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Kinetics of zeolite NaA crystallization in microgravity
Materials Letters 59 (2005) 2668 – 2672 www.elsevier.com/locate/matlet
Kinetics of zeolite NaA crystallization in microgravity Hongwei Songa, Olusegun J. Ilegbusia,*, Albert Sacco Jr.b,c a
Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, USA b Department of Chemical Engineering, Northeastern University, Boston, MA 02115, USA c Center for Advanced Microgravity Material Processing, Northeastern University, MA 02115, USA Received 12 October 2004; accepted 16 April 2005 Available online 12 May 2005
Abstract The paper describes a population balance-based study of zeolite NaA crystallization from solution for well-mixed batch operations both on ground and in low earth orbit. The model is based on heterogeneous and secondary nucleation and a size-dependent power law crystal growth mechanism. The predicted final particle size distribution is compared with the available experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Population balance; Heterogeneous nucleation; Secondary nucleation
1. Introduction The production of large, perfect zeolite crystals remains a technical challenge due to the complexity of the underlying crystallization mechanism [1,2]. It is generally agreed that zeolite synthesis involves a series of chemical reactions and phase transitions. When solutions of silicate and aluminate are mixed, an amorphous gel is formed. Upon hydrothermal treatment, the gel gradually dissolves into the solution. Crystal nuclei form at a certain solute supersaturation concentration, and grow by transport of soluble species through a solution phase. Zeolite crystallization can thus be described by the kinetics of the reactions and phase transitions mentioned above. After gel dissolution, nucleation and crystal growth play a crucial role in determining the crystallization behavior and hence the properties of the end products. In addition, factors such as temperature and additives will influence the crystallization process by changing the rates of gel dissolution, nucleation and crystal growth. We hypothesize that a fundamental understanding of the nucleation and growth mechanisms will be enhanced by modeling the kinetics of these processes and their effects
* Corresponding author. E-mail address: [email protected] (O.J. Ilegbusi). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.04.015
on the final size-distribution of crystals grown in low earth orbit. In order to optimize crystal size and reduce defect formation, low earth orbit has been used to suppress the density-driven thermal convection. This approach allows extraneous nucleation events to be effectively suppressed and the suspension of crystals in a nutrient pool [3], providing an ideal environment for synthesizing large crystals. Sacco and co-workers [4,5] have made several attempts to grow large, perfect zeolite crystals on the space shuttle. These crystals have exhibited substantial increase in size over those grown on the ground. For example, zeolite A, X and mordenite crystals grown on the USML-1 mission in 1992 were about 10– 50% larger than their earth-grown counterparts [4]. Similarly, zeolite A, X, Beta and silicalite crystals grown during the 16-day USML-2 mission in 1997 had a linear dimension of up to 70% larger than their terrestrial counterparts [5]. It is also worth noting that these space products exhibited broader particle size distribution and fewer structural defects than ground-based crystals. Although the growth of zeolite crystals in space has been of considerable interest [6,7], our fundamental understanding of the growth mechanism and the associated kinetics in the absence of gravity is still poor. Sand et al. [8] postulated that the reduction of particle settling and thermal convective currents in microgravity eliminates secondary nucleation
H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
and keeps the growing crystals suspended in their nutrientrich environment. In addition, the reduction of hydrostatic pressure and the prominence of surface force prevent nucleation at all solid surfaces. Furthermore, convectioninduced solute transport is suppressed and the crystals grow in a quiet fluid environment, implying a diffusion-controlled growth mechanism in space. Several nucleation mechanisms [9,10] have been considered to be important in the formation of zeolites. Most notable are heterogeneous nucleation in the solution, collision-bred secondary nucleation and autocatalytic nucleation as nuclei are released from the shrinking amorphous gel phase. Zeolite nuclei are largely formed in the gel and/or at the gel/liquid interface by the linkage of specific subunits during gel precipitation and/ or aging. The nuclei are unable to grow inside the gel matrix, therefore they are potential nuclei when hidden in the gel matrix. They can start to grow after being released as the gel dissolves during crystallization, when they are in full contact with the liquid phase. It is highly desirable to provide a fundamental understanding of the impact of microgravity on the crystallization behavior of zeolites since this environment virtually eliminates particle settling and natural convection. In this paper, a population balance-approach is used to study zeolite crystallization from solution both on the ground and in low earth orbit. A recent set of experimental data is interpreted with the theoretical model to determine the kinetics parameters of nucleation and growth. It is found that a heterogeneous nucleation model adequately represents the behavior of a batch crystallizer.
2. Formulation Given the nucleation and growth profiles of individual crystals, the crystallization profile can be established by means of a population balance approach. The kinetics parameters are predicted by comparing simulation results with the published experimental data. A number of cases were studied in an attempt to fit empirically the observed crystallization profiles in space experiments, taking into account the rapid increase and subsequent decrease in the rate of nucleation. A function was thereby established to describe the profiles of nucleation and growth for zeolite crystallization in space. The population balance for a constant volume, isothermal batch crystallizer, begins with the definition of the number density function, n(L,t), describing the evolving zeolite crystal-size distribution, which is found from the solution of the following equations: Bn BðnGÞ þ ¼0 Bt BL
ð1Þ
where G is the linear crystal growth rate, L is the linear crystal dimension, and t is the time. Appropriate initial and boundary conditions must be specified to fully describe the
2669
system. This partial differential equation can be transformed into a set of dynamic moment equations, resulting in a set of ordinary differential equations that is easier to solve than the original equation and is frequently just as useful in extracting information. The moments of the distribution are defined as Z V Li nð L; t ÞdL i ¼ 0; 1; 2; 3 . . . ð2Þ mi ¼ 0
where m i is the ith moment of crystal size distribution. Applying Eq. (2) to Eq. (1) and solving the moment transformations, the zeroth, first, second, and third moments can be related to total number, length, area, and volume of crystals in the suspensions thus: dm0 ¼B t
ð3Þ
dm1 ¼ m0 G t
ð4Þ
dm2 ¼ 2/a m1 G t
ð5Þ
dm3 ¼ 3qð/V =/a Þm2 G t
ð6Þ
where B is the nucleation rate, / a is the area shape factor, q is the zeolite crystal density, and / v is the volume shape factor. Eq. (1) assumes that there are no seeds present in the system at t = 0. To make accurate use of the above equation, it is essential to define nucleation and growth rates in terms of state variables as functions of time. The concentration of species in the different phases present is normally considered to be the most important of these variables. This conclusion implies a requirement to incorporate mass balance equations for different phases into the solution algorithm. Without the hydrodynamic or shearing effects on nucleation, the zeolite crystallization follows the classical homogeneous nucleation rate expression [10]: A B ¼ B0 Iexp 2 ð7Þ ln s where B 0 is the frequency factor, A is the interfacial energy parameter, and s is the relative supersaturation, defined as solute concentration divided by the equilibrium concentration, c / c s . The c s is the saturation concentration of solute species in equilibrium with zeolite crystals at the synthesis pH and temperature. Heterogeneous nucleation is thought to be responsible for much of the nucleation in many zeolite crystallization systems, by reducing the interfacial energy necessary for nucleation on foreign, extraneous, or heterogeneous surfaces. As such, heterogeneous nucleation can be expressed in a similar manner to homogeneous nucleation,
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H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
Table 1 Parameter Values used in zeolite crystallization under microgravity Items
Commercial
USML-1
3
4.0 10 4.0 10 5 4.5 10 12 5.0 1016 250
k1 (/s) k2 (m3/kg) kg (m4/kg) B0(/m3s) A
USML-2 3
4.0 10 4.0 10 5 5.0 10 10 5.0 1013 280
4.0 10 3 4.0 10 5 5.0 10 10 4.0 1013 280
but with a smaller value of the interfacial energy parameter A. The experimental conditions reported were static, and no hydrodynamic or shearing effects on nucleation were considered. With this simplification, the interfacial tension term r is related to A thus [11]: A¼
16pr3 Vm2 NA3 3ð RT Þ
ð8Þ
3
where V m is the volume of unit cell, N A is the Avogadro constant, R is the gas constant, and T is the temperature. The crystal growth is a process of transporting growth units from the solution to the crystal surface and then incorporating them into the crystal [12]. We propose the following growing mechanism: zeolite precursors of the pseudo-cells dissolve in the solution phase from the gel and then migrate to the surface of nuclei due to Brownian motion. On arriving at the crystal surface they integrate, thereby facilitating the crystal growth process. The crystal surface reaction has been postulated to involve the following elementary steps [13]: l1 l2
M @P
critical size beyond which the crystals grow slowly. A value of m = 1 has been suggested to represent a diffusioncontrolled growth rate while m = 0.5 implies a surface reaction-controlled growth rate [15]. The latter mechanism is due to a stepwise process in which surface reaction precipitates half a pseudo-cell at a time on the crystal surface until the supersaturation level reaches equilibrium. As a transient process, the nucleation and growth rates are both dependent on the solute concentration. Typically, the specific solute is released from the amorphous gel phase. Material balances on the amorphous gel phase and solute phase are required to completely represent the problem. These balances can be expressed in general thus: dc ¼ k1 c þ k2 ccT t
ð14Þ
dcT 1 ¼ k1 c k2 ccT M0 B qG/a m2 2 dt
ð15Þ
in which the first and second terms on the right side of Eqs. [(14) and (15) are the forward and backward dissolution rates, respectively. The consumption rates of pseudo-cells during nucleation and growth are represented by the third and fourth terms in Eq. (15). Dissolution rate constants k 1 and k 2 should ideally be measured independently. Because of the complexity of the precipitation system, however, isolated measurements are very difficult, and the parameter values are therefore estimated.
ð9Þ 3. Computational details
P þ N YN
T
ð10Þ
T 14
P þ N Y Nn
ð11Þ
In effect, M represents pseudo-cell precursor, P the pseudo-cell, N the zeolite nuclei, N* the intermediate species, and N n the crystal with n-1 pseudo-cells. The parameters l 1, l 2, l 3 and l 4 represent gel dissolution rate constant, reverse gel dissolution rate constant, intermediate species formation rate constant and crystal growth rate constant, respectively. Repetition and summation of the precipitation reactions over n unit cells result in the following overall equation: nP þ N YNn
ð12Þ
Based on the above mechanism the final crystal would contain n pseudo-cells and a nucleus. We use a power law size-dependence of linear crystal growth rate on supersaturation in the form [14]: m
G ¼ kg ðc ce Þ ð1 þ R=Rc Þ
b
ð13Þ
where k g is mass transfer coefficient, m is the order of growth, b is the order of size-dependence, and R c is the
The coupled ordinary differential equations are solved with the Runge – Kutta numerical technique embodied in the MATLAB software. We assume that after mixing the solutions of silicate and aluminate, amorphous gels simultaneously form and uniformly distribute in the 40
Gel Concentration (kg/m3)
l3
30
20
2, 3
10 1 0
0
0.5
1
Time (second)
1.5
2
x 104
Fig. 1. Predicted gel concentration curves for three cases: 1. Commercial (ground), 2. USML-1 (space), and 3. USML-2 (space).
H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
10
-8
2 30
20
2
1
Crystal Growth Rate (m/s)
Solute Concentration (kg/m3)
40
2671
3
10
0
2
1
0
3
4
10
-9
1
10
x 104
Time (second)
3
-10
0
Fig. 2. Predicted solute species concentration curves for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
reactor. Upon hydrothermal treatment, the gel gradually dissolves and solute appears in the solution. The crystal will form at a certain level of solute supersaturation, grow and settle down in the presence of gravity. The solution composition used in zeolite NaA syntheses on ground and in microgravity is 1.94 Na2O&Al2O3&0.84 SiO2&194 H2O. For simplicity, we use the amounts of reagents that can be transformed into zeolite crystals as the initial concentration of gel phases. Once the kinetics parameters of phase transitions are determined for a synthesis system, the crystallization curve and final particle size distribution can be simulated. Therefore, the model equations can be used to fit the experimental data of zeolite synthesis both on ground and in space, and in so doing, the values of these kinetics parameters can be determined. The values of the principal input parameters used in the computation are presented in Table 1.
4. Results and discussion
1
2
3
4
x 104
Time (second)
Fig. 4. Predicted crystal growth rates for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
and USML-2. These results are obtained from a numerical solution of Eqs. (14) – (15) with the initial conditions specified in Table 1. In this model, the gel dissolution is assumed to follow an exponential law and the rate constant is strongly dependent on gravity level. This assumption is based on the fact that the shear stress as a result of particle sedimentation can accelerate the gel dissolution process. The gel dissolution curves for both ground (commercial) and space syntheses are essentially a decaying exponential function. The rate constant in space decreases by about one order of magnitude relative to the value on ground. This result is due to the significant reduction in shear stress and convection in the low earth orbit of space. In gravity, the environment will become less uniform and possibly more dilute at the bottom of the vessel as the particles settle. In microgravity particles however stay suspended in the supersaturated solution of uniform composition. The evolution of solute species predicted for the three cases mentioned above is presented in Fig. 2. In the present model, the solute species is released from dissolved gel and consumed
Fig. 1 shows the predicted transient gel concentration for three experimental conditions: ground (commercial), USML-1
1
0.8
10
12
1
Crystallinity (%)
Nucleation Rate (#/m3s)
10
10
10
10
8
2
0.6
1
2
3
0.4
3
6
0.2 10
4
0
1
2
Time (second)
3
4
x 104
Fig. 3. Predicted heterogeneous nucleation rates for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
0 0
1
2
Time (second)
3
4
x 104
Fig. 5. Predicted crystallization curves for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
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H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
5. Conclusion
1.0
- USML-2 - USML-1 - Commercial
0.9
Normalized Volume
0.8
In summary, we have used the population balance-model to study zeolite A crystallization for well-mixed batch operations both on ground and in low earth orbit. This model assumes heterogeneous nucleation and diffusioncontrolled crystal growth mechanisms for ground synthesis and homogeneous nucleation and surface reaction-controlled crystal growth mechanism in microgravity. As expected, the model predicts that microgravity substantially suppresses nucleation and enhances crystal growth. The predicted final particle size distribution compares favorably with the available experimental data.
0.7 0.6 0.5 0.4 1 0.3
3 2
0.2 0.1 0.0 0
25
50
75
100
125
Geometric Diameter (µm) Fig. 6. Predicted (dished line) and experimental particle size distributions for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
for nucleation and crystal growth. Fig. 3 shows the corresponding nucleation rates predicted for three cases. It is seen that the frequency factor decreases by three orders of magnitude in low earth orbit relative to ground. The interface energy parameter increases only slightly under microgravity. This trend is due to the elimination of heterogeneous nucleation and autocatalytic nucleation in microgravity. Fig. 4 shows the crystal growth rates predicted for the three cases mentioned above. The growth rate constant increases by two orders of magnitude in low earth orbit relative to that on ground. In addition, the diffusion-controlled crystal growth mechanism also dominates zeolite A crystallization under microgravity. The corresponding crystallization curves are shown in Fig. 5. The curve is typically a sigmoid-shaped profile expected for a process of this type and is observed to be self-limiting at a volume fraction of 1. The fraction of transformed volume in low earth orbit is significantly slower than that on ground. This result may be explained as follows. The decrease in nucleation rate under microgravity induces a longer stage of nucleation event and thus delays crystallization. Fig. 6 shows the corresponding final crystal size distribution predicted for zeolite A crystallization both on ground and in space. For comparison, the observed experimental data are also presented in Fig. 6. The model predictions for the mean size appear to agree favorably with the experimental data both ground and space. For shape, however, the model over-predicts slightly the large crystals for the two space cases compared to experimental results.
Acknowledgement This work was supported by the NASA Center for Advanced Microgravity Materials Processing (CAMMP) at Northeastern University.
References [1] E.R. Geus, A. Mulder, D.J. Vischjager, J. Birlein, J. Am. Chem. Soc. 110 (1998) 2986. [2] J. Warzywoda, A.G. Dixon, R.W. Thompson, Albert Sacoo Jr., J. Mater. Chem. 5 (1995) 1019. [3] Albert Sacco Jr., IEEE Trans. Ed. 34 (1991) 27. [4] A. Sacco Jr., N. Bac, J. Warzywoda, I. Guray, R.W. Thompson, L.A. McCauley, Space Technology and Application International ForumAlbuquerque, NM, January 7 – 11AIP Conference Proceedings, vol. 361, 1996. [5] A. Sacco Jr., N. Bac, J. Warzywoda, I. Guray, M. Marceau, L. Sacco, M. Whalen, Zeolite Crystal Growth (ZCG) Flight on USML-2, Final Report, NASA Contract No.: NAS8-40260. [6] E. Pennis, Sci. News 140 (1992) 22. [7] Al. Sacco Jr., R. W. Thompson, A. G. Dixon, AIAA93-0473, 31st Aerospace Sciences Meeting and Exhibit, 1993, Reno, NV. [8] L.B. Sand, A. Sacco Jr., R.W. Thompson, A.G. Dixon, Zeolites 7 (1987) 387. [9] G. Scott, R.W. Thompson, A.G. Dixon, A. Sacco Jr., Zeolites 11 (1991) 178. [10] J. Warzywoda, R.W. Thompson, Zeolites 9 (1989) 341. [11] G.J. Myatt, P.M. Budd, C. Price, F. Hollway, S.W. Carr, Zeolites 14 (1994) 190. [12] C.T. Kerr, J. Phys. Chem. 70 (1966) 1047. [13] V. Nikolakis, D.G. Vlacho, M. Tsapatsis, Microporous and Mesoporous Mater. 21 (1998) 337. [14] H. Kacirek, H. Lechert, J. Phys. Chem. 80 (1975) 1291. [15] A.Y. Sheikh, A.G. Jones, P. Graham, Zeolites 16 (1996) 164.
Kinetics of zeolite NaA crystallization in microgravity Hongwei Songa, Olusegun J. Ilegbusia,*, Albert Sacco Jr.b,c a
Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, USA b Department of Chemical Engineering, Northeastern University, Boston, MA 02115, USA c Center for Advanced Microgravity Material Processing, Northeastern University, MA 02115, USA Received 12 October 2004; accepted 16 April 2005 Available online 12 May 2005
Abstract The paper describes a population balance-based study of zeolite NaA crystallization from solution for well-mixed batch operations both on ground and in low earth orbit. The model is based on heterogeneous and secondary nucleation and a size-dependent power law crystal growth mechanism. The predicted final particle size distribution is compared with the available experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Population balance; Heterogeneous nucleation; Secondary nucleation
1. Introduction The production of large, perfect zeolite crystals remains a technical challenge due to the complexity of the underlying crystallization mechanism [1,2]. It is generally agreed that zeolite synthesis involves a series of chemical reactions and phase transitions. When solutions of silicate and aluminate are mixed, an amorphous gel is formed. Upon hydrothermal treatment, the gel gradually dissolves into the solution. Crystal nuclei form at a certain solute supersaturation concentration, and grow by transport of soluble species through a solution phase. Zeolite crystallization can thus be described by the kinetics of the reactions and phase transitions mentioned above. After gel dissolution, nucleation and crystal growth play a crucial role in determining the crystallization behavior and hence the properties of the end products. In addition, factors such as temperature and additives will influence the crystallization process by changing the rates of gel dissolution, nucleation and crystal growth. We hypothesize that a fundamental understanding of the nucleation and growth mechanisms will be enhanced by modeling the kinetics of these processes and their effects
* Corresponding author. E-mail address: [email protected] (O.J. Ilegbusi). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.04.015
on the final size-distribution of crystals grown in low earth orbit. In order to optimize crystal size and reduce defect formation, low earth orbit has been used to suppress the density-driven thermal convection. This approach allows extraneous nucleation events to be effectively suppressed and the suspension of crystals in a nutrient pool [3], providing an ideal environment for synthesizing large crystals. Sacco and co-workers [4,5] have made several attempts to grow large, perfect zeolite crystals on the space shuttle. These crystals have exhibited substantial increase in size over those grown on the ground. For example, zeolite A, X and mordenite crystals grown on the USML-1 mission in 1992 were about 10– 50% larger than their earth-grown counterparts [4]. Similarly, zeolite A, X, Beta and silicalite crystals grown during the 16-day USML-2 mission in 1997 had a linear dimension of up to 70% larger than their terrestrial counterparts [5]. It is also worth noting that these space products exhibited broader particle size distribution and fewer structural defects than ground-based crystals. Although the growth of zeolite crystals in space has been of considerable interest [6,7], our fundamental understanding of the growth mechanism and the associated kinetics in the absence of gravity is still poor. Sand et al. [8] postulated that the reduction of particle settling and thermal convective currents in microgravity eliminates secondary nucleation
H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
and keeps the growing crystals suspended in their nutrientrich environment. In addition, the reduction of hydrostatic pressure and the prominence of surface force prevent nucleation at all solid surfaces. Furthermore, convectioninduced solute transport is suppressed and the crystals grow in a quiet fluid environment, implying a diffusion-controlled growth mechanism in space. Several nucleation mechanisms [9,10] have been considered to be important in the formation of zeolites. Most notable are heterogeneous nucleation in the solution, collision-bred secondary nucleation and autocatalytic nucleation as nuclei are released from the shrinking amorphous gel phase. Zeolite nuclei are largely formed in the gel and/or at the gel/liquid interface by the linkage of specific subunits during gel precipitation and/ or aging. The nuclei are unable to grow inside the gel matrix, therefore they are potential nuclei when hidden in the gel matrix. They can start to grow after being released as the gel dissolves during crystallization, when they are in full contact with the liquid phase. It is highly desirable to provide a fundamental understanding of the impact of microgravity on the crystallization behavior of zeolites since this environment virtually eliminates particle settling and natural convection. In this paper, a population balance-approach is used to study zeolite crystallization from solution both on the ground and in low earth orbit. A recent set of experimental data is interpreted with the theoretical model to determine the kinetics parameters of nucleation and growth. It is found that a heterogeneous nucleation model adequately represents the behavior of a batch crystallizer.
2. Formulation Given the nucleation and growth profiles of individual crystals, the crystallization profile can be established by means of a population balance approach. The kinetics parameters are predicted by comparing simulation results with the published experimental data. A number of cases were studied in an attempt to fit empirically the observed crystallization profiles in space experiments, taking into account the rapid increase and subsequent decrease in the rate of nucleation. A function was thereby established to describe the profiles of nucleation and growth for zeolite crystallization in space. The population balance for a constant volume, isothermal batch crystallizer, begins with the definition of the number density function, n(L,t), describing the evolving zeolite crystal-size distribution, which is found from the solution of the following equations: Bn BðnGÞ þ ¼0 Bt BL
ð1Þ
where G is the linear crystal growth rate, L is the linear crystal dimension, and t is the time. Appropriate initial and boundary conditions must be specified to fully describe the
2669
system. This partial differential equation can be transformed into a set of dynamic moment equations, resulting in a set of ordinary differential equations that is easier to solve than the original equation and is frequently just as useful in extracting information. The moments of the distribution are defined as Z V Li nð L; t ÞdL i ¼ 0; 1; 2; 3 . . . ð2Þ mi ¼ 0
where m i is the ith moment of crystal size distribution. Applying Eq. (2) to Eq. (1) and solving the moment transformations, the zeroth, first, second, and third moments can be related to total number, length, area, and volume of crystals in the suspensions thus: dm0 ¼B t
ð3Þ
dm1 ¼ m0 G t
ð4Þ
dm2 ¼ 2/a m1 G t
ð5Þ
dm3 ¼ 3qð/V =/a Þm2 G t
ð6Þ
where B is the nucleation rate, / a is the area shape factor, q is the zeolite crystal density, and / v is the volume shape factor. Eq. (1) assumes that there are no seeds present in the system at t = 0. To make accurate use of the above equation, it is essential to define nucleation and growth rates in terms of state variables as functions of time. The concentration of species in the different phases present is normally considered to be the most important of these variables. This conclusion implies a requirement to incorporate mass balance equations for different phases into the solution algorithm. Without the hydrodynamic or shearing effects on nucleation, the zeolite crystallization follows the classical homogeneous nucleation rate expression [10]: A B ¼ B0 Iexp 2 ð7Þ ln s where B 0 is the frequency factor, A is the interfacial energy parameter, and s is the relative supersaturation, defined as solute concentration divided by the equilibrium concentration, c / c s . The c s is the saturation concentration of solute species in equilibrium with zeolite crystals at the synthesis pH and temperature. Heterogeneous nucleation is thought to be responsible for much of the nucleation in many zeolite crystallization systems, by reducing the interfacial energy necessary for nucleation on foreign, extraneous, or heterogeneous surfaces. As such, heterogeneous nucleation can be expressed in a similar manner to homogeneous nucleation,
2670
H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
Table 1 Parameter Values used in zeolite crystallization under microgravity Items
Commercial
USML-1
3
4.0 10 4.0 10 5 4.5 10 12 5.0 1016 250
k1 (/s) k2 (m3/kg) kg (m4/kg) B0(/m3s) A
USML-2 3
4.0 10 4.0 10 5 5.0 10 10 5.0 1013 280
4.0 10 3 4.0 10 5 5.0 10 10 4.0 1013 280
but with a smaller value of the interfacial energy parameter A. The experimental conditions reported were static, and no hydrodynamic or shearing effects on nucleation were considered. With this simplification, the interfacial tension term r is related to A thus [11]: A¼
16pr3 Vm2 NA3 3ð RT Þ
ð8Þ
3
where V m is the volume of unit cell, N A is the Avogadro constant, R is the gas constant, and T is the temperature. The crystal growth is a process of transporting growth units from the solution to the crystal surface and then incorporating them into the crystal [12]. We propose the following growing mechanism: zeolite precursors of the pseudo-cells dissolve in the solution phase from the gel and then migrate to the surface of nuclei due to Brownian motion. On arriving at the crystal surface they integrate, thereby facilitating the crystal growth process. The crystal surface reaction has been postulated to involve the following elementary steps [13]: l1 l2
M @P
critical size beyond which the crystals grow slowly. A value of m = 1 has been suggested to represent a diffusioncontrolled growth rate while m = 0.5 implies a surface reaction-controlled growth rate [15]. The latter mechanism is due to a stepwise process in which surface reaction precipitates half a pseudo-cell at a time on the crystal surface until the supersaturation level reaches equilibrium. As a transient process, the nucleation and growth rates are both dependent on the solute concentration. Typically, the specific solute is released from the amorphous gel phase. Material balances on the amorphous gel phase and solute phase are required to completely represent the problem. These balances can be expressed in general thus: dc ¼ k1 c þ k2 ccT t
ð14Þ
dcT 1 ¼ k1 c k2 ccT M0 B qG/a m2 2 dt
ð15Þ
in which the first and second terms on the right side of Eqs. [(14) and (15) are the forward and backward dissolution rates, respectively. The consumption rates of pseudo-cells during nucleation and growth are represented by the third and fourth terms in Eq. (15). Dissolution rate constants k 1 and k 2 should ideally be measured independently. Because of the complexity of the precipitation system, however, isolated measurements are very difficult, and the parameter values are therefore estimated.
ð9Þ 3. Computational details
P þ N YN
T
ð10Þ
T 14
P þ N Y Nn
ð11Þ
In effect, M represents pseudo-cell precursor, P the pseudo-cell, N the zeolite nuclei, N* the intermediate species, and N n the crystal with n-1 pseudo-cells. The parameters l 1, l 2, l 3 and l 4 represent gel dissolution rate constant, reverse gel dissolution rate constant, intermediate species formation rate constant and crystal growth rate constant, respectively. Repetition and summation of the precipitation reactions over n unit cells result in the following overall equation: nP þ N YNn
ð12Þ
Based on the above mechanism the final crystal would contain n pseudo-cells and a nucleus. We use a power law size-dependence of linear crystal growth rate on supersaturation in the form [14]: m
G ¼ kg ðc ce Þ ð1 þ R=Rc Þ
b
ð13Þ
where k g is mass transfer coefficient, m is the order of growth, b is the order of size-dependence, and R c is the
The coupled ordinary differential equations are solved with the Runge – Kutta numerical technique embodied in the MATLAB software. We assume that after mixing the solutions of silicate and aluminate, amorphous gels simultaneously form and uniformly distribute in the 40
Gel Concentration (kg/m3)
l3
30
20
2, 3
10 1 0
0
0.5
1
Time (second)
1.5
2
x 104
Fig. 1. Predicted gel concentration curves for three cases: 1. Commercial (ground), 2. USML-1 (space), and 3. USML-2 (space).
H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
10
-8
2 30
20
2
1
Crystal Growth Rate (m/s)
Solute Concentration (kg/m3)
40
2671
3
10
0
2
1
0
3
4
10
-9
1
10
x 104
Time (second)
3
-10
0
Fig. 2. Predicted solute species concentration curves for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
reactor. Upon hydrothermal treatment, the gel gradually dissolves and solute appears in the solution. The crystal will form at a certain level of solute supersaturation, grow and settle down in the presence of gravity. The solution composition used in zeolite NaA syntheses on ground and in microgravity is 1.94 Na2O&Al2O3&0.84 SiO2&194 H2O. For simplicity, we use the amounts of reagents that can be transformed into zeolite crystals as the initial concentration of gel phases. Once the kinetics parameters of phase transitions are determined for a synthesis system, the crystallization curve and final particle size distribution can be simulated. Therefore, the model equations can be used to fit the experimental data of zeolite synthesis both on ground and in space, and in so doing, the values of these kinetics parameters can be determined. The values of the principal input parameters used in the computation are presented in Table 1.
4. Results and discussion
1
2
3
4
x 104
Time (second)
Fig. 4. Predicted crystal growth rates for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
and USML-2. These results are obtained from a numerical solution of Eqs. (14) – (15) with the initial conditions specified in Table 1. In this model, the gel dissolution is assumed to follow an exponential law and the rate constant is strongly dependent on gravity level. This assumption is based on the fact that the shear stress as a result of particle sedimentation can accelerate the gel dissolution process. The gel dissolution curves for both ground (commercial) and space syntheses are essentially a decaying exponential function. The rate constant in space decreases by about one order of magnitude relative to the value on ground. This result is due to the significant reduction in shear stress and convection in the low earth orbit of space. In gravity, the environment will become less uniform and possibly more dilute at the bottom of the vessel as the particles settle. In microgravity particles however stay suspended in the supersaturated solution of uniform composition. The evolution of solute species predicted for the three cases mentioned above is presented in Fig. 2. In the present model, the solute species is released from dissolved gel and consumed
Fig. 1 shows the predicted transient gel concentration for three experimental conditions: ground (commercial), USML-1
1
0.8
10
12
1
Crystallinity (%)
Nucleation Rate (#/m3s)
10
10
10
10
8
2
0.6
1
2
3
0.4
3
6
0.2 10
4
0
1
2
Time (second)
3
4
x 104
Fig. 3. Predicted heterogeneous nucleation rates for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
0 0
1
2
Time (second)
3
4
x 104
Fig. 5. Predicted crystallization curves for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
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H. Song et al. / Materials Letters 59 (2005) 2668 – 2672
5. Conclusion
1.0
- USML-2 - USML-1 - Commercial
0.9
Normalized Volume
0.8
In summary, we have used the population balance-model to study zeolite A crystallization for well-mixed batch operations both on ground and in low earth orbit. This model assumes heterogeneous nucleation and diffusioncontrolled crystal growth mechanisms for ground synthesis and homogeneous nucleation and surface reaction-controlled crystal growth mechanism in microgravity. As expected, the model predicts that microgravity substantially suppresses nucleation and enhances crystal growth. The predicted final particle size distribution compares favorably with the available experimental data.
0.7 0.6 0.5 0.4 1 0.3
3 2
0.2 0.1 0.0 0
25
50
75
100
125
Geometric Diameter (µm) Fig. 6. Predicted (dished line) and experimental particle size distributions for three cases: 1. Commercial, 2. USML-1, and 3. USML-2.
for nucleation and crystal growth. Fig. 3 shows the corresponding nucleation rates predicted for three cases. It is seen that the frequency factor decreases by three orders of magnitude in low earth orbit relative to ground. The interface energy parameter increases only slightly under microgravity. This trend is due to the elimination of heterogeneous nucleation and autocatalytic nucleation in microgravity. Fig. 4 shows the crystal growth rates predicted for the three cases mentioned above. The growth rate constant increases by two orders of magnitude in low earth orbit relative to that on ground. In addition, the diffusion-controlled crystal growth mechanism also dominates zeolite A crystallization under microgravity. The corresponding crystallization curves are shown in Fig. 5. The curve is typically a sigmoid-shaped profile expected for a process of this type and is observed to be self-limiting at a volume fraction of 1. The fraction of transformed volume in low earth orbit is significantly slower than that on ground. This result may be explained as follows. The decrease in nucleation rate under microgravity induces a longer stage of nucleation event and thus delays crystallization. Fig. 6 shows the corresponding final crystal size distribution predicted for zeolite A crystallization both on ground and in space. For comparison, the observed experimental data are also presented in Fig. 6. The model predictions for the mean size appear to agree favorably with the experimental data both ground and space. For shape, however, the model over-predicts slightly the large crystals for the two space cases compared to experimental results.
Acknowledgement This work was supported by the NASA Center for Advanced Microgravity Materials Processing (CAMMP) at Northeastern University.
References [1] E.R. Geus, A. Mulder, D.J. Vischjager, J. Birlein, J. Am. Chem. Soc. 110 (1998) 2986. [2] J. Warzywoda, A.G. Dixon, R.W. Thompson, Albert Sacoo Jr., J. Mater. Chem. 5 (1995) 1019. [3] Albert Sacco Jr., IEEE Trans. Ed. 34 (1991) 27. [4] A. Sacco Jr., N. Bac, J. Warzywoda, I. Guray, R.W. Thompson, L.A. McCauley, Space Technology and Application International ForumAlbuquerque, NM, January 7 – 11AIP Conference Proceedings, vol. 361, 1996. [5] A. Sacco Jr., N. Bac, J. Warzywoda, I. Guray, M. Marceau, L. Sacco, M. Whalen, Zeolite Crystal Growth (ZCG) Flight on USML-2, Final Report, NASA Contract No.: NAS8-40260. [6] E. Pennis, Sci. News 140 (1992) 22. [7] Al. Sacco Jr., R. W. Thompson, A. G. Dixon, AIAA93-0473, 31st Aerospace Sciences Meeting and Exhibit, 1993, Reno, NV. [8] L.B. Sand, A. Sacco Jr., R.W. Thompson, A.G. Dixon, Zeolites 7 (1987) 387. [9] G. Scott, R.W. Thompson, A.G. Dixon, A. Sacco Jr., Zeolites 11 (1991) 178. [10] J. Warzywoda, R.W. Thompson, Zeolites 9 (1989) 341. [11] G.J. Myatt, P.M. Budd, C. Price, F. Hollway, S.W. Carr, Zeolites 14 (1994) 190. [12] C.T. Kerr, J. Phys. Chem. 70 (1966) 1047. [13] V. Nikolakis, D.G. Vlacho, M. Tsapatsis, Microporous and Mesoporous Mater. 21 (1998) 337. [14] H. Kacirek, H. Lechert, J. Phys. Chem. 80 (1975) 1291. [15] A.Y. Sheikh, A.G. Jones, P. Graham, Zeolites 16 (1996) 164.