Periodical alternation between precipitation and dissolution of camphor solid film on an ethanol solution

Rhythmic behaviors are generally observed in nonlinear chemical reactions such as the Belousov-Zhabotinsky reaction and enzymatic reactions. Similarly, a simple phase change can also lead to rhythmic behavior. It has been reported previously that camphor solid films alternate between generation and disappearance on ethanol (EtOH) solution, and a phenomenological mechanism has been suggested for this. The evaporation of EtOH decreases the temperature on the surface of the solution via vaporization heat and induces precipitation in the camphor solid film. At this time, the film prevents evaporation, and thus, the surface temperature increases due to thermal diffusion from the atmosphere, resulting in dissolution of the solid film. To verify the previously suggested phenomenological mechanism, we controlled the evaporation rate of EtOH using a porous plastic cover. As a result, the period of oscillation increased with decreasing pore diameter, and finally, the oscillation did not occur without pore in the cover, where the camphor solid film was not observed. Additionally, a new mathematical model was proposed, and the numerical calculations agreed well with experimental observations. Linear stability and bifurcation analyses revealed the detailed mechanism of this phenomenon, which agreed well with the phenomenological explanation mentioned above.

Figure 1

Experimental setup and results on periodic alternation between precipitation and dissolution of camphor solid film. (a) Illustration of the experimental setup. A saturated EtOH solution of camphor was poured into a plastic Petri dish and was covered with a plastic plate. The plastic plate had pores at equal intervals (3.0 mm), and the diameter $d$ of the pores was changed from 0.0 to 2.5 mm. (b) Time series snapshots of camphor EtOH solution without the plastic cover. The solution became bright or dark with the generation or dissolution of the camphor film, respectively. (c) Profiles of the averaged brightness of the camphor EtOH solution. $t=0$ is the starting time of video recording, which was about 10 s after pouring the solution in the Petri dish. (d) Periods depending on the pore diameter $d$. Here, “open” means that no cover was used. The periods were estimated by averaging the initial five alternations. The error bar indicates the standard deviation of the five data points.

Figure 2

Illustration of the setup of our mathematical model.

Figure 3

Numerical result obtained from the three-variable ODE model in Eq. (4). The parameters are $\varepsilon =6.0\times {10}^{-4}, \kappa =$ (a) 2.0 and (b) 1.4, $\sigma =2.4, \eta =1.0\times {10}^{-4}, \delta =4.0\times {10}^{-4}, \alpha =0.1$, and $\beta =0.015$. The alternation between precipitation and dissolution is well-reproduced and the stability of the oscillation depended on the parameters, which is (a) stable and (b) unstable for long time. In the case of (b), the period gradually increases with time. This behavior agrees with the experimental observation, as shown in Fig. 1.

Figure 4

Numerical results obtained from the two-variable ODE model in Eq. (5). The blue and orange lines indicate the precipitate ($p$) and temperature ($u$), respectively. The alternation between precipitation and dissolution is well-reproduced and its period is constant with time. The parameters are $\varepsilon =6.0\times {10}^{-3}, \kappa =2.0, \sigma =2.4, \eta =1.0\times {10}^{-4}, \alpha =0.1$, and $\beta =$(a) 0.0120, (b) 0.0145, (c) 0.0150, and (d) 0.1200. (e) Phase diagram with change in the values of $\beta$ and $\sigma$. Green, blue, and orange regions indicate zero and nonzero $p$ of steady state and oscillatory state, respectively. (f) Amplitudes and (g) periods of oscillation of $p$ depending on the value of $\beta$.

Figure 5

(a) Nullclines obtained from $dp/dt=0$ (blue lines) and $du/dt=0$ (orange line). (b) Phase diagram with (i) $\sigma$ and $\beta$ and (ii) $\sigma$ and $\kappa$. The background color indicates the state obtained from numerical calculations [Fig. 4], where the blue, green, and the orange areas indicate the steady with nonzero $p^{*}$, the steady with zero $p^{*}$, and the oscillatory states, respectively. The orange line indicates $\sigma =1+\beta /{\alpha }^2$, where the transcritical bifurcation of EP1 occurs. The blue line is $\sigma =\kappa {(\kappa +1)}^2({\alpha }^2+\beta +{(\kappa -1)}^2/{\left(2\kappa \right)}^2)/(4{\kappa }^2{\alpha }^2+{(\kappa -1)}^2)$. It indicates the condition where EP3 is on the local maximum point of $p$-nullcline represented in Eq. (7). Other parameters are $\kappa =2.0$ and $\alpha =0.1$.

Figure 6

Bifurcation diagram with $\sigma =2.4$. The blue line indicates EP1, orange line indicates EP2, and green line indicates EP3. The inset is a magnified diagram. The solid and broken lines indicate the stable and unstable branches, respectively. Other parameters are $\kappa =2.0$ and $\alpha =0.1$.

Figure 7

Results of the bifurcation analysis with $\sigma =2.4$. (a) The equilibrium points $(p^{*},u^{*})$ which are plotted (i) against $\beta$ and (ii) in $p\text{−}u$ space (gray line). The background color of blue and red in (i) correspond to stable and unstable states, respectively. Similarly, solid and broken lines in (ii) indicate stable and unstable states, respectively. The yellow line in (ii) is nullclines. (b) The $\beta$ dependency of (i) real and (ii) imaginary parts of the eigenvalues $\lambda$. The insets are magnified diagrams in the $\beta$ range from 0.012 to 0.015. The other parameters were $\varepsilon =6.0\times {10}^{-4}, \kappa =2.0, \eta =1.0\times {10}^{-4}$, and $\alpha =0.1$.