^{1}and Alejandro D. Rey

^{1,a)}

### Abstract

The structure and dynamics of early stage kinetics of pressure-induced phase separation of compressible polymer solutions via spinodal decomposition is analyzed using a linear Euler–Cahn–Hilliard model and the modified Sanchez Lacombe equation of state. The integrated density wave and Cahn–Hilliard equations combine the kinetic and structural characteristics of spinodal decomposition with density waves arising from pressure-induced couplings. When mass transfer rate is slower that acoustic waves, concentration gradients generate density waves that cycle back into the spinodal decomposition dynamics, resulting in oscillatory demixing. The wave attenuation increases with increasing mass transfer rates eventually leading to nonoscillatory spinodal demixing. The novel aspects of acousto-spinodal decomposition arise from the coexistence of stable oscillatory density dynamics and the unstable monotonic concentration dynamics. Scaling laws for structure and dynamics indicate deviations from incompressible behavior, with a significant slowing down of demixing due to couplings with density waves. Partial structure factors for density and density-concentration reflect the oscillatory nature of acousto-spinodal modes at lower wave vectors, while the single maximum at a constant wave vector reflects the presence of a dominant mode in the linear regime. The computed total structure factor is in qualitative agreement with experimental data for a similar polymer solution.

This work was supported by a grant from Natural Science and Engineering Research Council of Canada (NSERC) and NOVA CHEMICALS. The authors gratefully acknowledge the invaluable contributions of Dr. Eric Cheluget and Dr. Majid Ghiass.

I. INTRODUCTION

II. MODEL OF PRESSURE-INDUCED PHASE SEPARATION KINETICS BY SPINODAL DECOMPOSITION IN BINARY COMPRESSIBLE POLYMER SOLUTIONS

A. Integrated model

B. Normal modes and modified Sanchez–Lacombe equation of state parametric data

III. RESULTS AND DISCUSSION

A. Mode classification, phase diagrams,acousto-spinodal couplings, and parametric sensitivity

1. Mode classification

2. Parametric phase diagrams

3. Acousto-spinodal couplings

4. Eigenvalue parametric sensitivity

B. Characterization of hydrodynamic variables

C. Growth rate and scaling laws

D. Structure factors

E. Mobility and interfacial tension

IV. CONCLUSIONS

### Key Topics

- Solution polymerization
- 29.0
- Eigenvalues
- 28.0
- Acoustic waves
- 19.0
- Polymers
- 16.0
- Equations of state
- 14.0

## Figures

Organization of the paper. (MSL: modified Sanchez–Lacombe EOS.)

Organization of the paper. (MSL: modified Sanchez–Lacombe EOS.)

(a) Phase diagram in terms of Mach number as a function of the Cahn number, for a wave vector *q* =1000. The merging modes that may emerge are monotonic growth (MG), monotoinic decay (MD), oscillatory growth (OG), and oscillatory decay (OD). The solid nearly horizontal line shows Δ = 0 (boundary between oscillatory and monotonic modes) and the vertical dashed line shows *b* _{0} = 0 (stability threshold). Diminishing inertia (*Ma*) and interfacial tension promotes oscillatory SD. (b) Schematic of corresponding eigenvalue plots. OG corresponds to a complex pair of eigenvalues and a positive real eigenvalue.

(a) Phase diagram in terms of Mach number as a function of the Cahn number, for a wave vector *q* =1000. The merging modes that may emerge are monotonic growth (MG), monotoinic decay (MD), oscillatory growth (OG), and oscillatory decay (OD). The solid nearly horizontal line shows Δ = 0 (boundary between oscillatory and monotonic modes) and the vertical dashed line shows *b* _{0} = 0 (stability threshold). Diminishing inertia (*Ma*) and interfacial tension promotes oscillatory SD. (b) Schematic of corresponding eigenvalue plots. OG corresponds to a complex pair of eigenvalues and a positive real eigenvalue.

Mach–Cahn number phase diagram for *q* = 500, 1000, 1500, and 2000. The solid line shows Δ = 0 (oscillatory-monotonic transition) and the dashed vertical line shows *b* _{0} = 0 (stability threshold).

Mach–Cahn number phase diagram for *q* = 500, 1000, 1500, and 2000. The solid line shows Δ = 0 (oscillatory-monotonic transition) and the dashed vertical line shows *b* _{0} = 0 (stability threshold).

Mach–Cahn number phase diagram for = −7.85, −8.2, −9.2, and −10.2, and *q* = 1000. Stronger coupling promotes SD since it augments the driving force for phase separation [Eq. (8)] and enhances oscillations through the feedback effect [Eq. (10)].

Mach–Cahn number phase diagram for = −7.85, −8.2, −9.2, and −10.2, and *q* = 1000. Stronger coupling promotes SD since it augments the driving force for phase separation [Eq. (8)] and enhances oscillations through the feedback effect [Eq. (10)].

Mach–Cahn number phase diagram for =11.5, 11.7, 11.83, and 12 and *q* = 1000. Increasing compressibility enhances instability but does not affect the monotonic/oscillatory threshold atΔ = 0. [Eq. (8)].

Mach–Cahn number phase diagram for =11.5, 11.7, 11.83, and 12 and *q* = 1000. Increasing compressibility enhances instability but does not affect the monotonic/oscillatory threshold atΔ = 0. [Eq. (8)].

Dimensionless coupling term [Eq. (11)] as a function of dimensionless time corresponding to the oscillatory mode (OG) for *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. The coupling introduces a standing wave into the Cahn–Hilliard equation (11) and SD is oscillatory.

Dimensionless coupling term [Eq. (11)] as a function of dimensionless time corresponding to the oscillatory mode (OG) for *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. The coupling introduces a standing wave into the Cahn–Hilliard equation (11) and SD is oscillatory.

Acoustic source [Eq. (10)] as a function of dimensionless time corresponding to the oscillatory mode (OG) for *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000.

Acoustic source [Eq. (10)] as a function of dimensionless time corresponding to the oscillatory mode (OG) for *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000.

Effect of Mach on the real part of complex eigenvalues (amplitude of oscillation) and imaginary part of complex eigenvalues (frequency of oscillation) when moving vertically on phase diagram (Fig. 2) for *Ca* = 10^{−4} and *q* = 1000.

Effect of Mach on the real part of complex eigenvalues (amplitude of oscillation) and imaginary part of complex eigenvalues (frequency of oscillation) when moving vertically on phase diagram (Fig. 2) for *Ca* = 10^{−4} and *q* = 1000.

Effect of Cahn on real part of complex eigenvalues (amplitude of oscillation) and imaginary part of complex eigenvalues (frequency of oscillation) obtained by moving horizontally on phase diagram for *Ma* = 10^{−5}, *q* = 1000.

Effect of Cahn on real part of complex eigenvalues (amplitude of oscillation) and imaginary part of complex eigenvalues (frequency of oscillation) obtained by moving horizontally on phase diagram for *Ma* = 10^{−5}, *q* = 1000.

Computed phase portrait of the three dimensionless hydrodynamic variables(ρ′(*x* = 0.1, *t*), v′(*x* = 0.1, *t*), ω′(*x* = 0.1, *t*)). (a) Phase trajectory in the OG region: *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. (b) Phase trajectory in the MG region: *Ma* =2 × 10^{3}, *Ca* = 10^{−4}, and *q* = 1000.

Computed phase portrait of the three dimensionless hydrodynamic variables(ρ′(*x* = 0.1, *t*), v′(*x* = 0.1, *t*), ω′(*x* = 0.1, *t*)). (a) Phase trajectory in the OG region: *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. (b) Phase trajectory in the MG region: *Ma* =2 × 10^{3}, *Ca* = 10^{−4}, and *q* = 1000.

The evolution of mass fraction as function of dimensionless time; (a) for a system located in the oscillatory region (OG): *Ma* = 10^{−5}, *Ca* = 10^{−4}, *q* = 1000 and (b) for a system located in the monotonic region (MG): *Ma* =2 × 10^{−3}, *Ca* = 10^{−4}, *q* = 1000.

The evolution of mass fraction as function of dimensionless time; (a) for a system located in the oscillatory region (OG): *Ma* = 10^{−5}, *Ca* = 10^{−4}, *q* = 1000 and (b) for a system located in the monotonic region (MG): *Ma* =2 × 10^{−3}, *Ca* = 10^{−4}, *q* = 1000.

The evolution of dimensionless velocity with dimensionless time for a system located in oscillatory region (OG) *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. As time passes three stages of oscillation, dead time, and monotonic are observed.

The evolution of dimensionless velocity with dimensionless time for a system located in oscillatory region (OG) *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. As time passes three stages of oscillation, dead time, and monotonic are observed.

The evolution of dimensionless pressure with dimensionless time for a system located in oscillatory region (OG) *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. As time passes three stages of oscillation, dead time, and monotonic are observed.

The evolution of dimensionless pressure with dimensionless time for a system located in oscillatory region (OG) *Ma* = 10^{−5}, *Ca* = 10^{−4}, and *q* = 1000. As time passes three stages of oscillation, dead time, and monotonic are observed.

The dimensionless growth rate *R*(*q*) vs dimensionless wave vector *q* for a simulation corresponding to *Ca* = 10^{−4} and three different Mach number: *Ma* = 10^{−5}, 8 × 10^{−4}, and 2 × 10^{−3}.

The dimensionless growth rate *R*(*q*) vs dimensionless wave vector *q* for a simulation corresponding to *Ca* = 10^{−4} and three different Mach number: *Ma* = 10^{−5}, 8 × 10^{−4}, and 2 × 10^{−3}.

The dimensionless growth rate *R*(*q*) vs dimensionless wave vector *q* for a simulation corresponding to *Ma* = 10^{−5} and three different Cahn number: *Ca* _{1} = 10^{−4}, 2 × 10^{−4}, and 3 × 10^{−4} _{.}

The dimensionless growth rate *R*(*q*) vs dimensionless wave vector *q* for a simulation corresponding to *Ma* = 10^{−5} and three different Cahn number: *Ca* _{1} = 10^{−4}, 2 × 10^{−4}, and 3 × 10^{−4} _{.}

The evolution of concentration, density, and concentration/density structure factors as a function of wave vector for a simulation corresponding to region (OG): *Ma* = 10^{−5} and *Ca* = 10^{−4}. The dotted line, dashed line, and solid line belong to *t* _{1} = 10^{−6}, *t* _{2} = 1.5 × 10^{−6}, *t* _{3} = 2 × 10^{−6}, respectively. The insets showing oscillations correspond to *t* _{2}.

The evolution of concentration, density, and concentration/density structure factors as a function of wave vector for a simulation corresponding to region (OG): *Ma* = 10^{−5} and *Ca* = 10^{−4}. The dotted line, dashed line, and solid line belong to *t* _{1} = 10^{−6}, *t* _{2} = 1.5 × 10^{−6}, *t* _{3} = 2 × 10^{−6}, respectively. The insets showing oscillations correspond to *t* _{2}.

(a) Total structure factor *S*(*q*,*t*) (solid line) at a dimensionless time of *t* = 1.2 × 10^{−6}, for 〈*b*〉 = 1.01 × 10^{10}cm^{−2}, Δ*b* = 6.206 × 10^{10}cm^{−2} (b) Experimental data (dotted line) for LLDPE in n-Pentane extracted from Ref. 8. The computed *S*(*q*,*t*) data were shifted vertically by a factor of +2 to fit the experimental data that corresponds to a different polymer solution.

(a) Total structure factor *S*(*q*,*t*) (solid line) at a dimensionless time of *t* = 1.2 × 10^{−6}, for 〈*b*〉 = 1.01 × 10^{10}cm^{−2}, Δ*b* = 6.206 × 10^{10}cm^{−2} (b) Experimental data (dotted line) for LLDPE in n-Pentane extracted from Ref. 8. The computed *S*(*q*,*t*) data were shifted vertically by a factor of +2 to fit the experimental data that corresponds to a different polymer solution.

Schematic of the analogous two spring–two mass mechanical model summarizing the effects operating during spinodal decomposition of compressible polymer solutions. The chemomechanical driving force for spinodal decomposition is and its resistance is *Ca*. Concentration fluctuations are a sound source term in the acoustic equation through the action of . A feedback effect from the density into the spinodal acts through . Decreasing the Mach number decreases the effective mass and gives rise to oscillatory spinodal decomposition.

Schematic of the analogous two spring–two mass mechanical model summarizing the effects operating during spinodal decomposition of compressible polymer solutions. The chemomechanical driving force for spinodal decomposition is and its resistance is *Ca*. Concentration fluctuations are a sound source term in the acoustic equation through the action of . A feedback effect from the density into the spinodal acts through . Decreasing the Mach number decreases the effective mass and gives rise to oscillatory spinodal decomposition.

*P* − ω phase diagram for a solution of hPBDM in *n*-hexane, *M* _{ w } = 49 000 g/mol at different temperature, computed using the MSL model (Ref. 7).

*P* − ω phase diagram for a solution of hPBDM in *n*-hexane, *M* _{ w } = 49 000 g/mol at different temperature, computed using the MSL model (Ref. 7).

Real eigenvalues as a function of Mach number in the growth region (*−b* _{0} > 0) of phase diagram, *Ca* = 10^{−4} and *q* = 1000. Under oscillatory growth OG (Δ > 0, starred line) the real part of the complex pair corresponds to oscillatory attenuation. Under monotonic growth MG (Δ < 0, dashed line) the positive real eigenvalues drives the SD which slows with increasing *Ma*.

Real eigenvalues as a function of Mach number in the growth region (*−b* _{0} > 0) of phase diagram, *Ca* = 10^{−4} and *q* = 1000. Under oscillatory growth OG (Δ > 0, starred line) the real part of the complex pair corresponds to oscillatory attenuation. Under monotonic growth MG (Δ < 0, dashed line) the positive real eigenvalues drives the SD which slows with increasing *Ma*.

Eigenvalues as a function of Mach mumbers for a case that moving vertically on decay region (*−b* _{0} < 0) of phase diagram, *Ca* = 4 × 10^{−4} and *q* = 1000. The solid lines belong to one real eigenvalues and the dotted lines belong to real part of two complex conjugate eigenvalues on OD region of phase diagram. The dashed line belongs to three real eigenvalues on MD region of phase diagram.

Eigenvalues as a function of Mach mumbers for a case that moving vertically on decay region (*−b* _{0} < 0) of phase diagram, *Ca* = 4 × 10^{−4} and *q* = 1000. The solid lines belong to one real eigenvalues and the dotted lines belong to real part of two complex conjugate eigenvalues on OD region of phase diagram. The dashed line belongs to three real eigenvalues on MD region of phase diagram.

Eigenvalues as a function of Cahn, moving horizontally on kinetics phase diagram: *Ma* = 10^{−3} and *q* = 1000. The point, dashed and solid lines, belongs to the MG region on phase diagram: MG: three real eigenvalues (unstable: with one positive eigenvalue) OD: three real eigenvalues (stable), and OD: one real and two complex conjugate eigenvalues.

Eigenvalues as a function of Cahn, moving horizontally on kinetics phase diagram: *Ma* = 10^{−3} and *q* = 1000. The point, dashed and solid lines, belongs to the MG region on phase diagram: MG: three real eigenvalues (unstable: with one positive eigenvalue) OD: three real eigenvalues (stable), and OD: one real and two complex conjugate eigenvalues.

## Tables

Solution classification and regime transitions.

Solution classification and regime transitions.

Maximal growth scaling laws.

Maximal growth scaling laws.

Polymer properties and thermodynamic parameters for the MSL EOS, Refs. 7 and 23.

Polymer properties and thermodynamic parameters for the MSL EOS, Refs. 7 and 23.

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