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Numerical simulation results of the nonlinear coefficient Q from FT-Rheology using a single mode pom-pom model
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10.1122/1.4754444
/content/sor/journal/jor2/57/1/10.1122/1.4754444
http://aip.metastore.ingenta.com/content/sor/journal/jor2/57/1/10.1122/1.4754444

Figures

Image of FIG. 1.
FIG. 1.

Simulation results for the single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, s a = 2, and q = 2. (a) G′ and G″ as a function of strain amplitude at a fixed frequency of 1 Hz. The SAOS data are obtained at smallest strain amplitude 0.01. (b) The normalized stress and strain as a function of time and (c) Fourier spectra as a function of frequency for the following strain amplitudes: γ 0 = 0.1, 1.0, 3.98, and 10 (noise level ≈ 10−11).

Image of FIG. 2.
FIG. 2.

Simulation of I 3/I 1 ( = I 3/1) as a function of strain amplitude at various frequencies (0.01, 0.063, 0.1, 0.63, 1, 6.3, 10, 63, 100 Hz) for a single mode differential pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, s a = 2 and (a) q = 2, (b) q = 4, and (c) q = 7. The parameter was calculated from (a) and plotted in (d), from (b) and plotted in (e), and from (c) and plotted in (f). is the asymptotic value at small strain amplitudes in (d), (e), and (f).

Image of FIG. 3.
FIG. 3.

(a) Values for G′/G 0(De) and G″/G 0(De) from SAOS simulations and (b) for Q 0(De) from LAOS simulations using the single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, and s a = 2 at various arm numbers, q, over a wide frequency range (10−3 to 103 Hz). The Deborah number is defined with respect to the orientation relaxation time as . (c) Values for G′/G 0(De) and G″/G 0(De) from SAOS simulations and (d) for Q 0(De) from LAOS simulations using the single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, and s a = 3 at various arm numbers, q, over a wide Deborah number range (10−3–103).

Image of FIG. 4.
FIG. 4.

Q 0(De) from LAOS simulations using the single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, and s a = 3 at various arm numbers, q = 2, 5, 6 over a wide Deborah number range. The filled symbols are obtained from simulation as the stretch function λ(t) set equal to 1, i.e., no stretch of backbone by arm polymers. The unfilled symbols are calculated from usual single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, and s a = 3 at various arm numbers, q = 2, 5, 6 like Fig. 3(d) .

Image of FIG. 5.
FIG. 5.

Q 0(De) from LAOS simulations for a single mode pom-pom model at various arm numbers (q) over a wide De range (10−3–103) with (a) s b = 10 and s a = 3, (b) s b = 10 and s a = 5, (c) s b = 25 and s a = 3, and (d) s b = 25 and s a = 4.

Image of FIG. 6.
FIG. 6.

(a) The parameter Q 0,max from LAOS simulations as a function of the number of arms for different arm lengths, s a = 2, 3 and 5, and a fixed backbone length s b = 10. (b) Master curve for Q 0,max as a function of the backbone volume fraction, , at a fixed backbone length s b = 10.

Image of FIG. 7.
FIG. 7.

(a) The master curve for Q 0,max as a function of backbone volume fraction at different backbone length. The modified Q 0,max for comb PS is obtained by multiplying the original experimental results of Hyun and Wilhelm (2009) by a factor 3/2 [ Wagner et al. (2011) ]. The comb PS data (including one linear PS) had backbone lengths of s b ≈ 16, and arm lengths of s a ≈ 0, 0.7, 1.5, and 2.8 and the number of arms given by q ≈ 30. More detailed information can be found in Hyun and Wilhelm (2009) . (b) The master curve of Q 0,max as a function of the effective number of entanglements along the backbone, , displays rather universal behavior.

Image of FIG. 8.
FIG. 8.

Normalized Q 0 (≡Q 0/Q 0,max) as a function of normalized frequency (≡ω/ω max) with varying backbone volume fraction at a fixed backbone length s b = 10. (a) Normalized Q 0 within the region of from 0 to 0.25, i.e., increasing Q 0,max with decreasing . (b) Q 0,max as a function of . (c) Normalized Q 0 within the region of from 0.25 to 1, i.e., decreasing Q 0,max with decreasing .

Image of FIG. 9.
FIG. 9.

(a) Q and (b) normalized Q/Q 0 as a function of strain amplitude at various De numbers for the single mode pom-pom model with s b = 10, s a = 3 and q = 3. (c) Q and (d) normalized Q/Q 0 as a function of strain amplitude at various De numbers for the single mode pom-pom model with s b = 10, s a = 3, and q = 4. (e) Q and (f) normalized Q/Q 0 as a function of strain amplitude at various De numbers for the single mode pom-pom model with s b = 10, s a = 3, and q = 5.

Image of FIG. 10.
FIG. 10.

Q 0 and a contour plot of the normalized Q/Q 0 as a function of De number and strain amplitude. In the contour plot, Q/Q 0 within the “blue” region is smaller than 1, Q/Q 0 within the “green” region is equal to 1, and Q/Q 0 within the “yellow and red” regions is larger than 1. The data were generated using the single mode pom-pom model with s b = 10, s a = 3, and (a) q = 2, (b) q = 3, (c) q = 4 (maximum values of Q/Q 0 ≈ 1.9) and (d) q = 5 (maximum values of Q/Q 0 ≈ 19 in the red region).

Image of FIG. 11.
FIG. 11.

The closed loop of stress as a function of strain (top, red color) and stress as a function of strain rate (bottom, blue color) at a strain amplitude of γ 0 = 10 and as a function of De number as well as Q 0 (circle plot) and Q at γ 0 = 10 (rectangular plot) as a function of De number for the single mode pom-pom model with (a) s b = 10, s a = 3 and q = 2, (b) s b = 10, s a = 3 and q = 5, and (c) s b = 10, s a = 3 and q = 7. The scaling range for Q 0 is the same for (a), (b), and (c).

Tables

Generic image for table
TABLE I.

The values of several times scales of Fig. 3(d) (the single mode pom-pom model with G 0 = 1 Pa, τ b = 1 s, s b = 10, and s a = 3 at various arm numbers, q).

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2012-10-03
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Numerical simulation results of the nonlinear coefficient Q from FT-Rheology using a single mode pom-pom model
http://aip.metastore.ingenta.com/content/sor/journal/jor2/57/1/10.1122/1.4754444
10.1122/1.4754444
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