Brownian transport of finite size particles in a periodic channel coexisting with an energetic potential Qun Chen, Bao-quan Ai, and Jian-wen Xiong Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 033119 (2014); doi: 10.1063/1.4891318 View online: http://dx.doi.org/10.1063/1.4891318 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Confined Brownian ratchets J. Chem. Phys. 138, 194906 (2013); 10.1063/1.4804632 Hydrodynamically enforced entropic Brownian pump J. Chem. Phys. 138, 154107 (2013); 10.1063/1.4801661 Transport of Brownian particles confined to a weakly corrugated channel Phys. Fluids 22, 122004 (2010); 10.1063/1.3527546 Transport properties of Brownian particles confined to a narrow channel by a periodic potential Phys. Fluids 21, 102002 (2009); 10.1063/1.3226100 Brownian escape and force-driven transport through entropic barriers: Particle size effect J. Chem. Phys. 129, 184901 (2008); 10.1063/1.3009621
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CHAOS 24, 033119 (2014)
Brownian transport of finite size particles in a periodic channel coexisting with an energetic potential Qun Chen, Bao-quan Ai,a) and Jian-wen Xiong Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
(Received 1 April 2014; accepted 14 July 2014; published online 28 July 2014) Transport of particles with different sizes moving in a two-dimensional periodic channel is studied in the presence of an unbiased external force and a periodic energetic potential. While particles are going through entropic barrier resulting from the geometric restraints, the transport is also influenced by the energetic potential. For the case of an unbiased external force, the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Particles move to the left when smaller than a critical radius and larger than another critical radius, whereas particles move to the right in the range of two critical radii. Therefore, the results we have presented can contribute further to the C 2014 AIP Publishing LLC. invention of machines for particle separation. V [http://dx.doi.org/10.1063/1.4891318]
Transport of Brownian objects in confined media plays a fundamental role in many transport phenomena. In this paper, we studied transport of particles with different sizes moving in a two-dimensional periodic channel coexisting with a periodic energetic potential and found that the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Particles move to the left when smaller than a critical radius and larger than another critical radius, whereas particles move to the right in the range of two critical radii.
field is imposed to a confined environment, the interplay should take into consideration, and the mutual effect markedly contributes to the directed motion of Brownian particles.17–19 In previous system,17–19 the radius of the particles is out of consideration, which exists in many other studies. However, it is found that the size of particles affects largely the Brownian motion.15,20–24 In this paper, based on the previous work, we focus on the question of how the interplay between energetic potential and entropic barrier affects the transport of Brownian particles in confined structures, when the radius of particles is considered.
II. MODEL AND METHODS I. INTRODUCTION
The study on the transport of Brownian particles has attracted considerable interest in theoretical and experimental physics. Generally speaking, the study falls into two major categories: the energetic transport1–4 and the entropic transport.6–11 In the former case, a spatial and temporal asymmetry, together with unbiased nonequilibrium perturbations, leads to the directed motion of particles.1–4 In most of researches, the periodic energetic potential are free, without boundary.3 However, it is not the identical situation in the entropic transport. When moving in a confined structure, Brownian particles could exhibit peculiar kinetic behavior. The spatially varying geometric restraints lead to the entropic barriers, which promote or hinder the transport of objects, yielding important effects exhibiting peculiar properties.6–14 Both of these two different cases are equally important, which has a large number of implications in processes such as osmosis, catalysis, and particle separation.15,16 Most of studies have referred to the energetic transport or the entropic transport but few focus on the interplay between both of them. When a periodic energetic potential a)
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We consider the Brownian particles moving in a twodimensional periodic channel (see Fig. 1(b)), and the particle is subjected to an unbiased external force and another varying force due to the presence of an energetic potential. Its
FIG. 1. Sketch of the two-dimensional periodic channel. (a) Energetic potential with period L ¼ 1.0. (b) Schematic diagram of the two-dimensional structure restraining the motion of different size of particles. For example, the radius of red ball is 0.7. The period of the structure is L, b is the halfwidth of the bottleneck, and the minimum width of the structure is 2b.
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C 2014 AIP Publishing LLC V
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overdamped dynamics can be described by the Langevin equation gR
pffiffiffiffiffiffiffiffiffiffiffiffiffi dx @U ð xÞ ¼ þ FðtÞ þ gR kB T nx ðtÞ; dt @x
(1)
dy pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ gR kB T ny ðtÞ; dt
(2)
gR
where t is the time, x and y are positions of the particle along the x, y axes, respectively. U(x) is a periodic energetic potential, which has the same period L with the channel. F(t) is an unbiased external force. gR is the friction coefficient of the particles, kB is the Boltzmann constant, and T is the temperature. Here, nx,y(t) is the standard Gaussian white noise and satisfies the following relations: hnm ðtÞi ¼ 0; hnm ðtÞnn ðt0 Þi ¼ 2dmn dðt t0 Þ;
(3) m; n ¼ x; y;
(4)
where h…i denotes an ensemble average over the distribution of the random forces. d(t) is the Dirac delta function. When the size of the Brownian particles is considered, the particle radius R (particles are spherical) has an effect on the friction coefficient gR, and it is described by Stokes’ law gR ¼ 6pR. For such system (see Fig. 1), its geometry is given by the boundary function x(x),23 which denotes the half-width of the structure x < C; b þ a1 x; (5) xðxÞ ¼ b þ a2 ðL xÞ; x C; where C ¼ La2/(a1 þ a2) corresponds to the location of the maximum half-width. a1 and a2 are the slopes of the boundary of the channel. x ¼ x mod L. Since the radius of particles is considered, an available space for particles is no longer the total width of the channel, and it can be described by the following equations:23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 0 x < Op ; b R2 x2 ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi > > < b þ a1 x R 1 þ a21 ; Op x < Cp ; pffiffiffiffiffiffiffiffiffiffiffiffiffi xeff ðxÞ ¼ b þ a2 ðL xÞ R 1 þ a22 ; Cp x < Lp ; > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : b R2 ð Lp x < L; x LÞ2 ; (6) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi where Lp ¼ L Ra2 = 1 þ a22 ; Op ¼ Ra1 = 1 þ a21 ; Cp pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ C þ R½ 1 þ a21 1 þ a22 =ða1 þ a2 Þ. We define a ¼ ða1 þ a2 Þ=2 and the asymmetry parameter Da ¼ (a1 a2)/2. For the sake of convenience, we can express the Langevin Eqs. (1) and (2) in a dimensionless form rffiffiffi d^ x 1 @ U^ ðx^Þ f ð^t Þ 1 ¼ þ þ n ð^t Þ; (7) r @^ x r r x^ d^t rffiffiffi d^ y 1 (8) ¼ n^ ð^t Þ; ^ r y dt
where the dimensionless variables are x^ ¼2 Lx ; y^ ¼ Ly ; ^ x Þ ¼ UðxÞ, f ð^t Þ ¼ FðtÞL ; ^t ¼ t ; s ¼ L gmax , and r ¼ R=b; Uð^ kB T s kB T kB T gmax ¼ 6pb. The friction coefficient is given by g ¼ rgmax. For simplicity, we shall omit the notation in all quantities and only the dimensionless variables are used. In this system, the corresponding Fick-Jacobs equation reads5,12–14,23 @Pð x; tÞ @ @P @j 0 ¼ D ð xÞ þ V ð xÞP (9) ¼ : @t @x @x @x From Eq. (10), we can find @P 0 þ V ðxÞP ; j ¼ D ð xÞ @x where DðxÞ ¼ 1r
1 ½1þx0eff ðxÞ2 1=3
(10)
is the position-dependent diffu-
sion coefficient. Note that Kalinay5 and Martens12 have given the rigorously derived expression for D(x). We checked the difference between these two expressions and found that in our systems the difference is very small. Here, jx0eff ðxÞj 1 is required for approaching to exact values. The free energy V(x) reads12–14,23 VðxÞ ¼ UðxÞ f ðtÞx TSðxÞ;
(11)
where TS ¼ln½2xeff ðxÞ is the entropic contribution, UðxÞ is the periodic energetic potential, e.g., ¼ U0 sin 2px L U(x þ L) ¼ U(x). U0 is the energetic potential height. f(t) is an external force, reading 8 > 1 > < f0 ; ns < t < ns þ s; 2 (12) f ðtÞ ¼ > > f0 ; ns þ 1 s < t < ðn þ 1Þs; ; : 2 where s is the period of the external force and f0 is the amplitude of the external force f(t). The relaxation 2times for longitudinal and transverse L2 L direction are sx ¼ 2Dx and sy ¼ 2Dy , respectively, where Lx,y are the maximal length of the x, y direction in one period, and D is diffusive constant. If F(t) changes very slowly with respect to t, namely, its period is longer than any other time scale of the system (s sx, s sy), there exists a quasistatic state. In the quasistatic state, we can get the average current J¼
jðf0 Þ þ jðf0 Þ ; 2
(13)
where the current j(f0) is obtained from Eq. (10) j ðf 0 Þ ¼
Ð1 0
1 exp½f0 Ð xþ1 eV ðyÞ dy eV ðxÞ dx x D ð yÞ
(14)
and the nonlinear mobility is defined as l ¼ j(f0)/f0. III. RESULTS AND DISCUSSION
In order to check the validity of the Fick-Jacobs approach, it is necessary to compare Brownian dynamics
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FIG. 2. Comparison of Brownian dynamics simulations (markers) based on Eqs. (1) and (2) with the Fick-Jacobs approach, Eq. (14) (lines). (a) The nonlinear a ¼ 1:25, Da ¼ 0.75, L ¼ 1.0, and mobility l versus f0 at r ¼ 0.3. (b)The nonlinear mobility l versus r at f0 ¼ 30.0. The other parameters are U0 ¼ 1.25, b ¼ 0.1 L.
simulations based on Eqs. (1) and (2) with Eq. (14). The compared results are shown in Figs. 2(a) and 2(b). From the figures, it is found that at lower values of f0 (f0 < 55.0) the analytical results match perfectly with the simulations whereas deviations occur at higher values of f0. The accuracy of the Fick-Jacobs description becomes worse for larger values of f0 because the assumption of equilibration in the transverse direction, which supports the elimination of the y coordinates, fails for higher values of the applied force. Thus, for not very large values of f0, we can use the FickJacobs approach to study the transport of Brownian particles in the confined structures. In addition, with increasing particle size the range of the applicability of the Fick-Jacobs method increases. In this paper, we set L ¼ 0.1 and b ¼ 0.1 L and all results presented in Figs. (3)–(7) are derived by the numerical evaluation of Eqs. (13) and (14).
FIG. 3. The current J versus energetic potential height U0 for different values of r. The markers in the inset represent the results obtained by Brownian dynamics simulations. The other parameters are a ¼ 1:25, Da ¼ 0.75, and f0 ¼ 30.0.
Figure 3 plots the current J as a function of energetic potential height U0 for different radius r at f0 ¼ 30.0. The current is a peaked function of U0. The current is positive for small values of U0, which indicates that the current is dominated by the entropic barrier. For very large values of U0, the barrier of the energetic potential is too high for particles to pass through. For intermediate potential height U0, e.g., U0 ¼ 1.25, the energetic potential competes with the entropic barrier, the radius r can determine the competition, particles of different sizes move in opposite directions. For example, particles of r ¼ 0.1 move towards the left, while particles of r ¼ 0.7 move towards the right. Therefore, one can separate particles of different radii by choosing a suitable U0. In order to further illustrate the competition between the energetic potential height U0 and the external force amplitude f0, we plot the current contours on the U0–f0 plane in Figs. 4(a) and 4(b). It is found that the current is always positive for U0 < 0.5 and goes to zero when U0 > 8.0. For a given f0, on increasing U0 from zero, the current first is positive, then becomes negative, and finally goes to zero. Due to the competition between U0 and f0, the current reversal can occur. Thus, particle separation may be realized. In addition, it seems that the zero-current contour line does not change when r is varied. In order to check this phenomenon, the zero-current contours on the U0–f0 plane are shown in Fig. 4(c) for different values of r. Obviously, the zero-current contour lines change with the radius r. However, this change is very small in the current contours on the U0–f0 plane. In addition, we have carefully checked the numerical results and found that j(f0) ¼j(f0) along the J ¼ 0 line for all parameter combinations (f0, U0). Along the separation line (J ¼ 0), j(f0) or j(f0) can not be equal to zero when f0 6¼ 0. Figure 5 shows the current J as a function of the external force amplitude f0 for different r at U0 ¼ 1.25. For small values of f0, the current is negative and demonstrates a valley shaped structure. There exists an optimum f0 at which the current takes its maximum negative value. As f0 increases, the current reversal will occur and the position of the current
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FIG. 4. (a) The current contours on the U0–f0 plane for r ¼ 0.1. (b) The current contours on the U0–f0 plane for r ¼ 0.7. The red lines and the blue lines denote a ¼ 1:25 the positive and negative current, respectively. (c) The zero-current contours on the U0–f0 plane for different values of r. The other parameters are and Da ¼ 0.75.
reversal is varying with radii. For example, when f0 ¼ 30.0, the force competes with the entropic barrier, the radius r takes over the leading role in particles transport, particles of
FIG. 5. The current J versus external force amplitude f0 for different values of r. The other parameters are a ¼ 1:25, Da ¼ 0.75, and U0 ¼ 1.25.
r ¼ 0.1 move towards the left, while particles of r ¼ 0.7 move towards the right. Therefore, we can separate particles and make them move in opposite directions by choosing a suitable value f0. To clearly see how the current J is influenced by the radius r, a plot of the current J as a function of r at U0 ¼ 1.25 and f0 ¼ 30 is presented in Fig. 6(a). From the figure, it is found that particles move to the left for r < ra or r > rb, whereas particles move to the right for ra < r < rb. In our system, we consider the case where the external force does not depend on the particle size. However, for some cases, the external forces depend on the particle size. In Fig. 6(b), we show the current J as a function of r for different cases. It is found that the curves are different for different particles. However, for all kinds of particles, current reversal can occur when r is varied. In this work, we mainly focus on forceindependent-size particles. Figure 7(a) shows the current contours on the r–f0 plane at U0 ¼ 1.25. When r is varied, the current is always negative for f0 < 23 and no current reversal occur. When f0 > 23, the current reversal appears on increasing r from 0 to 1. The green line corresponding to zero current shows the
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FIG. 6. (a) The current J versus the radius r of force-independent-size particles. (b) The current J versus the radius r for force-dependent-size particles: (I) a ¼ 1:25, and polymers (f0 / r1/4) (II) surface-charged colloids (f0 / r2), (III) charge or gravity (f0 / r3). The other parameters are U0 ¼ 1.25, f0 ¼ 30.0, Da ¼ 0.75.
FIG. 7. (a) The current contours on the r–f0 plane for U0 ¼ 1.25. (b) The red lines and the blue lines denote the positive and negative current, respectively. (b) a ¼ 1:25 and Da ¼ 0.75. The zero-current contours on r–f0 plane for different values of U0. The other parameters are
dependence of the threshold radii ra and rb on the external force amplitude f0. When f0 > 23, one can separate the finite size particles with the critical radii. Thus, it is very important to find the zero contours for particle separation. In Fig. 7(b), the zero-current contours on r–f0 plane are presented for different values of U0. It is found that the zero-current contours are sensitive to the parameter U0. On increasing U0, the zero current line moves to the right. For very large values of U0, this line disappears and particles cannot be separated. IV. CONCLUSION
In this paper, we studied the transport of finite size particles in a confined periodic structure. The geometric restraint results in the entropic barrier which has an important role in the particle transport. In this setup, the particle was subjected to an unbiased external force and there existed a periodic energetic potential. The dependence of the current J on the
potential height U0, the amplitude f0 of the external force, and the particle radius r is investigated based on the FickJacobs equation. It is found that the energetic potential competes with the entropic barrier for the case of the external force. All the particles move to the right when U0 < 0.5, since the current is dominated by entropic barrier for low energetic potential. When U0 > 2.0, particles move to the left, which indicates the large energetic potential determines the transport. However, there still exists a very small range of U0 for particles of different radius r moving to different directions. In addition, the current is negative for small f0. As the external force increases, the current reversal will occur. For the case of f0, due to the competition between the energetic potential and the entropic barrier, the direction of particles depends on the radius r. For example, when U0 ¼ 1.25 and f0 ¼ 30.0, particles move towards the left for r < ra and r > rb, while particles move towards the right for ra < r < rb. Therefore, the particles separation can be realized.
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The results we have presented contribute further to the invention of machines for particle separation. Some works have been shown that entropy can be used to sort Brownian particles according to their size. Reguera and coworkers23 presented a purely entropic particle splitting mechanism, which is able to separate particles of different sizes. Later, they25 found that a very efficient and fast separation with a practically 100% purity can be achieved by a proper optimization of the controllable variables. Martens and coworkers26 extended this separation mechanism and found that hydrodynamic flows can be used to achieve lower diffusivity of particles resulting in more efficient separation efficiency. In these separation devices, the biased force is necessary. In our setup, no biased force is needed, and the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Besides particle separation, the results we have presented may also have other applications, such as molecular motor movement through the microtubule in the absence of any net macroscopic force, ion transport through ion channels, transport in zeolites, and nanostructures of complex geometry. ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China (Grant No. 11175067), the PCSIRT (Grant No. IRT1243), and the Natural Science Foundation of Guangdong Province, China (Grant Nos. S2011010003323 and S2012010010661). 1
P. Pechukas and P. H€anggi, Phys. Rev. Lett. 73, 2772–2775 (1994). P. Reimann, R. Bartussek, R. H€aussler, and P. H€anggi, Phys. Lett. A 215, 26–31(1996); P. Reimann, Phys. Rep. 290, 149–155 (1997). 3 P. H€anggi and F. Marchesoni, Rev. Mod. Phys. 81, 387(2009). 2
Chaos 24, 033119 (2014) 4
S. Denisov, P. H€anggi, and J. L. Mateos, Am. J. Phys. 77, 602 (2009). P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006). D. Reguera, G. Schmid, P. S. Burada, J. M. Rubı, P. Reimann, and P. H€anggi, Phys. Rev. Lett. 96, 130603 (2006). 7 B. Q. Ai and L. G. Liu, Phys. Rev. E 74, 051114 (2006); B. Q. Ai, ibid. 80, 011113 (2009). 8 P. S. Burada, G. Schmid, P. Talkner, P. H€anggi, D. Reguera, and J. M. Rubı, BioSystems 93, 16 (2008). 9 A. M. Berezhkovskii, L. Dagdug, Y. A. Makhnovskii, and V. Y. Zitserman, J. Chem. Phys. 132, 221104 (2010). 10 V. Y. Zitserman, A. M. Berezhkovskii, A. E. Antipov, and Y. A. Makhnovskii, J. Chem. Phys. 135, 121102 (2011). 11 M. Borromeo, F. Marchesoni, and P. K. Ghosh, J. Chem. Phys. 134, 051101 (2011). 12 S. Martens, G. Schmid, L. Schimansky-Geier, and P. H€anggi, Phys. Rev. E 83, 051135 (2011). 13 L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012). 14 I. Pineda, J. Alvarez-Ramirez, and L. Dagdug, J. Chem. Phys. 137, 174103 (2012). 15 L. Gorre-Talini, J. P. Spatz, and P. Silberzan, Chaos 8, 650 (1998); D. Ertas, Phys. Rev. Lett. 80, 1548 (1998); T. A. J. Duke and R. H. Austin, ibid. 80, 1552 (1998). 16 M. C. Daniel and D. Astruc, Chem. Rev. 104, 293–346 (2004). 17 P. S. Burada, P. H€anggi, F. Marchesoni, G. Schmid, and P. Talkner, ChemPhysChem 10, 45–54 (2009). 18 P. S. Burada, Y. Li, W. Riefler, and G. Schmid, Chem. Phys. 375, 514 (2010). 19 P. Malgaretti, I. Pagonabarraga, and J. M. Rubı, Phys. Rev. E 85, 010105 (2012); J. Chem. Phys. 138, 194906 (2013). 20 C. Kettner, P. Reimann, P. H€anggi, and F. M€ uller, Phys. Rev. E 61, 312–323 (2000). 21 L. Dagdug, A. M. Berezhkovskii, Y. A. Makhnovskii, and V. Y. Zitserman, J. Chem. Phys. 129, 184706 (2008). 22 W. Riefler, G. Schmid, P. S. Burada, and P. H€anggi, J. Phys. Condens. Matter 22, 454109 (2010). 23 D. Reguera, A. Luque, P. S. Burada, G. Schmid, J. M. Rubı, and P. H€anggi, Phys. Rev. Lett. 108, 020604 (2012). 24 B. Q. Ai and J. C. Wu, J. Chem. Phys. 139, 034114 (2013). 25 T. Motz, G. Schmid, P. H€anggi, D. Reguera, and J. M. Rubı, e-print arXiv:1406.4708. 26 S. Martens, G. Schmid, A. V. Straube, L. Schimansky-Geier, and P. H€anggi, Eur. Phys. J.: Spec. Top. 222, 2453 (2013). 5 6
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.119.67.237 On: Tue, 18 Nov 2014 12:00:46
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CHAOS 24, 033119 (2014)
Brownian transport of finite size particles in a periodic channel coexisting with an energetic potential Qun Chen, Bao-quan Ai,a) and Jian-wen Xiong Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
(Received 1 April 2014; accepted 14 July 2014; published online 28 July 2014) Transport of particles with different sizes moving in a two-dimensional periodic channel is studied in the presence of an unbiased external force and a periodic energetic potential. While particles are going through entropic barrier resulting from the geometric restraints, the transport is also influenced by the energetic potential. For the case of an unbiased external force, the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Particles move to the left when smaller than a critical radius and larger than another critical radius, whereas particles move to the right in the range of two critical radii. Therefore, the results we have presented can contribute further to the C 2014 AIP Publishing LLC. invention of machines for particle separation. V [http://dx.doi.org/10.1063/1.4891318]
Transport of Brownian objects in confined media plays a fundamental role in many transport phenomena. In this paper, we studied transport of particles with different sizes moving in a two-dimensional periodic channel coexisting with a periodic energetic potential and found that the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Particles move to the left when smaller than a critical radius and larger than another critical radius, whereas particles move to the right in the range of two critical radii.
field is imposed to a confined environment, the interplay should take into consideration, and the mutual effect markedly contributes to the directed motion of Brownian particles.17–19 In previous system,17–19 the radius of the particles is out of consideration, which exists in many other studies. However, it is found that the size of particles affects largely the Brownian motion.15,20–24 In this paper, based on the previous work, we focus on the question of how the interplay between energetic potential and entropic barrier affects the transport of Brownian particles in confined structures, when the radius of particles is considered.
II. MODEL AND METHODS I. INTRODUCTION
The study on the transport of Brownian particles has attracted considerable interest in theoretical and experimental physics. Generally speaking, the study falls into two major categories: the energetic transport1–4 and the entropic transport.6–11 In the former case, a spatial and temporal asymmetry, together with unbiased nonequilibrium perturbations, leads to the directed motion of particles.1–4 In most of researches, the periodic energetic potential are free, without boundary.3 However, it is not the identical situation in the entropic transport. When moving in a confined structure, Brownian particles could exhibit peculiar kinetic behavior. The spatially varying geometric restraints lead to the entropic barriers, which promote or hinder the transport of objects, yielding important effects exhibiting peculiar properties.6–14 Both of these two different cases are equally important, which has a large number of implications in processes such as osmosis, catalysis, and particle separation.15,16 Most of studies have referred to the energetic transport or the entropic transport but few focus on the interplay between both of them. When a periodic energetic potential a)
Email: [email protected]
1054-1500/2014/24(3)/033119/6/$30.00
We consider the Brownian particles moving in a twodimensional periodic channel (see Fig. 1(b)), and the particle is subjected to an unbiased external force and another varying force due to the presence of an energetic potential. Its
FIG. 1. Sketch of the two-dimensional periodic channel. (a) Energetic potential with period L ¼ 1.0. (b) Schematic diagram of the two-dimensional structure restraining the motion of different size of particles. For example, the radius of red ball is 0.7. The period of the structure is L, b is the halfwidth of the bottleneck, and the minimum width of the structure is 2b.
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C 2014 AIP Publishing LLC V
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033119-2
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overdamped dynamics can be described by the Langevin equation gR
pffiffiffiffiffiffiffiffiffiffiffiffiffi dx @U ð xÞ ¼ þ FðtÞ þ gR kB T nx ðtÞ; dt @x
(1)
dy pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ gR kB T ny ðtÞ; dt
(2)
gR
where t is the time, x and y are positions of the particle along the x, y axes, respectively. U(x) is a periodic energetic potential, which has the same period L with the channel. F(t) is an unbiased external force. gR is the friction coefficient of the particles, kB is the Boltzmann constant, and T is the temperature. Here, nx,y(t) is the standard Gaussian white noise and satisfies the following relations: hnm ðtÞi ¼ 0; hnm ðtÞnn ðt0 Þi ¼ 2dmn dðt t0 Þ;
(3) m; n ¼ x; y;
(4)
where h…i denotes an ensemble average over the distribution of the random forces. d(t) is the Dirac delta function. When the size of the Brownian particles is considered, the particle radius R (particles are spherical) has an effect on the friction coefficient gR, and it is described by Stokes’ law gR ¼ 6pR. For such system (see Fig. 1), its geometry is given by the boundary function x(x),23 which denotes the half-width of the structure x < C; b þ a1 x; (5) xðxÞ ¼ b þ a2 ðL xÞ; x C; where C ¼ La2/(a1 þ a2) corresponds to the location of the maximum half-width. a1 and a2 are the slopes of the boundary of the channel. x ¼ x mod L. Since the radius of particles is considered, an available space for particles is no longer the total width of the channel, and it can be described by the following equations:23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 0 x < Op ; b R2 x2 ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi > > < b þ a1 x R 1 þ a21 ; Op x < Cp ; pffiffiffiffiffiffiffiffiffiffiffiffiffi xeff ðxÞ ¼ b þ a2 ðL xÞ R 1 þ a22 ; Cp x < Lp ; > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : b R2 ð Lp x < L; x LÞ2 ; (6) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi where Lp ¼ L Ra2 = 1 þ a22 ; Op ¼ Ra1 = 1 þ a21 ; Cp pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ C þ R½ 1 þ a21 1 þ a22 =ða1 þ a2 Þ. We define a ¼ ða1 þ a2 Þ=2 and the asymmetry parameter Da ¼ (a1 a2)/2. For the sake of convenience, we can express the Langevin Eqs. (1) and (2) in a dimensionless form rffiffiffi d^ x 1 @ U^ ðx^Þ f ð^t Þ 1 ¼ þ þ n ð^t Þ; (7) r @^ x r r x^ d^t rffiffiffi d^ y 1 (8) ¼ n^ ð^t Þ; ^ r y dt
where the dimensionless variables are x^ ¼2 Lx ; y^ ¼ Ly ; ^ x Þ ¼ UðxÞ, f ð^t Þ ¼ FðtÞL ; ^t ¼ t ; s ¼ L gmax , and r ¼ R=b; Uð^ kB T s kB T kB T gmax ¼ 6pb. The friction coefficient is given by g ¼ rgmax. For simplicity, we shall omit the notation in all quantities and only the dimensionless variables are used. In this system, the corresponding Fick-Jacobs equation reads5,12–14,23 @Pð x; tÞ @ @P @j 0 ¼ D ð xÞ þ V ð xÞP (9) ¼ : @t @x @x @x From Eq. (10), we can find @P 0 þ V ðxÞP ; j ¼ D ð xÞ @x where DðxÞ ¼ 1r
1 ½1þx0eff ðxÞ2 1=3
(10)
is the position-dependent diffu-
sion coefficient. Note that Kalinay5 and Martens12 have given the rigorously derived expression for D(x). We checked the difference between these two expressions and found that in our systems the difference is very small. Here, jx0eff ðxÞj 1 is required for approaching to exact values. The free energy V(x) reads12–14,23 VðxÞ ¼ UðxÞ f ðtÞx TSðxÞ;
(11)
where TS ¼ln½2xeff ðxÞ is the entropic contribution, UðxÞ is the periodic energetic potential, e.g., ¼ U0 sin 2px L U(x þ L) ¼ U(x). U0 is the energetic potential height. f(t) is an external force, reading 8 > 1 > < f0 ; ns < t < ns þ s; 2 (12) f ðtÞ ¼ > > f0 ; ns þ 1 s < t < ðn þ 1Þs; ; : 2 where s is the period of the external force and f0 is the amplitude of the external force f(t). The relaxation 2times for longitudinal and transverse L2 L direction are sx ¼ 2Dx and sy ¼ 2Dy , respectively, where Lx,y are the maximal length of the x, y direction in one period, and D is diffusive constant. If F(t) changes very slowly with respect to t, namely, its period is longer than any other time scale of the system (s sx, s sy), there exists a quasistatic state. In the quasistatic state, we can get the average current J¼
jðf0 Þ þ jðf0 Þ ; 2
(13)
where the current j(f0) is obtained from Eq. (10) j ðf 0 Þ ¼
Ð1 0
1 exp½f0 Ð xþ1 eV ðyÞ dy eV ðxÞ dx x D ð yÞ
(14)
and the nonlinear mobility is defined as l ¼ j(f0)/f0. III. RESULTS AND DISCUSSION
In order to check the validity of the Fick-Jacobs approach, it is necessary to compare Brownian dynamics
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FIG. 2. Comparison of Brownian dynamics simulations (markers) based on Eqs. (1) and (2) with the Fick-Jacobs approach, Eq. (14) (lines). (a) The nonlinear a ¼ 1:25, Da ¼ 0.75, L ¼ 1.0, and mobility l versus f0 at r ¼ 0.3. (b)The nonlinear mobility l versus r at f0 ¼ 30.0. The other parameters are U0 ¼ 1.25, b ¼ 0.1 L.
simulations based on Eqs. (1) and (2) with Eq. (14). The compared results are shown in Figs. 2(a) and 2(b). From the figures, it is found that at lower values of f0 (f0 < 55.0) the analytical results match perfectly with the simulations whereas deviations occur at higher values of f0. The accuracy of the Fick-Jacobs description becomes worse for larger values of f0 because the assumption of equilibration in the transverse direction, which supports the elimination of the y coordinates, fails for higher values of the applied force. Thus, for not very large values of f0, we can use the FickJacobs approach to study the transport of Brownian particles in the confined structures. In addition, with increasing particle size the range of the applicability of the Fick-Jacobs method increases. In this paper, we set L ¼ 0.1 and b ¼ 0.1 L and all results presented in Figs. (3)–(7) are derived by the numerical evaluation of Eqs. (13) and (14).
FIG. 3. The current J versus energetic potential height U0 for different values of r. The markers in the inset represent the results obtained by Brownian dynamics simulations. The other parameters are a ¼ 1:25, Da ¼ 0.75, and f0 ¼ 30.0.
Figure 3 plots the current J as a function of energetic potential height U0 for different radius r at f0 ¼ 30.0. The current is a peaked function of U0. The current is positive for small values of U0, which indicates that the current is dominated by the entropic barrier. For very large values of U0, the barrier of the energetic potential is too high for particles to pass through. For intermediate potential height U0, e.g., U0 ¼ 1.25, the energetic potential competes with the entropic barrier, the radius r can determine the competition, particles of different sizes move in opposite directions. For example, particles of r ¼ 0.1 move towards the left, while particles of r ¼ 0.7 move towards the right. Therefore, one can separate particles of different radii by choosing a suitable U0. In order to further illustrate the competition between the energetic potential height U0 and the external force amplitude f0, we plot the current contours on the U0–f0 plane in Figs. 4(a) and 4(b). It is found that the current is always positive for U0 < 0.5 and goes to zero when U0 > 8.0. For a given f0, on increasing U0 from zero, the current first is positive, then becomes negative, and finally goes to zero. Due to the competition between U0 and f0, the current reversal can occur. Thus, particle separation may be realized. In addition, it seems that the zero-current contour line does not change when r is varied. In order to check this phenomenon, the zero-current contours on the U0–f0 plane are shown in Fig. 4(c) for different values of r. Obviously, the zero-current contour lines change with the radius r. However, this change is very small in the current contours on the U0–f0 plane. In addition, we have carefully checked the numerical results and found that j(f0) ¼j(f0) along the J ¼ 0 line for all parameter combinations (f0, U0). Along the separation line (J ¼ 0), j(f0) or j(f0) can not be equal to zero when f0 6¼ 0. Figure 5 shows the current J as a function of the external force amplitude f0 for different r at U0 ¼ 1.25. For small values of f0, the current is negative and demonstrates a valley shaped structure. There exists an optimum f0 at which the current takes its maximum negative value. As f0 increases, the current reversal will occur and the position of the current
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FIG. 4. (a) The current contours on the U0–f0 plane for r ¼ 0.1. (b) The current contours on the U0–f0 plane for r ¼ 0.7. The red lines and the blue lines denote a ¼ 1:25 the positive and negative current, respectively. (c) The zero-current contours on the U0–f0 plane for different values of r. The other parameters are and Da ¼ 0.75.
reversal is varying with radii. For example, when f0 ¼ 30.0, the force competes with the entropic barrier, the radius r takes over the leading role in particles transport, particles of
FIG. 5. The current J versus external force amplitude f0 for different values of r. The other parameters are a ¼ 1:25, Da ¼ 0.75, and U0 ¼ 1.25.
r ¼ 0.1 move towards the left, while particles of r ¼ 0.7 move towards the right. Therefore, we can separate particles and make them move in opposite directions by choosing a suitable value f0. To clearly see how the current J is influenced by the radius r, a plot of the current J as a function of r at U0 ¼ 1.25 and f0 ¼ 30 is presented in Fig. 6(a). From the figure, it is found that particles move to the left for r < ra or r > rb, whereas particles move to the right for ra < r < rb. In our system, we consider the case where the external force does not depend on the particle size. However, for some cases, the external forces depend on the particle size. In Fig. 6(b), we show the current J as a function of r for different cases. It is found that the curves are different for different particles. However, for all kinds of particles, current reversal can occur when r is varied. In this work, we mainly focus on forceindependent-size particles. Figure 7(a) shows the current contours on the r–f0 plane at U0 ¼ 1.25. When r is varied, the current is always negative for f0 < 23 and no current reversal occur. When f0 > 23, the current reversal appears on increasing r from 0 to 1. The green line corresponding to zero current shows the
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FIG. 6. (a) The current J versus the radius r of force-independent-size particles. (b) The current J versus the radius r for force-dependent-size particles: (I) a ¼ 1:25, and polymers (f0 / r1/4) (II) surface-charged colloids (f0 / r2), (III) charge or gravity (f0 / r3). The other parameters are U0 ¼ 1.25, f0 ¼ 30.0, Da ¼ 0.75.
FIG. 7. (a) The current contours on the r–f0 plane for U0 ¼ 1.25. (b) The red lines and the blue lines denote the positive and negative current, respectively. (b) a ¼ 1:25 and Da ¼ 0.75. The zero-current contours on r–f0 plane for different values of U0. The other parameters are
dependence of the threshold radii ra and rb on the external force amplitude f0. When f0 > 23, one can separate the finite size particles with the critical radii. Thus, it is very important to find the zero contours for particle separation. In Fig. 7(b), the zero-current contours on r–f0 plane are presented for different values of U0. It is found that the zero-current contours are sensitive to the parameter U0. On increasing U0, the zero current line moves to the right. For very large values of U0, this line disappears and particles cannot be separated. IV. CONCLUSION
In this paper, we studied the transport of finite size particles in a confined periodic structure. The geometric restraint results in the entropic barrier which has an important role in the particle transport. In this setup, the particle was subjected to an unbiased external force and there existed a periodic energetic potential. The dependence of the current J on the
potential height U0, the amplitude f0 of the external force, and the particle radius r is investigated based on the FickJacobs equation. It is found that the energetic potential competes with the entropic barrier for the case of the external force. All the particles move to the right when U0 < 0.5, since the current is dominated by entropic barrier for low energetic potential. When U0 > 2.0, particles move to the left, which indicates the large energetic potential determines the transport. However, there still exists a very small range of U0 for particles of different radius r moving to different directions. In addition, the current is negative for small f0. As the external force increases, the current reversal will occur. For the case of f0, due to the competition between the energetic potential and the entropic barrier, the direction of particles depends on the radius r. For example, when U0 ¼ 1.25 and f0 ¼ 30.0, particles move towards the left for r < ra and r > rb, while particles move towards the right for ra < r < rb. Therefore, the particles separation can be realized.
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The results we have presented contribute further to the invention of machines for particle separation. Some works have been shown that entropy can be used to sort Brownian particles according to their size. Reguera and coworkers23 presented a purely entropic particle splitting mechanism, which is able to separate particles of different sizes. Later, they25 found that a very efficient and fast separation with a practically 100% purity can be achieved by a proper optimization of the controllable variables. Martens and coworkers26 extended this separation mechanism and found that hydrodynamic flows can be used to achieve lower diffusivity of particles resulting in more efficient separation efficiency. In these separation devices, the biased force is necessary. In our setup, no biased force is needed, and the competition between the energetic potential and entropic barrier leads to different transport direction of particles, which sensitively depends on the particles radius. Besides particle separation, the results we have presented may also have other applications, such as molecular motor movement through the microtubule in the absence of any net macroscopic force, ion transport through ion channels, transport in zeolites, and nanostructures of complex geometry. ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China (Grant No. 11175067), the PCSIRT (Grant No. IRT1243), and the Natural Science Foundation of Guangdong Province, China (Grant Nos. S2011010003323 and S2012010010661). 1
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