Scaling of Polymer Solutions as a Quantitative Tool

Abstract

Knowledge of interaction parameters and Kuhn length for a polymer/solvent pair is a foundation of polymer physics of synthetic and biological macromolecules. Here, we demonstrate how to obtain these parameters from the concentration dependence of solution viscosity. The centerpiece of this approach is the scaling relationship between solution correlation length (blob size) ξ = lgν /B and the number of monomers per correlation blob g for polymers with monomer projection length l. The values of parameter B and exponent v are determined by solvent quality for the polymer backbone, chain Kuhn length, and types and strength of monomer–monomer and monomer–solvent interactions. Parameter B assumes values B g, Bth , and 1 for exponent v = 0.588, 0.5, and 1, respectively. In particular, we take advantage of the linear relationship between specific viscosity ηsp in the unentangled (Rouse) regime and the number of correlation blobs Nw /g per chain with the weight average degree of polymerization, Nw , and g = B 3/(3ν – 1)(cl 3)1/(1 – 3ν) as a function of monomer concentration, c, and the corresponding B parameter. The values of the B parameters are extracted from the plateaus of normalized specific viscosity ηsp(c)/Nw (cl 3)1/(3ν – 1) or their locations as a function of the monomer concentration c in different solution regimes. The extension of the approach to entangled polymer solutions provides a means to obtain the chain packing number, Pe , and to complete the set of parameters {B g, Bth , Pe } (a system “fingerprint”) uniquely describing static and dynamic solution properties of a polymer/solvent pair. This approach is illustrated for solutions of poly­(ethylene oxide) in water, poly­(styrene) in tetrahydrofuran and toluene, poly­(methyl methacrylate) in ionic liquids, and sodium carboxymethylcellulose in water at high salt concentrations.

Quantifying the Effect of Multivalent Ions in Polyelectrolyte Solutions

“…In our analysis of the viscosity of semidilute salt-free polyelectrolyte solutions in the presence of multivalent counterions, we extend a scaling approach developed in recent publications. , This approach is based on the relationship between the correlation length (blob size) ξ and the number of the repeat units in it, g , with projection length l − The numerical coefficient B and exponent v are determined by the solvent quality for the polymer backbone and the type and strength of interactions at different length scales, starting from the solution correlation length down to the chain Kuhn length b and repeat unit projection length l . In semidilute polyelectrolyte solutions, the exponent v = 1, 0.588, 0.5, and 1, and the B -parameter is equal to B pe , B g , B th , and 1 in the different solution regimes, reflecting that correlation blobs are made of electrostatic blobs with sizes D e and number of repeat units g e , which in turn contain thermal blobs each of size D th and number of repeat units g th (Figure a).…”

“…In eq , P ̃ e is the number of entangled strands (packing number) required for a section of a chain with N e repeat units to entangle. We use to indicate that the system polydispersity is included in the definition of the packing number. , Accounting for the different concentration scaling of the number of repeat units per correlation blob g and strand degree of polymerization N e in the concentrated solution regime c > c ** requires rescaling the specific viscosity by a factor of λ = clb 2 = c / c ** for c > c ** and rewriting eq as The number of repeat units g per blob is calculated using eq for c ≤ c **, and g = B th –2 ( c **/ c ) 2 for c > c **.…”

“…Despite significant experimental, − ,,,− computational, − and theoretical ,− ,,− efforts, a quantitative description of polyelectrolyte solutions with divalent counterions continues to be desired. Here, we will address this issue by applying a recently developed data analysis technique , to evaluate scaling parameters, describing the conformations of polyelectrolyte and neutral chains at different length scales. The method , is based on the linear dependence of the solution specific viscosity on the weight-average degree of polymerization and the concentration dependence of the solution correlation length in the unentangled solution regime.…”

“…Here, we will address this issue by applying a recently developed data analysis technique , to evaluate scaling parameters, describing the conformations of polyelectrolyte and neutral chains at different length scales. The method , is based on the linear dependence of the solution specific viscosity on the weight-average degree of polymerization and the concentration dependence of the solution correlation length in the unentangled solution regime. In particular, we apply this scaling approach , to obtain the fraction of ionized groups, excluded volume, and Kuhn length in salt-free solutions of carboxymethylcellulose (CMC) with Na + , Ca 2+ , Mg 2+ , Mn 2+ , Co 2+ , and Ba 2+ counterions , and polystyrene sulfonate with Na + and Mg 2+ counterions. − This information is used to elucidate the differences and similarities in the system behavior depending on the counterion valence.…”

“…The method , is based on the linear dependence of the solution specific viscosity on the weight-average degree of polymerization and the concentration dependence of the solution correlation length in the unentangled solution regime. In particular, we apply this scaling approach , to obtain the fraction of ionized groups, excluded volume, and Kuhn length in salt-free solutions of carboxymethylcellulose (CMC) with Na + , Ca 2+ , Mg 2+ , Mn 2+ , Co 2+ , and Ba 2+ counterions , and polystyrene sulfonate with Na + and Mg 2+ counterions. − This information is used to elucidate the differences and similarities in the system behavior depending on the counterion valence. Furthermore, this approach , allows us to represent solution specific viscosity as a universal function of the number of repeat units per solution correlation length in a broad concentration range.…”