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Acousto-spinodal decomposition of compressible polymer solutions: Early stage analysis
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10.1063/1.3578175
/content/aip/journal/jcp/134/18/10.1063/1.3578175
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/18/10.1063/1.3578175

Figures

Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

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).

Image of FIG. 4.
FIG. 4.

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)].

Image of FIG. 5.
FIG. 5.

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)].

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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 × 103, Ca = 10−4, and q = 1000.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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 .

Image of FIG. 16.
FIG. 16.

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.

Image of FIG. 17.
FIG. 17.

(a) Total structure factor S(q,t) (solid line) at a dimensionless time of t = 1.2 × 10−6, for 〈b〉 = 1.01 × 1010cm−2, Δb = 6.206 × 1010cm−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.

Image of FIG. 18.
FIG. 18.

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.

Image of FIG. 19.
FIG. 19.

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).

Image of FIG. 20.
FIG. 20.

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.

Image of FIG. 21.
FIG. 21.

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.

Image of FIG. 22.
FIG. 22.

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

Generic image for table
Table I.

Solution classification and regime transitions.

Generic image for table
Table II.

Maximal growth scaling laws.

Generic image for table
Table III.

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

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2011-05-12
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Acousto-spinodal decomposition of compressible polymer solutions: Early stage analysis
http://aip.metastore.ingenta.com/content/aip/journal/jcp/134/18/10.1063/1.3578175
10.1063/1.3578175
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