The universal dependence of the reduced polydispersity index as a function of the reduced time τ = t/(t 1 N 2) [following an infinitesimal T-jump from N 0 = N(1 + ε) to N ∞ = N].
(a) The dependencies of the reduced mean chain lengths, N n /N and N w /N, on the reduced time τ = t/(t 1 N 2) for a strong T-jump leading to a significant decrease of the mean equilibrium chain length from N 0 = 100 N to N ∞ = N (α = 0.01). Mean lengths N n and N w are calculated using Eqs. (69), (70), and (67). Inset: the τ-dependence of the polydispersity index r = N w /N n . (b) The CLDs following the same T-jump as in (a): the mean chain length changes from N 0 = 100N (curve “0”, the CLD right upon the jump, at t = 0) to N (curve “∞”, t = ∞). Curve “*” corresponds to the peak of the polydispersity index r at τ = τ* ≈ 16.7. It is drawn according to Eqs. (65) and (67) with α = 0.01 at τ = τ*).
Chain length relaxations on a small T-jump, ε ≪ 1. Solid curves: the dependencies of the reduced perturbations of number and weight average chain lengths, (line n) and (line w), calculated using analytical equations (48), (49), (40), and (41) on the reduced time t/t f for N = 5 (t f = t 1/N). Dashed lines: the dependencies of the same quantities obtained numerically in Ref. 8 by solving master kinetic equations (as shown in Figs. 4 and 5 of Ref. 8).
The universal dependence of the reduced unimer concentration perturbation η = (N/ε)(1 − c 1(t)/c 1(∞)) for t ≫ t 1 after an infinitesimal T-jump (from N 0 = N(1 + ε) to N ∞ = N) on the reduced time τ = t/(t 1 N 2). Comparison between the analytical prediction of the present paper, the function η = η n (τ), Eq. (40) (solid line), and the dependence , Eq. (75), deduced from Eq. (32) of Ref. 24 (dashed line). Inset: the time dependence of the ratio .
The dependence of the reduced free unimer concentration c 1/c 1(∞) on the reduced time for N = 50, N 0 = 54 (ε = N 0/N − 1 = 8%). Solid curve—analytical (see equation in Ref. 37); dots— numerical results of Ref. 24 (see Fig. 3). (The right vertical axis shows the arbitrary units used in Fig. 3 of Ref. 24. The characteristic times corresponding to t f = t 1/N, t 1 and t s = 4N 2 t 1 are marked with vertical dashed green lines: f, 1, and s, respectively.)
The dependence of the reduced chain length, N n , on the reduced time for N = 50, N 0 = 54. Thick black solid curve: analytical prediction, Eq. (48); dots: numerical results of Ref. 24 (see their Fig. 5); thin purple solid curve: analytical results of Ref. 24 (see their Fig. 5). Vertical dashed green lines mean the characteristic times as in the previous figure.
The dependence of ln (1 − N n (t)/N) vs t for N 0 = 2, N = 12. Numerical results (black thick solid curve); asymptotic behavior according to Eq. (76) (red thin curve); the power law, N − N n (t)∝1/t proposed in Ref. 23 (blue dashed curve). The colored curves are shifted along the vertical axis to achieve the best agreement.
The numerically obtained dependence of N − N n (t) vs t for N 0 = 2, N = 12. Vertical dashed green lines indicate the characteristic times (as in Fig. 5) and show the cross-overs between four characteristic regimes of relaxation (note that here t f /t 1 ≃ N 0/N 2 ≈ 0.014, t s /t 1 ≃ 4N 2 ≈ 576 and t 1 = 1).
The longest relaxation time t max = 1/γ vs N: solid curve—Eq. (77), red squares—our numerical results for relaxation after a jump from N 0 = 2 to N = 6, 8, 10, 12.
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