^{1}, Amparo Galindo

^{2}and George Jackson

^{2,a)}

### Abstract

Polymers are naturally polydisperse. Polydispersity may have a large effect on the phase behavior of polymer solutions, in particular, on the liquid-liquid phase equilibria. In this paper, we determine the cloud and shadow curves bounded by lower critical solution temperatures for a number of systems where the polymer is polydisperse in terms of molecular weight (chain length). The moment method [P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem. Phys.116, 265 (2001)] is applied with the SAFT approach to determine cloud and shadow curves with continuous Schulz-Flory distributions. It is seen that chain length polydispersity always enhances the extent of liquid-liquid phase equilibria. The predicted cloud curves obtained for continuous distributions are very similar to those obtained for simple ternary mixtures with the same polydispersity index, while the corresponding shadow curves can be very different depending on the composition of the parent distribution. The ternary phase behavior can be used to provide an understanding of the shape of the cloud and shadow curves. Regions of phase equilibria between three liquid phases are found for ternary systems when the chain length distribution is very asymmetrical; such regions are not observed for Schulz-Flory distributions even in the case of a large degree of polydispersity.

One of the authors (P.P.) thanks the Modeling Programme of BP Chemicals for funding a studentship. We acknowledge further support from Engineering and Physical Sciences Research Council (EPSRC) of the UK (Grant Nos. GR/N20317, GR/N03358, GR/N35991, GR/R09497, and EP/E016340), the Joint Research Equipment Initiative (JREI) for computer hardware (GR/M94427), and the Royal Society-Wolfson Foundation for the award of a refurbishment grant.

I. INTRODUCTION

II. THEORY

A. SAFT (Wertheim TPT1) approach for polydisperse systems

B. Determination of the cloud and shadow curves

C. Moment method

III. RESULTS AND DISCUSSIONS

A. Ternary systems: polymer chains

B. Semicontinuous systems: polymer with Schulz-Flory length distribution

IV. CONCLUSION

### Key Topics

- Polymers
- 75.0
- Solvents
- 50.0
- Chemical potential
- 23.0
- Free energy
- 23.0
- Phase equilibria
- 20.0

## Figures

Cloud and shadow curves calculated with the SAFT (Wertheim TPT1) approach at a reduced pressure for the model (I) ternary mixture ( ). The thick continuous curve represents the cloud curve, and the dotted curve represents the corresponding shadow curve. The binodal curves (dot-dashed) of three different binary mixtures at the same pressure are shown for comparison: (a) ; (b) ; and (c) . The circles denote the critical points. is the total polymer weight fraction and is the reduced temperature.

Cloud and shadow curves calculated with the SAFT (Wertheim TPT1) approach at a reduced pressure for the model (I) ternary mixture ( ). The thick continuous curve represents the cloud curve, and the dotted curve represents the corresponding shadow curve. The binodal curves (dot-dashed) of three different binary mixtures at the same pressure are shown for comparison: (a) ; (b) ; and (c) . The circles denote the critical points. is the total polymer weight fraction and is the reduced temperature.

Number average chain length in the shadow phase as a function of the polymer weight fraction in the shadow phase obtained for the model (I) and (II) ternary mixtures at . The circles denote the critical points, and the square denotes a three-phase point.

Number average chain length in the shadow phase as a function of the polymer weight fraction in the shadow phase obtained for the model (I) and (II) ternary mixtures at . The circles denote the critical points, and the square denotes a three-phase point.

Cloud and shadow curves calculated with the SAFT (Wertheim TPT1) approach for the model (I) ternary mixture( ). The thick continuous curve represents the cloud curve, and the thin continuous curve represents the shadow curve. The circle denotes the critical point. is the total polymer weight fraction. The dot-dashed lines represent constant temperature slices corresponding to the ternary diagrams depicted in Fig. 4. The white squares denote cloud points, and the black squares are the corresponding shadow points.

Cloud and shadow curves calculated with the SAFT (Wertheim TPT1) approach for the model (I) ternary mixture( ). The thick continuous curve represents the cloud curve, and the thin continuous curve represents the shadow curve. The circle denotes the critical point. is the total polymer weight fraction. The dot-dashed lines represent constant temperature slices corresponding to the ternary diagrams depicted in Fig. 4. The white squares denote cloud points, and the black squares are the corresponding shadow points.

Ternary diagrams in weight fractions for the model (I) ternary mixture ( ) for a pressure and temperatures (a) , (b) , and (c) . The temperatures , , and correspond to the three constant temperature slices shown in Fig. 3. The thick continuous curves represent the coexistence curves, and the circles denote the critical points. The thin lines are tie lines. The dot-dashed lines represent a fixed ratio of composition of chains 2 and 3 that corresponds to the parent distribution , . The white squares denote cloud points, and the black squares are the corresponding shadow points.

Ternary diagrams in weight fractions for the model (I) ternary mixture ( ) for a pressure and temperatures (a) , (b) , and (c) . The temperatures , , and correspond to the three constant temperature slices shown in Fig. 3. The thick continuous curves represent the coexistence curves, and the circles denote the critical points. The thin lines are tie lines. The dot-dashed lines represent a fixed ratio of composition of chains 2 and 3 that corresponds to the parent distribution , . The white squares denote cloud points, and the black squares are the corresponding shadow points.

(a) Cloud and shadow curves calculated at pressure with the SAFT (Wertheim TPT1) approach for the model (II) ternary mixture ( ). The dotted rectangle is shown at higher magnification in (b). The thick continuous curve represents the cloud curve of the ternary system, and the thin continuous curve represents the shadow curve. The circle denotes the critical point. The white square is a three-phase point. (b) Inset of the region of the three-phase equilibria. The dot-dashed line represents the constant temperature slice at that corresponds to the ternary diagrams depicted in Fig. 6. The symbol cross denotes the cloud point at this temperature. The diamonds are the three-phase boundary points at .

(a) Cloud and shadow curves calculated at pressure with the SAFT (Wertheim TPT1) approach for the model (II) ternary mixture ( ). The dotted rectangle is shown at higher magnification in (b). The thick continuous curve represents the cloud curve of the ternary system, and the thin continuous curve represents the shadow curve. The circle denotes the critical point. The white square is a three-phase point. (b) Inset of the region of the three-phase equilibria. The dot-dashed line represents the constant temperature slice at that corresponds to the ternary diagrams depicted in Fig. 6. The symbol cross denotes the cloud point at this temperature. The diamonds are the three-phase boundary points at .

(a) Ternary diagram in weight fractions for the model (II) ternary mixture ( ) at and . The dotted rectangle is shown at higher magnification in (b). The dot-dashed lines represent a fixed ratio of composition of chains 2 and 3 that corresponds to the parent distribution , . (b) Inset of the region of three-phase equilibria. The thick continuous curves represent the coexistence curves, and the circle denotes the critical point. The thin lines are tie lines. The cross denotes a cloud point and corresponds to the same cloud point denoted by the cross in Fig. 5(b). The white triangles denote the compositions of the three coexistent phases. The diamonds are the intersection points between the boundaries of the three-phase region and the dot-dashed line (fixed parent distribution). These points correspond to the three-phase boundary points denoted as diamonds in Fig. 5(b).

(a) Ternary diagram in weight fractions for the model (II) ternary mixture ( ) at and . The dotted rectangle is shown at higher magnification in (b). The dot-dashed lines represent a fixed ratio of composition of chains 2 and 3 that corresponds to the parent distribution , . (b) Inset of the region of three-phase equilibria. The thick continuous curves represent the coexistence curves, and the circle denotes the critical point. The thin lines are tie lines. The cross denotes a cloud point and corresponds to the same cloud point denoted by the cross in Fig. 5(b). The white triangles denote the compositions of the three coexistent phases. The diamonds are the intersection points between the boundaries of the three-phase region and the dot-dashed line (fixed parent distribution). These points correspond to the three-phase boundary points denoted as diamonds in Fig. 5(b).

Fluid phase equilibria of polydisperse systems determined with Schulz-Flory distributions. (a) Parent Schulz-Flory distributions in terms of weight fractions for a number average chain length and polydispersity indices of and . (b) Cloud (thick) and shadow (thin) curves for a reduced pressure corresponding to the Schulz-Flory distributions depicted in (a). The circles represent the critical points. (c) Number average chain length in the shadow phase, as a function of polymer weight fraction in the shadow phase, calculated for the phase equilibria of (b). The circles represent the critical points.

Fluid phase equilibria of polydisperse systems determined with Schulz-Flory distributions. (a) Parent Schulz-Flory distributions in terms of weight fractions for a number average chain length and polydispersity indices of and . (b) Cloud (thick) and shadow (thin) curves for a reduced pressure corresponding to the Schulz-Flory distributions depicted in (a). The circles represent the critical points. (c) Number average chain length in the shadow phase, as a function of polymer weight fraction in the shadow phase, calculated for the phase equilibria of (b). The circles represent the critical points.

Cloud (continuous) and shadow (dotted) coexistence curves obtained for the continuous Schulz-Flory feed distribution (thick) and for the discrete (I) ternary mixture (thin). The circles denote critical points. For both systems, the number average chain length of the feed distribution is , and the polydispersity index is .

Cloud (continuous) and shadow (dotted) coexistence curves obtained for the continuous Schulz-Flory feed distribution (thick) and for the discrete (I) ternary mixture (thin). The circles denote critical points. For both systems, the number average chain length of the feed distribution is , and the polydispersity index is .

## Tables

Chain lengths and parent distributions in terms of the mole and weight fractions, for two polydisperse (bidisperse) polymers (I) and (II) with the same number average chain length and the same polydispersity index . Each polymer is represented by a mixture of two chain molecules 2 and 3 of lengths and .

Chain lengths and parent distributions in terms of the mole and weight fractions, for two polydisperse (bidisperse) polymers (I) and (II) with the same number average chain length and the same polydispersity index . Each polymer is represented by a mixture of two chain molecules 2 and 3 of lengths and .

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