Thermodynamics and statistical mechanics of chemically powered synthetic nanomotors

Figures & data

Figure 1. Relative concentration profiles of the fuel A along the axis $z$ of the Janus particle for three values of the Damköhler number. C and N denote the catalytic and non-catalytic hemispheres, respectively. The profiles are obtained for ${\bar{c}}_{\mathrm{B}}=0$.

Figure 2. Streamlines of the velocity field (22) around a Janus particle in the laboratory frame where the particle moves with the velocity $V_{\mathrm{s}\mathrm{d}}=2\mathrm{{\rm Y}}{a}_1/3$ for $\mathrm{R}\mathrm{e}\simeq 0.013$, $\mathrm{P}\mathrm{e}\simeq 0.17$, and $\mathrm{D}\mathrm{a}\simeq 6.2$ from [56]. The vertical axis $z$ corresponds with $\mathbf{u}$ of the Janus particle. The upper hemisphere is catalytic. The motor here behaves as a pusher with $\mathrm{{\rm Y}}{a}_2<0$. The blue arrow denotes the direction of particle motion and black arrows the fluid flow.

Figure 3. Janus particle subjected to an external force and magnetic field oriented in the $z$-direction: The random rescaled displacement $z_{\ast }=z\sqrt{D_{\mathrm{r}}/D_{\mathrm{t}}}$, the number of reactive events $n_{\ast }=n\sqrt{D_{\mathrm{r}}/D_{\mathrm{r}\mathrm{x}\mathrm{n}}}$, and the particle orientation $u_z$, versus the rescaled time $t_{\ast }=D_{\mathrm{r}}t$ for parameter values $\beta \mu B=2$, $W_{\mathrm{r}\mathrm{x}\mathrm{n}}/\sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}{D}_{\mathrm{r}}}=0.8$, and $\chi \sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}/D_{\mathrm{t}}}=0.8$ and (left) a zero external force, and (right) a rescaled external force equal to $f=\beta F\sqrt{D_{\mathrm{t}}/D_{\mathrm{r}}}=-2.5$.

Figure 4. Janus particle subjected to an external force and magnetic field oriented in the $z$-direction [61]: The mean values of the fluctuating rescaled velocities ${\dot{r}}_{\ast }=\dot{r}/\sqrt{D_{\mathrm{t}}{D}_{\mathrm{r}}}$ and rate ${\dot{n}}_{\ast }=\dot{n}/\sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}{D}_{\mathrm{r}}}$ versus the rescaled magnitude of the external force $f=\beta F\sqrt{D_{\mathrm{t}}/D_{\mathrm{r}}}$ for the parameter values $\beta \mu B=2$, $W_{\mathrm{r}\mathrm{x}\mathrm{n}}/\sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}{D}_{\mathrm{r}}}=0.8$, and $\chi \sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}/D_{\mathrm{t}}}=0.8$. The dots show the results of a numerical simulation with a statistics of ${10}^5$ trajectories integrated over the time interval $t_{\ast }=10$. $f_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}}$ denotes the rescaled stall force and $f_0$ the threshold between fuel synthesis and consumption. The Janus particle is propelled against the external force in the interval I. Fuel synthesis happens in the interval II.

Figure 5. Simulations with microscopically reversible kinetics of the Janus motor subjected to an external force $F_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and a magnetic field both oriented in the $z$-direction: Plots of the dependence on the rescaled force $f$ of the rescaled average motor velocity in the $z$-direction, $⟨{\dot{z}}_{\ast }⟩$ (left), and of the rescaled reaction rate, $⟨{\dot{n}}_{\ast }⟩$ (right) for systems with (a) ${\bar{c}}_{\mathrm{A}}=10$ and ${\bar{c}}_{\mathrm{B}}=9$ (circles) and (b) ${\bar{c}}_{\mathrm{A}}=10$ and ${\bar{c}}_{\mathrm{B}}=10$ (squares). The results for (a) and (b) systems were obtained from averages over 200 and 100 realizations of the dynamics, respectively. The fits to the data are indicated by (a) upper and (b) lower lines. See Huang et al. [62], for additional information.

Figure 6. Schematic representation of the different regimes of the active particle in the plane of the mechanical and chemical affinities for $\chi >0$ [53]. In domain I, self-diffusiophoretic mechanical work is powered by the reaction. In domain II, an external force of sufficient magnitude acting in a direction opposite to that of the Janus particle velocity can yield the synthesis of fuel from the product. For $\chi <0$, the slopes of the lines $⟨\dot{z}⟩=0$ and $⟨\dot{n}⟩=0$ are instead positive.

Figure 7. Schematic representation of the probability distributions $P(\pm n,t)$ of opposite fluctuations in the number $n$ of reactive events occurring during the time interval $[0,t]$ under nonequilibrium conditions.

Figure 8. Janus particle subjected to an external force and magnetic field oriented in the $z$-direction [61]: (a) Probability density $P(z_{\ast },n_{\ast };t_{\ast })$ with $n_{\ast }=n\sqrt{D_{\mathrm{r}}/D_{\mathrm{r}\mathrm{x}\mathrm{n}}}=2,1,0,-1,-2$ versus the rescaled displacement $z_{\ast }=z\sqrt{D_{\mathrm{r}}/D_{\mathrm{t}}}$ at the rescaled time $t_{\ast }=D_{\mathrm{r}}t=1$ for the parameter values $\beta \mu B=1$, $\beta F\sqrt{D_{\mathrm{t}}/D_{\mathrm{r}}}=-1$, $W_{\mathrm{r}\mathrm{x}\mathrm{n}}/\sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}{D}_{\mathrm{r}}}=0.4$, and $\chi \sqrt{D_{\mathrm{r}\mathrm{x}\mathrm{n}}/D_{\mathrm{t}}}=0.4$. (b) Verification of the mechanochemical fluctuation relation (46) in the same conditions. The probability ratio is calculated if $P(z_{\ast },n_{\ast };t_{\ast })$ and $P(-z_{\ast },-n_{\ast };t_{\ast })$ are larger than ${10}^{-4}$. The dots are the results of a numerical simulation with an ensemble of ${10}^7$ trajectories and an integration with the time step $d{t}_{\ast }={10}^{-3}$. The lines depict the theoretical expectations.

Figure 9. Time-dependent affinity $A_t$ of the finite-time fluctuation theorem for an immobile Janus particle of radius $R=1$ with surface reaction influenced by the diffusion of fuel and product from a spherical reservoir at the distance $L=10$ from the particle center. The asymptotic value of the affinity is equal to $A_{\mathrm{r}\mathrm{x}\mathrm{n}}=1$ (dotted line). The diffusion coefficients take the value $D=D_{\mathrm{A}}=D_{\mathrm{B}}=1$ and the rate constants $\kappa ={\kappa }_{+}={\kappa }_{-}$. The Damköhler number is given by $\mathrm{D}a=2\kappa R/D$.

Reigh SY, Huang MJ, Schofield J, et al. Microscopic and continuum descriptions of Janus motor fluid flow fields. Phil Trans R Soc A. 2016;374:20160140.

 Google Scholar

Gaspard P, Kapral R. Mechanochemical fluctuation theorem and thermodynamics of self-phoretic motors. J Chem Phys. 2017;147:211101.

 PubMed Web of Science ®Google Scholar

Huang MJ, Schofield J, Gaspard P, et al. Dynamics of Janus motors with microscopically reversible kinetics. J Chem Phys. 2018;149:024904.

 PubMed Web of Science ®Google Scholar

Gaspard P, Kapral R. Fluctuating chemohydrodynamics and the stochastic motion of self-diffusiophoretic particles. J Chem Phys. 2018;148:134104.

 PubMed Web of Science ®Google Scholar