Anal. Chem. 2008, 80, 8109–8114
Rotational Hydrodynamic Diffusion System To Study Mass Transport Across Boundaries Sai Sree Mamidi, Bo Meas, and Tarek R. Farhat* Department of Chemistry, Analytical Division, University of Memphis, Memphis, Tennessee 38152-3550 The design and operation of a new mass transport technique is presented. Rotational hydrodynamic diffusion system (RHDS) is a method that can be adapted for analytical laboratory analysis as well as industrial-scale separation and purification. Although RHDS is not an electrochemical technique, its concept is derived from hydrodynamic rotating disk electrode voltammetry. A diffusion advantage gained using the RHDS is higher flux of probe molecules across the boundary (e.g., membrane or porous media) with increased rotation rate compared to the static two-half-cell (THC) method. The separation concept of RHDS differs from pressurized, agitated, electrodialysis, and reversed osmosis systems in design and theory. The detection mechanism of the RHDS opens the possibility to study mass transport properties of a large variety of molecules using different types of ultrathin membranes. Therefore, the RHDS is a potential alternative to classical mass transport detection methods such as THC, impedance spectroscopy, and cyclic and rotating disk electrode voltammetry. Theoretical analysis on the rotational hydrodynamic flux is derived and compared to experimental flux measured using HCl, KCl, KNO3, Ni(NO3)2, LiCl, camphor sulfonic acid, and K3Fe(CN)6 ionic solutions. Values of effective diffusion coefficients of salts across Nucleopore membranes of thickness 6.0 and 10 µm with pore size 0.1 and 0.2 µm, respectively, are presented and discussed. The science of thin-film separation or filtration is gaining momentum in industry due to its economic impact, especially as a cost-effective process.1 As long as novel molecules of chemical, physical, and biological importance are being synthesized, the need for effective yields and high-throughput efficiency is major target for industrialists. At this juncture, our project focuses on developing a mass transport analytical method that use thin membranes mounted between two compartments,1,2 a concept shared by the “rotational hydrodynamic diffusion system” (RHDS) discussed here. The RHDS relies on the hydrodynamic rotational convective force to enhance the diffusion of probe molecules across thin membranes. In other techniques, the flux or permeation rate of solutes is induced by hydrostatic pressure, stirring, * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: 901-678-1423. Fax: 901-678-3447. (1) Scott, K., Hughes, R., Eds. Industrial Membrane Separation Technology, 1st ed.; Blackie Academic and Professional: New York, NY, 1996. (2) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962; Chapter 8. 10.1021/ac800889d CCC: $40.75 2008 American Chemical Society Published on Web 10/10/2008
concentration, and electrodialysis.1-3 Examples of commercial processes that fall under this category are reverse osmosis used to remove organic and inorganic contaminants from feedwater, ultrafiltration that relies on size exclusion to isolate proteins, and microfilteration for separating colloidal particles from biological species.1-3 There are similar concepts in the literature and commercial products that are related to the rotational hydrodynamic concept. Articles on rotary ultrafiltration that have addressed flow patterns usually tackle problems associated with colloidal suspensions that try to minimize fouling and particle retention.4-6 Other groups and Spintek technology described cross-flow dynamics in rotating disk filters4,6 that differs from the RHDS in design and theory. Irrespective of the type of flow occurring,4,5 the RHDS theory assumes a vortex flow with a stagnant diffusion layer near the surface.2,7 The RHDS is dedicated to study the diffusion or mobility of soluble ions or molecules in aqueous and organic media, a prelude to further research on membrane separation science. Another method that may appear similar but is different from the RHDS technique is the rotating cylinder electrode (RCE) method. The RCE, like the RDE, is an electrochemical technique dedicated to study mass transport in pipes or cylinders for corrosion of metals under controlled turbulent conditions.8 Friedlander and Spaeth worked on gas-liquid interfaces where hydrodynamic rotation was used to study the flux of respiratory gases into blood serum.9 This paper has three objectives: (a) derive a theory related to the ”rotational hydrodynamic diffusive effect”, (b) present the experimental set up of the RHDS, and (c) present experimental data on six aqueous ionic solutions to test the proposed theory. We aim to make the RHDS serve as a research analytical device to study the flux of probe molecules across porous and semipermeable membranes. That is, it should be a convenient alternative to electrochemical methods such as electrochemical impedance spectroscopy (EIS),7,10,11 cyclic voltammetry, and rotating disk (3) Cheryan, M. Ultrafiltration and Microfiltration Handbook; Technomic Publishing: Lancaster, PA, 1997. (4) Aubert, M. C.; Elluard, M. P.; Barnier, H. J. Membr. Sci. 1993, 84, 229– 40. (5) Reed, B. W. L.; Viadero, R.; Young, J. J. Environ. Eng. 1997, 123. (6) SpinTek. In Membr. Sep. Technol. News 1992, 10(12). (7) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; J. Wiley & Sons, Inc.: New York, 2001; Chapter 10. (8) Gabe, D. R.; Wilcox, G. D.; Gonzalez-Garcia, J.; Walsh, F. C. J. Appl. Electrochem. 1998, 28, 759–780. (9) Spaeth, E. E.; Friedlander, S. K. Biophys. J. 1967, 7 (6), 827–851. (10) Barreira, S. V. P.; Garcia-Morales, V.; Pereira, C. M.; Manzanares, J. A.; Silva, F. J. Phys. Chem. B 2004, 108, 17973–17982.
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electrode (RDE) voltammetry.12,13 Researchers apply electrochemical methods because they are easy to use and very sensitive. However, no quantities of the solute are collected because electrochemical methods work by consuming or perturbing selected electroactive molecules or ions at the surface of a nonporous electrode.7 Neither RDE nor EIS addresses design requirements when ultrathin films are deposited on porous supports. On the other hand, RHDS determines the flux of probe molecules across a selected membrane and, from an engineering point of view, predicts the output performance on an industrial scale. The hydrodynamic diffusion system RHDS combines two concepts: the hydrodynamic RDE voltammetry and the two-halfcell method.2,7 The rotating electrode is redesigned to become a rotating compartment that collects the sample solute under study. With RDE voltammetry, the electrochemical flux density across the boundary layer can be determined from current density measurements,14 but with RHDS, the physical flux density must be measured using a suitable analytical technique. Unlike RDE, the RHDS is not an electrochemical technique and can be used to measure the flux of a large variety of nonelectroactive molecular probes. For example, electrochemical methods dictate using molecular probes that are electrochemically active using the appropriate electrolyte. It is crucial that the electroactive probe yield a steady-state current-potential characteristic where the limiting currents can be easily measured in order to evaluate the flux.7,14 Such requirements are not needed in RHDS because it operates on a large variety of molecular probes (i.e., electroactive or not) in the absence of supporting electrolytes. Detection is not only electrochemical but can be chromatographic, spectroscopic, radioactive, and conductometric. The RHDS offers higher flux of probe molecules or ions across the membrane, generating higher sensitivity than static half-cells.1,2 Instead of making a single measurement of the flux as in static half-cells, the RHDS can obtain a characteristic showing the change of flux with rotation rate. From flux characteristics, physical parameters such as ionic diffusion coefficient (i.e., DM+, DX-), effective diffusion coefficient (i.e., DMX), and mobility can be accurately determined. At this juncture, we have used Nucleopore membranes of longitudinal microchannels to simplify the theory of RHDS molecular flux characteristics. Flux of simple ions across 0.1- and 0.2-µm pores that are 6.0 and 10.0 µm in length was studied. These ions are the ferricyanide, Fe(CN)63-(aq), lithium, Li+(aq), potassium, K+(aq), nitrate, NO3-(aq), nickel, Ni2+(aq), camphor sulfonate ion, Camph-SO3-, hydronium ion, H+(aq), and chloride ion, Cl-(aq) of known diffusion coefficients in aqueous solutions (except Camph-SO3-). Thus, the work presented here is analogous to determining the diffusion coefficients of probe ions with a voltammetric method using a plain platinum electrode not coated with any thin-film membrane. In a future paper, we address design requirements for a selected solid thin-film membrane (e.g., thin polymer gel membrane) on a porous support (e.g., Nucleopore or a perforated metal plate) and derive a more complex flux equation using the same ions. (11) Barsoukov, E., Macdonald, J. R., Eds. Impedance Spectroscopy: Theory, Experiment, and Applications, 2nd ed.; Wiley-Interscience: Hoboken, NJ, 2005. (12) Gough, D. A.; Leypoldt, J. K. Anal. Chem. 1979, 51, 439. (13) Ikeda, T.; Schmehl, R.; Denisevich, P.; Willman, K.; Murray, R. W. J. Am. Chem. Soc. 1982, 104, 2683. (14) Farhat, T. R.; Schlenoff, J. B. Langmuir 2001, 17, 1184–1192.
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Figure 1. (a) Schematic of the rotational hydrodynamic diffusion system showing an upper output cylinder that rotates within a lower input cylinder. A perforated 1.0-mm stainless steel disk supports a Nucleopore membrane (6-µm thickness, pore diameter 0.1 µm, pore density 4 × 108). Sample is injected or poured into the LIC. Volume of UOC is V ) 6.5 mL and exposed area of the membrane is A ) 1.23 cm2. (b) A schematic showing a vortex flow represented by L-shaped arrows with a stagnant diffusion layer near the surface. Rotation of the upper output cylinder causes a vortex with an upward velocity component υy. The convective currents hit the surface of the membrane that separates the UOC from the LIC and are diverted with a horizontal velocity component υr indicated by arrows.
EXPERIMENTAL SECTION Chemicals. All chemicals were used as provided by the manufacturer. Potassium chloride and KCl calibration conductivity standards from Alfa Aesar, lithium chloride, Camph-SO3H, and potassium ferricyanide from Aldrich; and hydrochloric acid and potassium and nickel nitrate from Fisher Scientific. Nucleopore membranes (6-µm thickness, pore diameter 0.1 µm; and 10-µm thickness, pore diameter 0.2 µm) were purchased from SPI supplies. Instruments. The RHDS is composed of two cylinders such that the upper output cylinder (UOC), volume 6.5 mL, freely rotates within a lower input cylinder (LIC) of volume 300 mL, Figure 1. The LIC contained the sample solution under study and was maintained within a temperature range of ∼20-22 °C. Ultrapure water, obtained from a Millipore Simplicity Simpak2 filtration unit, was injected with a hypodermic syringe into the UOC until it overflowed at the start of every experiment. The UOC module can be removed and installed for each run. Rotation of the UOC of the RHDS was fully controlled by an MSRX Speed control (1-10 000 rpm) from Pine Ins. Co. The upper cylinder UOC was made of transparent polycarbonate and specially designed to monitor liquid level, state, and flow. It was designed to support a perforated stainless steel disk and the porous membranes (i.e., Nucleopore filters). The porous membrane was firmly attached to a perforated stainless steel support of 1.0-mm pores, while surface area of the membrane in contact with the electrolyte was 1.23 cm2. In a different setup, the membrane was mounted using a water-resistant optical adhesive without the perforated stainless steel disk. The concentrations of the probes under study were determined by extracting 5.0-mL aliquots from the UOC with a hypodermic syringe after 2.0 or 4.0 min of rotation. Concentrations of ions and calibrations were determined using
∂ 2C -y2 ∂C ) B ∂y ∂y2
(4)
where B ) (Dω-3/2ν1/2)/0.51, C is the bulk concentration (mol cm-3), and D (cm2 s-1) is the diffusion coefficient of the probe molecule under study. For limiting current conditions when the surface concentration is equal to zero, the solution of the differential eq 4 is,
Figure 2. Schematic of the two diffusion regions RI and RII with different concentration gradients considered under steady-state conditions. Region RI is described by the Levich hydrodynamic law; ions in region RII are under Fick’s first law of diffusion. Stagnant diffusion layer of thickness ”δO”, L is the membrane thickness, C is the concentration of the sample under study.
an Oakaton CON11 conductivity meter/probe with a cell constant κ ) 1.0 and KCl calibration standards. Thus, conductance measurements refer to conductivity of solutions and were directly related to concentrations via calibration curves. All salt solutions were prepared with ultra pure water. RESULTS AND DISCUSSION Before proceeding to discuss the RHDS in detail, a brief description of the overall RHDS project is presented. A schematic of the RHDS shows a UOC that rotates within an LIC, Figure 1. Commercial porous membranes used in this study are mounted on a perforated metallic support while the probe molecules investigated are injected into the LIC. After rotating the UOC at a specific rotation rate for a fixed period, the concentration of the probe molecules collected in the UOC is measured. These concentrations are converted to flux values and plotted at various rotation rates. Experimental results were compared against the RHDS theory proposed in this research. Theoretical Flux Equation. To understand the theory of the hydrodynamic diffusion mechanism, consider the schematic of the rotating UOC illustrated in Figure 1a. The UOC hydrodynamic compartment rotates freely within the sample LIC compartment causing a vortex with an upward velocity component υy. The convective currents hit the surface of the supported membrane that separates the UOC from the LIC and are diverted with a horizontal velocity component υr as shown in Figure 1b. Both components depend on the rotation rate ω (rev s-1), kinematic viscosity ν (cm2 s-1), vertical distance y from the surface of the supported membrane, and radial distance r away from the center of the Nucleopore membrane such that7 υy ) -0.51ω3⁄2ν-1⁄2y2
(1)
Uo ) lim υy ) -0.88√ων
(2)
υr ) 0.51ω3⁄2ν-1⁄2ry
(3)
yf∞
Uo is the limiting velocity in the y direction. Our interest is only in the flux at r ) 0, where υr ) 0, and consequently, the complex differential equation of cylindrical coordinates7 (not shown) simplifies to
C)
( ∂C∂y )
1.6(Dω-3⁄2ν1⁄2)1⁄3
y)0
(5)
The flux density J (mol cm-2 s-1) is defined in terms of Fick’s first law of diffusion as the product of the concentration gradient ∂ C/∂ y and D such that
( ∂C∂y )
J)D
y)0
(6)
thus, substituting for ∂ C/∂ y, we obtain the Levich equation for the flux density,
J ) 0.62
CD2⁄3√ω ν1⁄6
(7)
The relation in eq 7 is important because it shows the flux, whether electrochemical or physical, increases with the square root of the rotation rate ω1/2. In this paper, we did not try to find a relation between the electrochemical flux of a solid commercial RDE and the physical flux JP of the RHDS. We focused only on the determination of the physical flux JP of the RHDS and its relation to the rotation rate ω. However, mass transport analysis of the RHDS depends on another crucial factor, the fluid dynamics within the UOC. It was observed that when the UOC was not full, a whirlpool developed inside the UOC that increased in depth as the rotation rate increased. The whirlpool grew due the friction between the liquid bulk and the inside walls of the rotating UOC in the presence of air. On the other hand, when the UOC was full, a very small cavity developed inside the UOC, one that did not increase in depth as the rotation rate is increased. The reason for a diminished whirlpool effect in a full and sealed rotating UOC was the absence of air and dominant vacuum cavitations that prevented the cone of the whirlpool from touching the surface of the porous membrane in the UOC. As a result, the stagnant diffusion layer remained intact. Ramping the rotation rate to 2500 rpm, the surface of the Nucleopore membrane exposed to the solution in the LIC did not develop a whirlpool effect or bubbles. In this project, two regions of diffusion RI and RII are assumed under steady-state conditions as shown in the schematic in Figure 2. The overall physical flux Jp of the probe molecules is a function of two diffusion lengths. There is the thickness of the diffusion (boundary) layer (δ) formed at the surface of the membrane from the LIC side and the length of the pores of the membrane of thickness L. Diffusion length δ is a function of porosity (p) and the rotation rate ω. Porosity is the ratio of the total area of the pores to the boundary area; hence, 0 < p < 1. Nucleopore membranes we used have a porosity of 0.031 and 0.094; thus δ is defined as Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
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δ ) δo(1 - p)
(8)
and according to the hydrodynamic concept,7
δO )
D1⁄3ν1⁄6 0.62ω1⁄2
(9)
To simplify, a profile of the steady-state regime that transforms from dotted line to a solid line was considered for analytical simulation, Figure 2. The hydrodynamic flux is an additive effect of the convective flux that brings the molecules to the surface of the diffusion layer and the diffusive flux that carries them toward the surface of the porous membrane. At the surface of the pore, another diffusive flux transports the molecule to the bulk of the UOC.1,7 The overall physical flux Jp is effectively a series mass transport resistance of concentration gradients across the diffusion layer and inside the pore as depicted in Figure 2 (not to scale). To simplify the mathematics, gradient line of (∆Co/L)p is extended to intercept the concentration axis at virtual concentration C* as shown by thick dotted line in Figure 2 such that, C* ) qC
(10)
where q > 1, unknown, and can be determined experimentally using KCl as a reference salt. Using Fick’s first law of diffusion on mass transport, flux JI in RI at steady state is, JI ) m(C* - CO)
(11)
and directly related to eq 7 where the mass-transfer coefficient m is, m)
D δ
(12a)
Substituting δ for from eqs 8 and 9,
m ) 0.62
D2⁄3 ω1⁄2 ν (1 - p)
(12b)
1⁄6
Flux across the pore is, JII )
D (C - Ci) L O
(13)
where Ci is set to zero because molecules or ions are considered to transport irreversibly immediately away from point K after permeation, Figure 2. Using concept of similar triangles yields,
CO )
C*L (δ + L)
(14)
By substituting for CO, δ, and m in eq 11 and eq 13, we find that JI and JII are equal, as expected. Therefore, the overall flux can be represented by
( )
Jt ) 0.62 8112
D2⁄3 1⁄2 * δ ω C δ+L ν1⁄6
(
)
(15)
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Substituting for C* and δ in eq 15 would redefine eq 15 in terms of D, A, L, ω, p, and C. Thus, the final form of the flux eq 15 is,
Jt )
[
]
0.62Dq ω1⁄2C D1⁄3ν1⁄6(1 - p) + 0.62Lω1⁄2
(16)
The value of q turned out to be ∼4 in all experiments, and we referred to it as cell constant. So far, the value of q is maintained at 4 even after changing the equipment (i.e., rotator and speed control) and the type of membrane (i.e., Nucleopore 0.1/6 vs 0.2/ 10 µm). The value of q together with other parameters was used to determine the effective diffusion coefficient DMX of each salt and the diffusion coefficient of each ion and vise versa, such that,
DMX )
DM+DXDM+ + DX-
(17)
DMX was applied in all calculations based on KCl as a reference salt where DCl- )1.02DK+. This way an ion with unknown diffusion parameters such as camphor sulfonate ion can be tested after proper analytical calibration of the RHDS. Experimental Flux Equation. The physical flux density JP of the RHDS is determined experimentally from concentration measurements over a fixed period of time ∆t using the setup shown schematically in Figure 1. For the physical flux Jp, a simple flux equation that describes the process in the absence of any kind of pore blocking or caking is1
Jp )
cV 1 ∆n ) Ae ∆t Fπr2A∆t
(18)
where Ae is the effective area calculated from pore density F, pore radius r, and exposed area of the membrane A. The amount of probe molecules collected during time ∆t is ∆n. The amount of probes is determined from the volume of the UOC V accommodating a concentration c of probe molecules. Volume V of ultrapure water was injected into the UOC at the start of the experiment. A value of ∆t was selected by testing durations of 1, 2, 4, 6, and 8 min, where the ratio of c/∆t was found constant. Values of ∆t ) 2 or 4 min were considered convenient because these durations are short enough to rule out the possibility that ions might diffuse from the UOC back to the LIC. Values of concentration c were determined from conductivity measurements. In all RHDS experiments, many parameters were set constant. Duration was set at ∆t ) 2.0 or 4.0 min, V ) 6.5 mL, and A ) 1.23 cm2. A large-volume (300.0 mL) salt solution (0.1 or 0.2 M) expressed in terms of ionic strength was placed in the LIC and maintained at a constant concentration C. Any changes in the physical flux JP with ω, ω1/2, and ω1/2C were examined to check if the proposed theory (i.e., eq 16) fits experimental results obtained using eq 18. Various and random rotation rates, usually in steps of 100 rpm, were applied. The UOC was immersed in the LIC at a depth of ∼0.5-1.0 cm. There was no change in the measured flux JP whether a partial or full immersion of the UOC in the LIC was made. Hence, the effect of the hydrostatic pressure on the flux was absent or negligible in this experimental setup. The effect of osmotic drag that may force pure water from the
UOC through the pores into the LIC was also checked. The osmotic drag was considered negligible because the level of pure water inside the UOC did not change and the conductivity (or TDS level) of the solution in the LIC remained constant during 5 min of operation. These experimental observations were attributed to a sealed UOC where the liquid inside it did not drip through the membrane. Minimizing or eliminating the above factors would leave the hydrodynamic system under diffusion control. Under static conditions, background diffusion or membrane resistance is represented by the flux JP at ω ) 1/60 rps (i.e., 1.0 rpm). In this study, testing the RHDS theory on an uncoated porous membrane is analogous to testing the “hydrodynamic RDE” theory on an uncoated or bare platinum electrode. Tests on ionic aqueous solutions have many advantages. First, ionic aqueous solutions allowed us to use the very sensitive conductivity technique (S m-1) for quantitative analysis, which minimized the error of variance. Conductivity measurements dictated converting all concentrations to activities to account for ionic strength. Second, the size of all ions used is on average ∼2000 times smaller than the pore size and mass transport inside the pore is essentially diffusive and controlled by the ionic mobility or the diffusion coefficient D. Thus, the effect of surface diffusion and multilayer diffusion is negligible.1,7,15 Third, membrane fouling by salt ions due to partial or complete pore blocking and cake formation is absent.1,15 For all samples, a calibration plot of conductivity against concentration was made in order to obtain a calibration constant. It was best to do the calibration on highly diluted solutions where the calibration constant was used to convert the conductivity measurements to concentrations. The flux characteristics of KCl are shown in Figure 3a,b. The theoretical solid line obtained using eq 16 showed a good fit to experimental flux measurements obtained using eq 18. The plot in Figure 3a shows a nonlinear increase in the flux Jp with the rotation rate ω as predicted by eq 16. At low rotation rates, the term D1/3A1/6(1 - p) . 0.62ω1/2L and the term in brackets may represent the slope of a linear plot of Jp versus ω1/2C, which can be used to determine D. At high rotation rates (i.e., ω f ∞ rpm), the term 0.62Lω1/2 . D1/3A1/6(1 - p) leaving Jp ) Jlimit ∼ qCD/ L, where Jlimit is the limiting flux that is independent of ω and asymptotic with the x-axis. Unlike the RDE theory, a plot of the flux Jp versus ω1/2 is nonlinear rather than linear (refer to Levich eq 7), Figure 3b, because the term ω1/2 is also a part of the slope of eq 16. In the RHDS technique, five parameters, the diffusion coefficient, the calibration constant, the pore density (i.e., pores per cm2), the membrane thickness, and the pore radius mainly affect the flux. If the value of the diffusion coefficient is to be determined accurately from the flux characteristic, the values of pore density, membrane thickness, and pore radius have to be measured accurately using instruments such as the scanning electron microscope. Values related to the Nucleopore membrane were obtained using the manufacturer’s specifications and our scanning electron microscope, Table 1. The concentration of the probes collected in the UOC (i.e., c in eq 18) was determined by multiplying the reciprocal of the calibration slope and the recorded conductivity values. In this case, we have compared the recorded calibration slopes to the tabulated CRC Handbook data to make (15) Sorensen, T. S., Ed. Surface Chemistry and Electrochemistry of membranes; Surfactant Science Series 79; Marcel Dekker, Inc.: New York, 1999.
Figure 3. Flux (mol cm-2 s-1) characteristics of KCl. The theoretical solid line obtained using eq 16 showed a good fit to experimental flux measurements obtained using eq 18. (a) A nonlinear increase in the flux Jp with the rotation rate ω (s-1) as predicted by eq 16. (b) Unlike the RDE theory, a plot of the flux vs ω1/2 (s-1/2) shows a nonlinear rather than a linear relationship because the term ω1/2 is part of the slope of eq 16. Background flux is 5.3 × 10-7 mol cm-2 s-1. T ∼ 22 °C, volume of UOC is V ) 6.5 mL, and exposed area of the membrane is A ) 1.23 cm2. Nucleopore membrane (thickness 6.0 µm, pore diameter 0.1 µm, pore density 4 × 108). Error bar set at 5%. Table 1. Values of Pore Density, Membrane Thickness, Porosity, and the Pore Radius Obtained Using Manufacturer’s Specifications and Our Scanning Electron Microscopea type Nucleopore Nucleopore
L (µm)
dp (cm-2)
rp (µm)
porosity
6 10
4 × 10 3 × 108
0.05 0.1
0.031 0.094
8
a rp is pore radius; dp is pore density (no. of pores cm-2); L is membrane thickness.
sure that all values were in the same range. In this study, we have also experimented on two Nucleopore membranes of different pore size (0.1 and 0.2 µm) and different thickness (6 and 10 µm). Using the same family of salts, values of D obtained in both cases were within experimental error and less than 2%. We have tested the RHDS technique and theory on hydrochloric acid, lithium chloride, potassium nitrate, nickel nitrate, camphor sulfonic acid, and potassium ferricyanide. We have obtained plots similar to those of KCl, which we have included for comparison in one graph, Figure 4a,b. As expected based on the literature, the ferricyanide ion Fe(CN)63-(aq) has the lowest diffusion coefficient, Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
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Figure 4. Flux characteristics of three different ions: the ferricyanide ion Fe(CN)63-(aq), the hydrated lithium ion Li+(aq), and K+(aq) ion (the chloride ion Cl-(aq) characteristic overlapped the K+(aq) ion). (a) A plot of the flux Jp vs the square root of rotation rate ω1/2 (s1/2) of the Fe(CN)63-(aq) ion (2) that showed the lowest diffusion coefficient, thus the lowest flux, followed by the Li+(aq) ion (O), and then the K+(aq) ion ([) [and chloride ion Cl-(aq)] with the highest flux. (b) A plot of the flux Jp vs the square root of rotation rate ω1/2 (s1/2) of the camphor sulfonate (aq) ion (9) that showed the lowest diffusion coefficient, thus the lowest flux, followed by the Ni2+(aq) ion ([), and then the NO3-(aq) ion (×) with the highest flux. T ∼ 22 °C, volume of UOC is V ) 6.5 mL, and exposed area of the membrane is A ) 1.23 cm2. Nucleopore membrane (thickness 10.0 µm, pore diameter 0.2 µm, pore density 3 × 108). Error bar set at 5%. Table 2. Values of Effective Diffusion Coefficients DMX of Selected Salts and Acidsa ion
DL × 10-5 (cm2 s-1)
DH × 10-5 (cm2 s-1)
ClK+ Li+ H+ Ni2+ NO3Fe(CN)63Camph-SO3
2.02 1.95 1.02 9.1 0.73 1.89 0.85 ?
1.95 ± 0.05 1.92 ± 0.05 1.1 ± 0.05 9.5 ± 0.05 0.68 ± 0.05 1.9 ± 0.05 0.83 ± 0.05 0.42 ± 0.05
DMX × 10-6 (cm2 s-1) KCl LiCl K3(FeCN)6 HCl KNO3 Ni(NO3)2 HCamph
9.75 6.97 5.82 15.8 9.6 5.0 4.02
a It compares D of ions obtained using rotational hydrodynamics DH compared to literature values DL. Camph-SO3 is camphor sulfonate ion; HCamph is camphor sulfonic acid.
Table 2, followed by the hydrated lithium ion Li+(aq), then K+(aq), and finally Cl-(aq). All values of D±(aq) fluctuated around literature values with average DCl-.16 In fact, data representing the Cl-(aq) nearly overlap those of K+(aq) and was not included in the plots of Figure 4 because DK+ ∼ DCl- ions. We did not find a literature value for camphor sulfonic acid but our experimental results showed (16) Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S. H. J. Phys. Chem. B 1998, 102, 4193–4204.
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Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
the lowest diffusion coefficient attributed to the large size of the camphor sulfonate ion (D ) 0.42 ± 0.05 cm2 s-1). However, the objective of this study is not focused on determining the values of diffusion coefficients of known ions but to demonstrate that the RHDS is a viable analytical technique to study mass transport across boundaries. That is, instead of getting a single data point of the flux as in static half-cells we can obtain a characteristic showing the change of flux with rotation rate analogous to characteristics of flux versus pressure, concentration, and electric field change. With RHDS, there is no need to apply differential pressure, concentration gradient, or electric field to study the flux. However, this is not meant to rule out a combination of RHDS with one or more of the above parameters in a future project. The flux theory of a bare support of cylindrical pores in RHDS is analogous to the flux theory of a bare platinum electrode in RDE voltammetry. In RDE, the flux is represented by the Levich flux eq 7, while in RHDS, the flux is represented by eq 16 accounting for pore diffusion. Therefore, the RHDS can be extended to study the diffusion of molecules through pores, especially if such molecules are made to interact with the pore walls or blocked by them. That is the basics of analytical separation. In RDE, coating a working electrode with a thin film requires a separate flux theory. In the same way, casting a thin film over the porous support in RHDS would also require flux equations different from eq 16 that we plan to investigate in the future using a variety of thin films such as layerby-layer and polymer gel membranes. To this end, we foresee that RHDS would have two types of applications. In analytical science, it would be used as a benchtop instrument to characterize the permeation properties of all types of molecules and ions across boundaries. The boundaries can be ion exchange, porous, gel, and inorganic composite membranes. In the industrial sector, the RHDS can be scaled up to separate and purify products from byproducts especially in pharmaceuticals and certain sectors of the chemical industry. CONCLUSION The RHDS is a new unconventional diffusion system to study molecular flux across boundaries. The RHDS is composed of two cylinders, a lower input and a rotating upper output cylinder. The smaller UOC rotates within the larger LIC to induce a mass transport process that is a combination of rotational hydrodynamic and Fick’s laws. A modified Levich model was used to derive a flux equation for a liquid-liquid interface across straight pores such as Nucleopore membranes. The RHDS technique was tested using seven ionic species represented by HCl, KCl, LiCl, Camph-SO3, KNO3, Ni(NO3)2, and K3Fe(CN)6 suitable for conductivity measurements. Values of the diffusion coefficients obtained by the RHDS were close to literature values, making the RHDS technique a viable method to study mass transport processes across boundaries. ACKNOWLEDGMENT The authors would like to thank Mr. John Daffron, department of physics machine shop, and Ms. Lou Boykins, Electron Microscopy Lab at the University of Memphis, for their technical contributions to the RHDS project. Received for review April 30, 2008. Accepted August 13, 2008. AC800889D
