^{1,a)}, Karl F. Freed

^{1}and Jack F. Douglas

^{2}

### Abstract

The newly developed lattice cluster theory (in Paper I) for the thermodynamics of solutions of telechelic polymers is used to examine the phase behavior of these complex fluids when effective polymer-solventinteractions are unfavorable. The telechelics are modeled as linear, fully flexible, polymer chains with mono-functional stickers at the two chain ends, and these chains are assumed to self-assemble upon cooling. Phase separation is generated through the interplay of self-assembly and polymer/solvent interactions that leads to an upper critical solution temperature phase separation. The variations of the boundaries for phase stability and the critical temperature and composition are analyzed in detail as functions of the number *M* of united atom groups in a telechelic chain and the microscopic nearest neighbor interaction energy ε_{ s } driving the self-assembly. The coupling between self-assembly and unfavorable polymer/solvent interactions produces a wide variety of nontrivial patterns of phase behavior, including an *enhancement* of miscibility accompanying the increase of the molar mass of the telechelics under certain circumstances. Special attention is devoted to understanding this unusual trend in miscibility.

The research is supported, in part, by NSF (Grant No. CHE-1111918).

I. INTRODUCTION

II. LATTICE CLUSTER THEORY: THE HELMHOLTZ FREE ENERGY AND SPINODAL CURVES

III. PHASE BEHAVIOR OF SOLUTIONS OF TELECHELIC POLYMERS

A. Spinodal curves

B. Critical temperature and critical composition

C. Why does the miscibility of solutions of telechelic polymers increase with their chain length?

IV. DISCUSSION

### Key Topics

- Polymers
- 31.0
- Self assembly
- 27.0
- Free energy
- 25.0
- Solution polymerization
- 23.0
- Solubility
- 20.0

##### F25

## Figures

Spinodal curves for solutions of short linear telechelic chains (*M* = 5) as computed from the LCT for the indicated sticky interaction energies ε_{ s }. The same exchange van der Waals energy ε = 100 K is used in all calculations presented in Figs. 1–17.

Spinodal curves for solutions of short linear telechelic chains (*M* = 5) as computed from the LCT for the indicated sticky interaction energies ε_{ s }. The same exchange van der Waals energy ε = 100 K is used in all calculations presented in Figs. 1–17.

Spinodal curves for solutions of linear telechelic chains as computed from the LCT for various numbers *M* of united atom groups in a single chain. The sticky interaction energy ε_{ s } and the exchange van der Waals energy ε are specified in the figure.

Spinodal curves for solutions of linear telechelic chains as computed from the LCT for various numbers *M* of united atom groups in a single chain. The sticky interaction energy ε_{ s } and the exchange van der Waals energy ε are specified in the figure.

Spinodal curves for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K), as computed from the LCT for various lengths (masses) *M* of telechelic chains.

Spinodal curves for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K), as computed from the LCT for various lengths (masses) *M* of telechelic chains.

Spinodal curves for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K) in the critical region, as computed from the LCT.

Spinodal curves for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K) in the critical region, as computed from the LCT.

Spinodal curves for solutions of weakly interacting linear telechelic chains (ε_{ s } = −600 K), as computed from the LCT for various chain lengths (masses) *M*, including the the critical length *M* _{ c }.

Spinodal curves for solutions of weakly interacting linear telechelic chains (ε_{ s } = −600 K), as computed from the LCT for various chain lengths (masses) *M*, including the the critical length *M* _{ c }.

The same as Fig. 5, but for solutions of linear telechelic chains interacting with a moderate sticky energy ε_{ s } = −1500 K.

The same as Fig. 5, but for solutions of linear telechelic chains interacting with a moderate sticky energy ε_{ s } = −1500 K.

The same as Fig. 5, but for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K).

The same as Fig. 5, but for solutions of strongly interacting linear telechelic chains (ε_{ s } = −3000 K).

The same as Fig. 5, but for solutions of very strongly interacting linear telechelic chains (ε_{ s } = −6000 K).

The same as Fig. 5, but for solutions of very strongly interacting linear telechelic chains (ε_{ s } = −6000 K).

The critical temperature *T* _{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the absolute sticky interaction energy |ε_{ s }| for different chain lengths *M*.

The critical temperature *T* _{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the absolute sticky interaction energy |ε_{ s }| for different chain lengths *M*.

The critical composition ϕ_{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the absolute sticky interaction energy |ε_{ s }| for different polymer chain lengths *M*.

The critical composition ϕ_{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the absolute sticky interaction energy |ε_{ s }| for different polymer chain lengths *M*.

The critical temperature *T* _{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the polymer length *M* for different sticky interaction energies ε_{ s }. Crosses indicate the critical value *M* _{ c }(ε_{ s }) above which *T* _{ c } begins to increase with *M*.

The critical temperature *T* _{ c } for the phase stability of linear telechelic polymer solutions, as computed from the LCT as a function of the polymer length *M* for different sticky interaction energies ε_{ s }. Crosses indicate the critical value *M* _{ c }(ε_{ s }) above which *T* _{ c } begins to increase with *M*.

The specific Helmholtz free energy β*f* for solutions of weakly interacting linear telechelic polymers (ε_{ s } = −300 K), as computed from the LCT as a function of the chain length *M* for fixed temperature *T* = 300 K and solution composition ϕ = 0.1. Variations of the individual components of β*f* from Eq. (12) are also displayed in the figure for comparison.

The specific Helmholtz free energy β*f* for solutions of weakly interacting linear telechelic polymers (ε_{ s } = −300 K), as computed from the LCT as a function of the chain length *M* for fixed temperature *T* = 300 K and solution composition ϕ = 0.1. Variations of the individual components of β*f* from Eq. (12) are also displayed in the figure for comparison.

The same as Fig. 12, but for solutions of strongly interacting linear telechelic polymers (ε_{ s } = −3000 K).

The same as Fig. 12, but for solutions of strongly interacting linear telechelic polymers (ε_{ s } = −3000 K).

The specific entropy *s*/*k* _{ B } for solutions of linear telechelic polymers, as computed from the LCT as a function of the polymer index *M* for fixed temperature *T* = 300 K and solution composition ϕ = 0.1. Solid and dashed curves refer to solutions of weakly (ε_{ s } = −300 K) and strongly (ε_{ s } = −3000 K) interacting linear telechelic polymers, respectively.

The specific entropy *s*/*k* _{ B } for solutions of linear telechelic polymers, as computed from the LCT as a function of the polymer index *M* for fixed temperature *T* = 300 K and solution composition ϕ = 0.1. Solid and dashed curves refer to solutions of weakly (ε_{ s } = −300 K) and strongly (ε_{ s } = −3000 K) interacting linear telechelic polymers, respectively.

The second derivatives of the specific free energies and *f* _{ s } (with respect to the polymer volume fraction ϕ) for solutions of linear telechelic chains, as computed from the LCT as functions of the chain index *M* for fixed temperature *T* = 300 K and ϕ = 0.1.

The second derivatives of the specific free energies and *f* _{ s } (with respect to the polymer volume fraction ϕ) for solutions of linear telechelic chains, as computed from the LCT as functions of the chain index *M* for fixed temperature *T* = 300 K and ϕ = 0.1.

The second derivatives of the athermal limit (*a*) and non-athermal (*b*′ = *b*/*T*) portions of the non-combinatorial free energy (with respect to the polymer volume fraction ϕ) for solutions of linear telechelic polymers, as computed from the LCT as functions of the chain length *M* for ϕ = 0.1.

The second derivatives of the athermal limit (*a*) and non-athermal (*b*′ = *b*/*T*) portions of the non-combinatorial free energy (with respect to the polymer volume fraction ϕ) for solutions of linear telechelic polymers, as computed from the LCT as functions of the chain length *M* for ϕ = 0.1.

The spinodal temperature *T* _{ sp } for the phase stability of solutions of linear telechelic polymers, as computed from Eq. (18) as a function of the chain index *M* for a fixed solution composition ϕ = 0.1 and various sticky interaction energies ε_{ s }.

The spinodal temperature *T* _{ sp } for the phase stability of solutions of linear telechelic polymers, as computed from Eq. (18) as a function of the chain index *M* for a fixed solution composition ϕ = 0.1 and various sticky interaction energies ε_{ s }.

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