A simulated annealing study of Si,Al distribution in the faujasite framework

~UTTERWORTH II~IE I N E M A N N

A simulated annealing study of Si,AI distribution in the faujasite framework Datong Ding, Baohui Li, Pingchuan Sun, Qinghua Jin, and Jingzhong Wang

Nankai University, Tianjin, PR China

Several important properties of faujasite are concerned with the Si,AI distribution in its framework. In the present study, a starting configuration of the distribution is generated with randomly positioned Si,Ah After the simulated annealing procedure, some orderings appear in the optimized configurations of the system and lead to a series of consequences, some of which have already been obtained by pioneer studies concerning ordering schemes. In optimized configurations, a certain degree of disorder reasonably is maintained because of the use of standard Monte Carlo finite temperature techniques, which bring about a realistic employ of the numerical simulation. The calculated AI distribution on the first shell of the Si site is assessed by zsSi M A S n.m.r, experiments to demonstrate the rationality of the present theory. The calculated AI distribution on the second shell of the AI site is related to the acidity, which leads to the proposition that the acid amount of faujasite is in proportion to the number of single AI four rings. The electrostatic energy calculation provides possible explanations of the observed discontinuity in the plot of unit cell parameter against AI content. Keywords: Faujasite; framework AI distribution; simulatedannealing; acidity of faujasite; 29SiMAS n.m.r.

INTRODUCTION In faujasites the substitutional content of AI for Si can be varied over a wide compositional range while preserving its structure type (1 < Si/A1 < 2.75: zeolites X and Y; Si/AI > 5: aluminum-deficient zeolites). For the low AI concentration (Si/AI > about 6), A l e ] - are isolated from each other because o f the dilution; all of the AIO~- exhibit strong acidity. As the AI content is high, the acid strength of an AIO~- may be weakened by the presence of other AI atoms in its local environment. Chemically, such a property may be interpreted as a self-inhibition effect of the aluminum sites.1 During the course of an accurate determination of the variation of the cubic lattice parameter with aluminum content in hydrated sodium faujasites, Dempsey et al. 2 found distinct discontinuities in what was expected to be a continuous linear relationship. At least three different phases might be occurring across the range of Si/AI variation, and the breaks corresponded to transitions from one type of Si,AI ordering to another. Those early experimental facts led t o the study of the character of Si,AI distribution u n d e r different aluminum content. In the faujasite lattice, every tetrahedral site is in three adjoining four rings. We may, in principle, distinguish four types of AI ions (AIO~-): those having 0, 1, 2, or 3 diagonally opposed A1 next nearest neighAddress reprint requests to Dr. Ding at the Dept. of Physics,

Nankai University, Tianjin 300017, Peoples Republic of China. Received 15 August 1994; accepted 26 January 1995 Zeolites 15:56,9-573, 1995 © Elsevier Science Inc. 1995 655 Avenue of the Americas, New York, NY 10010

bors; we denote them as {AI(mA1); m = 0-3}. Four degrees of acid strength of the associated protons can then be expected in order 0 > 1 > 2 > 3. 3 The relative populations of type AI(mA1) are state variables characterizing the Si/AI distribution of the system. The variation of the populations with A1 content is of much concern. The framework silicon atoms can be classified by environments having n AI nearest neighbors and denoted as {Si(nAl); n = 0--4} building units. 29Si n . m . r . permits the determination of the relative populations of those five silicon building units, which can be used for the determination of the Si/AI. 4 It becomes an assessment of the validity of different models of Si,AI distribution because the 29Si n . m . r , spectra provide information on the Si,A1 distribution on the first coordination shell of silicon atoms. A series of studies on Si,A1 framework distribution was based on the ordering schemes in [3-cages of faujasite. The favored schemes were assessed by their agreement with 29Si n . m . r , observations, lowest electrostatic energies, a n d crystal s y m m e t r y a r g u ments.2,~.6 Another series of studies was based on the constrained random models that AI atoms are randomly positioned in the framework subject to the constraint of Loewenstein's rule, which forbids the presence of A1-AI nearest neighbors, and superimposed on this a second weaker constraint that the number of AI-A1 next nearest neighbors is minimized. T h e Monte Carlo method is adequate to analyze problems o f substitutional disorder in solids, such as faujasite, for two 0144-2449/95/$10.00 SSDI 0144-2449(95)00005-Q

Simulated annealing study of faujasites: D. Ding et al.

reasons: the method is able to handle systems involved in large numbers of degree of freedom, and it simulates equilibrium properties of the system at nonzero temperature. 7-9 In the present work, the simulated annealing procedure suggested by Kirkpatrick et al) ° is employed, The idea is simply to start at a relative high enough temperature T s and slowly cool the system (faujasite with a given Si/AI ratio) with standard Monte Carlo finite temperature techniques down to the temperature of faujasite formation, TF. Although we extend the electrostatic energy calculation to next nearest neighboring A1-AI pairs, a simplification is made as far as possible to perform large Monte Carlo runs in reasonable computing time. The effort of the present work is aimed at a general theoretical course, based on the AI compositional d e p e n d e n c e of the 29Si n.m.r, spectra, in which the acidic properties of faujasite and the observed discontinuities in the plot of AI content-lattice parameter (of faujasite) can be explained comprehensively. EXPERIMENTAL Simulated annealing is a procedure of the Monte Carlo method in which there is a deep and useful connection between statistical mechanics and combinatorial optimization, l° During the simulated annealing procedure achieving the global optimum, Metropohs sampling is employed as the way of separation from local optimums. In the study of faujasite, the following conditions are defined, •

11

•

Configuration of

•

the system

Two unit cells composed of 384 tetrahedral sites with cyclic boundary conditions, that is, 16 complete sodalite cages, are considered a representative pottion of the faujasite framework for the simulation, This cyclic boundary condition ensures that the simulated results do not depend on representation pottion size. We assume that the numbers of A1 and Si atoms are N and 384-N, respectively, for a given Si/AI ratio. Any one microscopic configuration of the system is well defined by corresponding occupations of all of the Si,A1 atoms in the framework. A starting configuration is created by using a random number generator in which Si and AI atoms are randomly positioned on those 384 tetrahedral sites, O b j e c t i v e function In our present study, a simplified model is adopted in which the oxygens and compensating cations (for instance Na +) are ignored. The energy concerned with the Si,AI distribution configuration regards only the AI-AI repulsion and can be expressed approximately as:

E = (Jnn ~i,i ~i,Al~i',Al+ Jtnn E ~j,Al~j',AI j,j,

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C a n o n i c a l ensemble The system being studied is assumed in thermal contact with a heat bath of temperature T. A configuration of the system with energy E t°t" is weighted by factor exp(-Et°t'/KBT).

Metropolis algorithm The randomly generated starting configuration is optimized by repeated reconfigurations. In each reconfiguration, we randomly search the framework and find an AI-Si nn pair and then attempt to interchange them as a disturbance of the original configuration. Compare the E' of the new disturbed configuration with the E of the original one by Equation (1). According to Metropolis algorithm, i f E ' is lower than E, the interchange is accepted, and the new configuration is used as the starting one for the next step. For E' - E = AE > 0, the new configuration is accepted with the probability of P(AE) = e x p ( - A E / KBT). If the new one is rejected, the original one is kept for the next attempt. It is obvious that as T = 0, the probability for a new configuration with AE > 0 being accepted is zero. Although the zero-temperature Monte Carlo method is provided for a rapid rate of progress because of the unidirectional optimization (improvement-only rule), it often gets stuck in local optimums and leads to metastable states that are of no interest here. According to the Metropolis algorithm, the standard Monte Carlo finite temperature techniques, the acceptance of energy-unfavorable disturbances maintains nonzero probabilities, which is the way of getting separated from local optimums. Although the optimization gets somewhat retardatory, the system will not get caught in a metastable state, and the objective achieving the global optimum is ensured.

Annealing scheme

h

+JnnnE~k,Al~k,,A1)/2 k,k*

where the f r s t sum involves only the nearest neighbor pairs (i,i'), and the Kronecker ~ ensures that only A1-A1 pairs are counted. T h e interaction of the A1-AI nn pair is characterized by the p a r a m e t e r J . . . Analogously, the second term sums over only those next nearest neighbor pairs (j,j'), diagonally opposed on four rings. The interaction of such AI-A1 nnn pairs is characterized by J .t. . . The third term sums over only those nnn pairs (k,k'), metapositioned on six rings. The interaction of such AI-A1 nnn pairs is characterized by J~,,,. Those contributions from more distant AI-A1 pairs are omitted. The energy E, corresponding to a given configuration of the system, is determined by Equation (1) quantitatively. E is then defined as the objective function representing a quantitative measure of the goodness of the system or scaling the degree of optimization. The simulated annealing procedure is aimed at seeking the configurations with the lowest E values.

(1)

To equilibrate the system at the temperature of faujasite formation (TF = 350 K), a simulated annealing procedure is carried out. The temperature starts

Simulated annealing study of faujasites: D. Ding et al.

at T, ( - 5 . 5 TF),~ and then steps down to TF. In each step, T is reduced by a factor of 0.95, that is T(n) = 0.95"T,, until T(n) = T e. At each T(n), enough reconfigurations are attempted (either the number of attempts exceeds 10 s, or the cumulative number of accepted interchanges exceeds 104) for the equilibrium of the system,

1.01x "3 0.8 < .~ t ~ 0"61 o "' r N

/

iCOAL)

SiC4AL),~

~ ~

SiflAL) ~

/

~

_SiC2AL.)

/

< After the simulated annealing procedure the configuration of the system has been optimized, by which the canonical ensemble sample space is starting to build up as follows: 2 x 105 reconfiguration attempts are then performed. According to the Metropolis algorithm, the sample set either gets a new configuration or reiterates an original one for each attempt. In this way a canonical ensemble sample space is setup which consists of 2 x 105 configurations whose probability distribution approaches the Boltzmann distribution. T h e number of Si(nAl) and AI(mAI) can be counted for each configuration. Their average populations are obtained from the sample space. The setting values of parameters J , , , J~,,, and J~,,, influence the results of the numerical simulation. In the present study, we s e t J , , = IOKBTF (same as Herrero, see Ref. 8), by which AI-AI nn pairs are avoided in the simulation, and we regard J~,,~ and j t as interactions of an electrostatic nature. According to the ratio of distance between atoms diagonally opposed on four rings to that between atoms metapositioned on six rings, we have j h = jt/1.225. Therefore, jt is the only trial number in the simulation. The value of Jt~,,~ depends on the agreements between the calculated populations of Si(nAl) and 99Si n.m.r, data of Isi(,At) over a series of given Si/AI ratios. To assess the quality of the simulation, a parameter R, scaling the difference between the calculated normalized populations o f Si(nAl), Ical., and normalized 29Si n.m.r, intensities In.m.r. is defined as follows. 4 R = E

,,=0

n.m.r. _ ~Si(nAl) ] [Isi(nAl) al.

(2)

In comparing the numerical simulations with the n.m.r, data of 27 synthetic faujasite samples (Si/AI • rauo from 1.14 to 2.69 5--6 ' 12 - 14 ), we got corresponding R i values (j = 1-27). For all of the samples the best fit to the experimental data is found for J r , , = 1.2KBT F, which results in 1 / 2 7 ~ 2 7 I[Rj[ < 0.12. RESULTS AND DISCUSSION

Normalized populations of Si(nAl) Figure 1 shows that the normalized populations of

Si(nA1) calculated by the present numerical simulation for synthesized faujasites var~¢ with AI content, For comparison, several typical 2~Si n.m.r, data by which the ordering schemes gained proof of their rationality 6 are quoted as well. A randomly generated initial configuration of the

L

z°

~

'

~

0

~

~

. O0

FRAMEWORK ALUMINIUM

ATOMS/U.C.

Figure I Normalized Is~(.At) versus the number of AI atoms/unit cell obtained by simulated annealing at d t n . = 1.2KaTF. Symbols represent n.m.r, experiment observations obtained by Klinowski et al.e (A) /SHOAl); (BB) lsi(1Ai); (O) /Si(2Ai); (~7)ISiC~0; (V1) ISi(4AI)"

system is optimized by repeated reconfigurations of the simulated annealing procedure. In the optimized configurations, some ordered structures emerge consequentially, and a certain degree of disorder is teasonably maintained because of the finite temperature postulate. Therefore the finite temperature Monte Carlo simulation offers a better example of reality, which is close to the experimental observations (see Table I).

R e l a t i v e populations of AI(mAI) and acidity The number of type AI(mAI) can be counted for each configuration sample. The relative populations of AI(mAI) are obtained by taking an average from the sample space. Figure 2 shows that calculated populations of four types of AI(mAI) vary with AI content. The correlation between calculated {Si(nA1)} and {AI(mAI)} is set up from a reliable theoretical base since both of them take their source at the same canonical ensemble sample space of the Monte Carlo simulation. The former coincide with experimental 29Si n.m.r, observations and offer a support to the latter. We assume that each acid site makes a constant contribution to the sample independently. T h e contributions of type AI(mAI) are scaled by a corresponding parameter b=(m = 0-3) and ordered with b0 > b 1 > b2 > b3. Suppose that the average number of AI atoms/unit cell of the faujasite sample is NAI and the average numbers of AI(mAI) are n,,, respectively (NAl = no + nl + n2 + ha). The acid amount/unit cell B of the sample is then expressed as: (3) The efficiency parameter ot0 defined in faujasites 15 is considered as the ratio of acid amounts to bulk AI content (oro = B/NAIL of the sample. In comparing Equation (3) with the A1 content dependence of ~t0 found experimentally,15 the following consequences can be reasonably deduced. In the high AI content B = bon o + bin 1 + b2n9 + ban a.

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Simulated annealing study of faujasites: D. Ding et aL "Fable 1 Comparison between building units {Si(nAI)} q u o t e d f r o m n.m.r, observations and ordering schemes of Ref. 6 and that from present results Relative p o p u l a t i o n s of Si(nAI) ~ = o / s , ( . A , ) = 100 Si/AI

n = 0

1.14a 1.18 n 1.14c 1.39 a 1.40b 1.39c 1.71 ~ 1.67 n 1.71 c 1.98a 2.00 b 1.98c 2.46 a 2.42 b 2.46 ~

n .

1.9 7.7 4.5 2.2 0.0 2.8 4.9 0.0 1.5 5.7 0.0 2.6 5.2 5.9 7.3

1

n.

.

1.4 0.0 2.2 8.9 14.3 9.2 15.3 20.0 18.0 25.8 25.0 26.1 41.8 35.3 37.9

2

R= n

.

6,2 0,0 3,6 21,5 21.4 19.7 33.8 33.3 36.8 36.5 50.0 41.3 38.3 47.1 40.5

3

26.5 30.8 20.5 33.5 28.6 33.0 33.5 33.3 31.4 24.5 25.0 26.5 17.7 11.8 13.6

n

4

n.m.r, 32n4 = o [ Isi(nAI)

64.0 61.5 70.0 33.9 35.7 35.3 12.5 13.3 12.3 7.5 0.0 3.5 0.0 0.0 0.8

- -

tcal / SI(nAI)I I

20.2 18.0 14.4 4.6 11.1 11.4 28.0 14.2 18.9 10.1

a n.m.r, observation, Ref. 6. b Ordering scheme, Ref. 6. c Simulated annealing, present study.

limit, ot0 ---> 0 indicates that b3 = 0, a n d in the low A1 c o n t e n t limit, tx0 ~ 1 indicates that b0 = 1.16 T h e r e f o r e we have a r e d u c e d f o r m o f (4)

B = n o + b i n x + b2n 2

instead o f E q u a t i o n (3), a n d (5)

oto = (n o + b i n I + b2n2)/NAl.

Based o n calculated relative fractions o f {AI(mA1)}, (n,,/NA!), by t h e n u m e r i c a l s i m u l a t i o n m e n t i o n e d

above, the AI c o n t e n t d e p e n d e n c e o f Oto(NAl), Equation (5), is a f f e c t e d by p a r a m e t e r s b l a n d b2. T h e values o f b 1 = 0.64, b2 = 0.44 are d e t e r m i n e d by trial a n d e r r o r to m a t c h the ot0(NAl) with e x p e r i m e n t a l results 15 (See Figure 3). I n the faujasite f r a m e w o r k , every t e t r a h e d r a l site defines t h r e e adjoining f o u r rings. It is obvious that each AI(mAI) is a c c o m p a n i e d with 3 - m f o u r rings conmining a single a l u m i n u m a t o m . T h e total n u m b e r o f

-~ < 0.8

single A1 f o u r ring is 3n o + 2n 1 + n 2, or 3 × (n o + 0.67nl + 0.33n2). T h e r e f o r e , we a r g u e t h a t the acidity o f faujasite is r o u g h l y in p r o p o r t i o n to the n u m b e r o f the single AI f o u r r i n g s u g g e s t e d by the analogy between B ~ n o + 0.67n 1 + 0.33n 2 a n d the simulated result B = n o + 0.64n 1 + 0.44n 2.

Electrostatic energy calculation T h e electrostatic e n e r g y E is calculated a c c o r d i n g to Equation (1) a n d obtains by taking the a v e r a g e f r o m the canonical e n s e m b l e s a m p l e space. I n Figure 4 the value o f E is plotted as a f u n c t i o n o f AI content (NA0. It can be seen that t h e r e are f o u r distinct regions with d i f f e r e n t slopes. In an e x t r e m e l y dilute c o n c e n t r a t i o n , A1 a t o m s are isolated f r o m each other, a n d electrostatic interactions a m o n g t h e m are trivial. E holds constant in the r a n g e o f NAI = 0--19. I n the r a n g e o f NAI = 19--40, the slight increase o f

AL(3

0.8

E

0.6,

\

AIC?AI'I

/

LLI

N

~ 0.6 ,'-d

0.4

0.4

IK ¢.y

o Z

0.2 °'°o

0.2 40

8o

FRAMEWORK ALUMINIUM ATOMS/U.C. Figure 2 Calculated AI content dependence of normalized populations of [AI(mAI); m = 0-3].

572

Zeolites 1 5 : 5 6 9 - 5 7 3 , 1995

o,o;,

, , 40 , , 60 , , , 20 80 FRAMEWORK ALUHINIUH ATOMS/U.E.

,

96

Figure 3 Efficiency parameter e=o versus the number of AI ati ores/unit cell. (©) experimental results quoted from Ref. 15.

Simulated annealing study of faujasites: D. Ding et al.

I000

with a discontinuity at Si/AI ~ 2.0(NAI ~ 64). For compositions with Si/A1 < 2.0, an addition o f o n e AI atom/unit cell contributes m o r e t h a n twice that for the composition with Si/AI > 2.0. As NAI ~ 80, AI(1A1) dies out, a n d AI(2A1) --~ AI(3AI) is the only t r a n s f o r m left in the system. This might be the reason that brings a b o u t a n o t h e r discontinuity in the plot o f unit cell p a r a m e t e r at NAI ~ 80.

800

... 6 0 0 p~ 400 U.J

200

REFERENCES

C]l~,~_ - ~ ~ --

-

0

20

40

60

~0

Q6

FRAMEWORK ALUMINIUM ATOMS/U.C. Figure4

Calculated electrostatic energy as a function of AI con-

tent.

E results mainly f r o m the presence o f AI(IAI). In the range o f N ~

= 4 6 - 6 4 , the g r o w t h and decline o f

AI(2AI) a n d

AI(0AI) increase the rate o f change o f E with NAl. As the u p p e r b o u n d NA! reaches 64, the isolated A I ( 0 A I )

disappears. In the range OfNA! > 64,

the a p p e a r a n c e o f AI(SAI) causes a rate o f change o f E with NAI twice as large as that in the r a n g e OfNAI < 64 (see Figure 2). This composition coincides with the composition in which the discontinuity was f o u n d in the plot o f unit cell p a r a m e t e r against the n u m b e r o f the A1 atoms/unit cell. 2 Based o n the o r d e r i n g schemes, Klinowski et al. 6 f o u n d that the plot o f the calculated electrostatic energy against the n u m b e r o f the A1 atoms/unit cell clearly indicates two distinct types o f linear behavior

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