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Crystal population balance model for nucleation and growth of colloidal TPA- silicalite-1
I. Kiricsi, G. P~iI-BorbEly,J.B. Nagy, H.G. Karge (Editors) Porous Materials in Environmentally Friendly Processes
Studies in Surface Science and Catalysis, Vol. 125 9 1999 Elsevier Science B.V. All rights reserved.
117
C r y s t a l p o p u l a t i o n b a l a n c e m o d e l for n u c l e a t i o n and g r o w t h of colloidal T P A silicalite- 1 D. Creaser Dept. of Chemical Technology, Lulefi University of Technology, $971-87 Lulefi, Sweden
A model describing the crystallization of colloidal TPA-silicalite-1 is developed and tested. The nucleation rate is approximated from experimental data. Crystal growth is surfacereaction limited and the Gibbs-Thomson effect suppresses growth of small crystals, resulting in an induction period. Slow crystal growth during the nucleation period results in the narrow crystal size distribution typically observed for colloidal TPA-silicalite-1 syntheses. 1. INTRODUCTION The crystallization of zeolites consists of two general steps, nucleation and crystal growth [ 1]. Due to a supersaturated concentration of the reagent species, crystal nuclei form. During an induction period, the rate of growth may initially be slow, followed by a period of a nearly linear growth rate. Crystal growth continues until the reagent is reduced to an equilibrium concentration. An approach termed population balance modelling, which accounts for the crystal size distribution, has been used to mathematically model this general view of crystallization [2] and its application to zeolite crystallizations from homogeneous solutions has been investigated [3]. The objective of this work is to combine this approach with the observations of several recent publications [4-16] to develop and test a model that describes the general physicochemical processes believed to occur during colloidal TPA-silicalite-1 crystallizations from homogeneous solutions.
2. MODEL DEVELOPMENT
Figure 1 shows the particle size monitored by dynamic light scattering (DLS) during a synthesis of TPA-silicalite-I at 80~ The molar composition of the synthesis sol is 9 TPAOH: 25 SiO2:480 H 2 0 : 1 0 0 ethanol. Experimental details are given in a parallel publication [4]. Before hydrothermal treatment, but after the hydrolysis of the tetraethoxysilane silica source, the sol contained subcolloidal particles with an average diameter of 4.0 nm. The presence of the subcolloidal particles persisted during hydrothermal treatment with a constant diameter of about 4 nm. After 24 h, growing crystals were first detected. The subcolloidal population was detected for a short period after the appearance of the growing crystals, but later the presence of growing crystals made them undetectable. However, it has been found though by SAXS/WAXS [5-7] and cryo-TEM [8] analysis that the
118
80 70
model o subcolloidal particles )~ crystals
~
~( 9 ~(
)~
~ 6o L_ N 50 E
>
k g = 16000 nm crna / m i n mol
~
'~ 20
~
/
kA ..- 0"0! min"1
2'0
So
Time (h)
6'0
e'o
Figure 1. Average crystal diameter from DLS and model predictions subcolloidal particles are present throughout the crystallization. After an induction period, the crystals increased in diameter almost linearly until their growth rate began to slow down and by about 70 h, growth was completed. The ultimate average diameter of the crystals was 74 nm and they had a monomodal size distribution with a coefficient of variation (c.v.) of 8.6%. 2.1 Crystal Nucleation The measurements for Figure 1 were done ex-situ at room temperature and thus may not accurately represent the crystallization process during the induction period. With in-situ measurements it was found that the subcolloidal particles increased gradually in size and a deconvolution of the intensity data suggested the growing crystals originated from the subcolloidal particles [8]. It has also been reported that the subcolloidal particles possess entrapped TPA § cations and a form of short-range order by Raman and FTIR spectroscopy [9]. These findings suggest that the subcolloidal particles, already present in solution before hydrothermal treatment, may be the crystal nuclei, de Moor et al. [5-7] have proposed that nucleation occurs by aggregation of hydrated organic-inorganic composite species [ 10, l 1] into primary particles with a size of about 2.8 nm. In accordance with the findings above, in the model it shall be assumed that the diameter of the crystal nuclei is 3 nm. The fact that the product crystals have a monomodal and narrow size distribution suggests that the nucleation may consist of a sudden burst of nuclei upon the start of hydrothermal treatment. Another possibility is that the subcolloidal particles are the nuclei and a fraction of them begin to grow immediately upon the start of hydrothermal treatment and there is no distinct nucleation process at all. Recent work though using two-stage varying-temperature syntheses strongly indicates that a continuous nucleation process occurs, lasting over much of the induction period [4]. In this work so called two-stage syntheses were carried out involving a rapid increase in temperature at some point during the crystallization. As the duration of the
119
40 35 30 0 ,reX
25
" 20 .o 15 r 0
n
10
60oc
5 00
50
1O0
150
Time (h) 18 16 14 0
~ 9
12
~10 t-"-
.o
8
:= 6
r
0
a.
4
80~
2 0
0
~
a
6
Time (h)
8
10
period at the initial temperature was extended the crystal concentration and ultimate crystal size varied until they were equal to those obtained for a complete synthesis at the initial temperature. The period of time during which the crystal concentration varied was considered to represent the nucleation period and thus rate of nucleation. Figure 2 shows the crystal population during a synthesis at 60 and 80~ At 60~ the nucleation period is much longer, extending over about 100 h, and the larger number of measurements that could be made over this period indicate that the rate of nucleation is initially high and decreases exponentially. An exponential function to express crystal population was fitted to the data at 60~ and is shown by the solid curve. The nucleation period at 80~ was shorter, lasting from 4 to 6 h but the same type of exponential relation was considered to apply over the more limited amount of data. From this function the predicted nucleation rate at 80~ is
Figure 2. Crystal population during nucleation
N(t)
= ~dP _ 89.24 x lOt2e -~
(1)
dt
The above relation is empirically based, nucleation may be initiated due to a supersaturation of the reagent species, subsequently the degree of supersaturation and the rate of nucleation decreases due to depletion of the reagent caused by growth of the nuclei [1 ]. 2.2 Induction Period
If nucleation occurs over a substantial part of the induction period, the crystal growth rate during this period must be slow in order to yield a product with a narrow crystal size distribution. Den Ouden and Thompson [3], in their analysis of the crystallization of silicalite-1, assumed that the induction period stemmed from the time required for silica to convert by hydrolysis from an inactive amorphous form to the active precursor species. However, this must not be the case for the crystallization of silicalite-1 from a clear homogeneous solution using a tetraethoxysilane (TEOS) silica source. Hydrolysis of TEOS by the TPAOH solution occurs even before hydrothermal treatment, ensuring the presence of
120 monomeric silica for the crystallization. Figure 2 indicates that the rate of nucleation is high immediately upon starting hydrothermal treatment. Apparently, there is no delay during which a supersaturation of reagent species builds up to initiate nucleation. It is thus more likely that the induction period is due to the Gibbs-Thomson effect, which accounts for the higher solubility of smaller crystals. If the driving force for crystal growth is the difference between the bulk and equilibrium concentrations of precursor silica species, the GibbsThomson effect would suppress growth during the early stage of crystallization. The OstwaldFreundlich equation expresses the solubility of a crystal of diameter L (Ceqt)as a function of the solubility at an infinite diameter (Ceqc): Ceq L -- C eqG
exp/~,4 ( ~TL
)
(2)
2.3 Crystal Growth Recent studies of the crystallization of silicalite-1 have discussed various crystal growth mechanisms. Dokter et al. [12-14] proposed that the crystallization consisted of two aggregation steps. During nucleation primary aggregates formed which subsequently formed secondary aggregates with a size of about 10 nm. More recently, De Moor et al. [5-7] found that the secondary aggregation was not a necessary part of the growth mechanism. Instead it seemed that growth occurred mainly by the addition of the primary aggregates (about 2.8 nm) which presumably correspond to the subcolloidal particles, to the crystal surface. Although, Shoeman [ 15] has shown with the extended Derjaguin-Landan and Verwy-Oberbeek (DLVO) theory that the colloidal particles in TPA-silicalite-I synthesis sols are stable against aggregation which would rule out growth by aggregation mechanisms. In the model developed here a classical crystal growth mechanism shall be assumed to apply, i.e. the addition of nutrient silica species which are probably monomeric or perhaps oligomeric from solution to the growing crystals. The stability of the colloidal particles against aggregation in Shoeman's analysis is primarily provided by surface adsorbed TPA + cations and it has been shown that the subcolloidal particles and growing crystals contain these entrapped cations [9]. It is reasonable to assume that the structure-directing effects of the TPA § cations would provide suitable sites for the incorporation of incoming nutrient species into the zeolite crystal structure. According to the assumed reaction sequence amorphous silica (A) depolymerizes to form the nutrient species (P) that feed the growth of crystals (Z). Other species not involved in the zeolite growth (O), perhaps larger oligomeric silica, may also be present. Throughout the crystallization process a dynamic equilibrium exists between these species: A r P r Z %~ O Since little is known about the exact nature of these processes, the model used will be simplistic in this respect. The rate of change of the amorphous species concentration (CA) will be expressed as
dCA dt
~---
- k a ( C A - K C~ )
(3)
121 where Ke is the equilibrium between the amorphous and nutrient species. The rate of change of the nutrient species concentration (Cb) is then
d C-----2-b= dt
k a ( C a - K eC h ) -
Vmrr'm-------~2G 2
(4)
where the first term accounts for the formation of precursor species and the second term accounts for the consumption of precursor species due to crystal growth. The term m2, defined later, is the cumulative area of the crystals. These rate equations are of course primarily empirical because of course it is contradictory to use a dissolved concentration of amorphous species. In reality the nutrient concentration is the solubility of the amorphous silica. However, this solubility varies as conditions in the sol change, as the crystallization advances. These simple rate equations effectively express that a portion of the nutrient originally is in the amorphous form. Knowing the average diameter of the product crystals and the yield of silicalite-1, the total concentration of silica as silicalite-1 can be estimated (C~e,,). By a silica mass balance between starting and final conditions, the following expression for the equilibrium constant K~ is obtained:
Ke =
Cze"
-1
(5)
C b o -- C eqG
The solubility of the silicalite-1 in the solution (Ceqc) is not known. It is not the total quantity of silica unconverted into silicalite-1, because this silica could also be partly amorphous or as other soluble non-nutrient silica species. A value of 2.5• 104 mol cm -3 will be used for CeqG, a rough estimate of the solubility of monomeric silicate at high alkalinity (>pH 11) [17]. It should be emphasized that the uncertainty of this value is large. It can be seen from equation (5) that K~ also expresses the distribution of silica between amorphous and nutrient forms at the start of hydrothermal treatment, since the difference, Cbo- Ceqz, represents the quantity of silica originally in the nutrient form that is converted into silicalite-1.
2.4 Population Balance Model If the crystal population size distribution at any time is described by n = n(L, t), then the population balance for a closed homogeneous batch system can be described as: On ~gt
i) ,gL
~+~(Gn)
=0
(6)
The initial and boundary conditions needed to solve the population balance equation are:
n(L, 0) = 0
(7)
n(Lmi,, t) = N(t)IG(L, t)
(8)
where Lmi, is the diameter of nuclei (3 nm). The nucleation rate (N) is expressed by equation (1) and the growth rate function is (9) This growth rate function assumes that the surface reaction and not mass transport of nutrient to the crystal surface is rate limiting. A chronomal analysis and the value of the apparent
122
activation energy in the applicable temperature range, according to Shoeman et al. [16], supports this assumption. Equation (6) is difficult to solve, however the mathematics can be simplified if the population density function is expressed in terms of its moments:
mj = ~ , , U n dL
(10)
where the moments (mj's) represent the cumulative number, length, area and volume (j = O, 1, 2, 3...) of crystals respectively. When a moment transformation is performed on equation (6), the result is
dmj _ d---t--- L~,,N - kg j[Ceqcam j - (C h - bCeq a )mj_, ]
(11)
Now the partial differential equation (6) has been converted into a system of j+l coupled ordinary differential equations that can be solved more easily. Note that for the derivation of equation (11), the following linear estimate of the true Ostwald-Freundlich equation is used: Ceq L : Ceq G (b + aL )
(12)
A closed system of coupled differential equations is not obtained with equation (2) or even a Taylor-series expansion of it. To accurately simulate the Gibbs-Thomson effect, constant values of a and b are not used, instead they are continuously updated based on equation (2) as the system of differential equations (l l) are numerically solved.
3. RESULTS AND DISCUSSION Figure 1 shows the model predictions of particle size during the course of a synthesis at 80~ The values of the adjustable parameters (ka, kg, Ke and o') giving a good fit to the experimental data are also shown. The model correctly simulates the induction period of about 10 h, a period of nearly linear growth followed by a declining rate of growth. The predicted average final crystal size is 74 nm with a c.v. of 2.58%. The actual c.v. is much higher at 8.6%, but this difference is probably due to the fact that the model does not account for a size distribution of the nuclei. The predicted low c.v. indicates that the rate of growth during the nucleation period is sufficiently slow. Table 1 shows that increasing the crystal growth rate constant broadens the product size distribution. Table 1 Effect of growth rate constant on final c.v. of product
k~ (nm cm 3 mol -I min15500 16000 16500
I)
c.v. (%) 2.51 2.58 2.65
Table 2 Effect of interfacial energy on final c.v. of product o"
(• 10-7 J cm -2) 12.47 12.35 12.25
c.v.
(%) 2.58 3.06 3.24
123 Nearly the maximum value of the interfacial energy (12.48x10 -7 J cm -2) is used in the model in Figure 1, based on the values of other parameters. At the maximum interfacial energy the solubility of the nuclei is equal to the solubility of the silicalite-1 and no growth occurs. Thus the value used for the model gives the longest possible induction period9 Reducing interfacial energy, shortens the induction period and allows faster growth during the nucleation period which broadens the product size distribution, as Table 2 shows9 Interfacial energy is generally difficult to measure experimentally and few values are reported in the literature. The fact that it varies with particle size [18] and temperature [3] makes comparisons difficult. Iller [19] reports interfacial energy values for various forms of crystalline and amorphous silica, which are in the vicinity of 50 x 10 -7 J c m -2, although varying over a wide range9 This suggests that the values of interfaciai energy arrived at by the modelling are low. The value of the amorphous silica depolymerization rate constant (ka) used was sufficiently high to maintain a nearly constant ratio between CA and Cb, determined by the equilibrium constant Ke. Also, with the large value of Ke used, 99.8% of the silica before crystallization begins in the amorphous form. This results in only a small absolute change in the nutrient concentration over the course of crystallization which is a requirement in order to obtain the nearly linear rate of growth observed9 The small change in Cb, also permits the GibbsThomson effect to cause a sufficiently long induction period9 The model predictions are insensitive to increased values of Ke, since already a vast majority of silica starts in the amorphous form. Reduced values of Ke are unsuitable because then the Gibbs-Thomson effect would become less effective at providing the required induction period and the period of nearly linear growth would be reduced. Thus an important requirement for the suitability of the model is that a large majority of the silica is amorphous prior to crystallization. As discussed above, the uncertainty of the value of Ceqa is large, however this value is not crucial for evaluating the suitability of the model, since if Ceqa is varied the same model predictions could be obtained by compensating with adjustments in the rate constants. As demonstrated the model can satisfactorily simulate the experimentally observed features of colloidal TPA-silicalite-1 crystal growth, suggesting that its underlying assumptions are reasonable. However, it is of a semi-empirical nature and the development of models that can thoroughly represent the physicochemical processes occurring during a crystallization will require further experimental insights into these processes.
NOMENCLATURE
a,b CA Cb Cbo
Ceqa CegL Cze,, G Ke
ka
coefficients for linear estimate of Ostwald-Freundlich equation amorphous silica concentration mol c m -3 nutrient species concentration mol cm -3 nutrient species concentration at t = 0 mol c m -3 equilibrium nutrient species concentration for growth mol cm -3 equilibrium nutrient species concentration at crystal size L mol cm 3 concentration of silica as silicalite-1 mol cm -3 crystal growth rate nm mm amorphous silica/nutrient species equilibrium constant dimensionless amorphous silica depolymerization rate constant mln 9
9
-I
-]
124
kg mj
crystal growth rate constant jth moment of the crystal size distribution N nucleation rate n population density function L crystal diameter Lmin nuclei diameter P nuclei/crystal population t time Vm molar volume tr interfacial surface energy
n m c m 3 min t tool -~ number nm / cm -3 number cm -3 min -~ number nm-! cm 3 nm nm number cm -3 min tool nm 3 J cm -2
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
E.J.P. Feijen, J.A. Martens and P.A. Jacobs, Stud. Surf. Sci. Catal., 84 (1994) 3. A.D. Randolph and M.A. Larson, Theory of Particulate Processes, 2 nd Ed. Academic Press, San Diego, (1988). C.J.J. Den Ouden and R.W. Thompson, J. Colloid Int. Sci., 143 (1991) 77. Q. Li, D. Creaser and J. Sterte, in press Microporous and Mesoporous Materials (1999). P.-P.E.A. de Moor, T.P.M. Beelen, B.U. Komanschek and R.A. van Santen, Microporous Materials, 9 (1997) 117. P.-P.E.A. de Moor, T.P.M. Beelen, B.U. Komanschek, O. Diat and R.A. van Santen, J. Phys. Chem. B, 101 (1997) 11077. P.-P.E.A. de Moor, T.P.M. Beelen, B.U. Komanschek and R.A. van Santen, Microporous and Mesoporous Materials, 21 (1998) 263. B.J. Schoeman, Zeolites, 18 (1997) 97. B.J. Schoeman, Stud. Surf. Sci. Catal., 105 (1997) 647. S.L. Burkett and M.E. Davis, Chem. Mater., 7 (1995) 920. S.L. Burkett and M.E. Davis, Chem. Mater., 7 (1995) 1453. W.H. Dokter, T.P.M. Beelen, H.F. van Garderen, C.P.J. Rummens, R.A. van Santen and J.D.F. Ramsay, Colloids and Surf. A: Physicochem. Eng. Aspects, 85 (1994) 89. W.H. Dokter, T.P.M. Beelen, H.F. van Garderen, and R.A. van Santen, J. Appl. Cryst., 27 (1994) 901. W.H. Dokter, H.F. van Garderen, T.P.M. Beelen, R.A. van Santen and W. Bras, Agnew. Chem. Int. Ed. Engl., 34 (1995) 73. B.J. Schoeman, Microporous and Mesoporous Materials, 22 (1998) 9. B.J. Schoeman, J. Sterte and J.-E. Otterstedt, Zeolites, 14 (1994) 568. P. Caullet and J.L. Guth, ACS Symp. Ser. 398, Zeolite Synthesis, M.L. Occelli and H.E. Robson, Eds., (1989) 83. J. Heicklen, Colloid Formation and Growth, Academic Press, New York, (1976) 50. R.K. Iler, The Chemistry of Silica, John Wiley & Sons, New York, (1979) 54.
Studies in Surface Science and Catalysis, Vol. 125 9 1999 Elsevier Science B.V. All rights reserved.
117
C r y s t a l p o p u l a t i o n b a l a n c e m o d e l for n u c l e a t i o n and g r o w t h of colloidal T P A silicalite- 1 D. Creaser Dept. of Chemical Technology, Lulefi University of Technology, $971-87 Lulefi, Sweden
A model describing the crystallization of colloidal TPA-silicalite-1 is developed and tested. The nucleation rate is approximated from experimental data. Crystal growth is surfacereaction limited and the Gibbs-Thomson effect suppresses growth of small crystals, resulting in an induction period. Slow crystal growth during the nucleation period results in the narrow crystal size distribution typically observed for colloidal TPA-silicalite-1 syntheses. 1. INTRODUCTION The crystallization of zeolites consists of two general steps, nucleation and crystal growth [ 1]. Due to a supersaturated concentration of the reagent species, crystal nuclei form. During an induction period, the rate of growth may initially be slow, followed by a period of a nearly linear growth rate. Crystal growth continues until the reagent is reduced to an equilibrium concentration. An approach termed population balance modelling, which accounts for the crystal size distribution, has been used to mathematically model this general view of crystallization [2] and its application to zeolite crystallizations from homogeneous solutions has been investigated [3]. The objective of this work is to combine this approach with the observations of several recent publications [4-16] to develop and test a model that describes the general physicochemical processes believed to occur during colloidal TPA-silicalite-1 crystallizations from homogeneous solutions.
2. MODEL DEVELOPMENT
Figure 1 shows the particle size monitored by dynamic light scattering (DLS) during a synthesis of TPA-silicalite-I at 80~ The molar composition of the synthesis sol is 9 TPAOH: 25 SiO2:480 H 2 0 : 1 0 0 ethanol. Experimental details are given in a parallel publication [4]. Before hydrothermal treatment, but after the hydrolysis of the tetraethoxysilane silica source, the sol contained subcolloidal particles with an average diameter of 4.0 nm. The presence of the subcolloidal particles persisted during hydrothermal treatment with a constant diameter of about 4 nm. After 24 h, growing crystals were first detected. The subcolloidal population was detected for a short period after the appearance of the growing crystals, but later the presence of growing crystals made them undetectable. However, it has been found though by SAXS/WAXS [5-7] and cryo-TEM [8] analysis that the
118
80 70
model o subcolloidal particles )~ crystals
~
~( 9 ~(
)~
~ 6o L_ N 50 E
>
k g = 16000 nm crna / m i n mol
~
'~ 20
~
/
kA ..- 0"0! min"1
2'0
So
Time (h)
6'0
e'o
Figure 1. Average crystal diameter from DLS and model predictions subcolloidal particles are present throughout the crystallization. After an induction period, the crystals increased in diameter almost linearly until their growth rate began to slow down and by about 70 h, growth was completed. The ultimate average diameter of the crystals was 74 nm and they had a monomodal size distribution with a coefficient of variation (c.v.) of 8.6%. 2.1 Crystal Nucleation The measurements for Figure 1 were done ex-situ at room temperature and thus may not accurately represent the crystallization process during the induction period. With in-situ measurements it was found that the subcolloidal particles increased gradually in size and a deconvolution of the intensity data suggested the growing crystals originated from the subcolloidal particles [8]. It has also been reported that the subcolloidal particles possess entrapped TPA § cations and a form of short-range order by Raman and FTIR spectroscopy [9]. These findings suggest that the subcolloidal particles, already present in solution before hydrothermal treatment, may be the crystal nuclei, de Moor et al. [5-7] have proposed that nucleation occurs by aggregation of hydrated organic-inorganic composite species [ 10, l 1] into primary particles with a size of about 2.8 nm. In accordance with the findings above, in the model it shall be assumed that the diameter of the crystal nuclei is 3 nm. The fact that the product crystals have a monomodal and narrow size distribution suggests that the nucleation may consist of a sudden burst of nuclei upon the start of hydrothermal treatment. Another possibility is that the subcolloidal particles are the nuclei and a fraction of them begin to grow immediately upon the start of hydrothermal treatment and there is no distinct nucleation process at all. Recent work though using two-stage varying-temperature syntheses strongly indicates that a continuous nucleation process occurs, lasting over much of the induction period [4]. In this work so called two-stage syntheses were carried out involving a rapid increase in temperature at some point during the crystallization. As the duration of the
119
40 35 30 0 ,reX
25
" 20 .o 15 r 0
n
10
60oc
5 00
50
1O0
150
Time (h) 18 16 14 0
~ 9
12
~10 t-"-
.o
8
:= 6
r
0
a.
4
80~
2 0
0
~
a
6
Time (h)
8
10
period at the initial temperature was extended the crystal concentration and ultimate crystal size varied until they were equal to those obtained for a complete synthesis at the initial temperature. The period of time during which the crystal concentration varied was considered to represent the nucleation period and thus rate of nucleation. Figure 2 shows the crystal population during a synthesis at 60 and 80~ At 60~ the nucleation period is much longer, extending over about 100 h, and the larger number of measurements that could be made over this period indicate that the rate of nucleation is initially high and decreases exponentially. An exponential function to express crystal population was fitted to the data at 60~ and is shown by the solid curve. The nucleation period at 80~ was shorter, lasting from 4 to 6 h but the same type of exponential relation was considered to apply over the more limited amount of data. From this function the predicted nucleation rate at 80~ is
Figure 2. Crystal population during nucleation
N(t)
= ~dP _ 89.24 x lOt2e -~
(1)
dt
The above relation is empirically based, nucleation may be initiated due to a supersaturation of the reagent species, subsequently the degree of supersaturation and the rate of nucleation decreases due to depletion of the reagent caused by growth of the nuclei [1 ]. 2.2 Induction Period
If nucleation occurs over a substantial part of the induction period, the crystal growth rate during this period must be slow in order to yield a product with a narrow crystal size distribution. Den Ouden and Thompson [3], in their analysis of the crystallization of silicalite-1, assumed that the induction period stemmed from the time required for silica to convert by hydrolysis from an inactive amorphous form to the active precursor species. However, this must not be the case for the crystallization of silicalite-1 from a clear homogeneous solution using a tetraethoxysilane (TEOS) silica source. Hydrolysis of TEOS by the TPAOH solution occurs even before hydrothermal treatment, ensuring the presence of
120 monomeric silica for the crystallization. Figure 2 indicates that the rate of nucleation is high immediately upon starting hydrothermal treatment. Apparently, there is no delay during which a supersaturation of reagent species builds up to initiate nucleation. It is thus more likely that the induction period is due to the Gibbs-Thomson effect, which accounts for the higher solubility of smaller crystals. If the driving force for crystal growth is the difference between the bulk and equilibrium concentrations of precursor silica species, the GibbsThomson effect would suppress growth during the early stage of crystallization. The OstwaldFreundlich equation expresses the solubility of a crystal of diameter L (Ceqt)as a function of the solubility at an infinite diameter (Ceqc): Ceq L -- C eqG
exp/~,4 ( ~TL
)
(2)
2.3 Crystal Growth Recent studies of the crystallization of silicalite-1 have discussed various crystal growth mechanisms. Dokter et al. [12-14] proposed that the crystallization consisted of two aggregation steps. During nucleation primary aggregates formed which subsequently formed secondary aggregates with a size of about 10 nm. More recently, De Moor et al. [5-7] found that the secondary aggregation was not a necessary part of the growth mechanism. Instead it seemed that growth occurred mainly by the addition of the primary aggregates (about 2.8 nm) which presumably correspond to the subcolloidal particles, to the crystal surface. Although, Shoeman [ 15] has shown with the extended Derjaguin-Landan and Verwy-Oberbeek (DLVO) theory that the colloidal particles in TPA-silicalite-I synthesis sols are stable against aggregation which would rule out growth by aggregation mechanisms. In the model developed here a classical crystal growth mechanism shall be assumed to apply, i.e. the addition of nutrient silica species which are probably monomeric or perhaps oligomeric from solution to the growing crystals. The stability of the colloidal particles against aggregation in Shoeman's analysis is primarily provided by surface adsorbed TPA + cations and it has been shown that the subcolloidal particles and growing crystals contain these entrapped cations [9]. It is reasonable to assume that the structure-directing effects of the TPA § cations would provide suitable sites for the incorporation of incoming nutrient species into the zeolite crystal structure. According to the assumed reaction sequence amorphous silica (A) depolymerizes to form the nutrient species (P) that feed the growth of crystals (Z). Other species not involved in the zeolite growth (O), perhaps larger oligomeric silica, may also be present. Throughout the crystallization process a dynamic equilibrium exists between these species: A r P r Z %~ O Since little is known about the exact nature of these processes, the model used will be simplistic in this respect. The rate of change of the amorphous species concentration (CA) will be expressed as
dCA dt
~---
- k a ( C A - K C~ )
(3)
121 where Ke is the equilibrium between the amorphous and nutrient species. The rate of change of the nutrient species concentration (Cb) is then
d C-----2-b= dt
k a ( C a - K eC h ) -
Vmrr'm-------~2G 2
(4)
where the first term accounts for the formation of precursor species and the second term accounts for the consumption of precursor species due to crystal growth. The term m2, defined later, is the cumulative area of the crystals. These rate equations are of course primarily empirical because of course it is contradictory to use a dissolved concentration of amorphous species. In reality the nutrient concentration is the solubility of the amorphous silica. However, this solubility varies as conditions in the sol change, as the crystallization advances. These simple rate equations effectively express that a portion of the nutrient originally is in the amorphous form. Knowing the average diameter of the product crystals and the yield of silicalite-1, the total concentration of silica as silicalite-1 can be estimated (C~e,,). By a silica mass balance between starting and final conditions, the following expression for the equilibrium constant K~ is obtained:
Ke =
Cze"
-1
(5)
C b o -- C eqG
The solubility of the silicalite-1 in the solution (Ceqc) is not known. It is not the total quantity of silica unconverted into silicalite-1, because this silica could also be partly amorphous or as other soluble non-nutrient silica species. A value of 2.5• 104 mol cm -3 will be used for CeqG, a rough estimate of the solubility of monomeric silicate at high alkalinity (>pH 11) [17]. It should be emphasized that the uncertainty of this value is large. It can be seen from equation (5) that K~ also expresses the distribution of silica between amorphous and nutrient forms at the start of hydrothermal treatment, since the difference, Cbo- Ceqz, represents the quantity of silica originally in the nutrient form that is converted into silicalite-1.
2.4 Population Balance Model If the crystal population size distribution at any time is described by n = n(L, t), then the population balance for a closed homogeneous batch system can be described as: On ~gt
i) ,gL
~+~(Gn)
=0
(6)
The initial and boundary conditions needed to solve the population balance equation are:
n(L, 0) = 0
(7)
n(Lmi,, t) = N(t)IG(L, t)
(8)
where Lmi, is the diameter of nuclei (3 nm). The nucleation rate (N) is expressed by equation (1) and the growth rate function is (9) This growth rate function assumes that the surface reaction and not mass transport of nutrient to the crystal surface is rate limiting. A chronomal analysis and the value of the apparent
122
activation energy in the applicable temperature range, according to Shoeman et al. [16], supports this assumption. Equation (6) is difficult to solve, however the mathematics can be simplified if the population density function is expressed in terms of its moments:
mj = ~ , , U n dL
(10)
where the moments (mj's) represent the cumulative number, length, area and volume (j = O, 1, 2, 3...) of crystals respectively. When a moment transformation is performed on equation (6), the result is
dmj _ d---t--- L~,,N - kg j[Ceqcam j - (C h - bCeq a )mj_, ]
(11)
Now the partial differential equation (6) has been converted into a system of j+l coupled ordinary differential equations that can be solved more easily. Note that for the derivation of equation (11), the following linear estimate of the true Ostwald-Freundlich equation is used: Ceq L : Ceq G (b + aL )
(12)
A closed system of coupled differential equations is not obtained with equation (2) or even a Taylor-series expansion of it. To accurately simulate the Gibbs-Thomson effect, constant values of a and b are not used, instead they are continuously updated based on equation (2) as the system of differential equations (l l) are numerically solved.
3. RESULTS AND DISCUSSION Figure 1 shows the model predictions of particle size during the course of a synthesis at 80~ The values of the adjustable parameters (ka, kg, Ke and o') giving a good fit to the experimental data are also shown. The model correctly simulates the induction period of about 10 h, a period of nearly linear growth followed by a declining rate of growth. The predicted average final crystal size is 74 nm with a c.v. of 2.58%. The actual c.v. is much higher at 8.6%, but this difference is probably due to the fact that the model does not account for a size distribution of the nuclei. The predicted low c.v. indicates that the rate of growth during the nucleation period is sufficiently slow. Table 1 shows that increasing the crystal growth rate constant broadens the product size distribution. Table 1 Effect of growth rate constant on final c.v. of product
k~ (nm cm 3 mol -I min15500 16000 16500
I)
c.v. (%) 2.51 2.58 2.65
Table 2 Effect of interfacial energy on final c.v. of product o"
(• 10-7 J cm -2) 12.47 12.35 12.25
c.v.
(%) 2.58 3.06 3.24
123 Nearly the maximum value of the interfacial energy (12.48x10 -7 J cm -2) is used in the model in Figure 1, based on the values of other parameters. At the maximum interfacial energy the solubility of the nuclei is equal to the solubility of the silicalite-1 and no growth occurs. Thus the value used for the model gives the longest possible induction period9 Reducing interfacial energy, shortens the induction period and allows faster growth during the nucleation period which broadens the product size distribution, as Table 2 shows9 Interfacial energy is generally difficult to measure experimentally and few values are reported in the literature. The fact that it varies with particle size [18] and temperature [3] makes comparisons difficult. Iller [19] reports interfacial energy values for various forms of crystalline and amorphous silica, which are in the vicinity of 50 x 10 -7 J c m -2, although varying over a wide range9 This suggests that the values of interfaciai energy arrived at by the modelling are low. The value of the amorphous silica depolymerization rate constant (ka) used was sufficiently high to maintain a nearly constant ratio between CA and Cb, determined by the equilibrium constant Ke. Also, with the large value of Ke used, 99.8% of the silica before crystallization begins in the amorphous form. This results in only a small absolute change in the nutrient concentration over the course of crystallization which is a requirement in order to obtain the nearly linear rate of growth observed9 The small change in Cb, also permits the GibbsThomson effect to cause a sufficiently long induction period9 The model predictions are insensitive to increased values of Ke, since already a vast majority of silica starts in the amorphous form. Reduced values of Ke are unsuitable because then the Gibbs-Thomson effect would become less effective at providing the required induction period and the period of nearly linear growth would be reduced. Thus an important requirement for the suitability of the model is that a large majority of the silica is amorphous prior to crystallization. As discussed above, the uncertainty of the value of Ceqa is large, however this value is not crucial for evaluating the suitability of the model, since if Ceqa is varied the same model predictions could be obtained by compensating with adjustments in the rate constants. As demonstrated the model can satisfactorily simulate the experimentally observed features of colloidal TPA-silicalite-1 crystal growth, suggesting that its underlying assumptions are reasonable. However, it is of a semi-empirical nature and the development of models that can thoroughly represent the physicochemical processes occurring during a crystallization will require further experimental insights into these processes.
NOMENCLATURE
a,b CA Cb Cbo
Ceqa CegL Cze,, G Ke
ka
coefficients for linear estimate of Ostwald-Freundlich equation amorphous silica concentration mol c m -3 nutrient species concentration mol cm -3 nutrient species concentration at t = 0 mol c m -3 equilibrium nutrient species concentration for growth mol cm -3 equilibrium nutrient species concentration at crystal size L mol cm 3 concentration of silica as silicalite-1 mol cm -3 crystal growth rate nm mm amorphous silica/nutrient species equilibrium constant dimensionless amorphous silica depolymerization rate constant mln 9
9
-I
-]
124
kg mj
crystal growth rate constant jth moment of the crystal size distribution N nucleation rate n population density function L crystal diameter Lmin nuclei diameter P nuclei/crystal population t time Vm molar volume tr interfacial surface energy
n m c m 3 min t tool -~ number nm / cm -3 number cm -3 min -~ number nm-! cm 3 nm nm number cm -3 min tool nm 3 J cm -2
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