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ЭЛЕКТРОНИКА
UDC 517.958: 536.71
MATHEMATICAL MODELLING OF DIFFUSION-REACTION INSIDE MESOPOROUS SPHERICAL NANOPARTICLES. ANALYTICAL SOLUTION FOR THE CASE OF EXTREMELY FAST IRREVERSIBLE ADSORPTION
OLEINICK A.__________________________________
The process of solution remediation by an ensemble of functionalized mesoporous spherical nanoparticles consisting of nanotube bundles is considered. The physicochemical basis for the process at hand is diffusion mass transport of target molecules towards the interior of the nanotubes coupled with the adsorption by ligands lining the internal surface of the tubes. The analytical solution for the case of extremely fast irreversible adsorption is presented.
1. Introduction
Sol-gel synthesis of inorganic mesoporous materials of different forms (layers or spheres of nano/mesosized tubes) coupled with the recently succeeded functionalizing of this kind of materials with organic ligands opened a vast range of their applications, for example, in remediation of contaminated solutions, heterogeneous catalysis, highly selective electrodes etc. [1-6]. Recently in work [7] the mathematical model and qualitative analysis were presented for the case of the solution remediation process by an ensemble of microspheres consisting of tube bundles (see Figure 1). The filtration of contaminated solutions is performed by means of diffusion of target material inside the tubes followed by the adsorbtion at active sites lining the internal surface of the tubes. Depending on the relative values of the pore and particle radii and wall thickness (between adjacent pores) the mathematical model of the process at hand could be formulated in one or two spatial dimensions. It was noted that systems amenable to 1D formulation are more effective in terms of storage of target material, since they provide higher active surface area than systems with larger pores amenable to 2D formulation. Furthermore, for the case of irreversible adsorption (which is the inherent property of an effective system) in the 1D system a zone diagram was presented containing three characteristic zones (see Figure 2): I and III where diffusion or adsorption dominates, respectively, and zone II where these two processes compete with each other. Using this diagram it is easy to conclude that in terms of efficiency, i.e. storage and operative time parameters, zone I is the most promising
one, since in this zone the adsorption rate is extremely fast leading to a significant decrease of operative time. The purpose of this paper is to develop an analytical solution for this special but very important from the practical point of view case.
2. Mathematical model
The qualitative description of the process at hand is the following. The stirred volume VQ contains initial solution of target species of concentration Cq . At time t = 0 the Npart fully wetted spherical particles with the average radius Rpart are added into the solution. Target molecules diffuse with the diffusion coefficient Dbulk in the open solution, and upon entering the interior of the tubes their diffusivity becomes Dpore, which is generally much lower than Dbulk. Diffusion together with the stirring create a stagnant ‘diffusion layer’ of thickness 5 around each particle, which defines the rate of target species transport into nanotubes openings (concentration gradient of target species). Each nanotube is characterized by its length l, radius Rpore and the half-thickness of the wall ю separating adjacent tubes (Figure 1).
Figure 1. Schematic representation of a spherical mesoporous particle consisting of a dense bundle of nanopores. (a) Schematic view showing one nanopore placement inside the particle and its opening at the particle surface assuming a hexagonal packing, together with its diffusional “projection” into the solution. (b) Cross section of one nanopore and “its” diffusion layer along a plane containing the nanopore axis (shown by the central dashed line)
The interior surface of the tubes is covered with specific active sites with surface concentration rsite, which are capable of binding/releasing target species with the adsorption and desorption rate constants k ads and k des respectively. Adsorption of target molecules at the walls of the tubes depletes the external solution of target
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РИ, 2007, № 4
species thus decreasing the incoming flux. The steady-state is achieved when equilibrium between bulk solution and interior of the tubes is established.
Since the number of particles involved into the experiment as well as the number of tubes borne by each particle is extremely high, by virtue of the ergodic principle it is possible to consider the diffusion-reaction process in the averaged nanotube, i.e. the tube with the averaged characteristics, in order to describe the overall ensemble of the spherical particles acting in the contaminated solution. Generally the tube radius and wall thickness are experimentally available by means of gas adsorption techniques [4,5]. The averaged tube length was estimated in [7] based on the average particle radius (which can be obtained experimentally using light-scattering techniques) and is given by L = Rpart / 3 .
Figure 2. Kinetic zone diagram illustrating the different behaviors experienced by 1D-nanoporous system as a function of its main dimensionless parameters characterizing its dynamics (adapted from ref. [7], see below for definitions of parameters)
For the sake of clarity we give the mathematical model of the diffusion-adsorption process occurring in the interior of one tube which was derived in [7] for the case of extremely small pores with diameters comparable to the target ion or molecule size (hence amenable to 1D formulation) and conditions described above:
da dz d 2д = d —2 _ (X0S 0 )[(1 - 0)a - 0k] • 5y2 (1)
— = X 0[(1 -0)a -0k] ; dz (2)
dc Q . - = -9(c - ay=°), (3)
which is associated with the following initial conditions ( X = 0 ):
c = 1; (4a)
0 < y < 1: a = 0, 0 = 0; (4b)
y = 0: II О ocr CD II 0 (4c)
and boundary conditions (x > 0):
y = 0: If) =-31 (c - ay .0). W') y .0 Л ■ (4a)
y = 1: M = 0 J y=1 , (4b)
РИ, 2007, № 4
where equations (1), (2) describe the evolution of the dimensionless (with respect to the bulk solution) concentration of target species, a , and adsorption coverage, 0, inside the nanotube. The concentration inside the nanopore is coupled to that in the bulk solution, c = Cb / Cb, through Equations (3) and (4a). Variables t = Dt/rpore and y = x/L are the dimensionless time and spatial coordinate directed to the center of the particle with the origin at the pore entrance. The other dimensionless parameters used in the formulation (1)-(4) have the following meaning: Ц = Dpore/Dbulk is the ratio of diffusion coefficients; X 0 = kadsL2Cb /Dbulk is the dimensionless adsorption rate constant; k = Kdes/Cb is the dimensionless desorption equilibrium constant; Kdes = kdes/kads is the desorption equilibrium constant; E0 = 2rsite /RporeC0 is the dimensionless storage capacity of the pore; 0= 4nNpartRpartctL2/VbS is the
dimensionless time constant; ^ =1 + S/Rpart ; ф = у2стф2 (L/8) is the dimensionless parameter defining the mass transport rate at the nanotube entrance; V =1 + ю/Rpore ; У is the factor commensurable to unity, which depends on exact packing of the nanotubes.
3. Analytical solution for the case of extremely fast irreversible adsorption
As mentioned in Introduction, from the application point of view one is interested only in the case of very weak desorption versus adsorption rates, which correspond to к << 1. This fact allows neglecting the last term in Equation (2) so that it can be reduced to:
^ = X 0(1 -0)a. (5)
dx
This equation can be easily solved yielding
9 = 1 - exp[-X0 J adx]. (6)
The integration constant in Equation (6) was defined using the initial condition 9 (x = 0) = 0 .
In order to obtain the concentration profile let us assume that X0 is sufficiently high (X0 ^ да ) and the concentration at the pore entrance is maintained at unity value. The latter assumption is appropriate for the case of the 1D-model since significantly different diffusion rates prevail inside and outside the pore, due to the molecular and electrostatic interactions within the pore which diminish the diffusivity of the target species. Indeed, then the concentration gradient outside the pore is negligible compared to that inside. Thus the concentration drop in the diffusion layer around the particle (i.e. over the distance 8 ) is small and hence a(0, x) may be assumed equal to unity.
In this situation, based on the qualitative consideration, the concentration profile can be accurately represented by a linear decay, since according to our assumption (i.e. X0 ^ да) any incoming molecule will adsorb immediately upon reaching a free site at the tube wall. Since diffusion
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is the limiting process here and the concentration at the end of the diffusion layer inside the nanotube is zero due to instant adsorption the concentration profile should be linear within the diffusion layer (see the scheme at Figure 3 a)
a =
1 ^ y ^8tube
® tube
°, y ^ 8tube
(7)
where 8tube is the diffusion layer thickness inside the tube. Thus the concentration gradient is constant within the diffusion layer at a given time and is equal to
3a _ 1
8 tube
for 0 < y < 8tube and vanishes for larger values of y.
In order to determine the dependence of the diffusion layer thickness 8tube on time we do the following. Recalling that X ° is large we can replace the expression for adsorptive coverage, Equation (6), with a Heaviside step-function [8] (Figure 3b)
0= H(8 tubeM — y), (9)
since in this case the variation of 9 will necessarily be extremely sharp and localized to the vicinity of y = 8tube. Next, let us consider the mass conservation law for one nanotube in dimensioned variables
3C
^RporeDpore J Qx
dt =
x=0
0
or in the dimensionless form
L
= ^Rpore J Cdx + 2^RporeTsite |0 dx ,
0
(10a)
X
-hj
0
3a
3y
1 1
dx = J ady +»019 dy
y=0 0 0
(10b)
which equates the quantity of target species that has entered the tube over the duration x and the sum of the quantities of target species adsorbed at the walls and that contained in the solution inside the tube. Substitution of the concentration profile representation Equation (7) and its gradient Equation (8) into the latter expression yields:
0
1
8 tube(T)
8(XV
dx = J 1
0 V
л
y
8 tube(T) у
dy +
+ ^0 JH(8tubeM-y)dy = -^2^ + «0 Stub^^.
0
'tube!
2
10 ^tubew. (11)
Differentiation of Equation (11) results in the following ordinary differential equation
h _ Г1 + ~ j d8tube 8tube“ 12 “0 J dt
whose solution is
8 tube
4n
----1—x
1+2»0 .
(12)
(13)
This result shows that the considered diffusion-adsorption process is reminiscent of chronoamperometric electrochemical conditions with the apparent diffusion coefficient
a =
4q
1 + 2S 0
(14)
It is interesting to note that in the limit of low ^ (which is equivalent to low adsorptive capacity) the diffusion layer thickness becomes 8tube ^ ^/qx corresponding to uncomplicated linear diffusion along the pores whereas when »0 is large the apparent diffusion rate is severely limited by adsorption kinetics.
Figure 3. Schematic representation of (a) the concentration profile and (b) two coverage fronts for
Г2 > Г1 in the case of extremely fast adsorption
The expression (13) for the diffusion layer thickness allows now obtaining the final result for the wall coverage 9 . To this end, the integration in Equation (6) is carried out upon substitution of Equations (7) and (13):
f \
CD II 1 •ё 0 J y2/а 1 dC
= 1 - exp
-X
( I— 2)
x- 2w E + 3L
V a a
Vі J
(15)
where the lower integration limit corresponds to the time moment xmin = y2 / a when the diffusion layer reaches the pointy (the concentration a(y, x) is zero for all x < xmin). Since there are no adsorbed species in the part of the nanotube corresponding to y >Stube the expression for coverage along the tube length is
' f IT y2)
1 - exp 0 1 X - 2^f-+— , y <-\[ол
va a 1 J . (16)
0, y > л/ах
The latter expression for the coverage allows obtaining the following analytical result for the average coverage:
avg
: J 9 dy = л/ах - -2 erfЦ/^qx )
(17)
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РИ, 2007, № 4
where erf(.) is the error function. Note that this expression is valid only for t< (1 + 2 H0) / 4p and for longer times 0avg = 1. In case when X0 ^ да the second term tends to zero even for very small values of x, so that Equation (17) reduces to:
px
0avg=2fiШ0. (18)
4. Discussion
In Figure 4 we plot numerical results for the concentration, coverage and average coverage obtained with the constant boundary condition at the entrance of the pore a = 1 (which is consistent with our above assumption) and the corresponding analytical results.
Figure 4. Comparison between analytically obtained and computed characteristics: a) concentration; b) coverage; c) average coverage. The inset in c) details the variation around x = 104
Numerical results were obtained by means of the fully-implicit method [9] on a uniform grid in space and exponentially expanded grid in time [10]. The grid used involved 200 spatial subdivisions and the parameters of time grid were: the initial step Дх0 = 10_6 and expansion parameter p = 5 x 10_3 [10]. Parameter values for these calculations were p = 0.01, ф = 282.33, u = 9.35 x ю_4, X 0 = 10 and S0 = 200. It is clear from the figure that the agreement between numerical and analytical results is remarkable. The numerical and analytical results for the РИ, 2007, № 4
average coverage coincide along the growing part of the plot to within a few tenths of a percent. The accuracy decreases slightly near the kink of the plot (x «104) since at this moment the diffusion layer meets the bottom of the tube, but it does not exceed 0.3%.
The analytical solution derived here is based on the assumption that the unity dimensionless concentration is kept at the pore entrance. In fact this assumption introduces a certain limitation since the bulk concentration of target species does not remain constant during the experiment. Nevertheless the obtained expressions are valid with reasonable accuracy for the conditions when the bulk concentration reduces by less than 25%.
5. Conclusion
The diffusion-reaction processes in the interior of mesoporous spherical particles have been considered. The analytical expressions for the concentration, coverage and average coverage have been obtained in the limit of extremely fast irreversible adsorption with the additional assumption of constant concentration at the entrance of the tube. The comparison of the analytical results with numerical ones reveals excellent agreement between them, thus validating both the correctness of the analytical solution and accuracy of numerical results.
Acknowledgement. This work was supported by the Ukrainian Ministry of Education and Science in part of project M/11 -2008.
References: 1. C.T. Kresge, M.E. Leonowicz, W.J. Vartuli, J.S. Beck. Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism // Nature 1992, 359, 710712. 2. A. Walcarius, C. Delacфte. Mercury(II) binding to thiol-functionalized mesoporous silicas: critical effect of pH and sorbent properties on capacity and selectivity // Anal. Chim. Acta 2005, 547, 3-13. 3. A. Walcarius, M. Etienne, J. Bessiure. Rate of access to the binding sites in organically modified silicates. 1. Amorphous silica gels grafted with amine or thiol groups // Chem. Mater. 2002, 14, 2757-2766. 4. A. Walcarius, M. Etienne, B. Lebeau. Rate of access to the binding sites in organically modified silicates. 2. Ordered mesoporous silicas grafted with amine or thiol groups // Chem. Mater. 2003, 15, 2161-2173. 5. A. Walcarius, C. Delacфte. Rate of access to the binding sites in organically modified silicates. 3. Effect of structure and density of functional groups in mesoporous solids obtained by the co-condensation route // Chem. Mater. 2003, 15, 4181-4192. 6. A. Walcarius, E. Sibottier, M. Etienne, J. Ghanbaja. Electrochemically assisted self-assembly of mesoporous silica thin films // Nature Materials 2007, 6, 602608. 7. C. Amatore. Theoretical trends of diffusion-reaction into tubular nano- and mesoporous structures: A general physicochemical and physicomathematical modeling // Chem. Eur. J. in press. 8. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1970. 9. K. Fletcher, Computational methods in fluid dynamics, Volume 1, Moscow, Mir, 1991, p. 502. 10. C. Amatore, I. Svir. A new and powerful approach for simulation of diffusion at microelectrodes based on overlapping sub-domains: application to chronoamperometry at the microdisk // J. Electroanal. Chem., 2003, 557, 75-90.
Поступила в редколлегию 21.10.2007
Рецензент: д-р техн. наук, с.н.с. Свирь И.Б.
Oleinick Alexander, Ph.D., senior scientist, post-doc of Mathematical and Computer Modelling Laboratory of KNURE. Scientific interests: mathematical modelling of bio-and physicochemical processes, mathematical physics. Address: KNURE, Kharkov, 61166, 14 Lenin Avenue.
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