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