^{1}, Jacek Dudowicz

^{2}and Karl F. Freed

^{2,a)}

### Abstract

Particle association in “complex” fluids containing charged, polar, or polymeric molecular species often leads to deviations from the corresponding state description of “simple” fluids in which the molecules are assumed to have relatively symmetric interactions and shapes. This fundamental problem is addressed by developing a minimal thermodynamic model of activated equilibrium polymerizationsolutions that incorporates effects associated with the *competition* between van der Waals and associative interactions, as well as features related to molecular anisotropy and many-body interactions. As a dual purpose, we focus on thermodynamic signatures that can be used to identify the nature of dynamic clustering transitions and the interaction parameters associated with these *rounded*thermodynamic transitions. The analysis begins by examining “singular” features in the concentration dependence of the osmotic pressure that generically characterize the onset of particle association. Because molecular self-assembly can strongly *couple* with fluidphase separation, evidence is also sought for associative interactions in the behavior of the second and third osmotic virial coefficients. In particular, the temperatures and where and , respectively, vanish are found to contain valuable information about the relative strength of the associative and van der Waals interactions. The critical temperature for phase separation, the critical composition , and the rectilinear diameter , describing the asymmetry of the coexistence curve for phase separation, along with the average cluster mass and extent of polymerization at the critical point, further specify the relevant interaction parameters of our model. Collectively, these characteristicproperties provide a thermodynamic*metric* for defining fluid complexity and in developing a theoretically based corresponding state relation for complex fluids.

We thank Ferenc Horkay of NIH for many helpful discussions relating to the polyelectrolyte data shown in Fig. 5(a) and for providing these data in the new representation shown in Fig. 5(b). This research is supported, in part, by NSF Grant No. CHE-0416017.

I. INTRODUCTION

A. In search of a corresponding state relation for complex fluids

B. Interparticle potential shape versus many-body interactions

C. Measures of fluid complexity

II. ACTIVATED EQUILIBRIUM POLYMERIZATION MODEL FLUID WITH ASSOCIATIVE AND VAN DER WAALS INTERACTIONS

III. ILLUSTRATIVE CALCULATIONS FOR A MODEL COMPLEX ASSOCIATING FLUID

A. Activated equilibrium polymerization model

B. Singular equation of state features in associating fluids

C. Self-consistent determination of polymerization transition lines

D. Second and third virial theta temperatures, and

E. Strong coupling and evidence for the inversion of and

F. Other metrics of fluid complexity

IV. CORRESPONDING STATE DESCRIPTION OF ACTIVATED EQUILIBRIUM POLYMERIZATION

V. DISCUSSION

### Key Topics

- Polymerization
- 77.0
- Polymers
- 49.0
- Complex fluids
- 39.0
- Thermodynamic properties
- 30.0
- Solution polymerization
- 29.0

## Figures

Typical variation of the dimensionless osmotic pressure with the initial monomer composition for equilibrium polymer solutions at temperatures above the critical temperature. The curves in the figure are computed for the activated equilibrium polymerization model described in the text and specified by , , , , and (i.e., clusters are assumed to be rigid structures). Unless stated otherwise, the same free energy parameters , , , and , the exchange energy , the factor , and the factor (that quantifies the strength of three-body interactions) are used in computations summarized in Figs. 2–11.

Typical variation of the dimensionless osmotic pressure with the initial monomer composition for equilibrium polymer solutions at temperatures above the critical temperature. The curves in the figure are computed for the activated equilibrium polymerization model described in the text and specified by , , , , and (i.e., clusters are assumed to be rigid structures). Unless stated otherwise, the same free energy parameters , , , and , the exchange energy , the factor , and the factor (that quantifies the strength of three-body interactions) are used in computations summarized in Figs. 2–11.

The polymerization transition line (dashed line), coexistence curve (dotted line), and the spinodal curve (solid line) for a model fluid exhibiting activated equilibrium polymerization in the intermediate coupling regime (defined in the text) where the polymerization transition temperature at the critical composition and the critical temperature are comparable in magnitude. The critical point is indicated by a cross, while other symbols denote estimates of the polymerization line that are obtained from the composition variation of the average polymerization index (circles), the isothermal osmotic compressibility (diamonds), and the osmotic pressure (triangles). The coexistence curve (dotted line) is evidently asymmetric, reflecting the “polymeric” nature of the fluid at the critical temperature, and the rectilinear diameter (long dashed line) displays a large slope. The curvature of the rectilinear curve derives from the three-body interactions that generally arise from size anisotropy (between a solvent molecule and a monomer of the associating species) or from the molecular polarizability (see text).

The polymerization transition line (dashed line), coexistence curve (dotted line), and the spinodal curve (solid line) for a model fluid exhibiting activated equilibrium polymerization in the intermediate coupling regime (defined in the text) where the polymerization transition temperature at the critical composition and the critical temperature are comparable in magnitude. The critical point is indicated by a cross, while other symbols denote estimates of the polymerization line that are obtained from the composition variation of the average polymerization index (circles), the isothermal osmotic compressibility (diamonds), and the osmotic pressure (triangles). The coexistence curve (dotted line) is evidently asymmetric, reflecting the “polymeric” nature of the fluid at the critical temperature, and the rectilinear diameter (long dashed line) displays a large slope. The curvature of the rectilinear curve derives from the three-body interactions that generally arise from size anisotropy (between a solvent molecule and a monomer of the associating species) or from the molecular polarizability (see text).

The critical temperature for the phase separation of equilibrium polymerization solutions as a function of the polymerization energy and computed for the activated polymerization model described in the text. The existence of three different regimes in the qualitative behavior of provides a convenient criterion for defining the weak, strong, and intermediate coupling regimes of associative interactions, which are labeled in the figure by I, II, and III, respectively.

The critical temperature for the phase separation of equilibrium polymerization solutions as a function of the polymerization energy and computed for the activated polymerization model described in the text. The existence of three different regimes in the qualitative behavior of provides a convenient criterion for defining the weak, strong, and intermediate coupling regimes of associative interactions, which are labeled in the figure by I, II, and III, respectively.

(a) Temperature variation of the second and third virial coefficients for equilibrium polymer solutions in the weak coupling regime , as computed for the activated polymerization model described in the text. (b) Temperature variation of the second and third virial coefficients for equilibrium polymerization solutions in the strong coupling regime phase as computed for the activated polymerization model described in the text. (c) Temperature variation of the second and third virial coefficients for equilibrium polymerization solutions in the intermediate coupling regime between polymerization and phase separation as computed for the activated polymerization model specified in the text. The second and third virial coefficients as functions of the inverse temperature for (d) the weak, (e) the strong, and (f) the intermediate coupling regimes.

(a) Temperature variation of the second and third virial coefficients for equilibrium polymer solutions in the weak coupling regime , as computed for the activated polymerization model described in the text. (b) Temperature variation of the second and third virial coefficients for equilibrium polymerization solutions in the strong coupling regime phase as computed for the activated polymerization model described in the text. (c) Temperature variation of the second and third virial coefficients for equilibrium polymerization solutions in the intermediate coupling regime between polymerization and phase separation as computed for the activated polymerization model specified in the text. The second and third virial coefficients as functions of the inverse temperature for (d) the weak, (e) the strong, and (f) the intermediate coupling regimes.

(a) The mixing component of the osmotic pressure as a function of polymer volume fraction for polyacrylate gels in equilibrium with solutions of NaCl and . Different curves correspond to different salt concentrations, as in Fig. 6 of Ref. 87(a). (b) The second viral coefficient and the ratio (where is the third virial coefficient) as functions of the concentration of . The virial coefficients are obtained from fits of the FH expression for the osmotic pressure to the experimental data shown in (a).

(a) The mixing component of the osmotic pressure as a function of polymer volume fraction for polyacrylate gels in equilibrium with solutions of NaCl and . Different curves correspond to different salt concentrations, as in Fig. 6 of Ref. 87(a). (b) The second viral coefficient and the ratio (where is the third virial coefficient) as functions of the concentration of . The virial coefficients are obtained from fits of the FH expression for the osmotic pressure to the experimental data shown in (a).

Temperature variation of the second virial coefficient of equilibrium polymer solutions as computed for different values of the activation entropy that are specified in the figure .

Temperature variation of the second virial coefficient of equilibrium polymer solutions as computed for different values of the activation entropy that are specified in the figure .

(a) The characteristic temperatures of equilibrium polymerization solutions as functions of the energy of polymerization and computed for the activated polymerization model described in the text. The temperatures considered include the critical temperature, the theta temperature , and the temperature where the third virial coefficient vanishes. (b) The critical composition as a function of for the system analyzed in (c). Same as (a) but for a system with three-body interactions neglected . (d) Same as (b) but for a system with three-body interactions neglected .

(a) The characteristic temperatures of equilibrium polymerization solutions as functions of the energy of polymerization and computed for the activated polymerization model described in the text. The temperatures considered include the critical temperature, the theta temperature , and the temperature where the third virial coefficient vanishes. (b) The critical composition as a function of for the system analyzed in (c). Same as (a) but for a system with three-body interactions neglected . (d) Same as (b) but for a system with three-body interactions neglected .

Ratios of the characteristic temperatures of equilibrium polymer solutions as a function of the sticking energy [see Fig. 7(a)].

Ratios of the characteristic temperatures of equilibrium polymer solutions as a function of the sticking energy [see Fig. 7(a)].

The characteristic temperatures of equilibrium polymer solutions as a function of the exchange energy and computed for the activated polymerization model described in the text. The temperatures considered include the critical temperature, the theta temperature , and the temperature where the third virial coefficient vanishes.

The characteristic temperatures of equilibrium polymer solutions as a function of the exchange energy and computed for the activated polymerization model described in the text. The temperatures considered include the critical temperature, the theta temperature , and the temperature where the third virial coefficient vanishes.

Ratios of the characteristic temperatures of equilibrium polymer solutions that are analyzed in Fig. 9 as a function of the exchange energy .

Ratios of the characteristic temperatures of equilibrium polymer solutions that are analyzed in Fig. 9 as a function of the exchange energy .

The rectilinear diameter and the critical osmotic compressibility factor (normalized by the corresponding quantity for monomer-solvent systems) computed as functions of the polymerization energy for a solution exhibiting activated equilibrium polymerization. The free energy parameters are specified in the caption to Fig. 1, and the factor , describing the strength of ternary interactions, is set to zero (i.e., the ternary interactions are neglected in this illustrative example).

The rectilinear diameter and the critical osmotic compressibility factor (normalized by the corresponding quantity for monomer-solvent systems) computed as functions of the polymerization energy for a solution exhibiting activated equilibrium polymerization. The free energy parameters are specified in the caption to Fig. 1, and the factor , describing the strength of ternary interactions, is set to zero (i.e., the ternary interactions are neglected in this illustrative example).

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