Brownian Sieving Effect for Boosting the Performance of Microcapillary Hydrodynamic Chromatography. Proof of Concept

Microcapillary hydrodynamic chromatography (MHDC) is a well-established technique for the size-based separation of suspensions and colloids, where the characteristic size of the dispersed phase ranges from tens of nanometers to micrometers. It is based on hindrance effects which prevent relatively large particles from experiencing the low velocity region near the walls of a pressure-driven laminar flow through an empty microchannel. An improved device design is here proposed, where the relative extent of the low velocity region is made tunable by exploiting a two-channel annular geometry. The geometry is designed so that the core and the annular channel are characterized by different average flow velocities when subject to one and the same pressure drop. The channels communicate through openings of assigned cut-off length, say A. As they move downstream the channel, particles of size bigger than A are confined to the core region, whereas smaller particles can diffuse through the openings and spread throughout the entire cross section, therein attaining a spatially uniform distribution. By using a classical excluded-volume approach for modeling particle transport, we perform Lagrangian-stochastic simulations of particle dynamics and compare the separation performance of the two-channel and the standard (single-channel) MHDC. Results suggest that a quantitative (up to thirtyfold) performance enhancement can be obtained at operating conditions and values of the transport parameters commonly encountered in practical implementations of MHDC. The separation principle can readily be extended to a multistage geometry when the efficient fractionation of an arbitrary size distribution of the suspension is sought.

1 Numerical solution of the transport model 1

.1 Eluent flow
The axial component w(x, y) of the Stokes flow v = (u(x, y, z), v(x, y, z), w(x, y, z)) = (0, 0, w(x, y)) is the solution of the two-dimensional Poisson equation where ∇ ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the cross-sectional Laplacian operator, µ is the dynamic viscosity of the suspending solution, and ∂P ∂z = − P | z=0 − P | z=L L is the pressure gradient in the axial direction. Equation (1)  (1) has been obtained by a commercial finiteelement software (Comsol Multiphysics 5.5) by using an unstructured mesh corresponding to order 10 5 degrees of freedom.

Numerical approach for particle transport
The Euler-Maruyama algorithm for the integration of the Langevin equation (Eq. (4) of the main text) reads where ξ η, and ζ are normally distributed random variables.
Because of the presence of the diffusive terms, Eq. (S1) must be supplemented with boundary conditions at the internal baffles and the channel walls, which define the interaction between the solid boundaries and the advecting-diffusing particle whenever the particle center crosses the boundaries of the effective transport domain Ω dp . Here, elastic collisions are enforced, so that the z-component of the displacement is unaffected by the collision, whereas the cross-sectional component of the displacement (convective plus diffusive) that falls beyond the effective boundary ∂Ω dp is reflected along the local normal direction to the surface, as depicted in Fig. S1. Note that multiple reflections are possible, especially near the corners of ∂Ω dp . In the limit where ∆t → 0 in Eq. (S1), it can be shown that the elastic collision model becomes the Langrangian equivalent of the Neuman Boundary condition of vanishing normal diffusive flux at the boundary enforced on the particle number density φ(x, y, z, t) stemming from the Eulerian description of particle transport. A thorough comparison between the two approaches has been used to validate Brenner's macrotransport approach to describe transport of finite-sized particles in periodic patterns of impermeable ∂Ω a Figure S1: Reflection conditions for a particle whose center of mass crosses the boundary ∂Ω dp of the effective transport domain. Here x o = (x o , y o ) and x n = (x n , y n ) represent the position of the center of mass of the particle at time t and t + ∆t, respectively. ∂Ω o (depicted in blue) represents the physical boundary, i.e. the boundary as seen by a particle of vanishing size. Panel (a): simple reflection; Panel (b): double reflection.

Dynamics of monodispersed particle ensembles in BS-MHDC
The asymptotic behavior of the center of mass and the variance of an ensemble of particles of dimensionless diameter α obeys the linear scalings The constants K lines represent the best linear fit to the data, which yields W dp = 2.25 and 1/Pe eff a = 1.71 for the dimensionless average particle velocity and effective dispersion coefficient, respectively. Thus, the particles confined to the core channel travel more than twice as fast as the solvent molecules (whose dimensionless average velocity is unity). The data also provide a quantitative example of the dispersion enhancing effect resulting from the interaction of the non-uniform velocity profile and Brownian diffusion onto the channel cross-section: here, the dimensionless effective dispersion coefficient 1/Pe eff a is two orders of magnitude larger than the dimensionless bare diffusion coefficient 1/Pe a = 10 −2 .
As discussed in the main text, particles of radius a = 0.05, which can access the annular channel, display a different transient behavior of the marginal distribution F (z; t). Here, a sizeable transient can be identified before macrotransport regime is attained. Figure S3 shows the behavior of σ 2 (t) at the early stage of the transport process. The square variance  Figure S3: Short time behavior of the variance for the particle ensemble depicted in Fig. 6 of the main text. The red and blue lines line show the scalings σ 2 (t) = k t 2 , and σ 2 (t) = K (D) a + 2t/Pe eff a , respectively.

S6
follows an initial ballistic regime, where σ 2 (t) t 2 , followed by a transitional regime up to t 200, until macrotransport conditions are reached where the behavior of σ 2 (t) is consistent with Eq. (S2). The best fit to the data yields a value 1/Pe eff a = 11.8 for the dimensionless effective dispersion, to be compared with the value 1/Pe a = 2 × 10 −2 of the dimensionless bare particle diffusivity. Thus, for this particle size at the operating conditions chosen, the dispersion enhancement factor over the bare dimensionless particle diffusivity 1/Pe a is almost three orders of magnitude. Figure S4 shows the results of the comparison of separation performance at Pe 0.1 = 10, in the system depicted in Fig. 7 of the main text (this can be though of, e.g., as obtained by running the same experiment depicted in Fig. 7 of the main text when the eluent velocity is decreased by a factor ten, with all of the other parameters left unchanged). The comparison between Fig. 7-(a) and Fig. S4-(a) shows how the separation efficiency of the St-MHDC is weakly influenced by the mutated operating conditions, whereas that of the BS-MHDC system is sizeably enhanced (compare Fig. 7-(b) and Fig. S4-(b).). converges to macrotransport behavior at relatively short times (see Fig. S2). In the case of the BS-MHDC geometries, the analysis is more complex because the average particle velocity depends on time due to the non-uniform particle distribution onto the channel cross-section.

Comparison between St-MHDC and BS-MHDC
Here, both the numerator and the denominator of the resolution factor depend non-trivially on time.

S10
The separation mechanism proposed in this article can readily be generalized for the fractionation of an arbitrary size-distribution of particles. In this case, the efficient enforcement of the Brownian sieving principle can be accomplished through a device geometry where the width α of the communication slit between the core and the annular channel is made z-dependent. Specifically, one can define a set of increasing values α 1 < α 2 < · · · α n and assign a piecewise-constant dependence of α on prescribed intervals z 0 = 0, z 1 , z 2 , · · · z n = L of the channel axis, α 2 for z 1 ≤ z < z 2 · · · α n for z n−1 ≤ z < L (2) and accomplish the fractionation of particle of increasing diameters.