Figure 1

Numerical results for response of CLD to a small $T$-jump perturbation of amplitude $ϵ=0.1$ perturbing $T$-dependent parameters $v^{-}$ and $k^{+}$ in Eq. (1). Final equilibrium exponential (curve f, ${\overline{N}}_{eq}=1366$, $\theta =0.37$) is close to the initial equilibrium exponential (curve i). Along its path $i\to f$, the CLD deviates nonlinearly from both. Times in units of $v^{-}$ (for PAMS [1], $v^{-}\approx 0.1\mathrm{s}$). Time scales: $t^{*}\approx 400$, ${\tau }_{\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t}}\approx 3640$, ${\tau }_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}\approx 1.32\times {10}^5$, ${\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}\approx 1.86\times {10}^6$. Curve a, $t=3640$: A hole has appeared at small $N$. Curve b, $t=3\times {10}^4$: hole filling. Curve c, $t=2.3\times {10}^5$: final shape relaxation towards equilibrium, curve f. Inset: Relaxation of fast and slow variables. Arrows from left to right indicate $t^{*}$ ${\tau }_{\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t}}$, ${\tau }_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}$, and ${\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}$. Fast variables $\delta m_t/m_{eq}=-\theta \delta {\overline{N}}_t/{\overline{N}}_{eq}$ (solid line) decay exponentially after ${\tau }_{\mathrm{f}\mathrm{a}\mathrm{s}\mathrm{t}}$, and are then enslaved to the slow variable $-\delta {\varphi }_t(0)$ (dashed line). Note initial nonlinear increase of $-\delta {\varphi }_t(0)$ after $t^{*}$. Theory predicts $\delta {\varphi }_t(0)\sim t^{-1/2}$ during ${\tau }_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}<t<{\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}$ and $\delta {\varphi }_t(0)\sim e^{-t/{\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}}$ for $t>{\tau }_{\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{w}}$.Reuse & Permissions

Figure 2

A living polymer is an initiator (square) plus $N$ monomers. Monomers can add (subtract) from its live end (filled circle) at rates $v^{+}$ ($v^{-}$). The initiators of Refs. 5, 7 are bifunctional (one interior initiator plus two live ends).Reuse & Permissions

Figure 3

Viscosity relaxation measurements by Ruiz-Garcia and Greer [7] following small $T$-jumps ($|\Delta T|\approx 1°\mathrm{K}$) on PAMS in tetrahydrofuran initiated by sodium naphthalide. Reproduced from Fig. 1 of Ref. 7. Circles (triangles) refer to positive (negative) $\Delta T$. $m_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=0.29\mathrm{g}\mathrm{m}/{\mathrm{c}\mathrm{m}}^3$, ${\rho }_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{s}}\approx 2.7\times {10}^{-4}\mathrm{g}\mathrm{m}/{\mathrm{c}\mathrm{m}}^3$ (semidilute conditions).Reuse & Permissions