Abstract
The diffusion behavior of interacting particles determines the behavior of a large number of systems ranging from pedestrians crossing a road to ions passing through channels in living cells. Here we present a system in which the nature of the diffusion process varies with changes in the external conditions. We find this special behavior in a colloidal model system, consisting of micron sized particles which are confined to narrow channels and interact via induced magnetic dipoles. When the density of these particles is changed, diffusion alternates between normal Fickian behavior and singlefile diffusion. This anomalous behavior is induced by the order of the particles in the restricted geometry and does not depend on the exact nature of the interparticle interactions.
Introduction
The analysis of interesting nonequilibrium phenomena in confined geometries has become possible over the last few years, ranging from the dynamics of pedestrians in a pedestrian zone^{1} to the electron transport and thermopower effects in atomic wires^{2,3}, the biologically important size selectivity of transport in ion channels^{4}, the mixing in “labonchip” devices^{5} and the pattern formation of colloids in microchannels^{6,7} and electric fields^{8}.
In this context, colloids have received much attention, since they often serve as model systems^{9,10,11,12,13} due to the ease of imaging particle coordinates by light scattering methods. The theoretical analysis of colloids has put forward the development of Brownian dynamics methods^{14,15,16}.
Here we report on unexpected and important aspects of the diffusion of colloids in a confined geometry, comparing experiments with numerical simulations. Our work may have impact on many other fields, where the motion of individual particles is affected by constrictions, as mentioned above.
Results
Our system consists of colloidal particles confined to channels of width L_{Y} . The repulsive interaction between the particles at distance r_{ij} can be described by a dipole potential , with the magnetic dipole moments M = χ_{eff}B_{ext} of the particles^{18}, where B_{ext} represents the external applied magnetic field. Typical values for χ_{eff} are in the order of 7.88 10^{−11} Am^{2}/T^{19}. It is known that superparamagnetic colloidal particles in twodimensional channels order into layers along the channel walls^{22}. The ratio of potential to thermal energy is characterized by the parameter Γ = μ_{0}M^{2}/(4πk_{B}TR^{3})^{22}, which can be interpreted as an inverse temperature of the system, where is the layer spacing of a triangular lattice with density n. We introduce reduced (dimensionless) length scales A* = A/σ of length scales A, e.g. , where the channel width L_{Y} denotes the region, which is accessible for the particle centers, and n* = nσ^{2}. Times are given in units of τ_{D} = σ^{2}/D_{0}, where D_{0} is the diffusion coefficient. Please note that τ_{D} = 1 corresponds to 218 seconds in the experiment.
We choose an interaction strength of Γ ≈ 13, which is close to the crossover from a solid to a liquid phase. This is achieved by means of a fixed density of n* = 0.4 and a fixed external field of B_{ext} = 0.16 mT. In this case the structure inside the channel is only a function of the channel width L_{Y} . If the width is varied, the system switches between layering order and a more disordered state. In Figure 1 a snapshot of such equilibrium configurations is shown. For channels of width and , the particles are ordered in three and four layers, respectively. For , the crossover from three to four layers takes place.
The main goal of this work is to study the influence of this layered structure on particle diffusion. For channels with a width (see Figure 2), singlefile diffusion is found, similar to a one dimensional channel, since the mutual passing of particles is forbidden due to hardcore interaction. On very short time scales, the mean square displacement (MSD) 〈Δx(t)^{2}〉 is that of a free Brownian particle (see Figure 2 inset), though it soon evolves into a dependence appropriate for singlefile motion with F as the singlefile mobility^{23,24}. For channel width , the long time evolution of the MSD is again expected to be Brownian, since particles can pass each other. The crossover to normal diffusion with increasing channel width is usually described in terms of a hopping time τ_{H}, the average time a particle needs to pass its neighbor. On time scales smaller than τ_{H} singlefile diffusion with an exponent 0.5, and on time scales greater than τ_{H} normal diffusion with exponent 1 is observed. For channels with a width , the hopping time τ_{H} = ∞.
While for hard spheres, the hopping time τ_{H} decays exponentially with increasing channel width^{25}, our results for the crossover behaviour are influenced by the additional V ∝ 1/r^{3} repulsion arising from the dipole interaction in our system. For , this repulsion becomes important, since the dominant restriction for the mutual passing of particles is the layering structure and not hard sphere exclusion. A description using the hopping time is, therefore, problematic for the crossover phenomenon seen in our studies, since the singlefile exponent of 1/2 is reached only in the completely ordered case. Instead, we choose to describe this scenario with an apparent, L_{Y} dependent, intermediate diffusion exponent α_{inter}. We emphasise, 1/2 ≤ α_{inter} ≤ 1 is not a new subdiffusive exponent but rather a convenient way to characterise the time evolution of the MSD () in an intermediate time regime between the short (t < 0.15τ_{D}) and long time (t > 1.5τ_{D}) normal diffusive behaviors suggested by our studies.
We extract values for α_{inter} by two methods. In the first method, the slope of the MDS data versus time, as shown in Figure 2, is obtained by a linear regression in the region 0.15τ_{D} < t < 1.5τ_{D}, and this interval has been kept constant for all analysed data, corresponding to the last time decade for the experimental studies before ballistic effects set in, which are not topics of this work. In the second method, the logarithmic derivative of the MSD data, d(log(〈Δx(t)^{2}〉))/d(log(t)), has been computed by numerical forward difference method as a function of time, see Figure 3, and the minimum value has been chosen as estimator for α_{inter}. The estimators obtained by both methods are presented in Figure 4 as a function of the channel width, the same analysis method has been applied for the data from the simulations and from the experiment. The minimum values for α_{inter} obtained by both methods agree fairly well, the maximum values obtained by the regression method are slightly larger compared to the “minimum slope” method due to interval averaging.
We found that even for channel widths , for which an integer number of layers fits into the channel, the layering order drives the system back into singlefile motion on an intermediate time scale. In Figure 2 the MSD 〈Δx(t)^{2}〉 is shown in a loglog plot for the three different cases. All MSD curves start with the same shorttime normal diffusive behavior with the microscopic diffusion coefficient D_{0}, followed by a subdiffusive regime. For (Figure 2(inset)) this is already the longtime singlefile behavior. For we obtain a subdiffusive regime with an apparent exponent 0.5 < α_{inter} < 1 as an intermediate behavior. The value of α_{inter}, however, depends crucially on the extent of layer order (Figure 2). If , the particles are ordered into well defined layers (see Figure 1) and we obtain singlefile diffusion with α_{inter} = 1/2 over a rather long timescale spanning more than a decade. On the other hand, for disordered channel structures (e.g. ), α_{inter} initially begins to decrease but then approaches the longtime diffusive limit relatively quickly and the α_{inter} = 1/2 value is never reached.
In Figure 3 we show the logarithmic derivative of the MSD as function of time for various channel widths. A comparison of the data for with the single file diffusion case (), which has been included for comparison for the same line density as for , where three well separated layers are present (see Figure 1), reveals that the approach to the single file diffusion exponent (0.5) is achieved only at time scales as large as τ_{D}. At this time scale, particles typically have diffused over a distance of their particle diameter as well and thus can leave their layers in ydirection and enter the region of a neighbouring layer, such that normal diffusion sets in. As a result, the time region, in which the slope of log(MSD) is minimal, approaching the single file diffusion value, is small, but note again that τ_{D} is the time region typically limiting the experimental studies. The minimum value for the logarithmic derivative is taken as estimator for α_{inter} and is shown in Figure 4. From Figure 3 we can conclude that at small times the MSD values for a channel width of are slightly larger compared to the cases with layer formation, since repulsive interaction from particles in neighbouring layers hinder the diffusion in the latter case.
In Figure 4 the intermediate diffusion exponent is plotted as a function of dimensionless channel width L_{Y}/R. We see an oscillatory dependency for the simulation results, which is due to the layer structure. If the particles are ordered in layers, the singlefile exponent 0.5 can be observed in the intermediate time regime. Otherwise, the MSD is still subdiffusive in this time regime, but with an exponent greater than the singlefile exponent 0.5.
In experiments it is not possible to vary the channel width. To overcome this problem, one varies the density n, instead. Since the layer spacing is the only relevant length scale as long as the particle diameter is small compared to the layer spacing R/σ ≪ 1 and the interaction strength Γ is sufficiently large, the variation of the channel width can be compared with a variation of density by introducing the dimensionless channel width L_{Y}/R.
The experimental results come from a channel with a width of about 30 μm. The density in the channel is varied so that the number of layers is between three and four. The mean square displacement is calculated and an intermediate exponent is obtained as in the simulations. One can see a full oscillation of this exponent (Figure 4), while the number of layers vary from a three layers system over a disordered system to a four layer system. The experimental values for the maxima and minima of α_{inter} are slightly smaller than the values from the simulations. These differences are caused by the experimental set up, in which several parallel channels are present and a resulting small density reduction at the channel walls due to repulsive interactions with particles from neighboring channels reduces the total density in the channel and thus the value of .
Discussion
In our work we have shown in simulation and experiment that the diffusion of model colloids has a nonmonotonic dependency on the channel width. In the experimental accessible region the MSD can be described by a subdiffusive exponent between 0.5 and 1, which oscillates with the width of the channel. The effect on the exponent is stable in the experiment and is expected to be general for the diffusion of interacting particles in confined channel geometries. For example, a similar crossover from singlefile to normal diffusion has been observed to determine the kinetics of template assisted assembly of colloidal particles^{26}.
In general, the diffusion behavior of interacting particles determines the behavior of a large number of systems ranging from pedestrians crossing a road to ions passing through channels in living cells. Our studies of a special system consisting of colloidal particles in confined geometry, accessible both by experimental and theoretical methods, show that the nature of the diffusion process can be modified by changes in the external conditions.
Methods
In the simulation, a conventional Brownian dynamics algorithm is used to simulate colloidal particles in twodimensional microchannels with the pairwise dipole interaction relevant to the experimental system. The finite particle diameter σ has been taken into account by a hardcore repulsion. The channel is set up in the xyplane with periodic boundary conditions in xdirection and hard wall boundary condition in ydirection. Both particleparticle and particlewall hardcore interaction are realized in a second step as proposed by Heyes^{21}.
In the experiment, the particles are trapped in channels made by standard PDMS moulding technique created with SU8 UVlithography using a master mould^{17}. The dimensions of these channels are 2 mm in length, 30 μm in width and 7 μm in height, ensuring that the particles are completely confined laterally by the channel walls.
The channel itself is filled with superparamagnetic colloids (Dynal M450, diameter σ = 4.5 μm, ρ = 1500 kg/m^{2}) suspended in water. Gravity confines the particles down to the substrate surface and buoyancy leads to an effective mass of m_{eff} = 2.385 10^{−14} kg. In that case, the particle motion is limited to two dimensions inside the channel. All experiments in the previous sections were done by means of a standard video microscopy setup. The positions of the particles were tracked using particle tracking software which is able to locate the particles with submicron accuracy^{20}. The sample is aligned horizontally in the video setup to ensure that the particles undergo only Brownian motion. After that, the system is equilibrated for at least 5 h before the measurement is started. All particle trajectories contain at least 20000 data points with a time resolution of one frame per second. The mean square displacement is calculated directly from these trajectories.
References
 1.
Helbing, D., Molnar, P., Farkas, I. & Bolay, K. Selforganizing pedestrian movement. Environment and Planning B: Planning and Design 28, 361 (2001).
 2.
Dreher, M., Pauly, F., Heurich, J., Cuevas, J. C., Scheer, E. & Nielaba, P. Structure and conductance histogram of atomicsized Au contacts. Phys. Rev. B 72, 075435 (2005).
 3.
Pauly, F., Viljas, J. K., Bürkle, M., Dreher, M., Nielaba, P. & Cuevas, J. C. Molecular dynamics study of the thermopower of Ag, Au, and Pt nanocontacts. Phys. Rev. B 84, 195420 (2011).
 4.
Roth, R. & Gillespie, D. Physics of Size Selectivity. Phys. Rev. Lett. 95, 247801 (2005).
 5.
Squires, T. M. & Quake, S. R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977 (2005).
 6.
Köppl, M., Henseler, P., Erbe, A., Nielaba, P. & Leiderer, P. Layer Reduction in Driven 2DColloidal Systems through Microchannels. Phys. Rev. Lett. 97, 208302 (2006).
 7.
Henseler, P., Erbe, A., Köppl, M., Leiderer, P. & Nielaba, P. Density reduction and diffusion in driven twodimensional colloidal systems through microchannels. Phys. Rev. E 81, 041402 (2010).
 8.
Leunissen, M. E., Christova, C. G., Hynninen, A. P., Royall, C. P., Campbell, A. I., Imhof, A., Dijkstra, M., v. Roij, R. & v. Blaaderen, A. Ionic colloidal crystals of oppositely charged particles. Nature (London) 437, 235 (2005).
 9.
Wei, Q. H., Bechinger, C. & Leiderer, P. SingleFile Diffusion of Colloids in OneDimensional Channels. Science 287, 625 (2000).
 10.
v. Blaaderen, A. Colloidal Molecules and Beyond. Science 301, 470 (2003).
 11.
Poon, W. Colloids as Big Atoms. Science 304, 830 (2004).
 12.
Tkalec, U., Ravnik, M., Copar, S., Zumer, S. & Musevic, I. Reconfigurable Knots and Links in Chiral Nematic Colloids. Science 333, 62 (2011).
 13.
Means, J. C. & Wijayaratne, R. Role of Natural Colloids in the Transport of Hydrophobic Pollutants. Science 215, 968 (1982).
 14.
Kubo, R. Brownian Motion and Nonequilibrium Statistical Mechanics. Science 233, 330 (1986).
 15.
Han, Y., Alsayed, A. M., Nobili, M., Zhang, J., Lubensky, T. C. & Yodh, A. G. Brownian Motion of an Ellipsoid. Science 314, 626 (2006).
 16.
Li, T., Kheifets, S., Medellin, D. & Raizen, M. G. Measurement of the Instantaneous Velocity of a Brownian Particle. Science 328, 1673 (2010).
 17.
Xia, Y. & Whitesides, G. M. Soft Lithography. Angewandte Chemie International Edition 37, 550 (1998).
 18.
Zahn, K., Lenke, R. & Maret, G. TwoStage Melting of Paramagnetic Colloidal Crystals in Two Dimensions. Phys. Rev. Lett. 82, 2721 (1999).
 19.
Kreuter, C., Leiderer, P. & Erbe, A. Determination of potential landscapes using video microscopy. Colloid and Polymer Science 290, 575 (2012).
 20.
Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interface Science 179, 298 (1996).
 21.
Heyes, D. M. & Melrose, J. R. Brownian dynamics simulations of model hardsphere suspensions. J. NonNewtonian Fluid Mechanics 46, 1 (1993).
 22.
Haghgooie, R. & Doyle, P. S. Structural analysis of a dipole system in twodimensional channels. Phys. Rev. E 70, 061408 (2004).
 23.
Fedders, P. A. Twopoint correlation functions for a distinguishable particle hopping on a uniform onedimensional chain. Phys. Rev. B 17, 40 (1978).
 24.
Richards, P. M. Theory of onedimensional hopping conductivity and diffusion. Phys. Rev. B 16, 1393 (1977).
 25.
Sane, J., Padding, J. T. & Louis, A. A. Brownian dynamics simulations of model hardsphere suspensions. Faraday Discuss. 144, 285 (2010).
 26.
Mondal, C. & Sengupta, S. Singlefile diffusion and kinetics of templateassisted assembly of colloids. Phys. Rev. E 85, 020402 (2012).
Acknowledgements
The authors thank the SFBTR6 for support and the NIC for computer time. SS acknowledges support from the IndoEU project MONAMI.
Author information
Affiliations
Department of Physics, University of Konstanz, 78457 Konstanz, Germany
 U. Siems
 , C. Kreuter
 , P. Leiderer
 & P. Nielaba
HelmholtzCenter DresdenRossendorf, 01314 Dresden, Germany
 A. Erbe
Department of Physics, Technical University of Munich, 85748 Garching, Germany
 N. Schwierz
TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
 S. Sengupta
Centre for Advanced Materials, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
 S. Sengupta
Authors
Search for U. Siems in:
Search for C. Kreuter in:
Search for A. Erbe in:
Search for N. Schwierz in:
Search for S. Sengupta in:
Search for P. Leiderer in:
Search for P. Nielaba in:
Contributions
US performed the BD simulations, CK the experiments, AE, PL and PN wrote the text and are responsible for the project planning and the experimental (AE and PL) and theoretical (PN) prerequisites, NS and SS contributed to the theoretical analysis. All authors have discussed and analysed the results and reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to U. Siems.
Supplementary information
PDF files
 1.
Supplementary Information
Nonmonotonic crossover from singlefile to regular diff
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialShareALike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/3.0/
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

On the SelfSimilar, WrightFunction Exact Solution for EarlyTime, Anomalous Diffusion in Random Networks: Comparison with Numerical Results
International Journal of Applied and Computational Mathematics (2018)

Tagged Particle in SingleFile Diffusion
Journal of Statistical Physics (2015)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.