Surface Excess Energy Governs the Non-Monotonic Behavior of Active Diffusivity with Activity

TL;DR

This work presents a thermodynamically grounded framework that links interfacial surface energy to the enhanced diffusion of chemically active particles. By combining non-dissipative (surface-tension gradient) and dissipative (entropy production) perspectives, the authors derive an explicit dependence of active diffusivity on an interfacial surface energy $E_s^{(e)}$, and validate the theory against experiments on nanoscale Janus particles and enzyme-functionalized vesicles. The analysis reveals that surface excess energy contains multiple contributions (enthalpic, entropic, electrostatic, and thermophoretic) that can combine nonlinearly to yield non-monotonic diffusion with activity, with self-electrophoresis playing a key quadratic role in several regimes. The results provide a unifying framework for predicting and tuning mobility in synthetic active matter by controlling interfacial energy generation and dissipation.

Abstract

Self-propulsion of particles is typically explained by phoretic mechanisms driven by externally imposed chemical, electric, or thermal gradients. In contrast, chemical reactions can enhance particle diffusion even in the absence of such external gradients. We refer to this increase as active diffusivity, often attributed to self-diffusiophoresis or self-electrophoresis, although these mechanisms alone do not fully account for experimental observations. Here, we investigate active diffusivity in catalytic Janus particles immersed in reactive media without imposed gradients. We show that interfacial reactions generate excess surface energy and sustained interfacial stresses that supplement thermal energy, enabling diffusion beyond the classical thermal limit. We consistently quantify this contribution using both dissipative and non-dissipative approaches, assuming that the aqueous bath remains near equilibrium. Our framework reproduces experimentally observed trends in diffusivity versus activity, including the non-monotonic behaviors reported in some systems, and agrees with data for nanometric Janus particles catalyzing charged substrates as well as vesicles with membrane-embedded enzymes driven by ATP hydrolysis. These results demonstrate that chemical reactions can induce and sustain surface-tension gradients and surface excess energy, providing design principles for tuning mobility in synthetic active matter.

Figures (5)

• Figure 1: Illustration of a catalytic Janus particle undergoing a first-order reaction at its interface, where substrate $M$ is converted into product $N$ at rate $\dot{r}$ producing heat $\dot{r}\Delta H_r$, and thereby inducing an excess surface energy $E_{s}^{(e)}$. Here, the particle is depicted with an orientation $\mathbf{n}$, extending from the catalytic (golden color) to the non-catalytic side (grey color). The interface region $i$ is located between the inner section of the particle $p$ and the surrounding fluid $b$. From the interface, the heat flux $\dot{Q}_c$ is transferred to the fluid, while $M$ is being absorbed from the fluid. The surface of the particle is parametrized by the polar $\theta$ and azimuthal $\phi$ angles.
• Figure 2: Surface fields. a) Dimensionless substrate concentration $\hat{C}_M = C_M/C_0$, b) Dimensionless product concentration $\hat{C}_N=C_N/C_0$, c) Dimensionless temperature $\hat{T}=T/T_0$, d) Dimensionless electrostatic potential $\hat{\psi}=\psi q_0/k_BT$. These results are obtained for $\alpha^2 = 10^{-2}$, $\beta^2=10^{-3}$, $\lambda^2 = 10^{-7}$, $\omega^2=10^{-5}$. Particle dimensions are also dimensionless, $\hat{r}=r/R$.
• Figure 3: Dimensionless surface excess energy as a function of the mean substrate concentration. (a) Left y-axis: Contribution from variations in substrate and product concentrations: $\hat{E}_{s,C_i}^{(e)} = \frac{1}{4\pi R^2} \int_S \nabla_S \hat{C}_i \, dS$. Right y-axis: Contribution from temperature variations: $\hat{E}_{s,T}^{(e)} = \frac{1}{4\pi R^2} \int_S \nabla_S \hat{T} \, dS$. (b) Left y-axis: Contribution from the electrostatic potential: $\hat{E}_{s,\psi}^{(e)} = \frac{1}{4\pi R^2} \sum_i z_i \int_S \nabla_S \hat{\psi} \hat{C}_i \, dS$. Right y-axis: Entropic contribution from variations in substrate and product concentrations $\hat{E}_{s,S}^{(e)} = \frac{1}{4\pi R^2} \sum_i \int_S \ln{\hat{C}_i}\nabla_S \hat{C}_i \, dS$
• Figure 4: Non-dissipative and dissipative surface excess energy and active diffusivity of nanometric Janus particles as a function of the average reaction rate $\dot{r} = k_r\langle C_M \rangle$[nM/s]. (a) Negative surface excess energy $-E_s^{(e)}/k_BT$ computed from Eq.(\ref{['Excess']}). (b) Dissipative surface excess energy $E_{s}^{(d)}/k_BT$ computed from Eq.(\ref{['Diss_Excess']}). (c) Active diffusivity $D$ computed from Eq.(\ref{['Diff']}) (continuous black line), active diffusivity computed from dissipative approach $D^{(d)}$ from Eq. (\ref{['D_d']}) (dashed dark grey line) whereas the black dots with the error bars represent the experimental data Qin2017_self-thermo, all following a quadratic dependence with $\dot{r}$. The dotted light grey line corresponds to the approximation proposed in Ref. Qin2017_self-thermo, which considers only the self-thermophoretic effect, and follows a linear dependence on $\dot{r}$. The model results are shown for $1\times10^{-19}\le\lambda^2\le1.13\times10^{-17}$, $3\times10^{-4}\le\xi^2\le3.29\times10^{-2}$ and $\tau = (k_r\beta^2)^{-1}$.
• Figure 5: Non-dissipative and dissipative surface excess energy and active diffusivity of catalytic liposomes as a function of the average reaction rate $\dot{r} = k_r\langle C_M \rangle$[nM/s]. (a) Negative surface excess energy $-E_s^{(e)}$[J] computed from Eq.(\ref{['Excess']}). (b) Dissipative surface excess energy $E_{s}^{(d)}/k_BT$ computed from Eq.(\ref{['Diss_Excess']}).(c) Active diffusivity $D$ computed from Eq.(\ref{['Diff']}) (continuous black line), active diffusivity computed from dissipative approach $D^{(d)}$ from Eq. (\ref{['D_d']}) (dashed dark grey line) whereas the black dots with error bars correspond to experimental data from Ref.Ghosh2019. The dotted light grey line corresponds to a linear approximation without considering entropic effects. For $1\times10^{-15}\le\lambda^2\le2\times10^{-13}$,$3\times10^{-2}\le\xi^2\le2.9$ and $\tau=k_r^{-1}$.