Ratchet‐Driven Directed Nanoparticle Transport Along a Dipole Chain

Abstract

Spontaneous generation of directed nanoparticle fluxes near periodic one‐ or two‐dimensional structures in viscous media—the ratchet effect—occurs under specific conditions: particle interaction with a periodic structure, broken symmetry of the system, and weak fluctuations that drive it out of thermodynamic equilibrium. Within the framework of the small potential energy fluctuation model, we investigate the directed motion of a single nanoparticle near a periodic dipole chain. The induced flux was analyzed as a function of three parameters: environmental temperature, fluctuation frequency, and the particle’s distance from the chain. Two types of flux dependence on distance were observed. At high temperatures, the flux decreases monotonically with increasing distance. At lower temperatures, the dependence becomes non‐monotonic, revealing an optimal nanoparticle position relative to the chain where transport velocity is maximized. Analysis of flux maxima shows that their location is determined primarily by the temperature parameter, while frequency variations affect the flux amplitude but not the position of the maximum. These findings demonstrate that optimal ratchet operation (maximum nanoparticle velocities) can be achieved either by adjusting the particle—periodic structure distance at fixed temperature or, if distance cannot be physically controlled, by varying the temperature.

A chain of oriented dipoles can serve as an inorganic analog of biological filaments for molecular motors. We studied directed nanoparticle motion in the dipole chain potential V(x,z) under small fluctuations, as a result of the ratchet effect. Despite decreasing asymmetry and amplitude of the potential with distance, velocity shows non‐monotonic behavior, defining an optimal position for transport along the chain.

1. Introduction

The organization of motion at the nanoscale is a relevant multidisciplinary problem in modern nanotechnology, physics, chemistry, and microbiology (see, for example, the leading reviews [1, 8]). Such motion may involve specific tasks: moving or rotating individual particles or ensembles, generating stable fluxes, ensuring mass transfer, or converting different forms of energy into mechanical work [1, 4]; or more specific: size selective separation of nanoparticles [9, 10], controlled enhancement of diffusion [11], current reversal [12, 13], and the directed transport of self‐propelled (active) particles [5, 14]. It may also serve as part of a more complex functionality, for example, delivering a nanorobot to perform subsequent operations [15, 17].

The ratchet effect—spontaneous emergence of directed nanoparticle motion under nonequilibrium conditions without stationary external forces or concentration gradients—can be an effective, accessible, and original method for creating controlled transport of nano‐objects. This type of motion is diffusive in nature and requires a combination of several factors: an effective potential profile interacting with the particle, broken mirror symmetry in the system to define the direction of motion, and an unbiased fluctuation process that takes the system out of thermodynamic equilibrium and provides an input of energy [1, 3, 9, 18]. In the theoretical description of the ratchet effect, several main classes are differentiated [1, 3] depending on what exactly causes the system to be driven out of equilibrium. In rocking ratchets, directed motion arises under the action of a periodic external force with zero mean value [19]; in flashing ratchets, the potential is modulated [20, 21]; one can also distinguish active ratchets, which use the self‐motion of particles [14, 22]; and information ratchets [6], etc. In addition to one‐dimensional models, there are models that take into account two (2D) [ 23] or even three (3D) [24, 25] dimensions, which allow for a more detailed description of the ratchet effect. These theoretical concepts have been confirmed by numerous experiments, including the first realization of a flashing ratchet with colloidal particles [9], particle separation systems [26]; optical rocking ratchets with controlled current direction [13, 23]; a rotating ratchet in a ferrofluid [27]; observation of flux reversal in an optical ratchet [28]; a non‐Gaussian correlated ratchet with optimal correlation time [29]; and numerical modeling of active self‐propelled ratchets [14].

In biological systems, the ratchet effect is realized, for example, in cytoskeletal motor proteins (such as myosin, kinesin, and dynein), which move along filaments or microtubules [3, 15, 30, 34]. These filaments have a periodic structure with a period of about 8 nm, forming an asymmetric potential profile (a necessary condition for directed motion) and simultaneously defining the transport trajectory. Artificial nanoparticle motion can be induced along periodic structures, either natural (e.g., solid surfaces [35]) or engineered, such as digitated electrodes with anisotropically positioned barriers on glass substrates, used in particle separation devices [36, 37]. For example, [38] studied patterned surfaces in the form of crystals, stripes, and bubble states, and analyzed the resulting ratchet properties.

A chain of oriented dipoles can be used as a real basic object for artificially organized transport, similar to the inorganic analog of a filament for biological molecular motors. In biological systems, motion arises due to cyclic conformational changes in motor domains and sequential “weak binding to the microtubule wall, allowing one‐dimensional diffusion along the microtubule” [39]. This binding automatically sets the distance at which the protein approaches the microtubule. In artificial nanosystems, which are smaller in scale (OH‐group chains on SiO₂ surfaces with a period of ~1 nm [35]) and structurally simpler [2], motion results from physical interactions between nanoparticles and their surroundings.

The aim of this work is to investigate how the ratchet effect (i.e., the velocity of directed motion) depends on the distance between a nanoparticle and a dipole chain. A key question is whether there exists an optimal distance at which maximum velocity is achieved, or whether motion simply becomes faster as the particle approaches the potential source.

In a real system containing charged particles in a viscous medium near a surface containing dipole chains (or a single dipole chain) generating an electrostatic potential, diffusion proceeds in the volume. And model simulation requires at least two spatial dimensions to be taken into account, which allows for a complete description of the phenomenon. In general, the inclusion of the additional dimension gives rise to apparent effects beyond the 1D case. Thus, in [40, 41], it is shown that in two‐dimensional channels with variable width, even at a constant temperature, vortex currents and an effective gradient potential arise. Studies [42] and [43] demonstrate that in two‐temperature ratchets, transverse dynamics are decisive for the emergence of a directed flow. Similarly, the results of [44] and the experimental study of a plasmonic Brownian ratchet [45] confirm that taking the second dimension into account leads to non‐monotonic dependencies and current reversals that cannot be described by one‐dimensional models.

However, in order to answer the question of whether an optimal distance exists and to characterize the dependence of the longitudinal velocity on this distance, it is not necessary to solve the full two‐dimensional problem. The ratchet effect in the vicinity of the dipole chain will manifest itself only along the chain, because only in that direction is the periodic and asymmetric. Transverse diffusion, arising in a potential that decreases monotonically with distance from the chain, does not contribute to the ratchet effect and, consequently, will not affect the qualitative answer regarding the type of behavior. Therefore, for a fundamental study of the ratchet effect, it is sufficient to suppress transverse motion and consider a one‐dimensional model with a fixed distance from the chain.

An experimental realization of fixed particle distance from the chain and free motion along it is possible using the following ideas, which do not require the application of external electromagnetic fields. In a nanofluidic channel, the particle is confined within a gap of fixed height, which mechanically restricts its vertical movement [27]. A similar principle is realized by confining the particle between two solid planes or grooves, where the potential source is covered by a thin dielectric layer [46], and a second substrate is placed on top, defining a constant channel height. An alternative is provided by hydrodynamic traps, in which counter‐flowing liquid streams or inertial focusing in microchannels hold the particle in a specified plane without contact with the walls [47, 48]. Also possible are gravitational‐Archimedean trapping (compensating the particle weight by the solution density) [49] and the opposed‐jets method (hydrodynamic barrier) [50]. Thus, in all the cases listed, the particle’s transverse motion is suppressed, and its dynamics reduce to one‐dimensional motion along the chain. Below, we describe the potential created by such a chain and its properties. We then present the calculation method and the numerical procedure, after which we discuss the obtained results and formulate our conclusions.

Under certain conditions, an orientationally structured system of adsorbed polar molecules can form on the surface of a solid body [35, 51, 52]. In two‐dimensional dipole systems, the base state is determined by the type of adsorbate lattice and is realized either in a ferroelectric or antiferroelectric configuration. Such ordered structures on the surface can be interpreted as a set of one‐dimensional chains in which the dipole moments are collinearly oriented along the chain axis [51, 52]. Let us place the x‐axis along the chain, and the origin of coordinates coincides with one of the dipoles. Let L denote the distance between neighboring dipoles (the chain period). Then the electrostatic potential of this chain at a point with coordinate x at a height z above the surface can be written as a series [51, 52]:

$$\phi (x,z)=\frac{2\mu }{{\epsilon }_0{L}^2}\sum _{h=1}^{\mathrm{\infty }}h{K}_0\left(\frac{2\pi h}{L}z\right)\sin \left(\frac{2\pi h}{L}x\right)$$ (1)

where μ is the dipole moment of the adsorbed molecule, ε ₀ is the electric constant, K ₀(x) is the modified Bessel function of the second kind of order zero (also known as the Macdonald function) [53], and h is summation variable. Then, a nanoparticle with charge q, located near the dipole chain, has potential energy V(x, z) = qφ(x, z), the form of which is shown in Figure 1a. Let us consider its main properties. This energy is a periodic function of x with period L, matching the chain periodicity, and decays as the distance from the chain increases. Also, the function breaks mirror symmetry: its extrema are unevenly distributed, quantified by the asymmetry coefficient κ = (x _max − x _min)/L [1]. Figure 1b shows how the maximum value V _max(z) (left axis, in units of V ₀ = 2qμ/ε ₀ L ²) and κ (right axis) depend on the distance from the chain. The maximum value decreases rapidly with distance, whereas the asymmetry coefficient increases (i.e., the potential becomes more symmetric; for a symmetric potential κ = 0.5). Thus, the conditions for the ratchet effect strongly depend on the nanoparticle’s distance from the surface, on which the dipole chain is located. Since in the present work we study the dependence of the ratchet effect on distance, introducing it as a control parameter, we denote the fixed distance by z ₀ (in contrast to the variable z in Equation (1), which here is a parameter rather than a coordinate of motion). In further describing the one‐dimensional model, we assume that the particle moves only along the x‐axis with constant z = z ₀, and that transverse motion along z is suppressed.

FIGURE 1.

(a) A schematic representation of the system. A chain of oriented dipoles (red spheres denote positive poles and blue spheres – negative poles) is arranged along the x‐axis (labeled x/L and shown as a red line). The potential energy of interaction between a particle q and the dipole chain, V(x, z) [see Equation (1)], is shown as a colored surface. A semi‐transparent plane defines a fixed distance z = z ₀, at which nanoparticles (turquoise spheres) move along the dipole chain (the x‐axis). (b) Dependence of the maximum V _max(z) in unit V ₀ = 2qμ/ε ₀ L ² (left axis) and asymmetry coefficient κ on the distance between the nanoparticle and the dipole chain.

2. Methods

For a wide range of model ratchet systems, the dynamics of a Brownian particle‐motor in a viscous medium is described by the Smoluchowski equation [1, 54] in terms of the probability density function ρ(x, t) of finding a Brownian particle at point x at time t:

$$\begin{array}{cc}\frac{\partial }{\partial x}\rho (x,t)+\frac{\partial }{\partial x}J(x,t)& =0,\hspace{0.17em}J(x,t)\\ & =-D{e}^{-\beta U(x,t)}\frac{\partial }{\partial x}{e}^{\beta U(x,t)}\rho (x,t)\end{array}$$ (2)

Here, $D={(\beta \zeta )}^{-1}$ is the diffusion coefficient (ζ is the friction coefficient, $\beta ={(k_{B}T)}^{-1}$ is the inverse thermal energy, k _B is Boltzmann’s constant, and T  is the absolute temperature), U(x, t) is the potential energy of the particle, and J(x, t) is the emerging probability flux. Since we consider motion along a one‐dimensional periodic structure, the potential energy is treated as a periodic function with period L (U(x + L, t) = U(x, t)). This periodicity also applies to ρ(x, t) and J(x, t).

The established process of ratchet functioning is characterized by a constant average flux J ≡ 〈J(x, t)〉 [1, 54, 58], which allows us to find the searched value – the average velocity of nanoparticles – as 〈ν〉 = LJ.

To solve the problem, we used the well‐known approximation of a slightly fluctuating potential energy (SFPE) [55, 58]. This approximation belongs to the class of flashing Brownian motors [1, 20, 21], in which the non‐equilibrium condition is created by modulation of the potential itself, rather than by an applied external force. The small‐fluctuation regime is characteristic of artificial ratchet systems and is particularly important for precision measurements and for work in sensitive environments (biological objects, fragile nanostructures) [58].

Within the SFPE framework, the potential energy of the particle is written in an additive‐multiplicative form [55, 58]:

$$U(x,t)=u(x)+\sigma (t)w(x)$$ (3)

where the first term u(x) is time‐independent (stationary part) and the second term σ(t)w(x) introduces fluctuations, with w(x) << u(x) (values of the function σ(t) of order unity). We assume that the ratio $|u(x)|\text{/}{k}_{B}T$ is arbitrary, and $|\sigma (t)w(x)|\text{/}{k}_{B}T<<1$. This method of defining the potential energy is well suited to the aim of the work: the focus is on the substrate object (a chain of dipoles), and to its potential we add a small external influence (a weak perturbation) to initiate motion, without introducing additional effects. The stationary part u(x) in this work is taken as the potential of the dipole chain $V(x,z_0)$ given by (Equation 1). The function σ(t) defines the temporal dependence of the potential energy $U(x,t)$ and directly governs the fluctuation process; since the fluctuations are unbiased, the condition must be satisfied: $\langle \sigma (t)\rangle =0$ [56]. In this work we consider a stochastic dichotomous process, i.e., random switching between two states $\sigma (t)=\pm 1$ Thus, the potential energy (3) alternates between two states $u(x)\pm w(x)$ with average switching frequency γ, corresponding to a symmetric fluctuation process.

Figure 2 schematically illustrates the operation of our model: black curves show the stationary potential $u(x)$ for different fixed distances $z_0\text{/}L$ (cross‐sections of the landscape in Figure 1)—as the distance from the chain increases, the amplitude of the potential decreases. The red curve represents the fluctuating part $w(x)$ (a harmonic signal).

FIGURE 2.

Components of the potential energy of a particle U(x, t): black lines correspond to stationary dipole‐chain contributions (1) for different values of z ₀/L = 0.075;0.0825;0.1;0.1125;0.125 (solid, dotted, dashed, dash‐dotted, and long‐dashed lines, respectively), red – small fluctuation part w(x) defined by (7).

The representation of the potential energy in the form (3) makes it possible to develop a theory of small perturbations with respect to the small parameter w(x) using the apparatus of Green’s functions for nanoparticle diffusion [57, 58]. In Ref. [57] a numerical method was proposed for calculating the Green’s function and the ratchet characteristics by means of Fourier transformation for arbitrary dependences $u(x)$ and $w(x)$. For this purpose $u(x)$, $w(x)$ and functions

$${\rho }^{\left(0\right)}\left(x\right)=e^{-\beta u\left(x\right)}/{\int }_0^L{\mathrm{dxe}}^{-\beta u\left(x\right)},q\left(x\right)=e^{\beta u\left(x\right)}/{\int }_0^L{\mathrm{dxe}}^{\beta u\left(x\right)}$$ (4)

must be represented as Fourier series ($f(x)=\sum _p{f}_p{e}^{i{k}_{p}x},$ where$k_p=2\mathrm{\pi }p/\mathrm{L},\hspace{0.17em}p=0,\pm 1,\pm 2,\dots ;\hspace{0.17em}f\left(x\right)$ – each of these functions). The function ${\rho }^{(0)}(x)$ is the equilibrium distribution in the stationary potential $u(x)$; it is this distribution that is “pushed” by the fluctuating addition, giving rise to the flux in the leading order of perturbation theory. The resulting constant flux J is then written in Fourier component space as [57]:

$$J=-i{\left(\beta \mathrm{DL}\right)}^2\sum _{\mathrm{pp\prime }{p}_1{p}_2}{k}_{\mathrm{p\prime }}{k}_{-p-p_1}{k}_{-p_2+\mathrm{p\prime }}{q}_{p_1}{\rho }_{p_2}^{\left(0\right)}{S}_{\mathrm{pp\prime }}{w}_{-p-p_1}{w}_{-p_2+\mathrm{p\prime }}$$ (5)

where the elements $S_{\mathrm{pp\prime }}$, the matrix analog of the Green’s function, are found from equation [57]:

$$\sum _{\tilde{p}}[(D{k}_p^2+\text{Γ}){\delta }_{\mathrm{p}\tilde{\mathrm{p}}}+\beta D{k}_p{k}_{p-\tilde{p}}{u}_{p-\tilde{p}}]S_{\tilde{\mathrm{p}}\mathrm{p}\text{'}}=-\frac{1}{L}{\delta }_{\mathrm{pp\prime }}$$ (6)

by inverting the corresponding matrix. The value $\Gamma =2\gamma$ for a symmetric dichotomous process, and ${\delta }_{\mathrm{pp\prime }}$ – matrix elements of the unit operator $\widehat{1}$.

In this work, we use a harmonic function of the fluctuation w(x) with amplitude w ₀, the same period L, and phase shift λ ₀ [57]:

$$w(x)=w_0\cos [2\pi (x/L-{\lambda }_0)]$$ (7)

Fourier components of which are given by a formula containing Kronecker $\delta$‐symbols

$$w_p=\frac{w_0}{2}[e^{2\pi i{\lambda }_0}{\delta }_{p,\text{−}1}+e^{-2\pi i{\lambda }_0}{\delta }_{p,1}]$$ (8)

Substituting these into the flux representation (5) simplifies the expression to a double summation [57]:

$$\begin{array}{l}J=-i{\left(\pi \beta D{w}_0\right)}^2\sum _{\mathrm{pp\prime }}{k}_{\mathrm{p\prime }}{S}_{\mathrm{pp\prime }}[q_{-p+1}{Z}_{\mathrm{p\prime }}^{(+)}+q_{-p-1}{Z}_{\mathrm{p\prime }}^{\left(\text{−}\right)}]\\ Z_{\mathrm{p\prime }}^{\left(\text{±}\right)}=e^{\pm 4\pi i{\lambda }_0}{\rho }_{\mathrm{p\prime }\pm 1}^{\left(0\right)}-{\rho }_{\mathrm{p\prime }\mp 1}^{\left(0\right)}\end{array}$$ (9)

The computational procedure is based on Equations (9) and (6). First, an integer number of harmonics N is chosen, and one‐dimensional arrays of wave vectors $k_p=2\pi p/L$ are constructed for p = −N, …, N, along with the corresponding Fourier components of elements u _p. The stationary part corresponds to the dipole chain potential (1), with z ₀‐dependent Fourier components:

$$u_p=\frac{V_0}{2i}p{K}_0\left(k_p{z}_0\right)$$ (10)

Next, matrix elements $S_{\mathrm{pp\prime }}$ are computed using an auxiliary matrix $\widehat{A}$ [57]:

$$A_{\mathrm{p}\tilde{\mathrm{p}}}\equiv D{k}_p^2{\delta }_{\mathrm{p}\tilde{\mathrm{p}}}+\beta D{k}_p{k}_{p-\tilde{p}}{u}_{p-\tilde{p}},\widehat{S}=\text{ −}\frac{1}{L}{(\widehat{A}+\Gamma \widehat{1})}^{-1}$$ (11)

After that, one‐dimensional arrays of Fourier components of functions $q_p$ and ${\rho }_p^{(0)}$ are compiled and calculated via the numerical integration procedure. Finally, the flux J is calculated according to (9). In the next section we analyze the dependence of this flux on the control parameters.

3. Results and Discussion

To characterize the ratchet effect quantitatively, we introduce two dimensionless parameters similarly to [57]. The first the parameter $\alpha =\beta V_0$—the ratio of the value $V_0=2q\mu /{\epsilon }_0{L}^2$, which determines the magnitude of the potential barrier of the dipole chain, to thermal energy. Small α corresponds to high temperatures (the particle barely feels the potential), while large α corresponds to low temperatures (the particle is strongly trapped). The second is the product of the fluctuation frequency to the value ${\tau }_0=\zeta L^2/V_0$—the diffusion time of a particle along the potential slope between two dipoles (over the distance of the spatial period of the system). These parameters allow us to analyze the ratchet velocity in terms of the relevant physical scales.

Figure 3 presents curves showing the dependence of particle flux (in units of $J\text{*}=w_0^2/(V_0\zeta L^2)$) on z ₀/L, the distance between the particle and the dipole chain. In the calculations, the fluctuation frequency was fixed at $\gamma {\tau }_0=50$, and the phase shift of the fluctuating component was set to ${\lambda }_0=0.25$ to maximize the positive net flux. The curves correspond to different values of the temperature parameter α (indicated near each curve): high temperatures are shown in burgundy and red, gradually shifting to blue as the temperature decreases. Results for values α between 0.1 and 0.2 are summarized in the inset at the top right. Two types of flux behavior were obtained: monotonically decreasing dependencies at low parameter values (inset) and non‐monotonic dependencies in the main figure, where the flux maximum shifts farther from the chain as the temperature decreases.

FIGURE 3.

A set of curves representing the dependence of the particle flux J/J* on the distance between the particle and the dipole chain. The curves are calculated for different values of the temperature parameter α (indicated near the curves). The frequency parameter is $\gamma {\tau }_0=50$.

Figure 4 demonstrates a similar frequency dependence: a set of flux curves J/J* depending on $z_0/L$ were calculated for a fixed temperature parameter $\alpha =5$ and phase shift ${\lambda }_0=0.25$ for different values of the frequency parameter $\gamma {\tau }_0$ (are shown in the graph legend). All obtained dependencies are non‐monotonic; they have different heights and similar shapes, while the positions of the curve maxima differ insignificantly.

FIGURE 4.

Family of curves showing the dependence of particle flux $J/J\text{*}$ on the distance $z_0/L$ for different values of the frequency parameter $\gamma {\tau }_0$ (indicated in the graph legend). The temperature parameter is α = 5.

For a more detailed analysis, we examined the behavior of the flux at its maximum with respect to the parameter $z_{\mathit{0}}/L$. This reduces the problem to varying the two remaining parameters $\alpha$ and $\gamma {\tau }_0$. To investigate their interplay, we consider cross sections of the landscapes $z_{max}(\alpha ,\gamma {\tau }_0)$ and $J_{max}(\alpha ,\gamma {\tau }_0)$. Figure 5a shows the dependence of the maximum position (blue curves and axes) and the maximum flux J _max/J* (red curves and axes) on the temperature parameter $\alpha$ at a fixed $\gamma {\tau }_0=50$. Similar dependencies on $\gamma {\tau }_0$ are shown for the maximum value of ${\alpha }_{max}=3.5$ (Figure 5b). As a result, we obtained two sets of complementary curves z _max/L and J _max/J*. In both dependencies, the flux demonstrates non‐monotonic behavior within the selected parameter ranges.

FIGURE 5.

Dependencies of the maximum flux position z _max/L (blue curves and axes) and maximum flux value J _max/J* (red curves and axes) on the temperature parameter at a (a) fixed value $\gamma {\tau }_0=50$ and on the frequency parameter at a (b) fixed value ${\alpha }_{max}=3.5$.

The difference is observed in the behavior of the maxima positions (blue curves). When the environmental temperature changes, the maximum shifts significantly (from z _max/L = 0.08 to 0.31, see Figure 5a). In contrast, when the frequency parameter varies, the maximum position remains nearly constant (from 0.24 to 0.22, Figure 5b). Thus, the location of the maximum flux is determined primarily by the temperature parameter.

The non‐monotonic behavior of the flux with increasing distance from the dipole chain is particularly noteworthy. Although both the asymmetry and amplitude of the potential decrease (conditions that normally weaken the ‘motor’ effect), the particle velocity increases until a certain distance is reached, thereby defining an optimal position for transport along the chain. The mechanism of the ratchet’s functioning assumes the presence of asymmetrically located potential barriers that the particle can overcome: the optimal height of these barriers determines the stochastic resonance phenomenon and is characterized by the maximum movement velocity [1]. In our case, in addition to the temperature parameter (which increases the barrier height as temperature rises), the barrier height is also controlled by the dipole chain potential itself: its amplitude decreases rapidly with distance from the chain (see Figure 1b). Approaching the potential source is equivalent to a local decrease in temperature (an increase in barriers), conservation of particles in potential wells until their motion stops (a decrease in flux to zero when approaching the chain). Thus, the observed non‐monotonic behavior is essentially a manifestation of stochastic resonance caused by the distance‐dependent amplitude of the potential.

At very small values of the temperature parameter, the barriers are initially low (see inset in Figure 3). Their increase with decreasing distance is insufficient to produce non‐monotonic behavior within the studied range, so in this case only the decreasing part of the full curve is observed.

4. Conclusion

We considered the motion of a charged nanoparticle in a stationary electrostatic potential of a dipole chain with collinearly oriented dipoles. The study was conducted within the framework of a one‐dimensional approximation, in which the distance to the chain is treated as a fixed control parameter z ₀. Directed nanoparticle transport arised from the introduction of slight stochastic fluctuations—adding a fluctuating harmonic dependence $\pm w(x)$ with a certain frequency γ to the chain potential $u(x)=V(x,z_0)$.

Using the previously developed computational method within the framework of the approximation of a slightly fluctuating potential energy (SFPE) [57], we constructed a numerical procedure and studied the resulting flux of directed motion as a function of three parameters: temperature, fluctuation frequency, and the particle’s distance from the dipole chain. It was shown that at high temperatures, the particle velocity increases monotonically with decreasing distance to the chain, but when the temperature decreases, the nature of the dependence changes to non‐monotonic and a velocity maximum occurs, determining the optimal position of the nanoparticle for movement along the chain at a fixed temperature. This maximum shifts farther from the chain as the temperature decreases. Variation of the frequency parameter produces a characteristic non‐monotonic dependence of the induced flux, but the position of the ratchet effect maximum with respect to distance remains nearly unchanged.

The non‐monotonic dependence of the velocity of directed motion on the distance to the dipole chain has several practical and fundamental applications. Thus, it provides a direct guideline for the design of artificial transport systems: at a fixed temperature, by setting the position of the nanofluidic channel or solid‐state gap to the optimal value z ₀, the maximum particle velocity can be ensured without empirical fitting (or, conversely, to determine the optimal temperature when z ₀ is fixed). Furthermore, the non‐monotonic dependence opens the possibility of controlling the motion of particles in mixtures: particles with different charges (which will accordingly change V ₀ and α) will have different optimal distances and, consequently, different directed velocities. This makes it possible to create differential transport of mixture components in practical applications. Finally, the non‐monotonic dependence shows that the velocity does not simply increase as the distance decreases, thereby deepening the understanding of ratchet effects in such potentials and offering new prospects for controlled nanoscale motion.

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.