Confined Brownian ratchets Paolo Malgaretti, Ignacio Pagonabarraga, and J. Miguel Rubi Citation: J. Chem. Phys. 138, 194906 (2013); doi: 10.1063/1.4804632 View online: http://dx.doi.org/10.1063/1.4804632 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i19 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors THE JOURNAL OF CHEMICAL PHYSICS 138, 194906 (2013) Confined Brownian ratchets Paolo Malgaretti,a) Ignacio Pagonabarraga, and J. Miguel Rubi Department de Fisica Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain (Received 8 March 2013; accepted 29 April 2013; published online 21 May 2013) We analyze the dynamics of Brownian ratchets in a confined environment. The motion of the parti- cles is described by a Fick-Jakobs kinetic equation in which the presence of boundaries is modeled by means of an entropic potential. The cases of a flashing ratchet, a two-state model, and a ratchet under the influence of a temperature gradient are analyzed in detail. We show the emergence of a strong cooperativity between the inherent rectification of the ratchet mechanism and the entropic bias of the fluctuations caused by spatial confinement. Net particle transport may take place in situa- tions where none of those mechanisms leads to rectification when acting individually. The combined rectification mechanisms may lead to bidirectional transport and to new routes to segregation phe- nomena. Confined Brownian ratchets could be used to control transport in mesostructures and to engineer new and more efficient devices for transport at the nanoscale. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4804632] I. INTRODUCTION Modulations in the available explored region lead to gradients in the system effective free energy, by inducing a local bias Breaking detailed balance due to the presence of unbal- in its diffusion that can promote a macroscopic net velocity anced forces acting on a system causes rectification of thermal for aperiodic channel profiles12 or due to applied alternating fluctuations and leads to new dynamical behaviors, very dif- fields.20 ferent from those observed in equilibrium situations.1, 2 Typ- We will analyze the interplay between rectification and ical realizations of rectified motion include the transport of confinement, and will characterize the new features associ- particles under the action of unbiased forces of optical,3, 4 ated to confined Brownian ratchets (CBR). We show that mechanical,5, 6 or chemical7, 8 origin. Hence, the implications the presence of strong cooperative rectification21 between the of rectification has attracted the interest of researchers in a ratchet and the geometrical confinement may lead to rectifi- variety of fields, ranging from biology to nanoscience, due to cation even when none of them can rectify the particle current its relevance in the transport and motion at small scales.2, 9 on their own. Such an interplay strongly affects particle mo- Accordingly, the behavior of such small engines, referred to tion. To understand the mutual influence between both recti- as Brownian ratchets, have been deeply studied and several fying sources, we will analyze three different ratchet models, models that capture some of their main features have been namely, the flashing ratchet, the two-level ratchet, and a ther- proposed.1, 2, 10, 11 mal ratchet. In the first two cases, the equilibrium is broken by Given the small size of rectifying elements, it is likely the energy injected in the system through the intrinsic ratchet that their motion takes place close to boundaries or in confined mechanism as it happens in the case of molecular motors. In environments. Nevertheless, ratchet models usually assume the third one, the driving force is a thermal gradient that cou- that rectification develops in an unbound medium. Since rec- ples to the probability current, hence inducing a, local, Soret tification develops as an interplay associated to how a particle effect. experiences local forces while it explores the space around The article is distributed as follows. In Sec. II, we present it, confinement plays an important role because it signifi- the main features of entropic transport and formulate the ki- cantly reduces the number of allowed states of the rectifying netic framework, based on the Fick-Jacobs equation, that will elements. This restriction can be understood as an effective allow us to describe the evolution of the probability distribu- change of the entropy of the Brownian ratchet as it displaces tion of a particle in the presence of free energy barriers. The along the confined environment.12 ratchet models that will be analyzed in detail are described in The relevance of entropic barriers to promote entropic Sec. III. In Secs. IV– VII we discuss the different scenarios transport13, 14 in confined environments has been recognized generated by different symmetric/asymmetric ratchets and/or in a variety of situations that include molecular transport in channel shapes, and conclude in Sec. VIII where we draw the zeolites,15 ionic channels,16 or in microfluidic devices,17, 18 main conclusions and outlook of this piece of work. where their shape explains, for example, the magnitude of the rectifying electric signal observed experimentally.19 In fact, spatially varying geometric constraints provide them- selves an alternative means to rectify thermal fluctuations.12 II. PARTICLE DYNAMICS IN A CONFINED MEDIUM A Brownian ratchet, with diffusion constant D, under a)Author to whom correspondence should be addressed. Electronic mail: the action of a potential, V (r,t), moving in a confined en- [email protected] vironment characterized by a varying cross-section channel 0021-9606/2013/138(19)/194906/9/$30.00 138, 194906-1 © 2013 AIP Publishing LLC 194906-2 Malgaretti, Pagonabarraga, and Rubi J. Chem. Phys. 138, 194906 (2013) in terms of the maximum, hmax, and minimum, hmin chan- nel apertures. Therefore, ∂xA(x) is the driving force that con- A(x) tains entropic, ∂xS(x), and enthalpic, ∂x V (x), contributions. The range of validity of the Fick-Jacobs equation has been 13, 25 -kBTS(x) analyzed, and it has been found that introducing the vary- ing diffusion coefficient24 Δ φ V(x) = D0 D(x) α (8) + ∂h 2 1 ∂x FIG. 1. Brownian ratchet and entropic barriers. A Brownian motor moving in a confined environment will be sensitive to the free energy A(x) (solid) with α = 1/3, 1/2 for 3D,2D, respectively, with the reference generated by the ratchet potential V (x) (dotted) and the entropic potential diffusion D0 = kBT/γ (R) and γ (R)∝R, enhances the range of (dashed), −k TS(x), induced by the channel shape. B validity of the factorization assumption, Eq. (4). Although we will keep D(x) for completeness, the results do not change of width, h(x, z), such as the one depicted in Fig. 1, can be qualitatively if a constant diffusion coefficient, D0, is consid- characterized in terms of the probability distribution function ered instead. (pdf), P(r, t), which obeys the Smoluchowsky equation, The free energy difference over a channel period L ∂ =∇· ∇ + ∇ P (r,t) [βD W(r)P (r,t) D P (r,t)], (1) F = ∂x A(x)dx (9) ∂t 0 −1 where β = kBT is the inverse of the temperature, T,atwhich governs the particle current onset. Looking for the steady so- the particle diffuses, while kB stands for Boltzmann constant. lution of Eq. (5) in a periodic system, pst(0) = pst(L), we find = = Instead of being regarded as an explicit boundary condition, that a net current, J 0, arises only when F 0, which the geometrical constraint can be included, alongside any ad- can have both an enthalpic, L V (x)dx = 0, and entropic, 0 ditional potential the diffusing particle may be subject to, as L k T ln(h(x))dx = 0, origin. Indeed the picture of A(x)as an effective potential B 0 a free energy is suggestive: a net current sets only when the difference in free energy along the period is not vanishing. V (x), |y|≤h(x), & |z|≤Lz, W(r) = (2) Clearly, at equilibrium, periodic potentials, V (x), in periodic ∞, |y| >h(x)or|z| >L, z channels, h(x), do not give rise to any difference in the free where we have considered, without lack of generality, that the energy and consequently no current. The relative performance long axis of the channel coincides with the axis x, that par- of a ratchet in an uniform channel can be quantified in terms ticles cannot penetrate the confining channel walls, and that of the dimensionless parameter, the channel is periodic, W(r) = W(r + Lex ), of length L,as = Lv0 shown in Fig. 1, and has a finite section. If the channel width μ0 , (10) μF˜ 0 varies slowly, ∂xh 1, one can assume that the particle equi- librates in the transverse section on time scales smaller than defined as the ratio between the Brownian ratchet aver- = 1 L = the ones in which the particle experiences the variations in age speed,v ¯ L 0 J (x)dx with J(x) D[βp(x, t)∂xA(x) + channel section. It is then possible to factorize the pdf ∂xp(x, t)] derived from Eq. (5), and the average speed of ≡ − a particle with mobilityμ ˜ βD0 under the action of a uni- e βW(r) ≡ P (r,t) = p(x,t) , (3) form effective force, f0 F0/L. In the absence of intrinsic −βA(x) e ratchet rectification, F0 = 0 and μ0 remains 1.