Membraneless water filtration using CO2

Water purification technologies such as microfiltration/ultrafiltration and reverse osmosis utilize porous membranes to remove suspended particles and solutes. These membranes, however, cause many drawbacks such as a high pumping cost and a need for periodic replacement due to fouling. Here we show an alternative membraneless method for separating suspended particles by exposing the colloidal suspension to CO2. Dissolution of CO2 into the suspension creates solute gradients that drive phoretic motion of particles. Due to the large diffusion potential generated by the dissociation of carbonic acid, colloidal particles move either away from or towards the gas–liquid interface depending on their surface charge. Using the directed motion of particles induced by exposure to CO2, we demonstrate a scalable, continuous flow, membraneless particle filtration process that exhibits low energy consumption, three orders of magnitude lower than conventional microfiltration/ultrafiltration processes, and is essentially free from fouling.

Supplementary Figure 3 | Image sequence of particle (polystyrene, 0.5 µm) removal driven by CO 2 dissolution. The time between each frame is ≈0.2 s, which allows tracking of individual particles (indicated by arrows) flowing through the filtrate stream. Supplementary

Supplementary Discussion
Governing equations. We use a one-dimensional model to describe the coupled diffusion and reaction of dissolved species and the diffusiophoretic motion of particles. The parameters in the model are listed in Supplementary Table 1. We neglect diffusioosmosis due to the wall surface charge since a Poiseuille flow driven by back pressure cancels the diffusioosmotic flow, making a zero net fluid flow in any given cross-section of the channel 4 . Not only is this approximation valid for the stationary experimental case in Fig. 2, but it is also true for the continuous flow filtration device in Fig. 4 because the ion gradients are established in the direction transverse to the main flow direction. Furthermore, the large speed of the main flow (≈300 µm s −1 ) compared to diffusioosmotic flow, O(10 µm s −1 ), would allow one to neglect the diffusioosmotic flow when predicting the particle motion in the filtration devices.
Using the sign convention for x presented in Figure 1, CO 2 diffuses inward from x = ±L. Dissolved CO 2 , concentration c c (x,t), reacts with water according to the overall reaction 5,6 The transport equation for the anions and cations is where j ± is the ion flux and the subscripts refer to the sign of the ion's charge. The stoichiometry of reaction 1 makes r + = r − ≡ r. The ion fluxes are 7 where φ is the electric potential, z = 1 is the ion valence, e is the elementary charge, k B is the Boltzmann constant, and T is the absolute temperature. Under the assumption of local charge neutrality, we have c + = c − ≡ c i and j + = j − ≡ j i . The assumption of local charge neutrality can be justified whenever the Debye length is significantly smaller than the length scale of concentration gradients 8 ; at 136 kPa the Debye length is 25 nm, which satisfies this requirement. It follows that and the electric field where β = (D + − D − )/(D + + D − ). The ion flux therefore simplifies to and we obtain where is the concentration of anions or cations, and r = k f c c − k b c 2 i . The evolution of the particle number concentration n(x,t) follows where D p = k B T /(6π µR p ) and u dp = Γ p ∂ ln c i ∂ x is the diffusiophoretic speed of the particles. Accounting for the finite ratio of particle size and Debye length, we estimate the particle mobilities −691 µm 2 s −1 and 1001 µm 2 s −1 for negatively-and positively-charged particles, respectively, based on the Keh and Wei model 9 . We do not consider variation of the mobility with the local pH since the zeta potential of polystyrene does not change significantly within the range of interest 10 .
Exploiting the symmetry of the domain, we only solve for the CO 2 , ion, and particle concentrations for where c sat c is the CO 2 concentration in equilibrium with the applied CO 2 pressure (p CO 2 /K h where K h is the Henry's law constant for CO 2 in water). At x = 0 the boundary conditions are: The initial conditions are where c atm c and c atm i are the concentrations of carbon dioxide and ions in equilibrium with the atmospheric CO 2 partial pressure (40 Pa), respectively, and we take n 0 = 1.
To compute the trajectories of particles we integrate the positions x p (t) of several particles with evenly spaced initial positions x p (0) according to dx p dt = u dp (x p (t),t) (13) Effect of CO 2 pressure. Supplementary Figure 1 shows the computed displacements of particles under several applied CO 2 pressures. For comparison, CO 2 partial pressures in carbonated beverages range from 200 to 400 kPa 11 . Near the experimental conditions (p CO 2 = 136 kPa), the particle displacements depend weakly on the CO 2 pressure. Over the considered range of pressures, the ratios of the final and initial ion concentrations vary from 1.6 (at 0.1 kPa) to 112 (500 kPa). When the ratio is below approximately 10, which happens for 10 kPa, the displacements vary more strongly with the CO 2 pressure, though a ratio of 2 is sufficient to generate noticeable motion.
Additional species in solution. In addition to the equilibrium of dissolved CO 2 with water we have the reactions HCO 3 and where the equilibrium constants are K 1 = k f /k b = 4.24 × 10 −7 mol L −1 , K 2 = 4.7 × 10 −11 mol L −1 and K w = 10 −14 mol 2 L −2 . Using these equilibria, Henry's law for the relationship between the CO 2 partial pressure and the dissolved CO 2 concentration, and the constraint of electroneutrality, we can determine the concentrations of the species in solution as a function of the CO 2 pressure. The equilibrium concentrations of the dissolved species for several representative CO 2 pressures are given in Supplementary Table 2. At these conditions, the concentrations of H + and HCO 3 are effectively equal, and the concentrations of CO 3 2and OHare negligible. The equilibrium concentrations may therefore be estimated as The particle suspensions employed contain several solutes. The suspension of negatively-charged particles (Bangs Laboratories) contains 2 mM sodium azide (NaN 3 ) as an antibiotic, and 0.1% surfactant (Tween). Upon dilution by a factor of 100 to reach a ∼ 0.01% solids volume, the concentrations become 20 µM and 8 µM, respectively. These solute concentrations are neglected because Tween is nonionic and the NaN 3 concentration is 7 times smaller than the H + concentration in a solution saturated with CO 2 at the pressure employed. The details of the solution composition for the amine-modified, positively charged polystyrene particles are unavailable from the supplier (Sigma-Aldrich). The provided typical compositions for this product are 0.1-0.5% surfactant and 0.2% inorganic salt. The safety data sheet for product number L9654 indicates a NaN 3 concentration of 0.1-0.25% (which is assumed to be the main inorganic salt), which corresponds to 15 mM to 38 mM. Dilution by 200× (to ∼ 0.01% solids volume) yields 77 µM to 192 µM, which is 0.55 to 1.4 times the H + concentration at saturation. We estimate the effect of the additional ions on particle motion by writing where c s is the concentration of additional ions, which reduces the speed of the particles. This reduction is partially offset by a decrease in the Debye length and therefore an increase in the mobilities of the particles, which we estimate as 1066 µm 2 s −1 to 1107 µm 2 s −1 over the range of NaN 3 concentrations. The particle trajectories computed for a solution without additional ions, with 0.1% NaN 3 , and 0.25% NaN 3 are compared in Supplementary Figure 2. Due to reasonable agreement with the experimental results, the trajectories for 0.1% NaN 3 were presented in Figure 3d.