Non-Feynman Deterministic Pulsing Ratchet Based on a Symmetric

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Non-Feynman Deterministic Pulsing Ratchet based on a Symmetric Periodic Potential with Harmonic Time Modulation Semen N. Semenov, and Martin E. Schimpf J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06050 • Publication Date (Web): 22 Sep 2017 Downloaded from http://pubs.acs.org on September 26, 2017

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The Journal of Physical Chemistry

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NON-FEYNMAN DETERMINISTIC PULSING RATCHET BASED ON A SYMMETRIC PERIODIC POTENTIAL WITH HARMONIC TIME MODULATION Semen Semenov1*, Martin Schimpf2 1

Institute of Biochemical Physics RAS, 119334 Moscow, Kosygin St. 4, Russia, [email protected]

2

Boise State University, Boise, Idaho 83725, USA, [email protected]

ABSTRACT The deterministic mechanism of directed particle motion in a symmetric periodic potential with harmonic time modulation is theoretically examined. We show that directed particle motion can be established with a symmetric potential in the absence of thermal motion, provided the spatial particle shift is synchronized with the period of temporal change in potential and the characteristic time required for the particle to pass through the interval between neighboring extremes in potential is equal to the half-period of the time modulation. Ratchet behavior is demonstrated using a simple symmetric force potential consisting of a square wave with opposite but equal amplitude regions. By establishing spatiotemporal synchronization between the particle shift in space and the time modulation, non-reversible directed particle motion is provided by inertia that is maintained as the potential profile approaches zero in the time modulation process.

INTRODUCTION The directed motion in periodic systems with alternating parameters is known as ratcheting. Ratchet systems can be roughly classified as pulsing or tilting. Pulsing ratchets provide directed motion through some sort of potential oscillation. The Feynman (thermal) ratchet is a specific type of theoretical microscopic pulsing device that is capable of extracting work from random fluctuations without violating the second law of thermodynamics 1-2. Rather than provide a comprehensive review of ratchets, we refer the reader to a few selected papers that contain hundreds of referenced works on the problem of directed motion caused by the ACS Paragon Plus Environment


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deterministic or stochastic time modulation of a periodic potential or as a result of a stochastic or deterministic driving force 3-5. The physical mechanism behind such motion is used to explain the action of many biological and physicochemical systems. Our interest in thermal ratchets comes from recent work on microgears that serve as microengines when placed in a suitable liquid with a superimposed temperature gradient 6. In our study of these thermophoretically driven engines we conclude that in order to be effective in a constant temperature gradient the device surface must have a special distribution of physicochemical properties. Although well-developed methods exist in the microelectronics industry for making planar devices with an inhomogeneous surface, the design of non-uniform surfaces in microdevices having more complex geometries would require the development of rather exotic technology. Consequently, we turned to the possibility of using alternating temperature gradients as a means of driving thermophoretic microdevices having homogeneous surface properties. An alternating temperature gradient can be established, for example, by the motion of a tooth-shaped device. When such a device is heated, the summits and valleys of the teeth will have different temperatures, and the resulting temperature distribution will change as a function of the relative position of the moving teeth. In a Feynman ratchet the directed motion is caused by either a time modulation of spatially periodic potential or a stochastic driving force with zero mean value acting on a particle placed in such a potential. Stochastic ratchets represent a separate branch of Feynman ratchets and are widely studied using non-equilibrium thermodynamics (see, e.g., 7 and references therein). Our work concerns systems with deterministic time modulation superimposed on a periodic potential, which requires neither asymmetry in the periodic potential nor the presence of thermal noise. In such systems the potential is not shifting in space over time, rather its amplitude is changing with the same rate at all the points simultaneously. To show expressly the distinction of our model, we first look at a system where the harmonic time modulation of the

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symmetric periodic pulsing potential causes directed particle motion. More specifically, we examine a pulsing ratchet with a symmetric potential and symmetric deterministic time modulation. Although it is commonly assumed in the literature that the directed motion of particles in such a system cannot be achieved, we will show that directed motion can, in fact, be achieved provided the characteristic time of the particle passing through the interval between neighboring extremes in the potential is equal to the half-period of the time modulation. Consider a periodic potential U ( x ) (where x is the linear coordinate) with harmonic time modulation. The modulation starts at time t = 0 with the particle resting at the bottom of a potential well. The equation of motion for an overdamped particle in such a potential takes the form

dx 1 ∂U =− cos (ωt ) dt ξ ∂x

(1)

where x is the spatial coordinate, t is time, ξ is the hydrodynamic friction coefficient, and ω is the frequency of modulation. Eq. 1 can be simplified by introducing the velocity of the particle, as defined by

v ( x) = −

1 ∂U ξ ∂x

(2)

Here U ( x ) is the potential in absence of modulation. Eq. 1 is applicable over times that are much longer than the characteristic time of relaxation of particle inertia. Nevertheless, inertia effects will play a role in particle movement, as outlined below. Equations of motion similar to Eq.1 have been widely discussed in the literature 1-3, 8. Eq. 1 describes a pulsating ratchet, where the potential varies over time but has no change in spatial periodicity. However, it is usually argued that the pulsating ratchet works only with asymmetric potentials 9. The only ratchets in the literature where the symmetry of the potential plays no role ACS Paragon Plus Environment


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are travelling potential ratchets where the potential profile is shifting with time 1-3, 10-13. By contrast, the system considered here can provide directed particle motion in both symmetric and asymmetric potentials. Like the traveling potential ratchet, our system allows for particles to reside in a potential region where a positive force is active, provided certain conditions are met. The difference in our system from the travelling potential ratchet is that the particle does not always move together with the shifting potential, rather, it is always in a region where the acting force is positive, as a consequence of its precise location within the periodic potential during the time when the potential changes sign. Eq. 1 is a form of the generic equation of overdamped motion used in most works on Feynman ratchets. The exact Newton equation of motion describing these systems is

m

∂U  x ( t )  d 2x (t ) dx ( t ) =− cos (ωt ) − ξ 2 dt ∂x dt

(3)

where m is the particle mass. Usually, however, the time span under consideration is much larger than the characteristic relaxation time of acceleration:

τ=

m

ξ

(4)

In such cases, the particle acceleration and its inertia, respectively, play a minor role and Eq. 1 is valid. There are a few reports that consider the role of particle inertia. For example, in 9 the inertial term is employed in considering the symmetry break necessary for obtaining non-zero particle current in a system with a spatially periodic potential and a harmonic exciting force. In 3, 14, 15

the following Newton equation is used to study travelling potential ratchet:

m

∂U  x ( t ) − ut  d 2 x (t ) dx ( t ) =− −ξ 2 dt ∂x dt


(5)

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Here u is the velocity of travel of the potential profile. In those works, inertia is shown to be important for describing certain modes of the directed particle motion. In 11 an overdamped travelling potential ratchet with a stochastic exciting force is considered. In 12 the modes of particle motion in a separate travelling potential well and a periodic series of such wells are considered. The latter work demonstrates that in an overdamped mode both stationary and non-stationary capturing is possible. In stationary capturing, the captured particle moves together with the travelling potential well. In the non-stationary system, the captured particle moves together with the potential well only for a certain time and then rests again at some shifted position. In a periodic set of travelling potential wells the non-stationary captured particle moves with a velocity lower that the velocity of the travelling potential well. Accounting for inertia in that system extends the set of possible particle motion modes in the travelling periodic set of rectangular potential wells and changes the range of parameters that correspond to stationary and non-stationary capture. A sinusoidal travelling potential is considered in 13, where the system described by Eq. 5 is supplemented with a stochastic exciting force. It is shown that under certain conditions a travelling-wave force field can capture an underdamped Brownian particle. This effect is most efficient at zero temperature (i.e., without any stochastic exciting force, similar to the system considered in 12) and for wave speeds smaller than a threshold value independent of the damping constant. The system in 13 corresponds to a stationary capturing of the system described in 12, and the authors conclude that Brownian particles in the system are sensitive to both the amplitude of the force field potential, as well as possible asymmetries in its waveform. In contrast to 13, the authors of 12 note that an asymmetric potential is not absolutely required, nor are thermal fluctuations in the travelling potential ratchets, which is consistent with the conclusions made in 3.



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Although we will not study travelling potentials in this work, there are similarities in the resulting particle motion between our system and certain modes of traveling potential ratchets, as discussed below. Utilizing Eq. 2, the solution of Eq. 1 can be written as x

t

dx ' ∫x v ( x ') = ∫t cos (ωt ') dt ' 0 0

(6)

where ω is the angular frequency. At first glance, Eqs. 1 and 2 predict no directed particle motion, since the right-hand side is oscillating rather than acting as a driving force for directed motion. However, one should take into account that variable x in Eq.1 is not the coordinate that determines the potential distribution. Instead, it is the instant coordinate of the moving particle

x ( t ) . The more specific forms of Eqs. 1 and 2 are

dx ( t ) = v  x ( t )  cos (ωt ) dt

(7)

1 ∂U  x ( t )  ξ ∂x ( t )

(8)

v  x ( t )  = −

Eq. 7 does yield a non-zero time-averaged particle velocity

times much larger than the oscillation period

2π

ω

dx ( t ) dt

when averaged over

. However, time-averaged velocity is only

possible for certain types of particle motion. For example, Eq. 7 provides a time-averaged particle velocity when the particle motion includes a harmonic component ∆x ( t ) = xosc Cosωt , where xosc is the amplitude of oscillations. This situation is realized in the well-known Kapitza pendulum, where such oscillations yield a change in the equilibrium position of a mechanical system 14.


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Similar arguments have been made that the periodic character of a spatial distribution in the field potential with harmonic time modulation makes directed motion impossible. Nevertheless, it is intuitively clear that directed motion is possible when the particle spends more time in the spatial intervals where the positive force acts versus intervals where the force is negative. We will show that just this situation can be achieved by synchronization. Below, we demonstrate that the combination of a specific periodic potential distribution with a specific harmonic time modulation can provide directed motion of the particle. The negative force acting on the particle becomes zero and then changes its direction, becoming a positive force in equal intervals of time. Due to synchronization of the particle motion and the frequency of the time modulation, the particle moving through the region where the force is positive approaches the adjacent region with the negative force just at the moment when the force in the latter region reverses its sign to become positive. Due to inertial force, the particle penetrates the latter region despite the reversal. In this way, the particle moves through the regions with continual exposure to a positive force. In order to explain the principle that allows for directed motion, we begin with a simple solvable potential model. The difference from most works on Feynman ratchets is that we will not consider the establishment of a directed flux or current of particles. Instead, with the goal of applying our results to nano- or microengines described in 6, 15, we will restrict ourselves to the study of directed motion of a single particle, which could be the rotor of a nanoengine.

SYMMETRIC FORCE POTENTIAL A simple model for the system under discussion is a symmetric triangular force potential. The spatial profile of the particle velocity in such a system is illustrated in Fig. 1. An impermeable wall at the left side serves to establish the initial conditions for the system. In the absence of such a wall, a particle can reside anywhere from the bottom to the summit of the potential well at the start of the modulation, since it can move in both directions with equal



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probability. Thus, the direction of particle motion must be selected by an external factor such as an impermeable wall or some type of starting device. Consider a particle at t = 0 that is placed at position x = 0 at the bottom of the potential well. The particle is held at this position by a negative force pressing it to the wall. The depth of the potential well is decreased according to the function Cos (ωt ) , reaching zero at t =

Fig. 1a), then increases to its maximum value at ωt = π (Fig. 1b). At time t =

π (see 2ω

π , the particle 2ω

begins to move from left to right under the action of a positive force, and its motion is described by Eq. 6, which acquires the following form x

dx ' ∫0 −v0 =

x (t ) =

v0

ω

t

∫ cos (ωt ') dt '

(9)

π 2ω

[1 − sin(ωt )] at 0 ≤ x ≤ l ;

ωt ≥

π 2

(10)

Here, v0 is the overdamped particle velocity in the spatial region where the positive constant force acts and the time modulation is absent. If the velocity of the particle is high enough it can reach position x = l in this motion before t =

3π . At this position, the bottom of the adjacent 2ω

potential well is reached (see Fig 1b), provided the condition

v0 ≥ l is satisfied. 2ω

For reasons discussed below, a stable directed motion will only occur when

v0 =l 2ω

(11)

When this condition is satisfied the depth of the potential well approaches zero just at the moment when the particle reaches this position and the well becomes the potential hill increasing during the next period of the modulation process. At this moment, the potential force continues ACS Paragon Plus Environment

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to drive the particle through the interval l < x ≤ 2l from left to right in Fig. 1 until reaching position x = 2l . This process is continued, resulting in directed motion of the particle, as illustrated in Fig. 2.

Figure 1. Spatial profile of the particle velocity in a symmetric triangular force potential U(x) at different times (the respective triangular force potential profile U(x) is shown by dotted lines).

a) v ( x ) at 0 ≤ t

Non-Feynman Deterministic Pulsing Ratchet Based on a Symmetric

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