Initial position, x p (0)/L

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.4 ) xp().2 ) ( 2L 2 xp Dc ( Displacement, /L.2.4.5.

1 .4 ) xp().2 ) ( 2L 2 xp Dc ( Displacement, /L Initial position, x p ()/L Supplementary Figure Computed displacements of (red) positively- and (blue) negatively-charged particles at several CO 2 pressures:.,,, 36 (thicker dashed line), and 5 kpa in order of increasing displacement. All other parameters are as listed in Supplementary Table. The displacements are shown at t = 2L 2 /D c = 68s. n/n 5 n/n n/n 5 x/l x/l x/l Time (min) Time (min) Time (min) Supplementary Figure 2 Computed particle concentrations and trajectories of positively-charged particles in (a) pure water, (b).% NaN 3 solution, and (c).25% NaN 3 solution.

2 Supplementary Figure 3 Image sequence of particle (polystyrene,.5 µm) removal driven by CO2 dissolution. The time between each frame is.2 s, which allows tracking of individual particles (indicated by arrows) flowing through the filtrate stream. 2

3 Supplementary Table Parameters used in the calculations description variable quantity CO 2 diffusivity D c.9 9 m 2 s H + diffusivity D m 2 s HCO 3 diffusivity D.85 9 m 2 s ambipolar diffusivity D i m 2 s atmospheric CO 2 partial pressure p atm CO 2 4 Pa applied CO 2 pressure p CO kpa channel half length L 4 µm particle radius R p.25 µm and.5 µm particle zeta potential ζ 7 mv and 6 mv CO 2 Henry s law constant 2 K h 298 LkPamol forward reaction rate 3 k f.39 s backward reaction rate 3 k b Lmol s water viscosity µ.9 mpa s Supplementary Table 2 Concentrations of species in water in equilibrium with CO 2 at several pressures p CO2 (kpa) CO 2 (moll ) H + (moll ) HCO 3 (moll ) CO 2 3 (moll ) OH (moll ) ph

4 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. 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 ( 3 µms ) compared to diffusioosmotic flow, O(µms ), 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, 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 k f CO 2 + H 2 O H + + HCO 3 kb () with forward rate constant k f and backward rate constant k b. The net forward reaction rate is r = k f c c k b c H +c HCO3. The transport equation for CO 2 is therefore c c t = D c 2 c c 2 r (2) The transport equation for the anions and cations is c ± t = j ± + r ± (3) where j ± is the ion flux and the subscripts refer to the sign of the ion s charge. The stoichiometry of reaction makes r + = r r. The ion fluxes are 7 ( c± j ± = D ± ± c ze ± k B T ) φ where φ is the electric potential, z = 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 36 kpa the Debye length is 25 nm, which satisfies this requirement. It follows that ( ci j i = D + + c ze i k B T ) ( φ ci = D c ze i k B T ) φ (4) (5) 4

5 and the electric field E = φ/ is E = β k BT ze lnc i where β = (D + D )/(D + + D ). The ion flux therefore simplifies to ( ) 2D+ D ci j i = D + + D (6) (7) and we obtain c i t = D i 2 c i 2 + r (8) where D i = 2D + D /(D + + D ) is the ambipolar diffusivity, c i (x,t) 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 n t + (u dpn) 2 n = D p 2 (9) lnc where D p = k B T /(6πµR p ) and u dp = Γ i p x is the diffusiophoretic speed of the particles. Accounting for the finite ratio of particle size and Debye length, we estimate the particle mobilities 69 µm 2 s and µm 2 s 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. Exploiting the symmetry of the domain, we only solve for the CO 2, ion, and particle concentrations for x [,L]. The boundary conditions for c c, c i, and n at x = L are c c = c sat c, x c i =, x n = () where c sat c is the CO 2 concentration in equilibrium with the applied CO 2 pressure (p CO2 /K h where K h is the Henry s law constant for CO 2 in water). At x = the boundary conditions are: The initial conditions are c c = c i = n = () c c (x,) = c atm c, c i (x,) = c atm i, n(x,) = n (2) 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 (4 Pa), respectively, and we take n =. To compute the trajectories of particles we integrate the positions x p (t) of several particles with evenly 5

6 spaced initial positions x p () according to dx p dt = u dp (x p (t),t) (3) Effect of CO 2 pressure. Supplementary Figure shows the computed displacements of particles under several applied CO 2 pressures. For comparison, CO 2 partial pressures in carbonated beverages range from 2 to 4 kpa. Near the experimental conditions (p CO2 = 36kPa), 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.6 (at. kpa) to 2 (5 kpa). When the ratio is below approximately, which happens for 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 CO 2 + H 2 O K H + + HCO 3 (4) we have the reactions HCO K 2 3 H CO 3 (5) and H 2 O K w H + + OH (6) where the equilibrium constants are K = k f /k b = moll, K 2 = 4.7 moll and K w = 4 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 2 and OH are negligible. The equilibrium concentrations may therefore be estimated as c eq H + = c eq HCO = c eq 3 i = p CO2 K /K h (7) 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.% surfactant (Tween). Upon dilution by a factor of to reach a.% solids volume, the concentrations become 2 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. 6

7 .5% surfactant and.2% inorganic salt. The safety data sheet for product number L9654 indicates a NaN 3 concentration of..25% (which is assumed to be the main inorganic salt), which corresponds to 5 mm to 38 mm. Dilution by 2 (to.% solids volume) yields 77 µm to 92 µm, which is.55 to.4 times the H + concentration at saturation. We estimate the effect of the additional ions on particle motion by writing u dp = Γ p ln(c s + c i ) (8) 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 66 µm 2 s to 7 µm 2 s over the range of NaN 3 concentrations. The particle trajectories computed for a solution without additional ions, with.% NaN 3, and.25% NaN 3 are compared in Supplementary Figure 2. Due to reasonable agreement with the experimental results, the trajectories for.% NaN 3 were presented in Figure 3d. 7

8 Supplementary References. Haynes, W. M. (ed.) CRC Handbook of Chemistry and Physics (CRC Press, 25), 96th edn. 2. Yaws, C. L. Handbook of Properties for Environmental and Green Engineering (Gulf Publishing Company, 28), st edn. 3. Jolly, W. L. Modern Inorganic Chemistry (McGraw-Hill, New York, 99), 2nd edn. 4. Shin, S. et al. Size-dependent control of colloid transport via solute gradients in dead-end channels. Proc. Natl. Acad. Sci. U.S.A. 3, (26). 5. Ho, C. & Sturtevant, J. M. The kinetics of the hydration of carbon dioxide at 25. J. Biol. Chem. 238, (963). 6. Gibbons, B. H. & Edsall, J. T. Rate of hydration of carbon dioxide and dehydration of carbonic acid at 25. J. Biol. Chem. 238, (963). 7. Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems (Cambridge University Press, Cambridge, UK, 29), 3rd edn. 8. Jackson, J. L. Charge neutrality in electrolytic solutions and the liquid junction potential. J. Phys. Chem. 78, (974). 9. Keh, H. J. & Wei, Y. K. Diffusiophoretic mobility of spherical particles at low potential and arbitrary double-layer thickness. Langmuir 6, (2).. Ohshima, H. & Furusawa, K. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications (Dekker, New York, 998), 2nd edn.. Kuntzleman, T. S. & Richards, C. Another method for determining the pressure inside an intact carbonated beverage can (or bottle). J. Chem. Educ. 87, 993 (2). 8

EEC 503 Spring 2009 REVIEW 1

EEC 503 Spring 2009 REVIEW 1 1. Why are chemical reactions important to energy, environmental and process engineering? Name as many reasons as you can think of. 2. What is a chemical reaction? 3. What