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# Hydrodynamic and entropic effects on colloidal diffusion in corrugated channels

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved July 25, 2017 (received for review May 10, 2017)

## Significance

When a particle diffuses in a corrugated channel, the channel’s boundaries have a twofold effect of limiting the configuration space accessible to the particle and increasing its hydrodynamic drag. Analytical and numerical approaches well-reproduce the former (entropic) effect, while ignoring the latter (hydrodynamic) effect. Here, we experimentally investigate nonadvective colloidal diffusion in channels with periodically varying width. While validating the current theory for channels much wider than the particle radius, we show that, in narrow channels, hydrodynamic and entropic effects can be equally strong and that hydrodynamic effects can be incorporated into existing descriptions by using an experimentally measured diffusivity. These results significantly advance our understanding of diffusive transport in confined geometries, such as in ionic channels and nanopores.

## Abstract

In the absence of advection, confined diffusion characterizes transport in many natural and artificial devices, such as ionic channels, zeolites, and nanopores. While extensive theoretical and numerical studies on this subject have produced many important predictions, experimental verifications of the predictions are rare. Here, we experimentally measure colloidal diffusion times in microchannels with periodically varying width and contrast results with predictions from the Fick–Jacobs theory and Brownian dynamics simulation. While the theory and simulation correctly predict the entropic effect of the varying channel width, they fail to account for hydrodynamic effects, which include both an overall decrease and a spatial variation of diffusivity in channels. Neglecting such hydrodynamic effects, the theory and simulation underestimate the mean and standard deviation of first passage times by 40% in channels with a neck width twice the particle diameter. We further show that the validity of the Fick–Jacobs theory can be restored by reformulating it in terms of the experimentally measured diffusivity. Our work thus shows that hydrodynamic effects play a key role in diffusive transport through narrow channels and should be included in theoretical and numerical models.

Diffusive transport occurs prevalently in confined geometries (1, 2). Notable examples include the dispersion of tracers in permeable rocks (3), diffusion of chemicals in ramified matrices (4), and the motion of submicrometer corpuscles in living tissues (5, 6). The subject of confined diffusion is of paramount relevance to technological applications and for this reason, has been generating growing interest in the physics (1, 2), mathematics (7), engineering (3), and biology communities (5, 6, 8).

Spatial confinement can fundamentally change equilibrium and dynamical properties of a system via two different effects: limiting the configuration space accessible to its diffusing components (1) and increasing the hydrodynamic drag (9) on them. The former (entropic effect) has been extensively studied analytically and numerically in the case of quasi-1D structures, such as ionic channels (10), zeolites (4), microfluidic channels (11, 12), and nanopores (13). In these systems, transport takes place along a preferred direction, with the spatial constraints varying along it. Focusing on the transport direction, Jacobs (14) and Zwanzig (15), in the absence of advective effects, assumed that the transverse dfs equilibrate fast and proposed to eliminate them adiabatically by means of an approximate perturbation scheme. In first order, they derived a reduced diffusion equation, known as the Fick–Jacobs (FJ) equation, reminiscent of an ordinary 1D Fokker–Planck equation in vacuo, except for two entropic terms (2, 16⇓⇓–19). Predictions of the FJ equation have been extensively checked against Brownian dynamics (BD) simulations in different types of channels (16, 19⇓⇓⇓⇓⇓⇓⇓–27). Using the FJ theory and BD simulations, researchers have predicted a variety of novel entropy-driven transport mechanisms, such as drive-dependent mobilities (2, 18, 20), stochastic resonance (28, 29), absolute negative mobilities (30), entropic rectification (31, 32), and particle separation (33). Several of these predictions are presently recognized as being of both fundamental and technological importance.

While these previous studies (2, 14⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–33) significantly improved our understanding of the entropic effects of confining boundaries, they largely ignored the hydrodynamic effects, which are notoriously difficult to treat analytically and in simulations (9, 34). How will hydrodynamic effects change the established entropic picture? To address this important question, we turn to laboratory experiments (12, 35⇓⇓⇓–39) and measure the diffusive dynamics of micrometric colloidal particles through channels with systematically modulated cross-sections. Contrasting the experimental results with predictions obtained by FJ approximation and from BD simulation, we discover that, as the channel’s width shrinks toward the particle’s diameter, hydrodynamic effects (9, 34, 40⇓⇓–43) grow in strength and become comparable with the predicted entropic effects, thus indicating an unexpected breakdown of the standard FJ theory and BD simulation in narrow channels. We further show that hydrodynamic effects can be incorporated by using an experimentally measured local diffusivity. With such a phenomenological modification, the FJ theory and BD simulation accurately predict the experimental data.

## Results

Our channels were fabricated on a coverslip by means of a two-photon direct laser writing system, which solidifies polymers according to the preassigned channel profile, *A*, the quasi-2D channel has a uniform height of 2.5 μm (

After fabrication, channels were immersed in water with fluorescently labeled Polystyrene spheres of radius *A*). The entrances are barely wider than the particle diameter so as to create insurmountable entropic barriers (2), which prevent the particle inside the channel from escaping. Particle motion in the quasi-2D channel was recorded through a microscope at rate of 30 frames per second for up to 20 h (see Movie S1 for a short segment of typical data). The projected particle trajectory in the

Inside the channels, the particle diffuses in a flat energy landscape. To show that, we quantized the measured particle coordinates *B*, particle counts are uniformly distributed with a standard deviation (SD) of about 12% of the mean. Regions where the particle counts drop sharply to zero are inaccessible to the particle’s center and in Fig. 1*B*, are delimited by the black curves (Fig. 2*A*, *Inset*). The effective channel’s boundary [denoted by *B*,

### First Passage Time Statistics.

From acquired particle trajectories, we measured the first passage times (FPTs) (7, 44, 45). As in the FJ theory, we focus on the particle motion along the channel direction and measure the duration of the unconditional first passage events that start at *A*, *Inset*) and end at *A*, *Inset*), with no restriction on the transverse coordinate *A* and *B*; all distributions (for three *C*–*F* against the diffusing distance, *C* to *D* sharply increases the diffusion time. For instance, the mean FPT to the center of the adjacent cells, *C* to 900 s in Fig. 2*D*. A similar increase can be observed in the SDs, *E* and *F*. To this regard, we notice that, for both channels, the experimental curves

We next compare our experimental data with the predictions of the standard FJ theory and BD simulations. The channel geometry renders our experimental system effectively 2D; analytical and numerical studies were carried out in the same dimension. Following the FJ scheme and taking advantage of symmetry properties of our experiments, we calculate the analytical expression,**2**, except that the outer integral runs here from **2** and **4** can be found in *SI Appendix*.

To use Eqs. **2** to **4**, one needs to know the diffusivity **2** and **4** were then computed explicitly for the measured value of *C*–*F*) agree closely with each other for both the wide and narrow channels. The comparison with the experimental data, instead, is satisfactory only in the case of the wide channel (Fig. 2 *C* and *E*). For the narrow channel in Fig. 2 *D* and *F*, the experimental data with

### Diffusivity Measurements.

The theoretical and numerical predictions discussed so far assume a constant particle diffusivity,

To show such a hydrodynamic effect in our device, we measured the particle diffusivity inside the channel. At any given location, *SI Appendix*, the value chosen for *A* and *B*. In both, *C*. The spatial variabilities of

We corroborate the local diffusivity measurements with full hydrodynamic computations. The hydrodynamic friction coefficient in the *C* as curves and are in excellent agreement with the experimental data.

### Hydrodynamic Correction.

Fig. 4 depicts that particle diffusion through narrow bottlenecks can be significantly slower than in the wide region; moreover, the spatial modulation of the local particle diffusivity increases with decrease in the bottleneck width, which explains the results in Fig. 3. This spatial modulation is in clear contrast with the assumption of constant diffusivity that we adopted above when implementing the FJ formalism and the BD simulation code. To appreciate the effect of the spatial dependence of the local diffusivity, we replace the constant diffusivity, *C*, both in the theoretical treatment and in the numerical code. The analytical and numerical predictions are plotted in Fig. 2 *C*–*F* as green curves and symbols, respectively. Their agreement with the experimental data is excellent. Furthermore, we used the improved BD code to also compute, other than the first two FPT moments, the FPT distributions displayed in Fig. 2 *A* and *B*. Again, the close comparison obtained with the experimental data confirms the validity of our phenomenological approach.

## Discussion

The coincidence of approximate analytical predictions and simulation results occurs for any choice of the local diffusivity [i.e., **3** suggests a phenomenological factorization of entropic and hydrodynamic effects, with validity that is justified a posteriori by the reported close comparison with the experimental data. In conclusion, we have shown that, by making use of the measured diffusivity *C*, the FJ theory can be improved to accurately predict the FPT statistics of Fig. 2; the FJ approach thus remains a powerful analytical tool to investigate diffusion in complex channels.

As shown in Fig. 4 *A* and *B*, the local diffusivity, **6**, particle diffusivity in confined geometries is generally smaller than in an unbounded space. In our channels, the maximum diffusivity

In this work, we focused on the nonadvective diffusion of a single particle, although technological applications often involve many suspended particles driven by external fields (12, 35⇓⇓–38). Particle transport is certainly complicated by excluded volume and hydrodynamic interactions between nearby particles in dense suspensions. Moreover, external driving may prevent the system from equilibrating in the transverse directions and produce even more complex transport patterns (20); it can also cause additional hydrodynamic effects (36, 37, 50). The experimental setup and the data analysis methods presented here provide a promising framework for future systematic investigations of these important and challenging problems.

## Materials and Methods

### Channel Fabrication and Imaging Procedure.

Microchannels were fabricated with a two-photon direct laser writing system (*A*, *Inset*] to fabricate the desired structure. After the scanning is finished, the remaining liquid resin was removed by washing the structure with 4-methyl-2-pentanone and then acetone for 5 min. Then, channels were thoroughly cleaned with distilled water to prevent particles from sticking to the channel boundaries.

Fluorescently labeled Polystyrene particles were purchased from Invitrogen (catalog no. F13080). Particle motion was recorded through a 60× oil objective (N.A. 1.3) in an inverted fluorescent microscope (Nikon Ti-E). With the help of an autofocus function (Nikon Perfect Focus), we imaged the diffusion of a colloidal particle in the channel for up to 20 h at room temperature (27 °C).

### BD Simulation.

The motion of a colloid particle is governed by a 2D overdamped Langevin equation in simulations. The particle diffusivity varies spatially when the diffusivity function *SI Appendix* has more details.

### Finite Element Calculation.

We solved the Stokes equations in a typical setup shown in *SI Appendix*, Fig. S2*A*. No slip boundary conditions were imposed on the side walls, floor, and ceiling, and open boundary conditions were imposed at the channel openings. The geometry of the side wall was set to reproduce the inner channel boundary measured in the experiments (Fig. 1*A*, *Inset*). A sphere was driven with a constant speed, *SI Appendix* has more details.

## Acknowledgments

We thank Mingcheng Yang and Xiaqing Shi for useful discussions. We acknowledge the financial support of National Natural Science Foundation of China Grants 11422427 and 11505128 and Program for Professor of Special Appointment at Shanghai Institutions of Higher Learning Grant SHDP201301.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: hepeng_zhang{at}sjtu.edu.cn.

Author contributions: H.P.Z. designed research; X.Y., C.L., and H.P.Z. performed research; X.Y., Y.L., F.M., and P.H. contributed new reagents/analytic tools; X.Y., C.L., F.M., P.H., and H.P.Z. analyzed data; and X.Y., F.M., P.H., and H.P.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1707815114/-/DCSupplemental.

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