1988**, 3, 1321–1341. [Google Scholar] [CrossRef] - Daoud, M.; Cotton, J.P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, C.; Picot, C.; de Gennes, P.G. Solutions of Flexible Polymers. Neutron Experiments and Interpretation. Macromolecules
**1975**, 8, 804–818. [Google Scholar] [CrossRef] - King, J.S.; Boyer, W.; Wignall, G.D.; Ullman, R.R. Radii of Gyration and Screening Lengths of Polystyrene in Toluene as a Function of Concentration. Macromolecules
**1985**, 18, 709–718. [Google Scholar] [CrossRef] - Phillies, G.D.J. Complete Numerical Tables for Phillies’ Phenomenology of Polymer Solution Dynamics; Third Millennium Publishing: Tempe, AZ, USA, 2011. [Google Scholar]
- Oono, Y.; Baldwin, P.R. Cooperative Diffusion of a Semidilute Polymer Solution: A Preliminary Study. Phys. Rev. A
**1986**, 33, 3391–3398. [Google Scholar] [CrossRef] - De Gennes, P.G. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. J. Chem. Phys.
**1971**, 55, 572–579. [Google Scholar] [CrossRef] - Lin, T.-H.; Phillies, G.D.J. Probe Diffusion in Polyacrylic Acid: Water—Effect of Polymer Molecular Weight. J. Coll. Interf. Sci.
**1984**, 100, 82–95. [Google Scholar] [CrossRef] - Lin, T.-H.; Phillies, G.D.J. Probe Diffusion in Poly(Acrylic Acid): Water. Effect of Probe Size. Macromolecules
**1984**, 17, 1686–1691. [Google Scholar] [CrossRef] - Ullmann, G.S.; Phillies, G.D.J. Implications of the Failure of the Stokes–Einstein Relation for Measurements with QELSS of Polymer Adsorption by Small Particles. Macromolecules
**1983**, 16, 1947–1949. [Google Scholar] [CrossRef] - Ullmann, G.S.; Ullmann, K.; Lindner, R.M.; Phillies, G.D.J. Probe Diffusion of Polystyrene Latex Spheres in Poly-(ethylene oxide):Water. J. Phys. Chem.
**1985**, 89, 692–700. [Google Scholar] [CrossRef] - Ullmann, K.; Ullmann, G.S.; Phillies, G.D.J. Optical Probe Study of a Nonentangling Macromolecule Solution—Bovine Serum Albumin:Water. J. Coll. Interf. Sci.
**1985**, 105, 315–324. [Google Scholar] [CrossRef] [Green Version] - Phillies, G.D.J. Diffusion of Bovine Serum Albumin in a Neutral Polymer Solution. Biopolymers
**1985**, 24, 379–386. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ware, B.R. (University of Syracuse, Syracuse, NY, USA). Private Communication, 1988.
- Gisser, D.J.; Johnson, B.S.; Ediger, M.D.; von Meerwall, E.D. Comparison Of Various Measurements of Microscopic Friction in Polymer Solutions. Macromolecules
**1993**, 26, 512–519. [Google Scholar] [CrossRef] - Morris, R.L.; Amelar, S.; Lodge, T.P. Solvent Friction in Polymer Solutions and Its Relation to the High Frequency Limiting Viscosity. J. Chem. Phys.
**1988**, 89, 6523–6537. [Google Scholar] [CrossRef] - Minnick, M.G.; Schrag, J.L. Polymer-Solvent Interaction Effects in Oscillatory Flow Birefringence Studies of Polybutadienes and Polyisoprenes in Aroclor Solvents. Macromolecules
**1995**, 13, 1690–1695. [Google Scholar] [CrossRef] - Krahn, J.R.; Lodge, T.P. Spacial Heterogeneity of Solvent Dynamics in Multicomponent Polymer Solutions. J. Phys. Chem.
**1995**, 99, 8338–8348. [Google Scholar] [CrossRef] - Zwanzig, R. Langevin Theory of Polymer Dynamics in Dilute Solution. Adv. Chem. Phys.
**1969**, 15, 325–333. [Google Scholar] - Mazur, P.; van Saarloos, W. Many Sphere Hydrodynamic Interactions and Mobilities in a Suspension. Phys. A
**1982**, 115, 21–57. [Google Scholar] [CrossRef] [Green Version] - Phillies, G.D.J. The Second Order Concentration Corrections to the Mutual Diffusion Coefficient of Brownian Macroparticles. J. Chem. Phys.
**1982**, 77, 2623–2631. [Google Scholar] [CrossRef] - Ladd, A.J.C. Hydrodynamic Interactions and the Viscosity of Suspensions of Freely Moving Spheres. J. Chem. Phys.
**1989**, 90, 1149–1157. [Google Scholar] [CrossRef] - Phillies, G.D.J. Dynamics of Crowded Brownian Particles. Adv. Chem. Phys.
**2016**, 48, 277–358. [Google Scholar] - Freed, K.F. Excluded Volume Effect on Quasi-Elastic Neutron Scattering from Concentrated Polymer Solutions. J. Chem. Phys.
**1976**, 64, 5126–5131. [Google Scholar] [CrossRef] - Freed, K.F.; Metiu, H. Mean Field Theory of the Hydrodynamics of Concentrated Polymer Solutions. J. Chem. Phys.
**1978**, 68, 4604–4611. [Google Scholar] [CrossRef] - Freed, K.F. Incorporation of Excluded Volume into the Multiple Scattering Theory of the Concentration Dependence of Polymer Dynamics. Macromolecules
**1983**, 16, 1855–1862. [Google Scholar] [CrossRef] - Bernal, J.M.G.; de la Torre, J.G. Transport Properties of Oligomeric Subunit Structures. Biopolymers
**1981**, 20, 129–139. [Google Scholar] [CrossRef] - DeWames, R.E.; Holland, W.F.; Shen, M.C. On the Molecular Theories of Polymer Solutions. J. Chem. Phys.
**1967**, 46, 2782–2794. [Google Scholar] [CrossRef] - Zwanzig, R.; Kiefer, J.; Weiss, G.H. On the Validity of the Kirkwood-Riseman Theory. Proc. Natl. Acad. Sci. USA
**1968**, 60, 381–386. [Google Scholar] [CrossRef] - Yamakawa, H. Transport Properties of Polymer Chains in Dilute Solution: Hydrodynamic Interaction. J. Chem. Phys.
**1970**, 53, 436–443. [Google Scholar] [CrossRef] - Peterson, J.M.; Fixman, M. Viscosity of Polymer Solutions. J. Chem. Phys.
**1963**, 39, 2516–2524. [Google Scholar] [CrossRef] - Jackson, J.D. Classical Electrodynamics; John Wiley and Sons: New York, NY, USA, 1962; p. 567. [Google Scholar]
- Graessley, W.W. Molecular Entanglement Theory of Flow Behavior in Amorphous Polymers. J. Chem. Phys.
**1965**, 43, 2696–2703. [Google Scholar] [CrossRef] - Graessley, W.W. Viscosity of Entangling Polydisperse Polymers. J. Chem. Phys.
**1967**, 47, 1942–1953. [Google Scholar] [CrossRef] - Bird, R.B.; Saab, H.H.; Curtis, C.F. A Kinetic Theory for Polymer Melts. 3. Elongational Flows. J. Phys. Chem.
**1982**, 86, 1102–1106. [Google Scholar] [CrossRef] - Bird, R.B.; Saab, H.H.; Curtis, C.F. A Kinetic Theory for Polymer Melts. IV. Rheological Properties for Shear Flows. J. Chem. Phys.
**1982**, 77, 4747–4757. [Google Scholar] [CrossRef] - Raspaud, E.; Lairez, D.; Adam, M.M. On the Number of Blobs per Entanglement in Semidilute and Good Solvent Solution: Melt Influence. Macromolecules
**1995**, 28, 927–933. [Google Scholar] [CrossRef] - Rouse, P.E. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. II. A First-Order Mechanical Thermodynamic Property. J. Chem. Phys.
**1998**, 108, 4628–4633. [Google Scholar] [CrossRef] - Milas, M.; Rinaudo, M.; Knipper, M.; Schuppiser, J.L. Flow and Viscoelastic Properties of Xanthan Gum Solutions. Macromolecules
**1990**, 23, 2506–2511. [Google Scholar] [CrossRef] - Graessley, W.W.; Masuda, T.; Roovers, J.E.L.; Hadjichristidis, N. Rheological Properties of Linear and Branched Polyisoprene. Macromolecules
**1976**, 9, 127–141. [Google Scholar] [CrossRef] - Langevin, D.; Rondelez, F. Sedimentation of Large Colloidal Particles through Semidilute Polymer Solutions. Polymer
**1978**, 14, 875–882. [Google Scholar] [CrossRef] - Carter, J.M.; Phillies, G.D.J. Second-Order Concentration Correction to the Mutual Diffusion Coefficient of a Suspension of Hard Brownian Spheres. J. Phys. Chem.
**1985**, 89, 5118–5124. [Google Scholar] [CrossRef] - Kuhn, T.F. The Structure of Scientific Revolutions; University of Chicago Press: Chicago, IL, USA, 1962. [Google Scholar]
- Dreval, V.E.; Malkin, A.Y.; Botvinnik, G.O. Approach to Generalization of Concentration Dependence of Zero-Shear Viscosity in Polymer Solutions. J. Polym. Sci. Polym. Phys. Ed.
**1973**, 11, 1055–1076. [Google Scholar] [CrossRef] - Pesce, F.; Newcombe, E.A.; Seiffert, P.; Tranchant, E.E.; Olsen, J.G.; Grace, C.R.; Kragelund, B.B.; Lindorff-Larsen, K. Assessment of Models for Calculating the Hydrodynamic Radius of Intrinsically Disordered Proteins. Biophys. J.
**2023**, 122, 310–321. [Google Scholar] [CrossRef] [PubMed] - Pietzsch, R.M.; Burt, M.J.; Reed, W.F. Evidence of Partial Draining for Linear Polyelectrolytes; Heparin, Chondroitin Sulfate and Polystyrene Sulfonate. Macromolecules
**1992**, 25, 806–815. [Google Scholar] [CrossRef] - Phillies, G.D.J. Phenomenology of Polymer Solution Dynamics; Cambridge University Press: Cambridge, UK, 2011; Chapters 8–9. [Google Scholar]
- Wheeler, L.M.; Lodge, T.P. Tracer Diffusion of Linear Polystyrene in Dilute, Semidilute, and Concentrated Poly(vinyl methyl ether) Solutions. Macromolecules
**1989**, 22, 3399–3408. [Google Scholar] [CrossRef] - Lodge, T.P.; Markland, P.; Wheeler, L.M. Tracer Diffusion of 3-Arm and 12-Arm Star Polystyrenes in Dilute, Semidilute, and Concentrated Poly(vinylmethyl ether) Solutions. Macromolecules
**1989**, 22, 3409–3418. [Google Scholar] [CrossRef] - Brown, W.; Zhou, P. Dynamic Behavior in Ternary Polymer Solutions. Polyisobutylene in Chloroform Studied Using Dynamic Light Scattering and Pulsed Field Gradient NMR. Macromolecules
**1989**, 22, 4031–4039. [Google Scholar] [CrossRef] - Phillies, G.D.J.; Brown, W.W.; Zhou, P. Chain and Sphere Diffusion in Polyisobutylene–CHCl
_{3}: A Reanalysis. Macromolecules**1992**, 25, 4948–4954. [Google Scholar] [CrossRef] - Pearson, D.S. Recent Advances in the Molecular Aspects of Polymer Viscoelasticity. Rubber Chem. Technol.
**1987**, 60, 439–496. [Google Scholar] [CrossRef] - Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Rowe: New York, NY, USA, 1971; Chapter 6. [Google Scholar]
- Adam, M.; Delsanti, M. Dynamical Properties of Polymer Solutions in Good Solvent by Rayleigh Scattering Experiments. Macromolecules
**1977**, 10, 1229–1237. [Google Scholar] [CrossRef] - Stockmayer, W.H. Dielectric Dispersion in Solutions of Flexible Polymers. Pure Appl. Chem.
**1967**, 15, 539–554. [Google Scholar] [CrossRef] - Adachi, K.; Kotaka, T. Dielectric Normal Mode Relaxation. Progr. Polym. Sci.
**1993**, 18, 585–622. [Google Scholar] [CrossRef] - Adachi, K.; Okazaki, H.; Kotaka, T. Application of Scaling Laws to the Dielectric Normal Mode Process of Cis-Polyisoprene in Solutions of Infinite Dilution to the Bulk. Macromolecules
**1985**, 18, 1687–1692. [Google Scholar] [CrossRef] - Adachi, K.; Kotaka, T. Dielectric Normal Mode Process in Semidilute and Concentrated Solutions of Cis-Polyisoprene. Macromolecules
**1988**, 21, 157–164. [Google Scholar] [CrossRef] - Adachi, K.; Imanishi, Y.; Shinkado, T.; Kotaka, T. Dielectric Study of the Concentration Dependence of the End-To-End Distance and Normal-Mode Relaxation Time of Polyisoprene in Moderately Good Solvents. Macromolecules
**1989**, 22, 2391–2395. [Google Scholar] [CrossRef] - von Meerwall, E.D.; Amis, E.J.; Ferry, J.D. Self-Diffusion in Solutions of Polystyrene in Tetrahydrofuran: Comparison of Concentration Dependences of the Diffusion Coefficient of Polymer, Solvent, and a Ternary Probe Component. Macromolecules
**1985**, 18, 260–266. [Google Scholar] [CrossRef] - Pickup, S.; Blum, F.D. Self-Diffusion of Toluene in Polystyrene Solutions. Macromolecules
**1989**, 22, 3861–3968. [Google Scholar] [CrossRef] - Kosfeld, R.; Zumkley, L. Mobility of Small Molecules in Polymer Systems. Berichte Bunsenges. Phys. Chem.
**1979**, 83, 392–396. [Google Scholar] [CrossRef] - Viovy, J.L.; Monnerie, L.L. A Study of Local Chain Dynamics in Concentrated Polystyrene Solutions Using Fluorescence Anisotropy Decay. Polymer
**1986**, 27, 181–184. [Google Scholar] [CrossRef] - Tardiveau, N. Unpublished Work. Ph.D. Thesis, Universite Paris VI, Paris, France, 1980. [Google Scholar]
- Johnson, B.S.; Ediger, M.D.; Kitano, T.; Ito, K. Concentration and Temperature Dependence of Local Motions in Polyisoprene/Tetrahydrofuran. Macromolecules
**1992**, 25, 873–879. [Google Scholar] [CrossRef] - O’Connell, R.; Hanson, H.; Phillies, G.D.J. Neutral Polymer Slow Mode and Its Rheological Correlate. J. Polym. Sci. B Polym. Phys. Ed.
**2005**, 43, 323–333. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Phillies, G.D.J.
Review: Kirkwood–Riseman Model in Non-Dilute Polymeric Fluids. *Polymers* **2023**, *15*, 3216.
https://doi.org/10.3390/polym15153216

**AMA Style**

Phillies GDJ.
Review: Kirkwood–Riseman Model in Non-Dilute Polymeric Fluids. *Polymers*. 2023; 15(15):3216.
https://doi.org/10.3390/polym15153216

**Chicago/Turabian Style**

Phillies, George David Joseph.
2023. "Review: Kirkwood–Riseman Model in Non-Dilute Polymeric Fluids" *Polymers* 15, no. 15: 3216.
https://doi.org/10.3390/polym15153216