Single-File Diffusion of Driven Interacting Colloids - World Scientific

December 24, 2014 9:49 WSPC/S1793-0480 204-BRL 1440008

Biophysical Reviews and Letters Vol. 9, No. 4 (2014) 413–434 c World Scientific Publishing Company DOI: 10.1142/S1793048014400086

Single-File Diffusion of Driven Interacting Colloids

Edith C. Eu´ an-D´ıaz Department of Physics, University of Antwerpen Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Division of Sciences and Engineering University of Guanajuato, Loma del Bosque 103 Lomas del Campestre, 37150 Le´ on Guanajuato, Mexico [email protected] Salvador Herrera-Velarde Subdirecci´ on de Postgrado e Investigaci´ on Instituto Tecnol´ ogico Superior de Xalapa Secci´ on 5A Reserva Territorial s/n 91096, Xalapa, Veracruz, Mexico Vyacheslav R. Misko and Fran¸cois M. Peeters Department of Physics, University of Antwerpen Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Ram´ on Casta˜ neda-Priego Division of Sciences and Engineering University of Guanajuato, Loma del Bosque 103 Lomas del Campestre, 37150 Le´ on Guanajuato, Mexico ramoncp@fisica.ugto.mx Received 12 August 2014 Revised 7 November 2014 Accepted 14 November 2014 Published 5 December 2014 The dynamical properties of interacting colloids spatially restricted to move in onedimensional channels [J. Chem. Phys. 133, 114902 (2010)] and subjected to external periodic energy landscapes [Phys. Rev. E 86, 081123 (2012)] have been recently reported in terms of the long-time self-diffusion behavior. However, the full description of the mean-square displacement, ranging from short times to long times, is still missing. Thus, by means of Brownian dynamics computer simulations, we revisit the process known as single-file diffusion in driven interacting colloidal systems at all time scales. In particular,

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E. C. Eu´ an-D´ıaz et al. we review three different pair potentials, namely, Weeks–Chandler–Andersen, Yukawa and superparamagnetic potentials. We mainly focus on the importance of the correlation between particles via the coupling among hydrodynamic interactions and the external periodic field, resulting in nontrivial particle dynamics along the file in systems composed of repulsively interacting particles. Special Issue Comments: This article reviews results on the dynamical properties of interacting colloids in a single file when they are subjected to external periodic energy landscapes presented before in [J. Chem. Phys. 33, 114902 (2010)] and [Phys. Rev. E 86, 081123 (2012)]. We analyze different interactions that cover from short to long ranges of the interparticle potentials. We mainly focus on the importance of the correlation between particles via the coupling among the hydrodynamic interactions and the external periodic field. Keywords: Colloids; single file; modulated potential; hydrodynamics; Brownian dynamics; structure; dynamics.

1. Introduction Diffusion is a well-known process related with the transport of molecules suspended in an aqueous medium. The latter one, commonly referred to as solvent, is crucial to determine the particle dynamical behavior. One example is the key role played by the water during the transport processes occurring in biological membranes. In ionic channels and water-conducting pores (aquaporins), one-dimensional (1D) confinement in conjunction with strong surface effects change the physical behavior of water4 and such changes affect the fluxes of ions through the membranes, as observed in the cellular membranes of the frog’s skin.14 In the particular case of macromolecules, more specifically, colloids immersed in a low Reynolds number solvent, the diffusion process can be described by a power law given by W (t) ∝ tα , with W (t) being the mean-square displacement (MSD). At short times, when the colloid only feels the collisions with the solvent molecules, the exponent α takes typically the value of 1, i.e. normal diffusion,15 however, at intermediate times, when the colloid is escaping from the first-neighbor colloidal layer, the dynamics becomes sub-diffusive, which means that α < 1.15 At long times, when the colloid has traveled over several mean-interparticle distances, its dynamics (depending on the degree of confinement) can be either normal (α = 1) or sub-diffusive (α < 1).15 In particular, when the colloids are confined in highly restricted geometries, such as quasi-one-dimensional (q1D) and 1D geometries, i.e. narrow cavities with a width of the order of the colloid size,35,34 the long-time MSD exhibits a sub-diffusive non-Fickian behavior called single-file diffusion (SFD), whose fingerprint, in absence of any external perturbation, is given by a particular value for the exponent α = 0.5.32,37,7,11,27,33,38,9,42,31 However, this relation may change due to the with of the channel, when the particles have different diffusion constants and under certain conditions regarding the waiting times that the particle spends before diffusing a considerable distance (for more details see Refs. 17–19 and 20).



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H¨ anggi and Marchesoni showed that the diffusion of a single file of hard rods driven by an external force F of spatial periodicity L obeys the sub-diffusive law46,22 2 D(F )t 2 2 , (1) lim [x(t) − x(t) ] t→∞ ρ π with D(F ) given by D(F ) =

L2 t2 (L, F ) − t(L, F )2 2 t(L, F )3

(2)

22,16 Additionally, Lin and tn (L, F ) being the nth moment of the first passage time.

et al. showed that the mobility factor, F ≡

1 ρ

D(F ) π ,

of an infinite system of hard 1 HR = ρ D0 S(0) ,33 rods, in absence of hydrodynamic interactions (HI), reduces to F π where S(0) is either the value of the static structure factor evaluated at q → 0 or the normalized isothermal compressibility.41 Equation (1) is also applicable for the transport of driven interacting particles subjected to a periodic external potential, uext (x).22,5 In this work, we report a few cases of interacting colloidal systems that obey relation (1) and, more interestingly, some that do not, even when the external field is spatially periodic. The deviations from the behavior described by Eq. (1) are closely related to the dynamical coupling between particles determined by the correlation among the repulsive direct interaction potential, long-ranged hydrodynamic interactions and the spatially periodic external field, as we discuss in detail further below. After the present Introduction, the article is organized as follows. In Sec. 2, we briefly explain the Brownian dynamics simulation method used to numerically evaluate the diffusion of interacting particles along the file. We also introduce the interaction potentials between colloids and the expressions of the physical quantities measured during the simulations. In Sec. 3, a detailed analysis of the results is presented. Finally, the paper ends with a section of concluding remarks. 2. Simulation Method, Interparticle Potential and Dynamical Observables The Brownian dynamics simulation scheme implemented in this work was first reported in Ref. 15. However, a full description of the algorithm can be found explicitly in Refs. 24, 25, 29 and 28. Here, we briefly summarize its important aspects. The HI are included in the Rotne–Prager mobility tensor.13 We typically consider N particles; N ≈ 400 with HI and N ≈ 1000 without HI. Particles move in a line of length L, which is linked to the particle number density according to the expression ρ = N/L. When the external field is switched on, L has to be chosen to guarantee the continuity of it on the borders of the line; this point is discussed further below. The packing fraction is ϕ = σρ and the mean-interparticle distance is defined as d ≡ ρ−1 .24,25,29


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The time step used in the BD simulations is ∆t = 2×10−4 (ρ2 D0 )−1 , with D0 = being the free-particle diffusion coefficient of particles of radius a immersed in a solvent of viscosity η, kB the Boltzmann constant and T the absolute temperature. The maximum time window reached in the simulations is tmax = 500 (ρ2 D0 )−1 , i.e. 2.5 × 106 time steps. To facilitate the analysis and reduce the set of parameters in our study, we use the following scaling factors: d for the distance; d2 /D0 for the time; and kB T for the energy. kB T 6πηa

2.1. Brownian dynamics The dynamics of an overdamped system, i.e. colloids immersed in a low Reynolds number fluid, is held on time scales longer than the momentum relaxation time, m ; this is known as the diffusive temporal regime,39 with m being the t τm = 6πηa mass of the colloid, which is in agreement with the Brownian dynamics simulation scheme15,24,25,27,29 used in this work. We express the discrete version of the Langevin equation for the position of particle i as,39  N  ri (t + ∆t) = ri (t) + β µij · [−∇rj U (r{N } ) + Fext j ]  j=1

+

N

∇rj · µij

j=1

  

∆t + ξi (t),

(3)

which gives us the new position of particle i at time t + ∆t. Equation (3) depends on the particle position at a previous time, ri (t), the net force acting on the particle and the stochastic force due to the collisions with the solvent molecules. The net force on particle i has two contributions: the force due to the particle–particle interaction; U is the total pair potential energy, and its coupling with the external field, Fext i . Hydrodynamic interactions are also included through the Rotner–Prager mobility tensor,39,13 µii = D0 I =

µij =

kB T I, 6πηa

kB T 8πηrij

r⊗r I+ 2 rij

+ kB T a

2

3 /4πηrij

r⊗r I − 2 , 3 rij

(4)

where I is a 3 × 3 unit matrix (in the 3D case) and the symbol ⊗ denotes a dyadic product. One notes that this mobility tensor includes the lowest corrections of particle size over the Oseen tensor description,13 and corresponds to the case of an unbounded fluid. One thus should keep in mind that our treatment of HI is only approximate for particles which are separated by a distance on the order of a. One also notes that the mobility tensor introduces long-range interactions and couples distant particles. Equation (3) reduces to the standard Langevin equation



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without HI when µij → D0 I.24,25,29 For the specific case of 1D systems, only the contribution in the x-direction of the mobility tensor is taken into account because, as it has been shown previously in Refs. 40 and 28, when the particles are inline, the movements of the particles along different directions are not hydrodynamically correlated with the ones in the direction of interest; the reduced expression for the mobility tensor in the x-direction is µij = [kB T /(4πηxij )] − [kB T a2 /(6πηx3ij )]. The stochastic term, ξ(t), mimics the action of a thermal bath and obeys the fluctuation-dissipation relation,13 ξi (∆t)ξj (∆t) = 2µij ∆t,

(5) 39,1

which is numerically implemented by a Cholesky decomposition. We deal with the dynamics of nondeformable particles in an unbounded fluid. One can then check that ∇rj · µij = 0 for the Rotner–Prager mobility tensor.39,13 2.2. Structure factor and mean-square displacement The static structure factor, S(qx ), characterizes the variations in the local density at the spatial frequency q ≡ 2π/λ, where λ is the wavelength. S(qx ) is obtained by using the relation,23,24 N

2 N

2 1 S(qx ) = cos(qx · xi ) + sin(qx · xi ) , (6) N i=1 i=1 the angular brackets · · · denote an ensemble average (or time average in an ergodic system) and qx is the magnitude of the wave vector in the x-direction. The mean-square displacement is computed by using the expression,24,25,29 Wx (t) = ∆x(t)2 =

N 1 [xi (t) − xi (0)]2 . N i=1

(7)

Then, for the sake of the discussion, at long times we fit our simulation data according to the relation, Wx (t) = F tα .

(8)

Thus, at long times all the effects associated to the HI can be described in terms of α and the particle mobility factor F . In the following plots, the error bars are smaller than the symbol size used to represent each curve. For the MSD, the associated errors are statistical significant only at reduced times tD0 /d2 > 350, yet still, there is a clear separation between the MSD curves and no overlap between error bars is observed during the time window employed in this work. 2.3. Interparticle pair potentials For homogeneous systems, we simulate colloidal particles with a very short-range interaction by using the Weeks–Chandler–Andersen potential, which was used


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recently to mimic the properties of a hard-rod system.45

σ 48 σ 24 βu(r) = 4 − + 0.25 , r r

(9)

In this work, we use = 2.5. A system of charged colloidal particles is also considered. The particles have radius a and interact via a repulsive screened-Coulomb potential. For distances r < σ ≡ 2a, the interaction is hard core, but for r > σ, two colloidal particles separated by a distance r interact via the repulsive part of the Derjaguin–Landau– Verwey–Overbeek (DLVO) pair potential,10,48,24,29 2 βu(r) = Zeff λB

eκa 2 e−κr , 1 + κa r

(10)

−1

where β ≡ (kB T ) is the inverse of the thermal energy, Zeff is the effective charge, λB = e2 /4πkB T is the Bjerrum length (in international units),44 with e being the proton charge, the solvent dielectric permittivity, and κ the Debye screening parameter.10,48 We have used the same set of parameters as those in Ref. 24. In the particular case of Yukawa systems, the packing fraction, ϕ = σρ, is the control parameter. Additionally, paramagnetic colloids have served as excellent model systems to investigate fundamental properties that are related to the role of hydrodynamics, melting transitions, order-disordered transitions and the elastic behavior in twodimensional crystals.43,26 In the experiments, an external and constant magnetic field is applied in the perpendicular direction of the air–water interface.50 This leads to a tunable quasi-long range magnetic dipole–dipole interaction between colloids. Such an interaction can be mathematically described by the potential,43,26 βu(r) =

Γ , r3

(11)

where r is the reduced separation (in units of the mean particle distance) between µ0 )χ2eff B 2 d−3 is the average interaction energy in units of two colloids and Γ ≡ β( 4π the thermal energy, µ0 is the vacuum susceptibility, B is the applied magnetic field, and χeff is the magnetic susceptibility of the particles. For paramagnetic colloids, the potential strength, Γ, is the relevant parameter to be varied, which is equivalent to change the particle number density because an increase in Γ results in an increase of collision rates between particles.49,43,26 In order to better understand the nature of the aforementioned interparticle potentials, Fig. 1 displays their distance-dependence behavior. As it can be seen, the WCA potential describes short-range interactions, whereas the superparamagnetic potential is the interaction with the longest range. Additionally, the Yukawa potential allows us to describe intermediate correlations among particles. Thus, we also report the way in which the range of the potential affects the particle dynamics along the file.



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Fig. 1. (Color online) Interparticle potentials used in the present work. Solid, dashed, and dasheddot lines correspond to the Weeks–Chandler–Andersen (??), Yukawa (10) and superparamagnetic (11) potentials, respectively.

2.4. External potential On the other hand, optical traps, created by the interference of highly focused laser beams, are of a great help to study complex fluids.2,21,30 In the last few years, optical traps have also been implemented in colloidal systems to induce a periodic optical substrate,3 i.e. periodic energy landscapes.12,27,35,34 N The total external energy can then be written as U ext = i=1 uext (ri ), where uext (ri ) is the external potential, usually referred to as the substrate, acting on particle i, given by the expression,24,25,27

2πxi ext u (xi ) = V0 sin , (12) aL where V0 is the amplitude or strength of the external potential and aL its periodicity. Now, it is appropriate to define the commensurability factor as24,25 p≡

d N = , aL naL

(13)

where naL is the number of sinusoidal potential periods within the channel of length L. 3. Results and Discussion Our results cover a wide range of files: homogeneous and heterogeneous, which are characterized by the set of parameters displayed in Ref. 15. In the former case,


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the packing fraction explored for the WCA potential is 0.2 ≤ ϕ ≤ 0.7, whereas for the Yukawa and superparamagnetic particles is 0.288 ≥ ϕ ≤ 0.43 and 0.25 ≤ Γ ≤ 4.67, respectively. On the other hand, a broad discussion about the effect of the commensurability in heterogeneous systems is presented further below. In the latter case, the packing fraction is fixed: ϕ = 0.43 for charged colloids and Γ = 4.67 for superparamagnetic particles. The commensurability factor takes the values of p = 14 , 13 , 12 , 23 , 34 , 1 and the strength of the external field is varied between V0 = 0.2kB T and V0 = 4.0kB T in steps of ∆V0 = 0.2kB T . 3.1. Homogeneous files with short-range interactions; V0 = 0 We turn to analyze, using the Brownian dynamics algorithm described above, the case reported by Sané et al. for those 1D systems composed of particles interacting via a WCA potential.45 We should mention that here particles are confined, strictly speaking, to a 1D geometry and walls are not explicitly included; Cui et al. analyze a similar case including the effect of the walls in Ref. 8. Such confinement sharply screens the HI and creates negative velocity correlations which lead to a slowing down of the motion. The static structure factor, see Fig. 2, does not depend on the HI; results with and without HI are basically the same. This means that the HI have no effect on the structure of the system under homogeneous conditions, as expected. The MSD curves for the systems displayed in Fig. 2 are shown in Fig. 3. For the time window indicated as the “region of interest”, i.e. long times, is found that the long-time self-diffusion decreases with the particle concentration and the MSDs with HI are, in general, larger than the MSDs in the absence of HI, which means

Fig. 2. (Color online) Static structure factor, S(q), of 1D colloidal systems composed of particles interacting with a WCA potential (??). Stars and triangles represent results obtained with and without HI, respectively.



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Fig. 3. (Color online) Mean-square displacements of 1D colloidal systems composed of particles interacting with a WCA potential (??). Stars and open symbols represent results obtained with and without HI, respectively. The time window where the dynamic factors, α and F , are calculated, is indicated between vertical lines. Dashed line is only a guide to the eye.

that HI promote particle diffusion. However, the MSD shows a dependence on time given by Eq. (8) with α ∼ 0.5. The mobility factor is then extracted by means of two routes: from the fits to the

MSD curves and using the expression F = F HR = ρ1 D0 S(0) with the information π of S(0) provided by the evaluation of the structure factor (see Fig. 2). The results are explicitly shown in Fig. 4. In particular, by neglecting HI, the mobility factor is completely described by F HR .45 The simulation with HI gives a mobility factor that follows the same behavior, but when one fits the MSD curves at long times the results are larger than the ones using the static route. We attribute this to the absence of physical walls in our model, in contrast with the inclusion of walls in the model of Sané et al.45 ; close to walls the particle diffusion decreases, see e.g. Ref. 6. Thus, the SFD for the WCA systems is accurately determined with the approximation proposed by Kollmann: W (t) ∝ t1/2 .32 As discussed in Ref. 15, the enhancement of the particle mobility can be understood in terms of the long-range hydrodynamic coupling between particles, see e.g. Refs. 50 and 43. This coupling also enhances the diffusion due to the collective diffusion induced and mediated by the HI. Collective diffusion is a dynamic process related with the cooperative movements of many particles that lead and promote a collective and faster diffusion of the particles in the file. However, a similar collective diffusion is observed at long-times in finite-size systems, i.e. circular channels,


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Fig. 4. (Color online) Mobility factor of 1D colloidal systems composed of particles interacting with a WCA potential (??). Open and filled symbols are results obtained with and without HI, respectively. Two different routes for obtaining the mobility factor are shown, the first one by using Eq. (1), and the second one by fitting the simulation data with the expression W (t) ≈ F ∗ t1/2 .

composed of particles interacting with long-range potentials and where the HI are not fully taken into account.47,36 We should stress that the exponent α and the mobility factor F for longer range repulsive systems were reported in Ref. 15 to elucidate the effects of the HI on the SFD. The values of the parameters predicted with HI are larger than those where HI are disregarded, although they behave similarly. Moreover, α is 0.5 for the case without HI, in excellent agreement with the theoretical predictions of Kollmann,32 and it takes the value of 0.56 with HI and also independently of the interaction potential between particles.15 To conclude this section, Table 1 summarizes main results presented here and in the longRef. 15. It contains the numerical values of the dynamic factors α and F ∗ in time regime which were obtained either through the approximation F ∗ ≈ ρ1 D0 S(0) π or W (t) ≈ F ∗ tα . Table 1 gives also a clear overview of the interactions boarded, their ranges and the control parameters that were used in the simulations.

3.2. Heterogeneous files; V0 = 0 We now present a detailed discussion on the influence of the external sinusoidal potential (see Eq. (12)) in an extended regime of commensurabilities, i.e. 0 < p ≤ 1, and its consequences on the dynamical properties of particles moving in periodic energy landscapes.15



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Table 1. Dynamic factors, α and F ∗ of 1D homogeneous colloidal systems for different values of their own control parameter according to the interaction potential, i.e. the packing fraction (φ) for the WCA and Yukawa potentials, and the average interaction energy (Γ) for the superparamanetic ones. Results shown inside brackets, (· · ·), were obtained with the inclusion of HI. ∗ stands for F ∗ calculated through The subscripts of F ∗ indicate the calculation method; F→S(0) q S(0) ∗ while F→W and α are obtained through W (t) ≈ F ∗ tα . the approximation F ∗ ≈ ρ1 D0 π (t) ∗ The values of F→W (t) and α for the Yukawa and Superparamagnetic case are taken from Fig. 4 in Ref. 15. Errors that are smaller than the last significant digit of the numerical value are not shown. Interaction

∗ F→S(0)

Control parameter

∗ F→W (t)

α

Short: WCA φ 0.2 0.3 0.3 0.3 0.3 0.3

0.58 0.63 0.49 0.57 0.47 0.48

± ± ± ± ± ±

0.05 0.18 0.07 0.20 0.18 0.15

(0.59 (0.57 (0.56 (0.62 (0.40 (0.42

± ± ± ± ± ±

0.03) 0.06) 0.10) 0.16) 0.05) 0.12)

0.79 0.69 0.65 0.51 0.41 0.33

(1.10) (1.18) (0.94) (1.00) (0.90) (0.69)

0.50 0.50 0.50 0.50 0.50 0.50

(0.50) (0.50) (0.50) (0.50) (0.50) (0.50)

Intermediate: Yukawa15 0.288 0.317 0.333 0.362 0.400 0.418 0.430

— — — — — — —

0.45 0.32 0.26 0.22 0.16 0.15 0.14

(0.75) (0.54) (0.45) (0.37) (0.28) (0.26) (0.24)

0.51 0.50 0.50 0.50 0.50 0.50 0.50

(0.56) (0.56) (0.56) (0.55) (0.56) (0.56) (0.56)

Γ 0.23 0.66 1.10 1.70 3.30 4.03 4.67

— — — — — — —

0.38 0.32 0.29 0.23 0.17 0.14 0.13

(0.57) (0.50) (0.46) (0.39) (0.30) (0.26) (0.23)

0.51 0.51 0.51 0.51 0.51 0.51 0.51

(0.56) (0.56) (0.56) (0.56) (0.56) (0.56) (0.56)

Long: Superparamagnetic15

3.2.1. Yukawa particles The MSDs of Yukawa particles for low commensurability factors are now analyzed. The case p = 1/4 is shown in Fig. 5. At short times, the particle dynamics is basically the same with and without HI, as expected, but at both intermediate and long times a clear separation is observed. In fact, in the latter time-regime, the fits give the results α = 0.5 and α ≈ 0.6 without and with HI, respectively. This early separation in the particle diffusion is stronger than in the previous homogeneous case. Furthermore, particles experience faster diffusion due to their hydrodynamical coupling with the external potential.15 Interestingly, it seems that the diffusion does not depend on the strength of the external field because the curves seem to collapse onto a single master curve.


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Fig. 5. (Color online) Mean-square displacement of Yukawa particles for the commensurability p = 1/4 and different values of the external potential strength, βV0 . Open symbols: results without HI. Stars: results with HI. Dashed line is only a guide to the eye.

For higher values of p, in particular, for p > 1/2 (see Fig. 6 for p = 3/4) the same particle dynamics as in the previous case is found. The inset of Fig. 6 shows the long-time regime. There, one observes that the inclusion of HI promotes a faster diffusion leading to a value of α ≈ 0.6. In particular, we find that α is a nonmonotonic function of p.15 When HI are neglected, the standard SFD scenario is practically found for all values of V0 , i.e. α ≈ 0.5. In this case, the hydrodynamic coupling becomes unimportant and the full functional form of the MSD depends exclusively on both the strength of the external field and the direct interactions among particles.15 For the particular case of p = 1/2 (see Fig. 7), at short times, the diffusion follows a normal single-file time dependence, not being the case for intermediate times as mentioned above. Focusing on the long-time behavior, which is shown in the inset of Fig. 7, one can notice that the particles diffuse slower for weak couplings with the external field, i.e. V0 < 2kB T . For strong couplings with the external field, the MSD curves show a systematical drop as V0 increases, even without considering HI explicitly; this slow decay should be associated with the distribution of particle along the sinusoidal field.15 In this case, due to the commensurability factor, one can find in average two particles per minimum of the potential (see Fig. 5 in Ref. 15). As explained in Ref. 15, due to their repulsiveness (see Eq. (10)), they push away from each other while trying to occupy the lowest energy point. Both particles behave as an effective dimer whose center of mass is, on average, at the position of the potential



425



minimum. When V0 (< 2kB T ) is small, due to thermal fluctuations, the dimers overcome the energetic barrier induced by external field. For larger V0 (> 2kB T ), the dimers are more localized each one in a well created by the external potential and each particle of the effective dimer tries to occupy the potential minimum.


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Fig. 8. (Color online) Mean-square displacement of Yukawa particles for the commensurability p = 1 and different values of the external potential strength, βV0 . Open symbols: results without HI. Stars: results with HI. Dashed line is only a guide to the eye.

This competition is reflected in the drop of the values of the MSD and its scaling with tα at long times.15 Figure 8 shows the long-time behavior of the MSD for p = 1. One can notice that the MSD shows a nonmonotonic dependence with V0 ≤ 1kB T ; at weak couplings (V0 ≤ 1kB T ) it decreases, but it increases at stronger couplings until it reaches a time dependence that seems to be independent of V0 . In Ref. 15 more precise limits for the weak and strong coupling are presented and a detailed explanation of this dynamical behavior is explicitly addressed. In brief, one can summarize this behavior as follows: The contribution of the HI at very weak couplings leads to a faster dynamics than the case without HI (see the case V0 = 0.2kB T in Fig. 8). Although, contrary to all the previous cases, for weak and strong couplings the HI lead to smaller α-values (the power of the time dependence of the MSD) than the case without HI.15 It was shown by Cui et al. in Ref. 8 that when the walls are considered, the particles inhibit each other’s motion rather than aid it. In order to understand this phenomenon without walls, in Ref. 28 is stated that the anticooperative hydrodynamic correlation functions for few particles inline where each of them is highly localized in the minima of the external potential. Taking this as a reference, we may attribute to all these slower motion regimes to the nontrivial hydrodynamic coupling with: the external field, the direct interaction and collective effects. When extending this discussion for strong coupling, one can notice that when V0 increases, the hydrodynamic contribution becomes less important, i.e. the results with and without HI become very similar. Furthermore, the localization



427

becomes even stronger in every minimum of the potential, giving rise to a re-entrant SFD, the latter being reflected in the time dependence of the MSD, which reaches the typical SFD value (α = 1/2). By looking at the values of the parameter α reported in Ref. 15, one finds that for very weak couplings, HI describe a faster diffusion than in the situation without HI. For intermediate couplings the opposite happens: hydrodynamic coupling among particles leads to a slower diffusion, i.e. α is smaller.15 The latter is totally different than the cases with p < 1. This means that the HI promote an anti-cooperative dynamics,28 which can be associated with a possible increase of the energy dissipation through the solvent as explicitly discussed in Ref. 15. So far we have found that, except for p = 1, HI promote a faster particle diffusion relative to the free-HI SFD. The way in which particles are accommodated along the channel depends on the particular choice of p, which also plays an important role on the dynamical behavior at long times. In other words, by properly changing p, one is able of tuning the long-time diffusion. 3.2.2. Paramagnetic colloids We turn now to the case of superparamagnetic colloids, where particles interact each other with a long-range potential given by Eq. (11). Due to the strong repulsion, the superparamagnetic colloids show similar particle configurations along the file as the Yukawa particles (data not shown).15 The MSD curves for p = 1/4 are shown in Fig. 9. Interestingly, they basically exhibit the same dependence as the one discussed in Fig. 5. This means that the dynamical behavior for this particular choice of p is independent of the nature of the repulsive interaction potential among particles. This feature is also observed in the case of p = 1/3 (data not shown). Furthermore, the inclusion of HI promotes a faster diffusion, as it is shown at long times in Fig. 9. A similar behavior is observed in the MSD curves for p = 3/4, which are displayed in Fig. 10. They exhibit characteristics rather similar to the ones discussed in the corresponding case for Yukawa particles, see e.g. Fig. 6. For the particular case p = 1/2, whose MSD curves are depicted in Fig. 11, a slower dynamical behavior at long times is found because the parameter α decays at smaller values of V0 and its magnitude is also smaller than the one obtained for Yukawa particles.15 This behavior might be associated to the distribution of particles on the potential minima.15 From Fig. 8 in Ref. 15, one can see the effective dimers around some minima, however, some particles are located on the maxima (in this case the probability is higher) due to the stronger repulsion at short distances and the long-range spatial correlation between particles. This increases the contribution of the interparticle potential to the particle distribution along the sinusoidal substrate. In fact, we can argue that particles find meta-stable equilibrium positions that are not frequently observed when the interaction potential is of short range and, as a consequence, it makes more difficult the collective diffusion along the channel.


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Fig. 9. (Color online) Mean-square displacement of superparamagnetic particles for the commensurability p = 1/4 and different values of the external potential strength, βV0 . Open symbols: results without HI. Stars: results with HI. Dashed line is only a guide to the eye.




429


Fig. 12. (Color online) Mean-square displacement of superparamagnetic particles for the commensurability p = 1 and different values of the external potential strength, βV0 . Open symbols: results without HI. Stars: results with HI. Dashed line is only a guide to the eye.

The long-time behavior for the case p = 1 is displayed in Fig. 12. A clear nonmonotonic long-time self-diffusion is observed. Furthermore, with the complementary information given in Ref. 15, one can appreciate the following dynamical regimes: (1) very weak coupling (V0 0.2kB T ); shows a decrease in the diffusion,


430


(2) weak coupling (1.0kB T V0 2.0kB T ); exhibits an increase in the diffusion, (3) strong coupling (V0 > 2.0kB T ) with almost-normal SFD, i.e. α ≈ 1/2. The transition from weak to strong coupling is located at larger V0 (> 2kB T ) than the Yukawa particles. Interestingly, in the strong coupling region, the particle dynamics seems to be almost independent of the HI and the kind of interaction between particles (α ≈ 0.55). As the nonmonotonic behavior exhibits the same features as the Yukawa particles, the detailed explanation given above should be similar to this case with the difference that the range of the interparticle interaction slightly modifies the limits between the dynamical regimes as it is mentioned in Ref. 15. Finally, we should point out that the insets of Fig. 9 in Ref. 15 show the behavior of F for the different values of p and V0 . Specifically for p < 1/2 and 1/2 < p < 1, Table 2. Dynamic factors, α and F ∗ of 1D heterogeneous colloidal systems for different values of the amplitude of the sinusoidal external potential V0 . Results shown inside brackets, (· · ·), were obtained with the inclusion of HI. The subscript of F ∗ indicate the calculation method; ∗ ∗ and α are obtained through W (t) ≈ F ∗ tα . The values of F→W and α for the F→W (t) (t) Yukawa and superparamagnetic case are taken from Figs. 6 and 9 in Ref. 15. Errors that are smaller than the last significant digit. Interaction

V0 /kB T

p = 1/4

p = 1/2

Intermediate: Yukawa.15 φ = 0.43 0.2 1.0 2.0 3.0 4.0

p = 3/4

p=1

α 0.49 0.50 0.50 0.50 0.50

(0.60) (0.60) (0.59) (0.59) (0.59)

0.49 0.49 0.49 0.45 0.41

(0.60) (0.59) (0.59) (0.52) (0.45)

0.49 0.49 0.49 0.49 0.50

(0.60) (0.59) (0.54) (0.52) (0.52)

0.27 0.47 0.53 0.56 0.56

(0.33) (0.38) (0.51) (0.54) (0.56)

0.13 0.13 0.14 0.14 0.15

(0.22) (0.22) (0.23) (0.24) (0.25)

0.05 0.03 0.03 0.03 0.03

(0.06) (0.02) (0.02) (0.02) (0.02)

0.49 0.49 0.48 0.48 0.48

(0.58) (0.58) (0.53) (0.51) (0.52)

0.34 0.41 0.49 0.54 0.57

(0.39) (0.35) (0.47) (0.56) (0.56)

(0.23) (0.23) (0.24) (0.25) (0.24)

0.06 0.03 0.03 0.03 0.03

(0.08) (0.03) (0.03) (0.03) (0.03)

∗ F→W (t)

0.2 1.0 2.0 3.0 4.0

0.13 0.13 0.13 0.13 0.13

(0.22) (0.22) (0.22) (0.22) (0.22)

0.13 0.13 0.12 0.11 0.09

(0.22) (0.22) (0.20) (0.18) (0.14)

0.49 0.49 0.49 0.49 0.49

(0.58) (0.58) (0.58) (0.57) (0.57)

0.49 0.49 0.47 0.43 0.37

(0.58) (0.58) (0.57) (0.51) (0.42)

Long: Superparamagnetic.15 Γ = 4.67 α 0.2 1.0 2.0 3.0 4.0

∗ F→W (t)

0.2 1.0 2.0 3.0 4.0

0.14 0.14 0.14 0.14 0.14

(0.23) (0.23) (0.23) (0.23) (0.23)

0.14 0.14 0.13 0.11 0.08

(0.23) (0.23) (0.21) (0.16) (0.11)

0.14 0.14 0.15 0.15 0.16



431

there is an increase in the mobility factor, being more dramatic for V0 /kB T ≥ 2.0 and similar to the ones reported for the Yukawa particles. For p = 1/2, the decrease in the mobility factor for V0 /kB T ≥ 2.0 is larger than the decay of F shown by the Yukawa particles (see inset of Fig. 9(a) in Ref. 15). Thus, superparamagnetic colloids exhibit a stronger noncooperative behavior than the Yukawa particles. Moreover, as in the case of the Yukawa particles, for p = 1 one finds that F decays exponentially and its values are independent of the inclusion of HI (see inset of Fig. 9(b) in Ref. 15). This confirms that the behavior of F imposed by the external field is present regardless the kind of repulsive interaction potential between particles.29 Table 2 gives an overview of the most important cases discussed here and in Ref. 15. It shows numerical values of the dynamic factors α and F ∗ in the long time regime which were obtained through the approximation W (t) ≈ F ∗ tα . Table 2 also gives an overview of the interactions, their ranges, some amplitudes of the external potential and, also, some of the commensurablilities.15 The information offers the opportunity to the reader to make a direct numerical comparison between the numerical values obtained without and with HI. These data also give the opportunity to compare directly with experimental measurements. 4. Concluding Comments We have reviewed the effect of an external field and hydrodynamics on the singlefile diffusion of driven repulsively interacting colloids; three types of interparticle potentials are reviewed, namely, Weeks–Chandler–Andersen, Yukawa and superparamagnetic potentials. We performed Brownian dynamics simulations at different commensurable conditions and external potential strengths. We found that, in general, the complex dynamical scenario of Yukawa and superparamagnetic colloidal particles is basically the same because both potentials led to a correlation among particles at long interparticle separations. The mean square displacements exhibited a subdiffusive behavior at long times and it scales as a power law W (t) ∝ tα , with α < 1. In homogeneous systems interacting with an almost hard-core potential, the parameter α got the value α ≈ 0.5, which is in agreement with previous, theoretical, simulation and experimental results. We also observed that in the homogeneous case hydrodynamics interactions led to a value of α ≈ 0.6. However, when the external potential was switched on, the particle diffusion became sensitive to the strength and commensurability of the sinusoidal potential with the interparticle spacing. Last, but not least, we have to emphasize that our simulations gave us the opportunity to understand the role of hydrodynamic interactions on systems composed of interacting Brownian particles under one-dimensional modulated energy landscapes. Further studies on single-file diffusion in random energy landscapes should take into account the effect of the hydrodynamic interactions, since they affect considerably the long-time self-diffusion mechanisms.


432


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Single-File Diffusion of Driven Interacting Colloids - World Scientific

Single-File Diffusion of Driven Interacting Colloids - World Scientific

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