Channel-length dependence of particle diffusivity in confinement†

Understanding the diffusive behavior of particles and large molecules in channels is of fundamental importance in biological and synthetic systems, such as channel proteins, nanopores, and nanofluidics. Although theoretical and numerical modelings have suggested some solutions, these models have not been fully supported with direct experimental measurements. Here, we demonstrate that experimental diffusion coefficients of particles in finite open-ended channels are always higher than the prediction based on the conventional theoretical model of infinitely long channels. By combining microfluidic experiments, numerical simulations, and analytical modeling, we show that diffusion coefficients are dependent not only on the radius ratio but also on the channel length, the boundary conditions of the neighboring reservoirs, and the compressibility of the medium.


B. Photolithography
The detailed protocols of photolithography are summarized here. We used a positive photoresist (AZ 9260, AZ Electronic Materials GmbH) to fabricate access channels. The fabrication steps are the following: (i) Spin-coating with a spin speed of 2000 rpm and a ramp of 1000 rpm/s for 30 s. (ii) Bake at 115 • C for 3 min on a contact hotplate. (iii) UV light exposure (365-405 nm, 52 mW/cm 2 ) with a mask aligner. (iv) Development with a developer (AZ400k, MicroChemicals) diluted with water at a ratio of 4 : 1 for several seconds. The final thickness was approximately 10 µm.

C. Removal of residual PDMS
We found that the surface of the silicon mold used for PDMS replication became contaminated with an invisibly thin layer of PDMS. To remove the PDMS residual from the surface, we cleaned the silicon molds with tetrabutylammonium fluoride (TBAF) [1]. The protocol is summarized in the following. TBAF was diluted by 1% in propylene glycol methyl ether acetate (PMA). The silicon chip was immersed into this mixture of TBAF and PMA on a hotplate at the temperature of 50 • C for 15 min. Subsequently, the silicon chip was immersed into PMA at a temperature of 50 • C for 15 min and finally rinsed with IPA. If the surface is not clean enough, the procedure needs to be repeated.

II. SIMULATIONS OF DIFFUSION COEFFICIENTS
A. Hydrodynamic frictions and diffusion coefficients When a particle is off-centered in a channel, it experiences not only a hydrodynamic drag force but also a torque. In general, in the Stokes flow, the relation between the force F and torque T that a solid body experiences in fluid and its translational velocity U and rotational velocity Ω is described as [2] where the first matrix on the right side of the equation is called a 6 × 6 resistive matrix ξ, where ξ T R = ξ T RT . The diffusion matrix D is then calculated by the Einstein relation as [3] Note that here D T R = D T RT because ξ T R = ξ T RT , as also shown by Happel and Brenner [2]. Using the above terms, the local apparent diffusion coefficient along the x-axis at the radial position, β ≡ r/R, can be obtained as In Fig. S2, we show the simulated local diffusion coefficients of various λ (= a/R) and L/R. At small λ, both open and closed channels behave qualitatively the same; the local diffusion coefficients are large when the particle is near the center and become gradually smaller near the wall. In contrast, at high λ, the trend is reversed for a closed channel; the local diffusion coefficients near the channel wall are larger than at the center. In the open channels (e.g., L/R = 5, 10), the diffusion coefficients decrease near the channel wall even at high λ.

C. Effective radius of the channel
The effective radius of the square cross-sectional channel was obtained as follows. First, normalized diffusion coefficients in closed cylindrical channels were simulated for various confinement ratios with a step size of ∆λ = 0.01 (e.g., λ = ..., 0.5, 0.51, 0.52, ...). The values in between were interpolated linearly. Then, this data sets obtained by simulations and linear interpolation were fitted to the two experimental diffusion coefficients (356 nm & 505 nm) in the closed channels by the method of least squares. We obtained the effective channel diameter, 2R ≈ 922 nm, which yields the confinement ratios, λ ≈ 0.386, 0.548, for 2a = 356, 505 nm, respectively. We used these confinement ratios for the simulations in Fig. 2. The matrix elements in Eq. 2 derived by Bungay and Brenner [4] are summarized in the following. The first and third rows of the matrix are where a n , b n , c n , and d
Note that p and T are defined as a gauge pressure (≡ 'absolute pressure' − 'atmospheric pressure') in pascals and as a degree Celsius, while in the referred literature [5] the unit of p is [bar]. V 0 is the specific volume at p = 0 (at 1 atm) defined as Thus, for a small pressure p, Eq. S7 can be simplified to (S10) B is approximately 2.1789 × 10 9 Pa at T = 20 • C. Figure S3 shows the experimentally measured specific volumes [5], the empirical model (Eq. S7), and the simplified model for a small applied pressure (Eq. S10) as a function of pressure. As the pressure in our system is sufficiently small, we use the model of Eq. S10. C. Finite element simulation of a particle in a closed system Here, we show time-dependent simulations of a particle in a compressible fluid under the influence of a constant force using the arbitrary Lagrangian-Eulerian method [6] to verify the analytical model shown in the main manuscript. In Fig. S4, the simulation results of normalized hydrodynamic frictions were plotted as a function of nondimensional time, t/τ . The simulation results showed excellent agreement with the analytical solutions.