8310
J. Phys. Chem. 1993,97, 8310-8315
Statistical Mechanics of Atom Ordering in Ultramarines M. C. Gordillo and C. P . Herrero’ Instituto de Ciencia de Materiales, CSIC Serrano, 1 1 5 dpdo., 28006 Madrid, Spain Received: February 16, 1993; In Final Form: May 18, 1993
The distribution of Si and A1 atoms over the framework of the aluminosilicate ultramarine with a ratio Si/Al = 1 is studied by a Monte Carlo method, in the temperature range from 100 to 1800 K. For low temperatures ( T < 700 K), the atom arrangement shows sublattice ordering with Si and A1 atoms alternating in the network. For T > 750 K, the long-range order of the atom distribution disappears, but there remains a certain degree of short-range ordering, with partial avoidance of AI-0-A1 and S i - M i groups. The heat capacity derived from the M C simulations shows a peak at about 750 K, suggesting an order4isorder phase transition at this temperature. Configurational entropy and free energy results confirm that at temperatures of pyrolytic synthesis this atom distribution is disordered and violates Loewenstein’s rule, in agreement with results obtained earlier by Z9Si N M R spectroscopy.
1. Introduction Ultramarines form a family of closely related compounds routinely used as pigments because of their intense colors. They are, as tectoaluminosilicates, formed by the three-dimensional arrangement of tetrahedral A104 and Si04units in a framework similar to that of sodalite, a well-known natural aluminosilicate. The main difference between the structure of ultramarines and sodalites is the nature of the ions enclathrated inside the sodalite cages, which in the former case are the big chromophore groups. In blue ultramarine these groups are S3- anions, but there is a complete series of ions that give a variety of colors from mauve to pink.1-4 Apart from their industrial applications, these compounds are interesting from the point of view of the substitutional disorder of Si and A1 atoms over their frameworks. In fact, blue ultramarine was reported to present a disordered Si,Al distribution,3,4in contrast with results obtained for its natural counterpart (lazurite), in which Si and A1 atoms follow a longrange order scheme.5 This order-disorder problem is general in aluminosilicateswith three-dimensional networks, and has been analyzed by experimental techniques such as X-raydiffraction,3-5as well as and 29Si and z1A1 magic-angle-spinning nuclear magnetic resonance (NMR) ~pectroscopy.3~~J’~~ For hydrothermally synthesized aluminosilicates, it is now generally accepted that silicon and aluminum atoms are distributed over the network in such a way that A1-O-A1 linkages are avoided (which is known in mineralogy as Loewenstein’srules), irrespective of the consideredframework. In the case of blue ultramarines, however, the large chromophore ions present in the structure are unstable in aqueous media, and pyrolytic methods are necessary to synthesize these materials. Thus, the high temperatures associated with these methods will favor the disorder of the tetrahedral atoms over the framework. This causes, after cooling, the material to be trapped in a metastable state with a Si, A1 distribution more disordered than that corresponding to thermodynamic equilibrium at room temperature? as suggested by the NMR results of Klinowski et ~ 1 . 4
Substitutional disorder in aluminosilicates has been also investigated by theoretical procedures. Monte Carlo (MC) methods, in particular,are adequate for the study of order-disorder problems in solid materials because they allow us to treat a great number of degreesof freedom in simulationcellscontaining several hundreds of atoms.I0 Besides, by means of this computational method, it is possible to carry out simulations in equilibrium conditions at finite temperatures (T> 0), and thus to consider the effects of thermal disorder, which are nonnegligible for a
precise characterization of actual aluminosilicate^.^^-^^ Nevertheless, to obtain reliable results it is necessary to employ realistic interatomic potentials to describe the interactions in the material under consideration. For aluminosilicates, in particular, a satisfactory description of the Si, A1 ordering is obtained when the potential model includes long-range Coulomb interactions as well as atomic polarization effects. The reliabilityof such a model was checked by comparison between the atom distributions obtained by MC simulations and those derived from 29% NMR data.14 In this paper, we present the results of a MC simulation of the Si, A1 distribution in blue ultramarine. Both, structural (shortand long-range order parameters) and thermodynamic (configurational energy, entropy, and free energy) properties of the tetrahedral atom arrangement in this material are analyzed up to temperatures typical of pyrolytic synthesis.
2. Computational Method The unit cell of blue ultramarine has an ideal composition Na7.5Si6A160z4S4,5. The three-dimensional structure of this compound is formed by the arrangement of sodalite cages (truncated octahedra) in a cubic disposition by sharing their 4-membered rings with neighboring cages. Our simulation cell contains 27 unit cells and includes 324 tetrahedral (T) atoms. To check the convergence of the variables calculated from our MC simulations, as a function of the size of the simulation cell, we performed several simulations for different sizes, and found that the use of cells including more than 400 T-sites does not change appreciablythe mean values of the thermodynamic and structural parameters presented below. Atomic coordinates for standard blue ultramarine were taken from Tarling et aL3 We modified, however,the coordinates of the S atoms in the chromophoregroup since, according to the data given in ref 3, these atoms were located at about 1 A from one another, a distance too short for a S S b o n d . Thus,wehavelocated theSatomson“l2e”positions of the space group I43m, as in ref 3, but with a SS distance of 2.1 A, more adequate for a S3- ion. The SJ- anions and the exchangeable Na+ cations were distributed randomly over their corresponding positions. The interatomic potential used in our MC simulationsincludes long- and short-rangeinteractions. The former consist of Coulomb and polarization energies, and the latter is a contribution derived from repulsion-dispersion forces. For each atom configuration,
0022-3654/93/2097-83 10%04.00/0 0 1993 American Chemical Society
Atom Ordering in Ultramarines
The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8311
the electrostatic part is calculated by the Ewald method, whereas the polarization energy is given by N
1 or -1 if it is occupied by A1 or Si respectively. This variable was employed to calculate the pair correlation functions SIand S2,which quantify the short-range order of the atom distribution:
(3) where N is the total number of atoms in the simulation cell, (rk is the polarizability of atom k, and Ek is the electric field at site k. This polarization energy is nonnegligible only in the case of sulphur and oxygen centers, because of the high values of their polarizabilities. For the oxygen polarizability we took the value given byOomsela1.15(a(O2-)= 1.984A3),while thepolarizability of the S3- anions was approached as a sum of the polarizabilities of two S neutral atoms16 and a S-ion1’ (a(S-)was taken as an average of a(S)and a(Sz-)9). Hence, the average polarizability assigned to each sulfur center in the S3- anion was a(S) = 3.9 A3. The uncertainty in this value is not critical for our purposes, since changes of a(S) of about 20% do not modify appreciably the results presented below. Finally, the short-range pair interaction is given by a Buckingham potential
( ) -rij
V(rij)= Aijexp --
Pij
Cij 6
rij
whose parameters for a pair of atoms i a n d j were taken from ref 15. For a fixed framework geometry, this energy term gives a constant contribution to the lattice energy, since the numbers of S i 4 and A 1 4 bonds are independentof the T-atom distribution. An unknown parameter in this potential model is the difference between the silicon and aluminum charges in the ultramarine structure, 6 = qsi - qM, As shown elsewhere,14 changes of the configurational energy caused by the Si, A1 ordering depend on this parameter, 6, which appears as a quadratic factor in both Coulomb and polarization energies. Due to the lack of experimental and theoretical data on this charge difference in ultramarines, we took the value found for zeolite A from a comparison between MC simulation results and z9SiNMR data (6 = 0.26e; e, elementary charge).13 It is expected that silicon and aluminum atoms have similar effective charges in both compounds due to the similarity between their networks. The use of different 6 values in our MC simulations changes the temperature and energy scales presented below. In particular, higher values of 6 yield order4isorder transition temperatures higher than that obtained here (see section 3). MC simulations were performed in the canonical ensemble (NYT), by starting at a temperature of about 2500 K and slowly cooling the system down to 100 K. This procedure is known in the literature as “simulatedannealing”and reduces the probability that the system could be trapped in metastable states.’*J9 We analyzed the atom distribution in the temperature range from 100 to 1800 K. At each considered temperature, we generated 5.5 X 104 atom configurations, from which the first 5 X lo3 ones were performed for system equilibration and were not included in the statistical averages. The use of longer simulation runs does not change the average results. The configurational space was sampled by using the Metropolis method,I0J9 and atom configurations were obtained sequentially. Given an atom arrangement, we generate the next one by trying to interchange each A1 atom in the simulation cell with a randomly selected Si atom. Such atom interchanges are accepted if Udirf < 0, where &iff = Ut - Ui is the energy difference between the atom configurations after and before the interchange. On the other hand, if &iff > 0, the atom interchange is accepted only if the quantity exp(-&n/kBr) is greater than a random number between 0 and 1 (kB, Boltzmann constant; T, temperature). The ordering of Si and A1 atoms is characterized here by means of several order parameters. With this purpose, for each tetrahedral site i, we define the variable ut, which takes the value
(4)
SIgives the correlation between nearest-neighbor (nn) T-atoms and&, between second neighbors (sn). In our context, the angle brackets mean an average over the T-atoms of each configuration and over the configurations generated at a given temperature. The minimum possible value for S1 is -1, which is attained when each A1 atom is surrounded by four Si atoms and vice versa. If each atom in the network is surrounded on average by two A1 and two Si atoms, then S1is zero. Sz has a similar behavior, reaching its maximum (+1) when all second neighbors of a T-atom are of the same kind as the reference atom (ui = uj for all pairs i, J in eq 4). In the ultramarine framework, one can define two sublattices of tetrahedral sites, hereafter called TI and T2, in such a way that each TI-site is surrounded by four Tz-sites and vice versa. This division is useful to measure the long-range ordering of the atom distribution by means of the parameter L defined as
Here, the angle brackets mean averages over the sites of each sub-lattice TI and T2, and the bars indicate absolute value. The extreme values of L are 1 when all T-atoms of the same type (A1 or Si) occupy the same sublattice, T I or Tz, and zero when one finds on average the same number of A1 and Si atoms in both sublattices. Whenever L is around 1, there is long-range order in the atom distribution, while this atom ordering disappears when L = 0. Another quantity that will be employed to characterize the Si, A1 distribution is the average number of A1-O-A1 linkages per A1 antisite, which will be denoted by y. We call A1 antisites the A1 atoms located on thesilicon sublattice, and similarly,Si antisites are Si atoms on the A1 sublattice. For our purpose, these two kinds of antisites are equivalent, since for equal mole fractions of A1 and Si, each A1 antisite implies the existenceof a Si antisite, and the number of AI-0-A1 groups equals the number of SiM i linkages for any atom distribution. For this reason, in the following we will focus only on the A1 antisites, whose number, NA, is related with the long-range order parameter L by the expression
N A = ‘/4NT( 1 - L) where NT is the number of tetrahedral sites per simulation cell. The transition between a long-rangeorder pattern and an atom arrangement with only short-rangeorder can be also characterized by the fluctuationsof severalquantities, e.g., the order parameters or the lattice energy. In the canonical ensemble employed in our MC simulations, the energy fluctuations are related with the heat capacity cDby the expression
(7) where the square fluctuations of the configurational energy along a MC trajectory at temperature T a r e given by
( A m 2 = (V2) - (V)’ (8) We employed the heat capacity calculated by using eq 7 to obtain the configurational entropy(S,) of the system in the temperature range T = 10CL1800K. This wasdonebymeansofanintegration along a reversible path between the ground state at T = 0 and
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Gordillo and Herrero
The Journal of Physical Chemistry, Vol. 97, No. 31, 1993
L
1.2
7 5
I!
I
I
-1.2 0
600
1200
I
------c
I
600
0
1800
i
\
1200
1800
Temperature (K)
Temperature (K) Figure 1. Long-range order parameter L (full circles) and its mean fluctuation AL (full squares)vs temperature. A horizontalarrowindicatas the value of AL corresponding to a random T-atom distribution in our
finite simulation cell.
Figure 2. Temperature dependence of the short-rangeorder parameters SI(full line) and S2 (dashed line),derived from Monte Carlosimulations. e,
c)
v1
the studied state at temperature T, using the well-known thermodynamic expression
(9)
Y
c ‘ ; J 3
;I: n cr
2
hi l
where is the configurational entropy of the ground state, in which A1 and Si occupy different sublattices. Making use of the computed entropy and the lattice energy, we evaluated the configurational Helmholtz free energy (F,)of the system at temperature T as Fc = U - TS,. -0
3. Results The temperature dependence of the long-range order parameter
L obtained from our MC simulations is shown in Figure 1 (black circles). For increasing temperature, one finds a sharp decrease of L from nearly 1 (sublattice ordering) to 0 (absence of long range ordering) at about 750 K, revealing the presence of a welldefined order4isorder transition at this temperature. This means that at high temperatures, Si and A1 atoms occupy both sublattices T1and T2 with the same probability; Le., the space group associated to the crystal structure of the material will change from P43n (low-temperature state; two sublattices crystallographically different) to Z43m.’ In Figure 1, we present also the average fluctuations of L,given by AL = ((Lz) - (L)2)1/2(full squares). This quantity presents a rather narrow peak at about the same temperature as the L decrease. Note that in the high temperature region, ALdoes not go to zero. However, one expects AL = 0 for a random atom distribution on an infinite lattice, and the nonzero value of AL in the high-temperature tail obtained here is caused by the finite size of the simulation cell. For a random atom distribution in a cell containing NT sites, one has AL = N T - I / ~ , ~which O in our case gives AL = 0.056 (arrow in Figure 1). Figure 2 shows the temperature dependenceof the short-range order parameters S1 (full line) and S 2 (dashed line). In the lowtemperature region, these pair correlation functions take their extreme values -1 and 1, respectively, as expected for strict alternation of Si and A1 atoms over the framework. Near the transition temperature one observes an increase (decrease) in the value of S1 (&), although not so sharp as that found in the case of the long-rangeorder parameter L. At high temperatures, these correlation functions approach zero asymptotically, which is the value corresponding to a random Si, A1 distribution. The order parameter SIis closely related to the number of A l U A I linkages
500 750
1000
1250
1500 1750
Temperature (K) Figure 3. Number of A1-O-AI groups per A1 antisite in the ultramarine framework (y), as a function of temperature. Note that for T > 750 K the term A1 antisite is only conceptual,sinceboth sublattiwarc equivalent. per simulation cell, Np. In fact, from eq 3 and the definition of ui one has
Np = ‘/2N, (SI+ 1) (10) The last structural parameter we present here is the ratio y between the number of A1-O-A1 groups and that of A1 antisites, which is shown in Figure 3 versus the temperature. Since the number of A1 antisites, NA, is related to the long-range order parameter L (see eq 6), the ratio y = N p / N ~ will give us valuable information about the Si, A1 distribution. The plot starts at T = 500 K, the temperature at which one has on average one A1 antisite in the simulation cell. At 500 K, we find y = 3.4, and this parameter decreases slowly until T 700 K, where a fast fall to a minimum at T 750 K is observed. At this temperature, the y vs Tcurve apparently suffers a sudden change of slope, and goes up continuouslyfor T > 750 K toward the value corresponding to a random atom distribution (y = 2). The behavior of the curve in the high-temperature region can be understood by taking into account that, above the order-disorder transition temperature, both sublattices, T1 and Tz are equally occupied (L= 0), and NA means that in this temperature region y = 4Np/ = N T / ~ This . NT;Le., y is a linear function of Npwith SlOpe4/N~.Remembering the relation between Npand SIgiven by eq 10, one has y = 2(S1 + l), which agrees with the behavior of S1 and y shown for T > 750 K in Figures 2 and 3 . We now turn to the thermodynamic properties of the atom distribution. In Figure 4 we present the configurational energy per unit cell (12 T-sites) versus the temperature. We take as zero energy that of the ground state. The long-range ordering causes an appreciable stabilization in the network, since the
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The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8313
Atom Ordering in Ultramarines
0
1200
600
0
I800
Temperature (K) Figure 4. Configurational energy of theSi, Al distributionvstemperature. The dashed line indicates the energy of a random atom distribution. Energies are given for a unit cell of ultramarine,containing 12 tetrahedral sites. The energy of the ground state (long-rangeorder pattem) is taken
600
1200
I800
Temperature (K) Figure 6. Heat capacity versus simulation temperature. Data points were obtained from energy fluctuations by means of cq 7.
as zero. I
s
u o 0
1
2
3
4
5
6
A1-0-AI groups per unit cell Figure 5. Energy e necessary to introduce an A1-O-Al group in the ultramarine framework as a function of the number of A1-O-A1 linkages already present. Data points are average values obtained along MC runs at different temperatures. A vertical arrow indicates the value correspondingto the transition temperature,which coincideswith the minimum
of
c.
difference between the energies at 900 and 600 K, above and below the transition temperature, respectively, is about 30 kJ/ mol. However, the energy change is not so sharp as the change of the parameter L in that temperature region. For higher temperatures, the energy goes slowly to that of a random distribution (dashed line). At 1800 K we find that the configurational energy is 14.5 kJ/mol lower than that corresponding to a random atom distribution,indicatingthat short-rangeordering plays an important role in the energeticstabilization of the system, even at such high temperatures. In connection with the configurational energy of the atom distribution, an interesting quantity is the energy necessary to create an A1-O-A1 group in the ultramarine framework, which will be called e. For a given Si, A1 arrangement, we define e as the energy required for the endothermic process 2Si-0-A1 ---c A1-O-A1 + Si-oSi. This energy depends on the actual atom distribution and, in particular, on the number of A1-O-A1 groups per unit cell, np, present in the framework, as shown in Figure 5 . The e vs np curve was obtained from the change of the configurational energy versus the number of A1-O-A1 linkages for the equilibrium atom distributions in the range T = 1002500 K. For low np values, e decreases with increasing np, and reaches a minimum at np = 2.3, the number of A1-O-A1 groups per unit cell found at the order-disorder transition temperature, T,(arrow in Figure 5 ) . For np > 2.3, which corresponds to the
I , / , , , , I 0
600
1200
1800
Temperature (K) Figure 7. Temperature dependence of the configurational entropy associated with the Si, Al distribution. A dashed line indicates the entropy of a random atom distribution. The entropy values are normalized to a unit cell containing 12 tetrahedral sites. temperature region of only short-range order, e grows up and finally reaches the value e = 12.2 kJ/mol for a random atom distribution, where np = 6 (one has on average 6 A1-O-AI, 6 Si-OSi, and 12 Si-0-A1 linkages per unit cell). Another thermodynamicindicationof the existence of an orderdisorder transition in the Si, A1 distribution is given by the temperature dependence of the heat capacity, c, (Figure 6). The sharp peak around 750 K is in agreement with a change from long-rangeto short-range ordering at this temperature. In Figure 7,we display the configurational entropy of the system versus the temperature, calculated from the MC simulations by means of eq 9 (full line). S,presents a big increase around T,, as expected for the disappearance of the sublattice ordering at this temperature. The dashed line in this figure shows the configurational entropy of a random atom distribution, obtained from the expression S,= ke In Q,where Q is the number of distinguishable atom configurations of the system (Q = NT!/[(NT/~)!]~). Note that at temperatures as high as 1800 K the thermal energy is not yet enough to yield a random distribution, and the dashed and continuous lines are still separated by about 5 J/K mol. Figure 8 shows the temperature dependence of the Helmholtz free energy, F,,calculated by using the configurational energy and the entropy presented above (closed circles). The free energy of a completely ordered Si, A1 distribution is taken as zero and is displayed as a horizontal dashed line. For comparison,the free energy corresponding to a random atom distribution is also presented as a function of temperature (dashed4otted line). At low temperatures, the free energy derived from our MC simulations is that of a totally ordered atom arrangement, and for
Gordillo and Herrero
8314 The Journal of Physical Chemistry, Vol. 97, No. 31, 1993
0
600
I200
I800
Temperature (K) Figure 8. Configurational free energy versus temperature. Data points were obtained from MC simulationsby the quation F, = CJ- TS,.Dashed
and dashed-dotted lines represent the free energy for a long-range-ordered and a random atom distribution, respectively. These two straight lines cross at a temperature of about 950 K.
Figure 9. Sketch of the ultramarineframework,showing different typs of defects appearing in the long-range-orderedatom distribution at low temperatures. Oxygen atoms, located between T atoms, are not shown. White and black circles represent silicon and aluminum atoms in their corresponding sublattices. Open squares and crossed circles symbolize A1and Si antisites, respectively. Four kinds of defects are shown: Those labeled A and B contain one A1 antisite, whereas types C and D include two A1 antisites (open squares).
temperatures higher than Tc,it approaches asymptotically to the line of the random distribution. The transition temperature Tc can be roughly estimated from the crossing point of the dashed and dashed-dotted lines presented in Figure 8, although the resulting value ( T 950 K) is higher than that obtained by more precise ways (maxima of c, and AL).
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4. Discussion
The structural parameters computed from the MC simulations give us a picture of the details of the substitutional disorder in the ultramarine framework. Our results indicate that a t low temperatures, the main feature of the Si, A1 distribution is the tendency to avoid atoms of the same type in adjacent tetrahedra, as observed also for other aluminosilicate structures.13J4 For increasing temperatures, different kinds of point defects appear in the atom distribution. As noted above, at T 500 K, one finds on averageoneA1 antisite in the simulation cell. In principle, in our MC simulations each A1 atom can jump to any site of the Si sublattice to create four AI-0-A1 groups (Figure 9; defect B), but it is energetically more favorable when an A1 atom interchanges its position with a nearest-neighbor silicon atom with the consequent creation of three AI-O-A1 linkages (Figure 9; defect A). The energies associated with these defects are 43.4 kJ/mol in case B and 3 1.9 kJ/mol in case A, Le., a difference of 2.8 kgT
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at 500 K. In order to understand the value of the ratio y found in the MC simulations it is necessary to take into account the degeneracy of each type of defects. If the atom jump takes place only between nearest neighbors, there are, in the ground state, four accessible configurations per A1 atom, while if an A1 atom interchanges its position with a further Si atom, one has N ~ / 2 - 4 possible configurations. This makes that, despite the large energy difference between these two types of defects, they are present in the framework with a concentration ratio A:B equal to 1:2.9 at 500 K. With these defect concentrations, one expects a ratio y = 3.7, which is slightly higher than the value found from the MC simulations (y = 3.4 f 0.2). This indicates that, a t this temperature, atom configurations with two or more antisites already appear. As shown in Figure 3, for higher temperatures the ratio y decreases slowly, until a temperature of about 700 K. At T 600 K, this parameter falls below 3, which indicates that part of the A1 antisites are disposed in arrangements like defects C or D in Figure 9 that contain a number of AI-0-A1 groups per A1 antisite lower than 3. Note that the energy required to create one of these defects is lower than that necessary to obtain two independent A1 antisites (36.3 kJ/mol for type C and 50.7 kJ/ mol for defect D, vs 63.8 kJ/mol for two distant A1 antisites of type A). From around 700 K up to the disappearance of long-range ordering, y shows a sharp decrease, coinciding with an increase in the number of antisites, which reaches its maximum value at Tc (NA= NT/4 for L = 0). This decrease in y can be understood if each A1 antisite tends to be surrounded by four Si antisites. This implies the appearance of ordered domains of antisites in the framework, with A1-O-A1 linkages only in their borders. Thus, the ratio y decreases as the ratio between the surface of the domains and their volume, and for big domains, it should tend to zero at the order-disorder transition. The minimum value of y obtained here (yhn > 0) is a consequence of the finite size of our simulation cell. For larger cell sizes, one expects the ratio y to approach to zero at Tc. Another aspect of the atom distribution that can be explained by the appearance of domains below Tcis the behavior of the energy E versus np, shown in Figure 5. When one has big domains, the A1-O-A1 groups are mainly located in the walls between domains. This causes a stabilization of the system, since the lattice energy is lower than that corresponding to the same number of A1-O-A1 linkages disposed at random over the network. This is the reason why the E vs np curve presents a minimum at the number of A1-0-A1 groups corresponding to the transition temperature. For T > Tc these domains are broken, in agreement with the behavior of the parameter y, which shows a continuous increase above the transition temperature (Figure 3). At very high temperatures ( T > 1700 K), the atom distribution is close to random and the energy required to create an A1-O-A1 linkage reaches the value of 12.2 kJ/mol at 1800 K. The correlation function SI gives us information about the number of AI-0-A1 groups in the unit cell as a function of temperature (see eq 10). It indicates that at temperatures characteristic of pyrolytic synthesis (T > 1000 K) the atom distribution does not follow Loewenstein’s rule, because S1 is far from thevalue associated to a perfectly ordered distribution (-1). This is in agreement with the results of Klinowski et al.4 for pyrolytically-synthesized blue ultramarine with Si/AI = 1.1 1, whose 29SiNMR spectrum indicates that the T-atom distribution is nearly random. This violation of Loewenstein’s rule can be explained from our MC simulations by the high thermal energy at the synthesis temperature, which is enough to overcome the effective repulsion between atoms of the same type (Si or Ai) in contiguous tetrahedra, thus allowing the presence of high concentrations of A1-O-A1 and S i - M i linkages. Apart from theinfluenceofthesynthesis temperatureon the atomdistribution, the chromophore ions present in the cavities of the ultramarine
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Atom Ordering in Ultramarines
The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8315
structure alsocondition theSi, AI arrangement on the tetrahedral sites. The Coulomb interaction between T-atoms favors the AlM i groups versus A1-O-A1 and Si-oSi linkages,14but the S3polarization energy is reduced by increasing the number of A1 atoms in contiguous tetrahedra. This energy term makes that the transition temperature T, be slightly lower than when it is neglected. Thus, the presence of the big chromophore ions favors the disorder in the Si, A1 distribution, as already discussed elsewhere.9 A final point that is not addressed in this paper is the question of the order of the phase transition discussed here. The absence of hysteresis in the computed structural and thermodynamic variables for increasing and decreasingtemperature runs, suggests that this is a second-order transition.21 Moreover, preliminary results on the critical exponents, obtained by finite-size scaling from MC simulationsof the Si, Al distribution for eight different cell sizes up to 432 T-sites, indicate that this orderdisorder transition belongs to the same universalityclass as the Ising model of ferromagnetism in three dimensions.22.23 These qualitative results require more precise calculations and this point will be addressed in further simulation work. On the other side, the experimental evidence for this orderaisorder transition is indirect, since it is based on the observed change of space group depending on the synthesis temperature, as indicated above. It would be interesting to analyzethis problem experimentally by spectroscopic and diffraction techniques to check whether this temperaturedriven transition in fact occurs and to determine precisely the temperature T,. 5. Conclusions
MC simulations allow us to present a microscopic description of the problem of substitutional disorder in aluminosilicate structures. Our simulation results for blue ultramarine with Si/ A1 = 1 show the way in which the sublattice ordering found at low temperatures is broken for increasing T. First, there appear point defects associated with antisites, which become ordered domains of antisites for T > 600 K. At T 750 K,these domains are large enough to break the sublattice ordering, and both sublattices of T-sites are crystallographicallyequivalent for higher temperatures, One expects thus at low T the space group P43n for the crystal structure of these compounds, versus I43m at high T,in line with experimental results. The value of the transition
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temperature presented here depends on the atomic charges employed in the calculation, and in particular, on the difference 6 = qsi - qd. Changes of this parameter would modify the value of T,,but the orderdisorder picture discussed here would remain unaffected. Acknowledgment. We thank R.Ramirez for assistance with the computer facilities, and L. Utrera for critical reading of the manuscript. This work was supported by CICYT (Spain) under Contract MAT91-0394. M.C.G. thanks the Ministerio de Educaci6n y Ciencia (Spain) for a postgraduate fellowship. References and Notes (1) Wells, A. F. Structural Inorganic Chemistry, 4th ed.; Clarendon Press: Oxford, 1975. (2) Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harper and Row: New York. 1983. (3) Tarling, S.E.; Barnes, P.; Klinowski, J. Acta Crystallogr., Sect. B 1988,844, 128. (4) Klinowski, J.; Carr, S.W.; Tarling, S.E.; Barnes, P. Nature 1987, 330, 56. ( 5 ) Hassan, I.; Peterson, R. C.; Grundy, H. D. Acta Crystallogr., Sect. C 1988, C41, 827. (6) Engelhardt, G.; Michel, D.High Resolution Solid Stare NMR of Silicates and Zeolites; Wiley: New York, 1987. (7) Klinowski, J. Prog. Nucl. Magn. Reson. Spectrosc. 1984, 16, 237. (8) Loewenstein, W. Am. Mineral. 1954, 39, 92. (9) Gordillo, M. C.; Herrero, C. P. Chem. Phys. Lett. 1992, 200, 424. (10) Binder, K.; Heermann, D. W. Monte Carlo Simulation in Statistical Physics; Springer-Verlag: Berlin, 1988. (1 1) Herrero, C. P.; Utrera, L.; Ramirez, R. Phys. Reo. B 1992,46,787. (12) Herrero, C. P.; Utrera, L.; Ramirez, R. Chem. Phys. Lett. 1991,183, 199. (13) Herrero, C. P. J. Phys. Chem. 1993, 97, 3338. (14) Herrero, C. P.; Ramirez, R. J . Phys. Chem. 1992, 96, 2246. (15) Ooms, G.; van Santen, R. A.; den Ouden, C. J. J.; Jackson, R. A.; Catlow, C. R. A.; J. Phys. Chem. 1988,92,4462. (16) Miller, T. M.; Bederson, B. Ado. At. Mol. Phys. 1978, 15, 1. (17) httel, C. Introduction to Solid Stare Physics, 6th ed.; Wiley: New York, 1986, (18) Kirkpatrick, S.;Gelatt, C. D., Jr.; Vecchi, M.P. Science 1983, 220, 671. (19) Heermann, D. W. Computer simulation methods; Springer: Berlin, 1986. (20) Herrero, C. P. J. Phys.: Condens. Matter 1993, in press. 121) Landau. D. P.: Binder. K. Phvs. Reo. B 1985. 31. 5946. (22j Barber,M. N.;Peanod, R. B.;?oussaint, D.;Richardson, J. L. Phys. Rev. B 1985, 32, 1720. (23) Creswick, R. J.; Farach, H. A.; Poole. C. P.. Jr. Introduction to Renorkaliration Group Methods in Physics; Wiley: New York, 1992.
