Brownian Ratchets and Diffusion Pumps.

The Brownian ratchet is a device composed of a saw-tooth wheel connected to a pawl mechanism that allows it rotate in only one direction. Introduced by Smoluchowski and popularized by Feynman, it is striking because a sufficiently small version of the device would seem to transform the kinetic energy of random thermal fluctuations into unidirectional rotational motion of the wheel that can do work. This would violate the Second Law of Thermodynamics and careful analysis indeed shows that in actuality the device cannot do work. The – by now – numerous experimental realizations of Brownian ratchets invariably involve some form of irreversible consumption of free energy. For example, In this letter we focus on a Brownian ratchet type diffusion pump that seemingly operates without free energy consumption.

The system is a polymer gel with a solvent that is a binary fluid. The majority component of the fluid is a good solvent for the gel while the minority component is a poor solvent. The solvent is initially in a homogeneous equilibrium state that is transformed into a supersaturated solution by a temperature jump. Minority phase droplets start to form by thermal activation. If the radius of such droplet exceeds the critical radius, which is proportional to the surface tension and inversely proportional to the degree of supersaturation, then the droplet size grows by diffusive transport while droplets with radii less than the critical radius disappear, thereby releasing minority phase molecules that will diffuse away. Liquid-liquid phase separation in binary liquids wei2020modeling normally is characterized by Oswald ripening, or “coarsening”, which refers to the growth of larger droplets of the minority phase and a corresponding shrinkage of smaller droplets, which is driven by capillary action. Oswald ripening is not observed in liquid-liquid phase-separation in gels. Instead, a stationary dispersion is observed composed of droplets of various sizes. This is attributed to the fact that both the cytoplasm of cells and the interior of the cell are permeated by a network of filamentous proteins such as actin, microtubules or DNA/nucleosome fibers that prevent the droplets from growing.

Cellular phase separation events are complex but the same phenomenon has been observed in a well-characterized model system composed of a polymeric gel in a solvent that is a binary liquid. The majority phase of the binary liquid is a good solvent for the gel polymers and the minority phase is a poor solvent. The solvent is initially in a homogeneous mixed state that is thermodynamically stable. After a discontinuous drop in temperature (or composition?), the homogeneous mixed state turns metastable. Minority phase droplets can form by thermal activation. If the radius of the droplet exceeds a critical radius, which is proportional to the surface tension and inversely proportional to the degree of supersaturation, then the droplet size can grow by diffusive transport while droplets with radii less than the critical radius disappear, thereby releasing minority phase molecules that will diffuse away. Assume the gel is placed in a solvent reservoir that is in a homogeneous metastable phase. If the mesh-size of the gel exceeds the size of the critical radius then thermally activated minority phase droplets can grow to at least the mesh-size. Further growth generates elastic deformation energy of the gel that eventually stops the growth of the droplet. The chemical potential of these minority phase droplets is the same as that of the reservoir while the free energy of the drop in the stressed polymer mesh can be lower or higher than that of the relaxed mesh without the droplet. Thermal fluctuations of such a “restrained” drop can cause the drop the change location to an adjacent mesh but it can also cause it to disappear. Attach an Ising variable to each mesh of the gel with “spin down” indicating the absence of a drop inside the mesh location and “spin up” the presence of a drop. Diffusive transport inside the gel drives minority phase transport from mesh locations where a drop has just disappeared to a mesh site where the drop is growing. The statistical physics of such a system could be reasonably described by an Ising model in a magnetic field and does not appear to be exceptional. But now assume that the gel is a sufficiently small system so that it contains, on average, either no drop or only one drop. When a droplet appears somewhere in that system, droplet growth requires diffusive transport of minority phase molecules from the reservoir outside the gel to the gel. When a droplet disappears, the diffusive transport would be reversed. It would seem that there is an alternating diffusion current to and from the gel system, somewhat like a diffusive form of breathing. Moreover, as discussed below, the disappearance of a drop involves momentum exchange between the drop and the reservoir, which might seem to allow for active Brownian motion of the gel system in the reservoir. The metastable reservoir could be viewed as a store of free energy so the First Law of Thermodynamics is not violated if this stored free energy is used to power the alternating diffusion current and active Brownian motion. In that case, however, the dynamics of the system should violate the fluctuation-dissipation theorem. In the next section we develop a simplified pair of coupled equations that describes the thermal fluctuations of the radius and location of the minority phase droplet. It allows us to check whether or not the fluctuation-dissipation theorem is violated.

First consider the formation of minority-phase droplets without the gel. The minority phase molecules are assumed to form an ideal solution in a solvent of majority phase molecules. They have a chemical potential

$$\mu=k_{B}T\ln\left(\frac{c}{c_{0}}\right)$$

(4)

with $c_{\infty}=N/A$ the area density of the minority phase (A is the area of the system and $N$ is the total number of minority phase molecules). The solvation concentration $c_{s}$ is defined as the solution concentration at which the minority phase molecules would be in thermodynamic equilibrium with a pure liquid of minority phase molecules. The chemical potential $\mu_{s}$ of minority phase molecules in the liquid state is then

$$\mu==k_{B}T\ln\left(\frac{c_{s}}{c_{0}}\right)$$

(5)

The Gibbs free energy of a two-dimensional circular droplet of radius $R$ is then

$$\begin{split}\Delta G(R)=&\pi R^{2}c_{l}(\mu-\mu_{s})+2\pi R\tau\\ &=\pi R^{2}c_{l}k_{B}T\ln\left(\frac{c_{\infty}}{c_{s}}\right)+2\pi R\tau\end{split}$$

(6)

with $c_{l}$ the area density of liquid state molecules and with $\tau$ the line tension of the drop. The critical radius of the drop corresponds to the maximum of $\Delta G(R)$ at

$$R^{*}=\tau/\left(c_{l}k_{B}T\ln\left(\frac{c_{\infty}}{c_{s}}\right)\right)$$

(7)

The gel is represented as a two-dimensional potential energy landscape $u(\vec{x})$ with a distribution of maxima. The potential energy of a drop with center at $\vec{r}$ is then

$$U(\vec{r})=\int_{|\rho|<R}{d^{2}}\vec{\rho}\medspace u(\vec{r}+\vec{\rho})$$

(8)

Transport is by diffusion where the diffusion flux is $\vec{J}$ equals $-n\nu\nabla\mu$ with $\nu$ the mobility. This relation is actually the diffusion equation:

$$\partial_{t}n=-\nabla\vec{J}=D\nabla^{2}n$$

(9)

in two-dimensions. The solution will be:

$$n(r)=Alnr+B$$

(10)

where $A$ and $B$ are undetermined constants.

Now we focus on the equilibrium condition of the diffusion. In this condition the chemical potential of minority phase is balanced with that of mixed phase (ideal solution as we assumed): $\mu_{in}=\mu=k_{B}Tln(\frac{n^{*}}{n})$, in which $n^{*}$ is an arbitrary number density of minority phase. Here we further set the number density at infinite region is a fixed value $n_{\infty}$, which plays a role of a reservoir of minority phase. If those two regions relaxes separately as steady currents, we can get an equation series by plug them in equation 35:

$$n(R)=AlnR+B=n^{*};n(\rho)=Aln\rho+B=n_{\infty}$$

(11)

here $R$ is the radius of the droplet and $\rho$ is the location at infinity. Then by solving this, the undetermined constants can be expressed by the constants we have:

$$A=\frac{n_{\infty}-n^{*}}{ln\frac{\rho}{R}};B=\frac{n^{*}}{ln\frac{\rho}{R}}ln\rho-\frac{n_{\infty}lnR}{ln\frac{\rho}{R}}$$

(12)

now we plug them back to the equation 36, then take gradient respect to $r$. The constant $B$ directly vanished and the current on the droplet $R$ will be a simple expression:

$$J(R)=-\frac{D}{R}\frac{n_{\infty}-n^{*}}{ln\frac{\rho}{R}}$$

(13)

Now we can determine the dynamics of radius $R$ by assuming the incoming current, the flux equation 38 times $2\pi R$, equals to the area change rate:

$$\frac{da}{dt}=2\pi R\frac{D}{R}\frac{n_{\infty}-n^{*}}{ln\frac{\rho}{R}}=2\pi R\dot{R}$$

(14)

where we applied $a=\pi R^{2}$. Then the radius change rate can be obtained by reorganizing 39:

$$\dot{R}=\frac{D}{R}\frac{n_{\infty}-n^{*}}{ln\frac{\rho}{R}}$$

(15)

then the number density of minority phase can be expressed by the equilibrium condition we mentioned above: $n^{*}=n_{0}e^{\frac{\mu_{in}}{k_{B}T}}$. Since $\rho>>R$ is always true in our assumption, so we could further set $ln\frac{\rho}{R}$ as a constant $\tilde{D}$. Thus the radius change rate can be reorganized as:

$$\dot{R}=\frac{\tilde{D}}{R}(n_{\infty}-n_{0}e^{\frac{\mu_{in}}{k_{B}T}})$$

(16)

here the heart of our model is the chemical potential $\mu_{in}$ is dependent on the interaction energy:

$$\mu_{in}^{{}^{\prime}}=\mu_{in}+\alpha\frac{U(\vec{x},R)}{R^{2}}$$

(17)

one can regard the last term as the excess Gibbs free energy offered by interaction potential up to a parameter $\alpha$. $\frac{1}{R^{2}}$ here represent unit area, alpha combining $\frac{1}{R^{2}}$ represent the number of molecules. So $\alpha$ is the density of minority phase, which is the intrinsic property of different material. In this equation it represents how sensitive the droplet against interactions, thus we name $\alpha$ as the ”dewetting parameter”. Noticed that, to make equation 42 reasonable, we assume that in the condensed phase the Gibbs free energy is nearly identical to Helmholtz free energy. We also drop the entropy term.

We found that the parameter $\alpha$ controls the dynamical properties of the droplet. For $\alpha$ between six and ten, the droplet interacts strongly with the filament. They may evaporate shortly after the simulation start of the simulation or undergo a number of shrink-and-growth cycles, producing the characteristic ”shark fin shape” dependence of radius on time. Eventually, the droplet will evaporate when there is a sufficiently large thermal fluctuation. For $\alpha$ between 1.3 and 6, the system is in a quasi-equilibrium steady-state regime with the radius fluctuating around a mean value. Finally, for $\alpha$ smaller than 1.3, the droplet interacts only weakly with the surroundings. The radius will grow in time proportional to $t^{\frac{1}{2}}$, permeating the surrounding filaments. The changes that take place in the growth characteristics also affect the mobility of droplets. In both the evaporation regime and the steady-state regime, the droplet will be localized most of the time in a minimum energy location, with rare jumps to neighboring minima. In the diffusive regime of small $\alpha$ where the droplet permeates the surrounding filaments, the droplet will be much more mobile. Below is fig.5 are displacement of the droplet over time with different $\alpha$.

It is interesting to compare these regimes with the equilibrium phase diagram that was recently obtained by the Kosmrlj group for this problem ronceray2021liquid. The phase diagram obtained in that paper is shown in figure 8.

The regime of micro-droplets would correspond to the regime of large $\alpha$ parameter while the regime permeation corresponds to the regime of small $\alpha$. The so-called permeo-elastic number is roughly inversely proportional to $\alpha$. There is no analog in our theory for cavitation where the droplet pushes the filaments out of the way and it would be interesting to pursue that.

With our current settings, the fluxes are driven by the gradient of chemical potential, and the growth rate is identical to the incoming flux. This setting is under the assumption that the diffusion is large enough such that the concentration field along this transportation can be soon vanished, $\nabla^{2}c=0;\frac{\partial c}{\partial t}=0$.

However, in actuality, the low and high-density/concentration domains are separated by an interface. The relaxation of the concentration field is not instantaneous as we assumed. Instead of analytically solving the Cahn-Hilliard equation yeung1992phase for this diffusive system, we use Einstein relation costigliola2019revisiting to add a radius fluctuation to the dynamics of $R$ as a proxy of the concentration field. Specifically, we add a noise term to the flux term which corresponds to the time evolution of the concentration profile:

$$\frac{\partial c}{\partial t}=-\vec{\nabla}\vec{J};\vec{J}=-\nu c\vec{\nabla}\mu+\vec{J_{n}}$$

(48)

Again, here $\nu$ is the mobility, and $\vec{J_{n}}$ is the noise term we add to the flux. Specifically, a path integral around the perimeter of the sphere was added to integrate fluxes around it:

$$\dot{R}=\frac{1}{R}(\dot{R_{0}}-\dot{R_{1}}e^{\frac{\alpha U(\vec{x},R)}{k_{B}TR^{2}}+\frac{\Gamma a}{Rk_{B}T}})+\frac{a}{2\pi}\int_{0}^{2\pi R}dsJ_{r}^{n}$$

(49)

where s is the path along the perimeter and $a$ again is the molecular area, and this integral can be represented as a cartoon in figure 9:

We now consider this correction term together as $\zeta(t)=\frac{a}{2\pi}\int_{0}^{2\pi R}dsJ_{r}^{n}$. Then the autocorrelation function of $\zeta(t)$ is:

$$<\zeta(t)\zeta(t^{\prime})>=(\frac{a}{2\pi})^{2}\int_{0}^{2\pi R}ds\int_{0}^{2\pi R}ds^{\prime}<J_{r}^{n}(s,t)J_{r}^{n}(s^{\prime},t^{\prime})>$$

(50)

it can be easily estimated as $2\pi R(\frac{a}{2\pi})^{2}\Delta\frac{1}{l}\delta(t-t^{\prime})$ up to an undetermined constant $\Delta$. That relies on equal time correlation of Helmholtz free energy and kinetics of the concentration change: $\frac{\partial\delta c}{\partial t}=D\nabla^{2}\delta c-\vec{\nabla}\vec{J_{n}}$. Here we only want audiences to notice the radius would fluctuate up to a noise due to such relaxation of the matter transfer. As we are not turning on this noise term in our later investigation, here We only leave the result as $\Delta=2Dc(R)$.

Before we turn on the fluctuations of dynamics of radius and displacement, we first simplified the system into a 1-dimensional model by coarse-graining the net force around the droplet.

This simplification can be done by assuming several Gaussian potentials around the droplet exert an effective harmonic force on it, thus we could regard all point mass on the droplet as feeling such potential. For a location center of mass of the droplet $\vec{r}$ and an arbitrary point $\vec{\rho}$ deviates from the center, this integrated potential is:

$$U_{eff}=\int_{\vec{\rho}<|R|}d^{2}\vec{\rho}\frac{1}{2}u_{0}(\vec{r}+\vec{\rho})^{2}=\frac{1}{2}u_{0}(r^{2}\pi R^{2}+\frac{\pi}{2}R^{4})$$

(51)

here $u_{0}$ is the effective spring constant of the harmonic, we measure this constant by simulating the 2D model with a small deviation, to make it comparable with the original model.

After obtaining the effective potential, we just need to replace the $U_{int}$ in equation 44 by this. The resulting expression has a clean look:

$$\dot{R}=\frac{1}{R}(\dot{R_{0}}-\dot{R_{1}}e^{\frac{\alpha u_{0}(r^{2}\pi+\frac{\pi}{2}R^{2})}{k_{B}T}+\frac{\Gamma a}{Rk_{B}T}})$$

(52)

Together with the equation of motion $\gamma\frac{d\vec{r}}{dt}=-\frac{\partial U}{\partial r}$ of the droplet, and with turning off both noise terms, the result differential equation series can be regarded as a nonlinear dynamical system. This deterministic description offers us insight into the dynamics. Here we use the resulting flow lines of the vector field to illustrate it.

When we turn off the interaction energy, all vectors point toward the center location dominated by a regular harmonic potential. With $\alpha=5$, a stable fixed point emerges at $r=0,R=R^{*}$ in which $R^{*}$ is the saturated radius, the horizontal vectors indicate at almost all finite displacements the radius tends to converge to the saturated radius. Two unstable fixed points result from surface tension when the size of the droplet is small enough, i.e. $R\approx 0$ the mechanical interaction forces the evaporation at a finite displacement. However, at a later stage $\alpha=9.2$, almost all vectors strictly rain down towards negative $R$, indicating the droplet will constantly evaporate even with a finite interaction with the potential.