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Langmuir 2001, 17, 7710-7711
Comments Comment on “Direct Analysis of Intraparticle Mass Transfer in Silica Gel Using Single-Microparticle Injection and Microabsorption Methods”
In their article, the authors introduced and evaluated a novel technique to directly observe intraparticle sorption and desorption kinetics in individual porous, spherical particles of silica gel 60.1 Methylene blue (MB) and rhodamine 6G (R6G) cationic dyes were used as sample solutes in an aqueous solvent. The authors evaluated their rate data by numerical simulation of eq 1, for radial diffusion:2,3
(
)
∂CP(r,t) ∂2CP(r,t) 2 ∂CP(r,t) ) DP + ∂t r ∂r ∂r2
(1)
where CP(r,t) is the concentration inside the particle, t is time, and r is the radial coordinate. The intraparticle diffusion coefficient (DP) was a fitting parameter. The capability of this technique to directly observe sorption and desorption processes on a single silica gel particle was clearly demonstrated, and intraparticle diffusion was unequivocally identified as the rate-determining sorption process. The authors also suggest that surface diffusion along the walls lining the pores is the mechanism by which the intraparticle diffusion occurs. The purpose of the present communication is to show that pore diffusion, through the liquid filling the pores, rather than surface diffusion, is the more probable mass transfer process. In terms of eq 1, this distinction resides in the form of the expression for DP. The processes of pore diffusion and surface diffusion occur in parallel. The general expression for DP is2,4-6
DP )
DwHw τw(1 + R)
+
DSR τS(1 + R)
(2)
In eq 2, Dw is the diffusion coefficient in solvent (water), and DS is the intrinsic surface diffusion coefficient. The unitless quantities H, τ, and R represent, respectively, the hindrance parameter, the tortuosity, and the ratio of moles of solute sorbed on the pore walls to the moles dissolved in the solution in the pores. The first term on the right-hand side is for sorption-retarded pore diffusion, and the second is for surface diffusion. The quantities 1/(1 + R) and R/(1 + R) are the fractions of the time in the particle that an average solute molecule spends unadsorbed and sorbed, respectively. Generally, the second term in eq 2 is more difficult to obtain, so it is common (1) Nakatani, K.; Sekine, T. Langmuir 2000, 16, 9256-9260. (2) Grathwohl, P. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/Desorption and Dissolution Kinetics; Kluwer Academic Publishers: Boston, 1998; Chapters 2 and 3. (3) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, 1975; Chapters 9 and 14. (4) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley & Sons: New York, 1984; Chapters 5 and 6. (5) Li, J.; Cantwell, F. F. J Chromatogr., A 1996, 726, 37-44. (6) Komiyama, H.; Smith, J. M. AIChE J. 1974, 20, 1110-1117.
to obtain the first term4,6 and then subtract it from the experimentally measured value of DP in order to get the second term. This is done as described below. To obtain the first term in eq 2, estimates of Dw, Hw, τw, and R are required. Values of Dw ) (5.2 ( 0.6) × 10-6 cm2/s for MB and Dw ) (3.8 ( 0.5) × 10-6 cm2/s for R6G are calculated from the Wilke-Chang equation using a temperature of 298 K, a viscosity of 0.90 cP for an aqueous solution of 0.1 M KCl, and molal volumes of 325 mL/(g mol) for MB and 538.6 mL/(g mol) for RhG. The molal volumes are obtained as prescribed by Wilke and Chang.7 The error is based on the nominal average deviation of 12%.7 The hindrance parameter Hw (constrictivity factor) has values between 0 and 1, with 1 indicating no hindrance.2,5,8,9 Hw values of 0.58 ( 0.1 for MB and 0.52 ( 0.1 for R6G are calculated using the Renkin equation,8,9 employing a pore diameter of 65 Å for silica gel 601 and molecular diameters of 7.9 ( 0.2 and 9.3 ( 0.2 Å for MB and RhG, respectively. The uncertainty in Hw is based on visual inspection of the results of Beck and Schultz.8 The molecular diameters are calculated, assuming a spherical shape, from the molecular volumes, which in turn are calculated in HyperChem 6.02 (Hypercube Inc., Gainesville, FL). The QSAR function, using 40 points on cube side and the van der Waals surface option, is used to calculate volumes in Å3, after geometry optimization with Amber 3. Although the Renkin equation strictly applies in the absence of electrostatic interactions between the diffusing species and the pore walls, the interactions between the cationic MB and R6G and the negatively charged silica surface are considered to be negligible. This is because of the small charge on the silica surface at pH ) 2 and the high ionic strength (0.11 mol/L) of the electrolyte. It is necessary to know the value of the dimensionless particle porosity P in order to calculate τw, and R. Particle porosity is the ratio of the volume of the pores in a particle to the total volume of the particle. It can be calculated as10
P )
VP VP + 1/FSilica
(3)
where the specific pore volume of silica gel 60 is VP ) 1.15 cm3/g and the density of the solid matrix is FSilica ) 2.2 g/cm3. The resulting value of P ) 0.72 is in agreement with values reported for a variety of chromatographic silica gel particles.10,11 A number of theories2 relate τw to particle porosity by an expression of the form
τw ) (P)1-m
(4)
(7) Wilke, C. R.; Chang, P. AIChE J. 1955, 1, 264-270. (8) Beck, R. E.; Schultz, J. S. Biochim. Biophys. Acta 1972, 255, 273303. (9) Deen, W. M. AIChE J. 1987, 33, 1409-1425. (10) Unger, K. K. Porous Silica. Its Properties and Use as Support in Column Liquid Chromatography; Elsevier Scientific Publishing: Amsterdam, 1979; Chapter 5.
10.1021/la010950d CCC: $20.00 © 2001 American Chemical Society Published on Web 11/03/2001
Comments
Langmuir, Vol. 17, No. 24, 2001 7711
where the constant m has been estimated to lie in the range of 1.5-3. Recent experimental measurements yielded τw ) 2 for Baker Bond PEI 300 silica gel,12 and values obtained for similar porous materials range from 1.7 to 2.7,13-16 suggesting that it is very reasonable to assume τw ) 2 ( 0.5 for silica gel 60. R is the ratio of the moles of solute sorbed on the pore walls (nP,S) to the moles of solute dissolved in the pore water (nP,W).3 Since the units of concentration used in ref 1 are moles per liter of particle, for both sorbed solute (CP,S) and solute in the pore water (CP,W), R is also a ratio of concentrations:
R)
nP,S CP,S ) nP,W CP,W
(5)
The total concentration of solute in the particle, in units of moles per liter of particle, was experimentally measured and reported in ref 1. It is related to the sorbed and dissolved intraparticle concentrations by
CP,EQ ) CP,S + CP,W
(6)
Also measured and reported in ref 1 are the values of the distribution coefficients, KCP,∞, for MB ()429) and R6G ()156) in the linear regions of their sorption isotherms, plotted as CP,EQ versus CW (not as CP,S versus CW):
KCP,∞ t
CP,EQ CW
(7)
Here, K and CP,∞ are parameters from a Langmuir equation and CW is the dissolved concentration in units of moles per (11) Unger, K. K.; Kinkel, N.; Anspach, B.; Giesche, H. J. Chromatogr. 1984, 296, 3-14. (12) Coffman, J. L.; Lightfoot, E. N.; Root, T. W. J. Phys. Chem. B 1997, 101, 2218-2223. (13) Koone, N.; Shao, Y.; Zerda, T. W. J. Phys. Chem. 1995, 99, 1697616981. (14) Hejtmanek, V.; Schneider, P. Chem. Eng. Sci. 1994, 49, 25752584. (15) Lorenzano-Porras, C. F.; Carr, P. W.; McCormick, A. V. J. Colloid Interface Sci. 1994, 164, 1-8. (16) Lorenzano-Porras, C. F.; Annen, M. J.; Flickinger, M. C.; Carr, P. W.; McCormick, A. V. J. Colloid Interface Sci. 1995, 170, 299-307.
liter of water and is related to CP,W by the expression
CW ) CP,W/P
(8)
Since we are dealing with equilibrium quantities, the concentration CW is the same in the pore water as in the water outside the particle. Equations 5-8 may be combined to give the following expression for R in the linear region of the isotherm:
R)
KCP,∞ -1 P
(9)
The resulting values of R are 595 for MB and 215 for R6G. When the above values of Dw, Hw, τw, and R are inserted in the first term of eq 2, the calculated values are (2.5 ( 0.9) × 10-9 cm2/s for MB and (4.6 ( 1.6) × 10-9 cm2/s for R6G. These values, for the first term in eq 2, are in very good agreement with, and fully account for, the values of DP ) 2-3 × 10-9 cm2/s for MB and DP ) 4-5 × 10-9 cm2/s for R6G, as measured in the sorption-rate experiments reported in ref 1. The fact that sorption-retarded pore diffusion fully accounts for the experimentally observed overall diffusion coefficient implies that surface diffusion is not significant for both MB and R6G in silica gel 60, under the experimental conditions employed in ref 1. One issue remaining to be addressed is the fact that the reported aqueous concentrations of the dyes (CW ) 4 × 10-6 mol/L) lie somewhat above the linear regions of the sorption isotherms of MB and R6G, which makes their R values not strictly constant during the kinetic experiments. The effects of this were modeled using the guidelines given by Grathwohl2 and Crank3, and it was found that they are negligible. That is, the deviation from the linear sorption isotherm is small enough to be ignored. Robert Bujalski and Frederick F. Cantwell*
Department of Chemistry, University of Alberta, Edmonton, Alberta, T6G 2G2, Canada Received June 22, 2001 In Final Form: September 18, 2001 LA010950D
