^{1,a)}, Jack F. Douglas

^{2}and Karl F. Freed

^{1}

### Abstract

We examine the rheological and dielectric properties of solutions of equilibrium self-assembling particles and molecules that form polydisperse chains whose average length depends on temperature and concentration (free association model). Relaxation of the self-assembling clusters proceeds by motions associated either with cluster rotations, with diffusive internal chain dynamics, or with interchain entanglement interactions. A hierarchy of models is used to emphasize different physical effects: Unentangled rodlike clusters, unentangled flexible polymers, and entangled chains. All models yield a multistep relaxation for low polymer scission rates (“persistent polymers”). The short time relaxation is nearly exponential and is dominated by the monomeric species and solvent, and the long time relaxation is approximately a stretched exponential, , a behavior that arises from an averaging over the equilibrium chain length distribution and the internal relaxation modes of the assembled structures. Relaxation functions indicate a *bifurcation* of the relaxation function into fast and slow contributions upon passing through the polymerization transition. The apparent activation energy for the long time relaxation becomes temperature dependent, while the fast monomeric relaxation process remains Arrhenius. The effective exponent , describing the long time relaxation process, varies monotonically from near unity above the polymerization temperature to a low temperature limit, , when the self-assembly process is complete. The variation in the relaxation function with temperature is represented as a function of molecular parameters, such as the average chain length, friction coefficient, solvent viscosity, and the reaction rates for particle association and dissociation.

This research is supported, in part, by NSF Grant No. CHE-0749788 and by the Joint Theory Institute which is funded by the University of Chicago and Argonne National Laboratory. We are grateful to Jacek Dudowicz for many helpful discussions.

I. INTRODUCTION

II. STRESS RELAXATION IN EQUILIBRIUM POLYMERS

A. Rodlike model for equilibrium polymers

B. Rouse model for equilibrium polymers

1. Effect of internal chain modes on relaxation

2. Single mode approximation

C. Reptation model for equilibrium polymers

III. DIELECTRIC RELAXATION IN THE EQUILIBRIUM POLYMER MODELS

IV. CONCLUSIONS

### Key Topics

- Polymers
- 59.0
- Stress relaxation
- 49.0
- Dielectric relaxation
- 40.0
- Reptation
- 35.0
- Relaxation times
- 32.0

## Figures

Concentration dependence of characteristic polymerization temperatures: (a) “Onset” , (b) inflection point , and (c) “saturation” temperatures for the quasichemical free association model of equilibrium self-assembly. is the “polymerization” temperature for the solution having a reference concentration . The two filled circles correspond to the states chosen in Fig. 2, and the solid line denotes the cooling history for obtaining in Figs. 3 and 6. The dotted line passes through the polymerization transition. The inset depicts the degree of association as a function of temperature.

Concentration dependence of characteristic polymerization temperatures: (a) “Onset” , (b) inflection point , and (c) “saturation” temperatures for the quasichemical free association model of equilibrium self-assembly. is the “polymerization” temperature for the solution having a reference concentration . The two filled circles correspond to the states chosen in Fig. 2, and the solid line denotes the cooling history for obtaining in Figs. 3 and 6. The dotted line passes through the polymerization transition. The inset depicts the degree of association as a function of temperature.

Multistep stress relaxation function for polydisperse mixtures (a) of unbreakable rodlike clusters with average length and at a low temperature and (b) of living rods with average length and at a higher temperature (open squares). is the rotational relaxation time of monodisperse rods at fixed . The short time relaxation constant is , and the ratio is taken as 1:3. The solid lines are SE fits with (a) and (b) , respectively, to the long time portion of the relaxation function as deduced from the steady-state compliance for the same temperatures. The inset depicts the long time contribution to for the rodlike model of unbreakable rods at . The model parameters used are , , and . The polymerization temperature is determined from the inflection point of as .

Multistep stress relaxation function for polydisperse mixtures (a) of unbreakable rodlike clusters with average length and at a low temperature and (b) of living rods with average length and at a higher temperature (open squares). is the rotational relaxation time of monodisperse rods at fixed . The short time relaxation constant is , and the ratio is taken as 1:3. The solid lines are SE fits with (a) and (b) , respectively, to the long time portion of the relaxation function as deduced from the steady-state compliance for the same temperatures. The inset depicts the long time contribution to for the rodlike model of unbreakable rods at . The model parameters used are , , and . The polymerization temperature is determined from the inflection point of as .

The sigmoidal temperature dependence of the fits of to the low frequency portion of for the scission-rodlike model of dynamics and aggregation. The model parameters for the scission and activation energies are the same as in Fig. 2.

The sigmoidal temperature dependence of the fits of to the low frequency portion of for the scission-rodlike model of dynamics and aggregation. The model parameters for the scission and activation energies are the same as in Fig. 2.

The temperature dependencies of the terminal relaxation time for the rodlike model of unbreakable equilibrium polymers and the relaxation time of the monomers (both normalized to ) as a function of the inverse reduced temperature . The segmental and monomer relaxation times are set equal. The model parameters for the scission energy and activation energies are the same as in Fig. 2. The inset displays the apparent activation energy divided by the high temperature value as a function of the reduced temperature for different scission energies : (a) , (b) , and (c) .

The temperature dependencies of the terminal relaxation time for the rodlike model of unbreakable equilibrium polymers and the relaxation time of the monomers (both normalized to ) as a function of the inverse reduced temperature . The segmental and monomer relaxation times are set equal. The model parameters for the scission energy and activation energies are the same as in Fig. 2. The inset displays the apparent activation energy divided by the high temperature value as a function of the reduced temperature for different scission energies : (a) , (b) , and (c) .

Normalized dynamic viscosity for self-assembling systems with average chain length as a function of the dimensionless frequency . The solid lines correspond (a) to unbreakable Rouse chains and to living polymers with MM reactions with (b) and (c) . The dotted lines are their SE fits, which yield (a) , (b) , and (c) . The normalizing coefficients are , where equals (a) 164.5, (b) 22.5, and (c) 7.51. Real components of the shear viscosity are displayed in (a), while imaginary components are presented in (b).

Normalized dynamic viscosity for self-assembling systems with average chain length as a function of the dimensionless frequency . The solid lines correspond (a) to unbreakable Rouse chains and to living polymers with MM reactions with (b) and (c) . The dotted lines are their SE fits, which yield (a) , (b) , and (c) . The normalizing coefficients are , where equals (a) 164.5, (b) 22.5, and (c) 7.51. Real components of the shear viscosity are displayed in (a), while imaginary components are presented in (b).

Stress relaxation modulus of unbreakable polydisperse Rouse chains with average chain length (open circles), its SE fit with obtained from the steady-state compliance (solid line), and the single mode approximation (dotted line). The inset displays the same curves on different scales but with the same line legends.

Stress relaxation modulus of unbreakable polydisperse Rouse chains with average chain length (open circles), its SE fit with obtained from the steady-state compliance (solid line), and the single mode approximation (dotted line). The inset displays the same curves on different scales but with the same line legends.

The sigmoidal temperature dependence of the fits of to the low frequency portion of for the scission-Rouse model. The inset presents the temperature dependence of the average chain length. The model parameters are and . The polymerization temperature is determined from the inflection point of as .

The sigmoidal temperature dependence of the fits of to the low frequency portion of for the scission-Rouse model. The inset presents the temperature dependence of the average chain length. The model parameters are and . The polymerization temperature is determined from the inflection point of as .

Normalized dielectric loss as a function of dimensionless frequency for equilibrium populations of chains with the same average chain lengths and different relaxation dynamics. The solid lines are for (a) unbreakable Rouse chains and living polymers that follow MM reaction kinetics with (b) and (c) . The dotted lines are the SE fits with (a) , (b) , and (c) .

Normalized dielectric loss as a function of dimensionless frequency for equilibrium populations of chains with the same average chain lengths and different relaxation dynamics. The solid lines are for (a) unbreakable Rouse chains and living polymers that follow MM reaction kinetics with (b) and (c) . The dotted lines are the SE fits with (a) , (b) , and (c) .

Normalized Cole–Cole plot for polydisperse mixture of unbreakable chains undergoing reptation dynamics at . The multistep dielectric relaxation includes (a) the high frequency relaxation of solvent and monomer particles and (b) the low frequency contribution associated with chain relaxation. The ratio of the relaxation strengths is taken as . The dotted line is the SE fit of the low frequency process with .

Normalized Cole–Cole plot for polydisperse mixture of unbreakable chains undergoing reptation dynamics at . The multistep dielectric relaxation includes (a) the high frequency relaxation of solvent and monomer particles and (b) the low frequency contribution associated with chain relaxation. The ratio of the relaxation strengths is taken as . The dotted line is the SE fit of the low frequency process with .

Differences between imaginary parts of the normalized dielectric permittivities in the scission-Rouse and the SE or CD models as a function of dimensionless frequency for equilibrium chain population with equal average chain lengths and different relaxation dynamics: (a) unbreakable Rouse chains and (b) living polymers with MM reaction kinetics and . The parameters for each model are presented in the legends. The exponent , which was presented first in (a) and (b), is evaluated analogous to , while the second value in the legends is obtained using the relations of Lindsey and Patterson (Ref. 64).

Differences between imaginary parts of the normalized dielectric permittivities in the scission-Rouse and the SE or CD models as a function of dimensionless frequency for equilibrium chain population with equal average chain lengths and different relaxation dynamics: (a) unbreakable Rouse chains and (b) living polymers with MM reaction kinetics and . The parameters for each model are presented in the legends. The exponent , which was presented first in (a) and (b), is evaluated analogous to , while the second value in the legends is obtained using the relations of Lindsey and Patterson (Ref. 64).

## Tables

SE parameters for monodisperse and polydisperse systems with relatively slow and rapid reaction kinetics.

SE parameters for monodisperse and polydisperse systems with relatively slow and rapid reaction kinetics.

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