Modelling chemical ordering in aluminosilicates and
subsequent prediction of cation immobilisation efficiency
John L. Provis, Peter Duxson, Grant C. Lukey and Jannie S. J. van Deventer *
Department of Chemical and Biomolecular Engineering
The University of Melbourne
Victoria 3010
AUSTRALIA
Abstract
A statistical thermodynamic model is developed for description of aluminium-silicon
ordering in aluminosilicate materials, with particular reference to geopolymeric and
glassy materials. The ‘Loewenstein aluminium avoidance principle’ is examined, and
found not to hold strictly in these materials. These findings, in agreement with NMR
data on glasses and natural and synthetic minerals and with quantum chemical
calculations, show that the tendency for formation of Si-O-Al linkages in preference to
Si-O-Si and Al-O-Al, leading to a significant degree of chemical ordering in these
materials, is based on energetic considerations. The model outlined in this paper is
developed by minimisation of the Gibbs energy of mixing in an amorphous
aluminosilicate phase. Predictions of the degree of ordering in a range of geopolymeric
and glassy materials are compared to quantitative data obtained from solid-state NMR.
Formation of Al-O-Al bonds will prove very important in the utilisation of both
geopolymerisation and vitrification as methods of immobilising toxic or radioactive
metals in cationic form. These bonds are significantly weaker than Si-O-Al bonds with
respect to the energy required to remove an Al atom from the material structure, and so
any degree of Al-O-Al formation will be significantly detrimental to immobilisation
efficiency over a prolonged period. Analysis of the relative degrees of Al-O-Al
formation observed in geopolymers and in aluminosilicate glasses suggests that, on this
basis and in addition to significant advantages in processing, geopolymerisation will in
many instances prove a more effective method of waste immobilisation than
vitrification.
Keywords: Geopolymer, aluminosilicates, short-range ordering, immobilisation,
cations
*
Author to whom correspondence should be addressed: jannie@unimelb.edu.au – Phone +61 3
8344 6619 – Fax +61 3 8344 7707
1. Introduction
Geopolymers are a class of high-performance mineral binders obtained by the reaction
of an aluminosilicates powder with alkali metal hydroxide or silicate solutions at
ambient or slightly elevated temperature (Davidovits 1991). The composition of the
geopolymeric binder is represented as (SiO2)x(MAlO2)1-x, where M is Na+ and/or K+,
and 0.52<x<0.70 for the samples investigated here. Geopolymers are believed to have a
primarily gel-like structure, are essentially X-ray amorphous and thus have no clearly
defined network topology. Recently, 29Si MAS-NMR spectra of geopolymers with a
range of compositions have been presented (Duxson et al. 2005). Deconvolution of
these spectra provides meaningful information regarding the distribution of the
tetrahedral silicon and aluminium sites forming the geopolymeric gel network structure
(Duxson et al. 2005), as has also been observed in aluminosilicate glasses (Lee &
Stebbins 1999) and minerals (Phillips et al. 1992).
Geopolymers have been identified as having significant potential in the area of
radioactive waste treatment (Davidovits & Comrie 1988, Khalil & Merz 1994, Bao et
al. 2003, Chervonnyi & Chervonnaya 2003), as they are able to strongly encapsulate a
range of cations. In particular, immobilisation of Cs + and Sr2+ is of significance in the
control and cleanup of contaminated sites and wastewater. These cations can each be
strongly bound into the geopolymeric matrix structure, which is therefore a much more
stable form of immobilisation than the encapsulation or vitrification methods currently
in use (Siemer 2002). Therefore, understanding of the degree of chemical ordering in
geopolymers, which will play a significant role in determining long-term chemical
stability of these systems, is critical in the widespread utilisation.
The notation introduced by Engelhardt et al. (1974) will be followed in this work, with
Qn(mAl) (0 ≤ m ≤ n ≤ 4) representing a silicate centre bonded to n other centres, of
which m are aluminium and (n-m) are silicon. Similarly, qn(m’Al) (0 ≤ m’ ≤ n ≤ 4) will
be used to represent an aluminium centre coordinated to n other centres, of which m are
aluminium and (n-m) are silicon. The vast majority of tetrahedral sites within a
geopolymeric network have been observed by NMR to have cross-link density n = 4
(Rahier et al. 1996, Duxson et al. 2005). Therefore, n = 4 will be treated as constant
throughout this investigation.
2. Model Development
The standard starting point in analysis of short-range order in aluminosilicates is
Loewenstein’s rule (Loewenstein 1954), which states that no two aluminium ions can
occupy the centres of tetrahedra linked by one oxygen. This ‘rule’ is often assumed to
be obeyed strictly in aluminosilicate structures. However, there is no firm
thermodynamic basis for strict application of aluminium avoidance, but rather an
energetic preference towards avoidance of Al-O-Al bond formation. Theoretical
calculations have shown that this tendency is due primarily to the exothermicity of the
reaction (1) both in solution and in the solid state.
 Si - O - Si 
  Al - O - Al    2  Si - O - Al 
(1)
Free energy minimization considerations may therefore be used to interpret the
observed tendency towards Al-O-Al avoidance in aluminosilicate structures. The energy
penalty associated with formation of Al-O-Al bonds in solution is believed to be
responsible for the observed short-range ordering in hydrothermally synthesised
aluminosilicates (Catlow et al. 1996). Increasing temperature has been shown to
increase the extent of disorder (Gordillo & Herrero 1992, Lee & Stebbins 2000), and
this must also be reflected in any model formulation.
The basic formulation of the model presented here was originally applied by
Efstathiadis et al.(1992) to ordering within an amorphous Si-C-H alloy. Here, the model
will be presented on a 2-component basis, describing the ordering of tetrahedral SiO 4/2
and AlO-4/2 centres within an aluminosilicate network. The choice of this particular
model was largely determined by the fact that all other existing models for Al-Si
ordering rely on an understanding of the network topology. This is currently unavailable
for geopolymeric materials due largely to their X-ray amorphous nature, meaning that
more complex models cannot yet be applied to this system.
As was noted previously, the energetic basis of chemical ordering in geopolymers is due
primarily to the exothermicity of Equation (1). The heat released by this reaction is
dependent on several factors including the cations present and the positions of the
centres within the network structure. This energy is therefore treated as a model
parameter denoted and defined by Equation (2), where E(I-O-J) represents the energy
of a bond linking tetrahedral centres I and J
 = 2E(Si - O - Al) - E(Si - O - Si) - E(Al - O - Al)
(2)
 is known to be relatively independent of x (Bosenick et al. 2001). However,  will
depend on the nature of the charge-balancing cations present, and this will be
considered in the application of the model to the different geopolymers under
investigation. Next-nearest neighbour (NNN) effects have been calculated to contribute
approximately ±20% to the value of  (Palin et al. 2001). However, incorporation of
this effect causes significant complications in modelling and it is therefore often
neglected. For the sake of simplicity, NNN effects will not be considered explicitly in
this model, but rather will be discussed later as a potential source of error.
The Gibbs free energy of mixing (network formation), G M, may be expressed in terms
of the entropy and enthalpy of mixing:
GM = HM – TSM
(3)
The entropy of mixing is given by the Boltzmann equation, Equation (4):
SM = kB ln 
(4)
where  is the number of distinct states the system may occupy.  is then given
approximately by SiAl, where I is the number of possible configurations of
tetrahedral sites containing framework cation I (Onabe 1982, Efstathiadis et al. 1992).
Due to the equivalency of each of the 4 bonds linked to a given tetrahedral site,
Equations (5) and (6) for Si and Al respectively may be obtained. These expressions
relate to a system containing N tetrahedral sites, N(Si) = xN of which are silicon and
N(Al) = (1-x)N are aluminium, with NI,J defined as the concentration of centres of type I
bonded via I-O-J bonds to sites of type J (Efstathiadis et al. 1992). It must also be noted
that NAl,Si = NSi,Al .
 N Si ! 

Si  

 N Si,Si ! N Si, Al ! 
4
 N Al ! 

Al  

 N Al,Al ! N Si, Al ! 
(5)
4
(6)
Substituting Equations (5) and (6) into Equation (4) and simplifying by use of Stirling’s
approximation, Equation (7) is obtained. This expression is given in terms of the
‘normalised bond concentrations’ nI,J of each type of bond (Efstathiadis et al. 1992),
where nI,J = NI,J/N(Si), and is asymptotically correct in the limit of large N(Si).
1  x  x 

S M  4k B N Si  nSi,Si ln nSi,Si   nAl,Al ln nAl,Al   2nSi,Al ln nSi,Al  
ln 

x
 1  x 

(7)
The enthalpy of mixing for a system of N atoms may be expressed in terms of heats of
formation of free tetrahedra H0(I) and bond energies E(I-O-J) by Equation (8).
1 x
H 0  MAlO 4 / 2   2nSi ,Si E Si  O  Si   4nSi , Al E Si  O  Al 
x
 2n Al , Al E  Al  O  Al ]
(8)
H M  N Si  [H 0 SiO 4 / 2  
Equations (3), (7) and (8) may then be combined to give an expression for G M in terms
of the normalised bond concentrations. Differentiation with respect to nSi,Si, with the
observation that n Al , Al  1 and nSi, Al  1 , then allows the use of Equation (2) to
n Si,Si
n Si,Si
replace the combination of E(I-O-J) terms from Equation (8) by . Setting G M  0
nSi,Si
and simplifying then gives Equation (9).
nSi2 , Al
nSi,Si n Al , Al
  

 exp 
 2k BT 
(9)
Combining Equation (9) with a mole balance (Equations (10) and (11)) and solving
algebraically, Equation (12) is obtained. This is an analytical expression for the degree
of Si-O-Al bonding in terms of x, , and T. A geopolymer synthesis temperature of
40°C is used throughout this work.
nSi,Si + nSi,Al = 1
(10)
nSi,Al + nAl,Al = 1  x
x
(11)
nSi,Al 
  
1
 
exp 
x
 2k BT 
2
 1


  
   
   2  exp 
  2   4 exp 
  1
 x

 2k BT 
 2k BT  




   
  1
2 exp 
 2k BT  

(12)
Equations (10)-(12) may then be used to calculate the total number of each type of bond
present in the system for any given x, and . This allows the calculation of the
geopolymer composition in terms of tetrahedral centres Q 4(mAl) and q4(mAl).
Application of a random bond distribution model (ie. neglecting NNN effects) then
allows the calculation of FSi(m) and FAl(m), the fraction of all tetrahedra present that are
Q4(mAl) and q4(mAl) respectively, as presented in Table 1. The coefficients 1,4,6,4,1
are due to the equivalency of all possible different arrangements of m Al and 4-m Si
centres around a given tetrahedral site, giving each value of m a degeneracy factor
according to the binomial distribution (Yin & Smith 1991).
Table 1. Tetrahedron fractions calculated from random bond distribution.
Si sites
FSi(m)
Al sites
FAl(m)
Q4(0Al)
xnSi4 ,Si
q4(0Al)
x4
n4
3 Si , Al
1  x 
Q4(1Al)
4 xnSi3 ,Si nSi, Al
q4(1Al)
4
x4
n3 n
1  x 3 Si, Al Al , Al
Q4(2Al)
6 xnSi2 ,Si nSi2 , Al
q4(2Al)
6
x4
n2 n2
1  x 3 Si, Al Al , Al
Q4(3Al)
4 xnSi,Si nSi3 , Al
q4(3Al)
4
x4
n n3
1  x 3 Si, Al Al , Al
Q4(4Al)
xnSi4 ,Al
q4(4Al)
x4
n4
1  x 3 Al , Al
Only silicon sites are directly observable by 29Si MAS NMR, so the calculated quantity
available for comparison to experimental data will be F Si(m)/x, the fraction of the silicon
present that is coordinated to m aluminium sites.
In application of this model to the metakaolin-based geopolymer system, a correction
for the unreacted material present must be made. Metakaolin has a layered structure,
and so tends to preferentially release aluminium in the early stages of dissolution as the
relatively weak Al-O-Al bonds are broken. Therefore, the remnant unreacted material,
which is estimated to comprise approximately 10% of the total metakaolin added, will
be somewhat dealuminated with respect to its original composition. An empirical
correction assuming 90% reaction and a remnant metakaolin composition of x = 0.600
was therefore applied to the experimental data, and provides a very good fit of the
model to the experimental data. The application of the model to the data of Duxson et
al. (2005) is shown by Figures 1-3, for Na, K, and mixed Na/K systems respectively.
Percentage of Si sites
The energy penalty  was found to be 40 ± 5 kJ/mol for Na systems and 30 ± 2 kJ/mol
for K systems. The mixed (1:1) Na/K system was found to be adequately described
simply by taking the mean value of the two endmembers, 35 kJ/mol.
100%
Q4(0Al)
90%
Q4(1Al)
80%
Q4(2Al)
70%
Q4(3Al)
Q4(4Al)
60%
50%
40%
30%
20%
10%
0%
Exp Model
0.535
Exp Model
0.589
Exp Model
0.632
Exp Model
0.669
Exp Model
0.699
xcorr
Percentage of Si sites
Figure 1. Comparison of model predictions to the data of Duxson et al. (2005) with compositions
corrected for 10% unreacted metakaolin. Na geopolymers,  = 40 kJ/mol.
100%
Q4(0Al)
90%
Q4(1Al)
80%
Q4(2Al)
70%
Q4(3Al)
Q4(4Al)
60%
50%
40%
30%
20%
10%
0%
Exp Model
0.535
Exp Model
0.589
Exp Model
0.632
xcorr
Exp Model
0.669
Exp Model
0.699
Figure 2. Comparison of model predictions to the data of Duxson et al. (2005) with compositions
corrected for 10% unreacted metakaolin. K geopolymers,  = 30 kJ/mol.
Percentage of Si sites
100%
Q4(0Al)
90%
80%
Q4(1Al)
Q4(2Al)
70%
Q4(3Al)
Q4(4Al)
60%
50%
40%
30%
20%
10%
0%
Exp Model
0.535
Exp Model
0.589
Exp Model
0.632
xcorr
Exp Model
0.669
Exp Model
0.699
Figure 3. Comparison of model predictions to the data of Duxson et al. (2005) with compositions
corrected for 10% unreacted metakaolin. Mixed (1:1) Na/K geopolymers,  = 35 kJ/mol
Table 2 provides a comparison between the predicted degree of Al-O-Al bond formation
for geopolymers with x = 0.50 and the experimentally observed proportion of Al-O-Al
bonds formed in glassy aluminosilicates of comparable composition. It can be seen from
these data that the proportion of the relatively unstable Al-O-Al bonds in geopolymeric
materials is much lower than is observed in glasses of similar composition. This
expected also to be true at the higher Si/Al ratios which are of primary interest in a
practical sense, but the difficulties inherent in experimentally identifying Al-O-Al bonds
in high-Si systems mean that comparisons are most readily conducted in systems of
relatively high Al content.
Table 2. Proportions of Al-O-Al bonds present in different amorphous aluminosilicates
Geopolymer composition (T = 40°C)
KAlSiO4
Na0.5K0.5AlSiO4
NaAlSiO4
Glass composition (Tg ~ 780°C)
NaAlSiO4
LiAlSiO4
Fraction Al-O-Al (Model predictions)
2.65 %
1.67 %
1.05 %
Fraction Al-O-Al (Experimental, Lee & Stebbins 2000)
9.5 ± 0.5 %
10.6 ± 0.5 %
It is also seen that the use of sodium as the charge-balancing cation within a
geopolymeric structure gives the lowest extent of Al-O-Al formation of any system
investigated. This may mean that sodium-containing geopolymers are the most
appropriate for use in waste immobilisation, which is advantageous considering the high
levels of sodium present in many low-level radioactive waste streams (Siemer 2002).
However, much further work remains to be done in this area, particularly with regard to
the relative reactivities and leachabilities of different alkali metal cations within the
geopolymeric structure.
3. Conclusions
A statistical thermodynamic model describing the ordering of tetrahedral Si and Al
centres within an X-ray amorphous geopolymeric binder has been formulated, and
shown to fit experimental data for metakaolin-based geopolymers. The need to
minimise the presence of relatively unstable Al-O-Al linkages in materials used in
radioactive waste immobilisation to prevent leaching of immobilised cations suggests
that geopolymers may be a valuable tool in the treatment of radioactive waste streams,
in preference to vitrification or simple encapsulation methods.
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Acknowledgements
This work was funded through the Particulate Fluids Processing Centre, a Special
Research Centre of the Australian Research Council.