Structuring of polymer solutions upon solvent evaporation

The morphology of solution-cast, phase-separated polymers becomes finer with increasing solvent evaporation rate. We address this observation theoretically for a model polymer where demixing is induced by steady solvent evaporation. In contrast to what is the case for a classical, thermal quench involving immiscible blends, the spinodal instability initially develops slowly and the associated length scale is not time invariant but decreases with time as $t^{-1/2}$. After a time lag, phase separation accelerates. Time lag and characteristic length exhibit power-law behavior as a function of the evaporation rate with exponents of $-2/3$ and $-1/6$. Interestingly, at later stages the spinodal structure disappears completely while a second length scale develops. The associated structure coarsens but does not follow the usual Lifshitz-Slyozov-Wagner kinetics.

Figure 1

Evaporation-induced structure development in a polymer solution, for interaction parameter $\chi =1.0$, degree of polymerization, $N=10$, and evaporation rate $\tilde{\alpha }=5.5\times {10}^{-4}$ scaled to the diffusion time scale. The color (gray scale) coding indicates differences in polymer volume fraction $\varphi$. At a lag time $t={\tau }_{\mathrm{L}}$ (see main text), a phase-separated structure becomes discernible (see also Figs. 2 and 3).

Figure 2

The volume fraction, $\varphi$, indicated on the left vertical axis in green (light gray) and dimensionless wave number, $\tilde{q}$, indicated on the right in red (dark gray), as a function of time $t$ relative to the lag time ${\tau }_{\mathrm{L}}$ for the parameter values in Fig. 1. The spinodal is crossed at $t=0$, for $t/{\tau }_{\mathrm{L}}\approx 1$ the binodal concentrations are approached (solid circles), and the structure coarsens. At $t/{\tau }_{\mathrm{L}}\approx 4.8$ the high-concentration branch of the binodal is reached by evaporation and the polymer redissolves. The morphology is initially characterized by a wave number (solid triangles) predicted from linearized theory in Eq. (3) (line). After the lag time a new length scale appears (open triangles).

Figure 3

Structure factor, $S\left(q\right)$, for different times running from $t/{\tau }_{\mathrm{L}}=0$ (light grey) to $t/{\tau }_{\mathrm{L}}=4.5$ (black) corresponding to Figs. 1 and 2. The inset highlights the subsequent appearance and disappearance of two structure peaks.

Figure 4

The dimensionless predominant wave number ${\tilde{q}}_{*}$ (main figure) and lag time ${\tilde{\tau }}_{\mathrm{L}}$ (inset) as a function of the dimensionless evaporation rate $\tilde{\alpha }$ for various interaction parameter values $\chi$ and degree of polymerization, $N$. The numerical (symbols) results are curve fitted from linearized theory (solid lines) and approach limiting power laws (dashed lines) for slow evaporation.