Dynamics of critical fluctuations in polymer solutions

Using dynamic light scattering we have investigated the time dependence of fluctuations near the critical point of phase separation in solutions of polystyrene in cyclohexane with polymer molecular weights ranging from $196000\text{to}11.4\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$. At the lowest polymer molecular weight the dynamic correlation function follows a single-exponential decay with a decay rate that can be represented by the mode-coupling theory of critical dynamics but with a mesoscopic viscosity that characterizes the hydrodynamic environment of the polymers in the solution. At all higher polymer molecular weights two distinct dynamic modes are observed, a slow and a fast mode, that originate from a coupling of the critical concentration fluctuations with viscoelastic relaxation of the polymer chain in solutions. This coupling causes an additional slowing down of the fluctuations on top of the well-known critical slowing down expected in the absence of a coupling between the two modes. From an analysis of the time dependence of the experimental dynamic correlation functions in terms of a theory of coupling of dynamic modes we are able to determine the viscoelastic properties of the polymers in the solution. These viscoelastic properties diverge in the theta-point limit of infinite polymer molecular weight.

Figure 1

Schematic representation of the light beams and the optical cell. The laser beams are opened in sequence by computer-controlled shutters.

Figure 2

Normalized intensity correlation function $g_2\left(t\right)-1$ for sample PS1 with a polystyrene molecular weight of $M_{\mathrm{w}}=195900\mathrm{g}{\mathrm{mol}}^{-1}$ at $T-T_{\mathrm{c}}=0.02\mathrm{K}$, plotted as a function of the delay time $t$. The data have been obtained at a scattering angle $\theta =90°$ $(q=0.0197\times {10}^{-2}{\mathrm{nm}}^{-1})$. The symbols represent the experimental data; the solid curve represents a fit to $g_2\left(t\right)-1=B_0\exp (-2t∕{\tau }_q)$ with ${\tau }_q=5.2\mathrm{ms}$ [47]. For convenience of representation here and in the following figures the correlation functions $g_2\left(t\right)-1$ have been normalized so that $B_0=1$.

Figure 3

Diffusion coefficient $D\left(q\right)$ of a solution (sample PS1) of polystyrene $(M_{\mathrm{w}}=195900\mathrm{g}{\mathrm{mol}}^{-1})$ in cyclohexane as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$. The symbols represent the experimental data obtained at three scattering angles: $\theta =30°$: open squares, $\theta =150°$: open circles, and $\theta =90°$: crossed squares. The curves represent the values calculated from the mode-coupling theory with an apparent (mesoscopic) viscosity represented by the solid curve in Fig. 4.

Figure 4

Apparent (mesoscopic) viscosity of a solution (sample PS1) of polystyrene $(M_{\mathrm{w}}=195900\mathrm{g}{\mathrm{mol}}^{-1})$ in cyclohexane as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$ obtained by fitting the experimental light-scattering data to the theoretical prediction of the mode-coupling theory of critical dynamics. The data for the three scattering angles are represented by the same symbols as in Fig. 3. The solid curve is a fit to the data. The dashed curve represents the viscosity of the solution [14] and the dotted curve represents the viscosity of the solvent (cyclohexane) [54].

Figure 5

Normalized intensity correlation functions $g_2\left(t\right)-1$ as a function of the delay time $t$ for sample PS5 with a polymer molecular weight of $M_{\mathrm{w}}=11.4\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$ at various values of the dimensionless temperature difference $\epsilon =(T-T_{\mathrm{c}})∕T$. (a) $\theta =30°$, and (b) $\theta =150°$. From left to right the value of $\epsilon$ associated with the curves decreases from ${10}^{-1}$ to ${10}^{-5}$.

Figure 6

(a) Three-dimensional equal-area representation of the Laplace inversions $H\left(\tau \right)$ as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$ for sample PS5 with a polymer molecular weight of $M_{\mathrm{w}}=11.4\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$ at $\theta =30°$. (b) Two-dimensional projection of the decay-time distributions $H\left(\tau \right)$ in a grayscale indicating the magnitude of $H\left(\tau \right)$. The solid curves represent the relaxation times ${\tau }_{+}$ and ${\tau }_{-}$ of the two coupled modes calculated from Eq. (12). The dashed curve represents the theoretical prediction for the original diffusive relaxation time ${\tau }_q$ of the critical concentration fluctuations as calculated from Eq. (17); the dotted curve represents the theoretical prediction for the original relaxation time ${\tau }_{\mathrm{ve}}$ of the viscoelastic mode of the polymers in the solution as calculated from Eq. (16).

Figure 7

Upper part of figure: $g_2\left(t\right)-1$ as a function of the delay time $t$ for sample PS5 with a polystyrene molecular weight of $M_{\mathrm{w}}=11.4\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$ at $\theta =30°$ and $T-T_{\mathrm{c}}=0.00254\mathrm{K}$. The symbols indicate the experimental data. The dotted curve represents single-exponential decay. The solid curve represents the values calculated from Eq. (15). Lower part of figure: Symbols show deviations of the experimental data from a sum of two stretched exponentials as given by Eq. (15); dashed curve shows deviations of the experimental data from a sum of two exponentials as given by Eq. (11).

Figure 8

Exponents ${\beta }_{+}$ (crossed circles) and ${\beta }_{-}$ (open circles) as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$ for sample PS5 at $\theta =150°$, indicating the distribution of decay times for the coupled fast and slow modes, respectively, when the correlation-function data are fitted to Eq. (15) as shown in Fig. 7.

Figure 9

Amplitudes $f_{+}$ and $f_{-}$ indicating the intensity of the coupled fast and slow modes as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$ for sample PS5 at $\theta =150°$. The symbols represent the values of $f_{+}$ (crossed circles) and $f_{-}$ (open circles) when the correlation-function data are fitted to Eq. (15). The solid curves represent the values calculated from Eq. (13).

Figure 10

Distributions of the relaxation times as a function of $\epsilon =(T-T_{\mathrm{c}})∕T$ for sample PS5 with a polystyrene molecular weight of $M_{\mathrm{w}}=11.4\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$ at $\theta =30°$ (a) and $\theta =150°$ (b). The grayscale represents the magnitude of the distribution as in Fig. 6b. The symbols represent the values for the two coupled modes extracted from a fit of the correlation-function data to Eq. (15). The solid curves represent the relaxation times ${\tau }_{+}$ and ${\tau }_{-}$ of the two coupled modes calculated from Eq. (12). The dashed curves represent the theoretical predictions for the original diffusive relaxation time ${\tau }_q$ of the critical concentration fluctuations as calculated from Eq. (17) and the dotted curves the theoretical predictions for the original relaxation time ${\tau }_{\mathrm{ve}}$ of the viscoelastic mode of the polymers in the solution as calculated from Eq. (16).

Figure 11

Same as Fig. 10, but for sample PS4 with a polystyrene molecular weight of $M_{\mathrm{w}}=3.90\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$.

Figure 12

Same as Fig. 10, but for sample PS3 with a polystyrene molecular weight of $M_{\mathrm{w}}=1.95\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$.

Figure 13

Same as Fig. 10, but for sample PS2 with a polystyrene molecular weight of $M_{\mathrm{w}}=1.124\times {10}^6\mathrm{g}{\mathrm{mol}}^{-1}$. Fit to a single stretched exponential: diamonds.

Figure 14

Viscoelastic length ${\xi }_{\mathrm{ve}}$ (crossed circles, left axis), viscoelastic relaxation time ${\tau }_{\mathrm{ve},\mathrm{c}}\simeq {\tau }_{\mathrm{ve}}$ (solid circles, left axis), and the mesoscopic viscosity ${\eta }_0$ (open circles, right axis) in terms of the solvent viscosity ${\eta }_{\mathrm{s}}$ as a function of the polymer molecular weight $M_{\mathrm{w}}$. The dashed lines show slopes of 0.51, 0.64, and 1.3, respectively. The square indicates the macroscopic viscosity of the polymer solution with $M_{\mathrm{w}}=195900\mathrm{g}{\mathrm{mol}}^{-1}$ as reported by Lao et al. [14].