American Mineralogist, Volume 96, pages 125–137, 2011
First-principles calculation of the elastic moduli of sheet silicates and their application to
shale anisotropy
B. Militzer,1,2,* H.-R. Wenk,1 S. Stackhouse,1 and L. Stixrude3
1
Department of Earth and Planetary Science, University of California, Berkeley, California 94720, U.S.A.
2
Department of Astronomy, University of California, Berkeley, California 94720, U.S.A.
3
Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K.
Abstract
The full elastic tensors of the sheet silicates muscovite, illite-smectite, kaolinite, dickite, and nacrite
have been derived with first-principles calculations based on density functional theory. For muscovite,
there is excellent agreement between calculated properties and experimental results. The influence
of cation disorder was investigated and found to be minimal. On the other hand, stacking disorder is
found to be of some relevance for kaolin minerals. The corresponding single-crystal seismic wave
velocities were also derived for each phase. These revealed that kaolin minerals exhibit a distinct
type of seismic anisotropy, which we relate to hydrogen bonding. The elastic properties of a shale
aggregate was predicted by averaging the calculated properties of the contributing mineral phases
over their orientation distributions. Calculated elastic properties display higher stiffness and lower
p-wave anisotropy. The difference is likely due to the presence of oriented flattened pores in natural
samples that are not taken into account in the averaging.
Keywords: Elasticity, clay, ab initio calculations, sheet silicates, seismic anisotropy
Introduction
Sheet silicates are among the minerals with highest elastic
anisotropy. Aggregates containing oriented sheet silicates, such
as schists and shales, also display very high anisotropies and
this has important implications for interpreting seismic travel
times through such rocks. Shales are the most abundant rocks in
sedimentary basins and relevant for petroleum deposits (e.g., Johansen et al. 2004) as well as for carbon sequestration (Chadwick
et al. 2004). Their physical properties are of great importance
in exploration geophysics (e.g., Mavko et al. 1998; Wang et al.
2001). Elastic properties of shales have been measured with
ultrasound techniques (e.g., Hornby 1998; Hornby et al. 1994;
Johnston and Christensen 1995), because of their importance for
seismic prospecting of hydrocarbon deposits (e.g., Banik 1984).
In metals and igneous and metamorphic rocks, anisotropic aggregate properties can be satisfactorily predicted by averaging the
single-crystal elastic properties over the orientation distribution
(Bunge 1985; Barruol and Kern 1996; Mainprice and Humbert
1994). In the case of shale, anisotropy can be calculated with
this method, but absolute values of predicted elastic stiffness
coefficients are over a factor of two higher than the measured
results (Valcke et al. 2006; Voltolini et al. 2009; Wenk et al. 2008).
Currently only empirical models are available to describe the
elasticity of shales that do not rely on measured microstructural
properties (e.g., Bayuk et al. 2007; Ougier-Simoni et al. 2008;
Ponte Castaneda and Willis 1995; Sayers 1994). There are various reasons why the simple averaging schemes are limited in
* E-mail: militzer@berkeley.edu
0003-004X/11/0001–125$05.00/DOI: 10.2138/am.2011.3558
125
the case of shales. One reason is that porosity and grain contacts
are inadequately accounted for. Another reason is the uncertainty
in the single-crystal elastic properties of sheet silicates (Chen
and Evans 2006).
Despite their importance, the single-crystal elastic moduli of
clay minerals have not been measured experimentally, because of
technical difficulties associated with the small grain size. There
is also considerable uncertainty in atomic force microscopy
(Prasad et al. 2002). For illite, elastic constants of the analog
mineral, muscovite, have traditionally been used (Vaughan and
Guggenheim 1986; Zhang et al. 2009) but considerable uncertainty arises from using such analogies. For kaolinite, values
from first-principles calculations exist for the ideal structure
(Sato et al. 2005; Mercier and Le Page 2008). However, the
structure of kaolin minerals in natural samples is expected to
differ considerably from the ideal kaolinite. Moreover, other
equally important sheet silicates have not yet been examined
by experiment or theory, including illite-smectite. In view of
this, in the present work, we have calculated the single-crystal
elastic properties of illite-smectite and the kaolin minerals kaolinite, dickite, and nacrite, which have structures more similar
to those found in nature. In addition, to establish and test our
method, we have computed the elastic properties of muscovite,
for which analogous experimental data exists. The computed
elastic properties are then used to predict the elastic properties
of the classic Kimmeridge Clay shale (age 151–156 Ma) from
Hornby (1998) for which experimental elastic properties are
available, as well as orientation distributions of component
phases (Wenk et al. 2010).
126
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
→
Computational methods
We used the Vienna ab initio simulation package (VASP) (Kresse and Hafner
1993; Kresse and Furthmüller 1996) to perform density functional calculations
using either the local density approximation (LDA) or the Perdew-Burke-Ernzerhof
(Perdew et al. 1996) generalized gradient approximation (GGA) in combination
with the projector augmented-wave method (Blöchl 1994). The wave functions of
the valence electrons were expanded in a plane-wave basis with an energy cut-off
of 586 eV or higher and the Brillouin zone was sampled using 4 × 2 × 2 and 6 ×
4 × 4 k-points grids (Monkhorst and Pack 1976) depending on the size of the unit
cell. All structures were relaxed until all forces were <10–3 eV/A.
To calculate the elastic constants we used the finite strain method (Karki et al.
↔
→
1997, 1997b, 2001). The lattice vectors of the strained unit cell, (a→′, b ′, →
c ′) ≡ A′,
↔
↔
↔
→
↔ ↔
were obtained from the unstrained cell, (a→, b , →
c ) ≡ A, using A′ = (1 + ε ) A, where
↔
↔
1 is the unit matrix and ε is the strain tensor. The diagonal and off-diagonal strain
tensors were of the form (Ravindran et al. 1998)

  δ 0 0
ε1 =  0 0 0

 0 0 0

 ;



 0 0
0
 
ε4 =  0 0 δ / 2

 0 δ / 2 0

 ,



where the indices are given in Voigt notation. The remaining strain tensors are
obtained by index permutations. For each strain, we performed two calculations
using δ = ± 0.005 and determined the corresponding stresses. The elastic constants
were then determined from simple stress-strain relations. Consistent results were
also obtained for other values of δ. Mainprice et al. (2008) performed similar
calculations for talc.
The error associated with our computed elastic constants, due to controlled
approximations, is estimated to be <±2 GPa, with the major part of this associated
with the careful, but slightly imperfect, structural relaxation and use of a finite
strain. In addition to this, we must also take into account the uncontrolled approximation of adopting a specific exchange-correlation functional. While LDA
and GGA have been shown to predict many properties of a variety of materials
with remarkable accuracy, neither approach is perfect. LDA tends to overbind the
0
sheets in sheet silicates, and zero-pressure densities predicted with LDA (ρP=
LDA)
are too high (by 3% for muscovite). In contrast, GGA tends to underbind the
P=0
sheets, and zero-pressure densities predicted with GGA (ρGG
A) are too low (by
5% for muscovite). Since the computed elastic constants are affected by changes
in density, we can improve the accuracy of the predicted constants by adopting
the experimental density (ρPEX=0P) and relaxing the structure at fixed volume. In this
approach it is important to account for uncertainties in the volume, structure, and
composition of the experimental sample and differences between it and the model
system. By adopting the experimental value of the volume, we remove one of the
main sources of error in elastic constants computed with DFT.
A similar strategy was adopted by White et al. (2009) who studied the geometry
of ideal kaolinite, comparing predictions for several different functionals, but with
the lattice parameters fixed at the experimental values. Following the suggestion
of one of the referees, we investigated the difference between the elastic constants
calculated with LDA, keeping all lattice parameters fixed at their experimental
value, and using the cell parameters obtained by optimizing the cell at the experimental density. The resulting elastic constants were found to be very similar, which
is not surprising, because the lattice parameters obtained from optimization at the
experimental volume agreed well with the experimental values.
Imposing the experimental density affects the interlayer separations, but leads
to little change in the geometry of the sheet itself. In clays, bonding between layers
is rather weak, which is the reason why it is difficult to describe it with density
functional methods. Consequently, at the zero pressure densities, we find significant
differences in the LDA and GGA predictions of elastic behavior in the direction
perpendicular to the sheets, while that within the sheet is less affected. Karki et
al. (2001) have shown that LDA leads to good agreement with experiment for a
wide variety of mantle phases. In view of this, while we will compare LDA and
GGA results in the paper, our preferred elastic constants come from LDA at ρPEX=0P.
It should also be noted that we compute isothermal elastic constants at 0 K,
whereas experimental studies determine adiabatic elastic constants at room temperature. However we expect the difference between the two to be small and direct
comparison between theoretical and experimental values to be reasonable.
Optimized structures, for the different phases, are available as CIF files at
the Inorganic Structural Database. To facilitate comparison with previous results,
elastic constants are reported for two crystallographic settings: (1) The conventional
setting (e.g., Standards on Piezoelectric Crystals 1949; Nye 1959; Matthies and
→
→
→
→ →
Wenk 2009) with an orthogonal coordinate system defined by Z||c
, Y||c
×a , and X
→ →
→
= Y×Z and (2) a setting where the Z is perpendicular to the tetrahedral planes (001)
→
→ →
→ → → → →
in both monoclinic and triclinic crystal structures: X||a , Z||a ×b , and Y = Z×X. The
latter choice preserves the direction of C33 when comparing different stackings in
kaolinite polytypes that have different →
c axes.
Model systems
Muscovite
Muscovite KAl2[AlSi3O10](OH)2 is a 2:1 sheet silicate, with
each layer comprising an octahedral sheet fused between two
tetrahedral sheets (Fig. 1). Due to isomorphic substitution, it
exhibits structural disorder in the tetrahedral sheets (Collins and
Catlow 1992), with 25% of sites occupied by Al3+ cations and
75% by Si4+ cations. This leads to the layers having a net negative charge, which is balanced by K+ anions residing between
the layers.
Since it is possible that cation disorder will affect the elastic
properties of the phase, we performed first-principles calculations
to determine the most favorable arrangements.
Isomorphic substitution of Si4+ by Al3+, in the tetrahedral
sheets, creates a local negative charge in the region of the Al3+
cation. It is therefore reasonable to assume that Al3+ cations
within the same tetrahedral layer, will tend to repel one another.
Our calculations support this. For an 84 atom unit cell, placing
two Al3+ cations in the same tetrahedral sheet increases the energy
by 1–2 eV, while placing all four Al3+ cations in the same sheet
increases it by 7 eV. Calculations for heterogeneous distributions
in larger super cells showed a similar trend. This provides a strong
argument for having similar Al3+ cation concentrations in each
layer, to reduce unfavorable interactions between them.
If one considers the conventional unit cell, assuming an equal
Al3+ concentration in each layer, there is only one Al3+ cation to
distribute among four sites. Every site is energetically equivalent
and leads to the formation of a Si4+-Al3+ chain parallel to the b
lattice vector (Fig. 2a). However, a more favorable cation distribution is obtained if one considers a simulation cell comprising two unit cells (Fig. 2b). This allows Al3+ ions to be placed
across the silicate rings and thus decrease Al3+-Al3+ interactions.
Monte Carlo simulations using larger super cells, with up to 36
substitution sites, found the same arrangement of cations to be
most favorable. In these calculations the optimization criteria was
to maximize the distance between the Al3+ sites or alternatively
minimize the Ewald energy.
However, doubling the size of the simulation cell would
make first-principles calculations of the elastic constants of
muscovite prohibitively expensive. We consequently used the
configuration shown in Figure 2a for all calculations of elastic
properties. Even for this configuration, there is still considerable
structural freedom, because one can choose a different site for
the Al3+ cation in each layer. However the energy differences
arising from such permutations is relatively small (~0.07 eV per
unit cell) compared to that possible with different concentrations
in the layers (1–2 eV), because of the increased separation and
screening of intermediate layers.
Out of the 44 ways to distribute four Al3+ ions equally among
the four layers, we selected eleven configurations that either
had a very low Ewald energy or the Al3+ ion separations were
especially large. For those eleven geometries, we relaxed the
MILITZEr ET AL.: AB InITIO CALCULATIOn OF ELASTIC MODULI OF SHEET SILICATES
127
P=0
atomic positions and unit cell with GGA at ρGG
A. Six of the
eleven relaxed structures were low in energy, within 0.07 eV per
unit cell, which is a range that one would expect for a disordered
mineral. For all six structures, we obtained very similar elastic
constants, reported in Table 1. This leads us to conclude that
cation disorder has very little impact on the elastic properties of
muscovite. The largest deviations are seen in model 4 (Fig. 2),
which has lower elastic constants mainly because the relaxation
at zero-pressure led to a slightly lower density.
Illite-smectite
Various structural models have been proposed for illitesmectite minerals (Moore and reynolds 1989; Sakharov et al.
1999; Plançon et al. 1985; Drits et al. 1994; Watanabe 1988).
They contain mixed layers of illite and swelling layers of smectite
that cause systematic peak shifts in X-ray powder patterns. The
interstratification of illite and smectite changes from random
alternation to full segregation. Contrary to muscovite, illitesmectites exhibit a wide range of compositions with significant
na, Ca, and Mg interlayers. Their exploration goes beyond the
scope of this work; instead we adopted an idealized structural
model put forth by Stixrude and Peacor (2002).
Table 1. Elastic constants of muscovite calculated
at zero pressure
→ →
with GGA, in the conventional setting, Z ||c
Model
1
2
3
4
5
6
0
0.007
0.016
0.042
0.054
0.072
E-E1 (eV)
2.683
2.688
2.675
2.690
2.689
ρ (g/cm3) 2.682
172.7
172.8
173.1
167.9
169.7
170.6
C11
166.7
166.9
167.7
158.6
167.0
165.6
C22
54.8
54.8
55.1
53.0
55.0
54.7
C33
14.2
14.3
14.7
9.4
14.9
15.2
C44
17.2
17.3
17.4
15.5
16.8
16.5
C55
67.6
67.6
67.8
66.5
67.5
67.5
C66
48.8
48.8
49.2
43.8
48.0
48.6
C12
20.1
20.2
20.4
17.7
20.6
20.5
C13
–17.7
–17.7
–17.6
–18.4
–17.3
–16.7
C15
17.4
17.5
17.9
14.2
18.0
18.5
C23
–1.7
–1.7
–1.7
–4.2
–1.8
–1.5
C25
–3.3
–3.2
–3.2
–4.9
–3.0
–2.5
C35
–4.4
–4.4
–4.5
–5.0
–4.1
–4.0
C46
Notes: Each column corresponds to a model with a different tetrahedral Al3+/Si4+
cation arrangement. The energy differences with respect to model 1 are also
given on line 2 in eV per 84 atom unit cell. One can see that cation disorder has
little effect on the elastic properties of the phase.
Figure 1. Muscovite. Comparison of Al3+ disorder models 1 and
4 reported in Table 1. The light and dark tetrahedra represent the Al3+
and Si4+ cation locations, respectively. The structure has been extended
beyond the unit cell (marked by thin lines) for clarity but the Al3+:Si4+
ratio is the same in both structures. The spheres denote K+ ions and the
octahedra mark the remaining Al3+ cations.
Figure 2. Sheets of tetrahedral cations in muscovite. The dark and
light spheres indicate the Al3+ and Si4+ ions, respectively. The left panel
shows the lowest energy configuration for a 1:3 Al:Si ratio within the
conventional unit cell (rectangular box). On the right, the lowest energy
configuration among all unit cells.
128
MILITZEr ET AL.: AB InITIO CALCULATIOn OF ELASTIC MODULI OF SHEET SILICATES
Illite-smectites are similar in structure to muscovite, in that
they are composed of aluminosilicate layers, formed from an
octahedral sheet fused between two tetrahedral sheets (2:1 sheet
silicate; Fig. 3), however, the arrangement of cations in the tetrahedral layers is quite distinct. Just as one would assume from their
name, illite-smectites are interstratified minerals that comprise
alternate layers of illite and smectite. In the present work, we
investigate one of the simplest illite-smectites known as rectorite,
where the illite layer has a structure similar to muscovite and
the smectite layer has the pyrophyllite structure. Two types of
interstratification have been considered by Stixrude and Peacor
(2002): layer centered (model A) and interlayer centered (model
B). In model A, alternate layers of illite and smectite, identical to
those found in the mineral end-members, are stacked in sequence.
However in model B, each layer is identical and comprises one
aluminum-poor and one aluminum-rich tetrahedral sheet; the updown alternation of these layers leads to two different interlayer
environments: one enclosed by low-charge tetrahedral sheets
and the other by high-charge tetrahedral sheets, where potassium
anions reside. Theoretical calculations have shown that interlayer
centered interstratification is more favorable than layer centered
models (Stixrude and Peacor 2002). For our calculations of elastic
properties we adopted the interlayer centered structure model B
of Stixrude and Peacor (2002), with a chemical composition of
K0.5Al2[Al0.5Si3.5]O10(OH)2, which corresponds to a 1:1 ratio of
illite and smectite, with each layer possessing one aluminum-free
tetrahedral sheet and one where 25% of sites are filled by Al3+
cations. This model differs from the natural smectite mineral
rectorite (e.g., Zhang et al. 2009) in that all interlayer cations
are assumed to be potassium, as oppose to a mixture of sodium,
calcium, and potassium and no interlayer water is included.
Kaolin minerals
Kaolinite is a 1:1 sheet silicate, i.e., tetrahedral layers are only
on one side of the octahedral sheet. Its ideal structural form is
triclinic with two Al2Si2O5(OH)4 formula units per unit cell and
a simple 1M stacking sequence (Bish 1993; neder et al. 1999)
that is shown in Figures 3 and 4. However, natural kaolinites, in
sedimentary rocks exhibit considerable stacking polytypism and
disorder (Bailey 1963, 1988; Kogure et al. 2002, 2005). Stacking polytypism of hydrous sheet silicates has been analyzed in
detail by Bailey et al. (1988) who identified 12 standard types.
Stacking polytypism in kaolin minerals has also been investigated
by Zvyagin and Drits (1996). Artioli et al. (1995) went beyond
the 12 standard types by developing a random stacking model,
which significantly improved agreement with powder diffraction
patterns of natural kaolinites. Dera et al. (2003) studied pressureinduced changes in the stacking behavior. A comprehensive
recent review of the stacking polytypism in kaolin minerals
is given by Mercier and Le Page (2008), who also provide an
extensive set of structural data derived from GGA calculations.
Sato et al. (2004) and Balan et al. (2005) have also investigated
structural details with ab initio calculations.
In this paper, we begin our calculations of elastic constants
with the ideal kaolinite structure and compare with the work by
Sato et al. (2005). To estimate the effects of stacking polytypism we then compare these with values for three polytypes of
kaolinite that have two layers in their unit cell: dickite, nacrite,
and an alternate kaolinite structure.
Dickite exhibits a 2M stacking sequence, where the vacancy
in octahedral site alternates between two sites leading to a
monoclinic two-layer supercell (Bailey et al. 1988). Such an
arrangement leads to increased interlayer binding. According
to our DFT calculations, it lowers the energy by 0.01 eV per
formula unit relative to the ideal kaolinite structure. For our second example of a two-layer mineral, we chose to look at nacrite
because it occurs in nature and its structure has been determined
◄Figure 3. (left) Illite-smectite
model with a chemical composition
of K 0.5Al 2[Al 0.5Si 3.5]O 10(OH) 2, which
corresponds to a 1:1 illite-smectite ratio.
Each layer is identical and comprises
one aluminum-free tetrahedral sheet
and one where 25% of sites are filled
by Al3+ cations. The up-down alternation
of these layers leads to two different
interlayer environments, one enclosed
by low-charge tetrahedral sheets and the
other by high-charge tetrahedral sheets,
where potassium anions reside. (right)
Ideal kaolinite structure where the unit
cell has been doubled in →
c direction.
notation is the same as Figure 1.
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
129
a
b
c
d
Figure 4. Stacking polytypism in ideal kaolinite (a), dickite (b), nacrite (c), and an additional structure (2M*, d). Only the Si4+ ions (dark)
and the Al3+ (light) are shown for simplicity. The view direction is perpendicular to the sheets and a shading has been added to the lower Si4+ layer.
The thin lines denote the unit cell.
experimentally (Zheng and Bailey 1994; Zhukhlistov 2008). As
our final example of two-layer stacking, we generated a new
structure (labeled 2M* throughout this paper) by doubling the
unit cell of ideal kaolinite, rotating the new layer by 120° and
shifting it by a/3. We selected this particular structure, among
other possible shifts and rotations, because it shows favorable
interlayer hydrogen bonding, which is confirmed by the computed energy that is again lowered by 0.01 eV per formula unit,
relative to ideal kaolinite.
Nacrite stands out because it has the largest offset between
adjacent layers (Fig. 4), which weakens the hydrogen bonding.
The accurate characterization of hydrogen bonds is difficult
experimentally because hydrogen is a weak X-ray scatterer but
also challenging for density functional theory because of its
weak binding forces. Zheng and Bailey (1994) pointed out that
one of the three OH groups may not participate in the interlaying bonding and that is what we found when we relaxed the
structures with LDA and GGA regardless which experimental
geometry we started from. During the relaxation, the outer OH
group tilts into the plane of the layer, preventing it from forming
a bond with upper layer. Balan et al. (2005) identified this tilt and
explained it as a result of the shorted Si-O bond in nacrite that
was observed theoretically and experimentally. This tilt was not
reported by Sato et al. (2005) but it was investigated by Benco
et al. (2001) who performed finite temperature DFT molecular
dynamics simulations of kaolinite to compare calculated and
measured OH stretching frequencies.
The stacking sequence in all four structures is illustrated in
Figure 4 where, for clarity, we only show the positions of the
tetragonal Si4+ and octahedral Al3+ cations along with the unit cell.
Studying larger structures would lead to prohibitively expensive
first-principles calculations.
Results
Structural parameters
For muscovite, we obtained fairly good agreement with
the lattice parameters determined in X-ray diffraction experi-
130
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
ments (Table 2). Vaughan and Guggenheim (1986) reported: a
= 5.1579(9), b = 8.9505(8), c = 20.071(5) Å, and β = 95.75(2)°
.007
(0)
while with GGA at ρGP=G0A we find a = 5.238+0
–0.003, b = 9.131 –0.014,
+0.0 30
+0.15
c = 20.723 –0.045, and β = 95.56 –0.65, where we give the results for
our lowest energy structure and report the deviations from them,
by the remaining five structures from Table 1 as uncertainty. The
Si-Al disorder has little impact on the structural parameters, as
expected. The disagreement with experiment is largest for the
lattice parameter c because of the relatively weak binding in this
direction. LDA shows better agreement with experiment than
GGA. It should be noted that the sample used by Vaughan and
Guggenheim (1986) exhibited a slightly different composition,
K0.9Al2.8Si3.1O12H2.1 compared to the ideal composition that we
adopted. Computations at ρPEX=0P are consequently performed at
the experimental unit-cell volume rather than the mass density
but we still refer to as ρPEX=0P for consistency. The agreement of
the lattice parameter then becomes very good, confirming that
the effect on ionic substitution in the Vaughan-Guggenheim
sample is small.
For illite-smectite, we cannot directly adopt an experimental
density because no explicit experimental data exist for our idealized rectorite model. We thus estimate the density to be 2.825
g/cm3 by averaging of the experimental values for the unit-cell
volume of muscovite and pyrophyllite (Table 3).
For ideal kaolinite, the calculated lattice parameters in Table
4 are in reasonable agreement with the measurements of Neder et
al. (1999) although the zero-pressure density is underestimated
by 4% with GGA and overestimated by 4% with LDA.
Our structural parameters for dickite also slightly deviate
from the theoretical work of Sato et al. (2004). As seen for ideal
kaolinite in Table 4, we again predict a smaller sheet separation,
but larger lattice parameters within the sheets. Our value for the
angle β of 96.62° is in much better agreement with measured
values of 96.48° by Bish and Johnston (1993) and 96.77° by Dera
et al. (2003) than 104.57° reported by Sato et al. (2004).
In the case of nacrite, the lattice parameters computed with
P=0
GGA at ρGG
A are in reasonable agreement with measurements
by Zhukhlistov et al. (2008): a = 8.910, b = 5.144, c = 14.593
Å, and β = 100.50° and again the density is underestimated by
4%. It should be noted that, relative to kaolinite, the a and b axis
are swapped. The computed energy is 0.01 eV per formula unit
lower than that for ideal kaolinite.
Elastic properties
Table 2 compares our elastic constants calculated for muscovite with LDA and GGA with the experimental results of Vaughan
and Guggenheim (1986). Elastic constants determined with LDA
at ρPLD=0A are stiffer than those calculated at the measured density,
ρPEX=0P, predominantly because LDA overestimates the density.
However, elastic constants from LDA calculations at ρPEX=0P are
in very good agreement with experiment. The interlayer spacing, and consequently stiffness, in the direction perpendicular
to the layers, C33, is most sensitive to changes in density. GGA
yields good agreement with experimental elastic constants, when
P=0
P =0
calculated at ρGG
A, but overestimates values at ρEXP.
The computed elastic constants for our illite-smectite model
are shown in Table 3. Similar predictions are obtained for C11
P=0
and C22 with LDA at ρPLD=0A and GGA at ρGG
A, but the values for
C33 differ substantially (i.e., LDA: 67.9 and GGA: 19.6 GPa).
This reflects the fact that LDA overestimates the density and the
interlayer binding while GGA underestimates both. The elastic
constants from LDA and GGA computations for the estimated
density of 2.825 g/cm3 are also compared in Table 3. We consider
that LDA calculations, e.g., C33 = 27.2 GPa, to be our most accurate prediction for elastic properties of illite-smectite because,
for muscovite, GGA overestimated C33 when the experimental
density was used. In general, it will remain challenging for existing density functional techniques to predict C33 accurately for
weakly bonded sheet silicates in cases where the composition and
structure of the interlayers is variable and poorly defined.
Tables 4 and 5 summarize the computed elastic constant
for the kaolin group minerals. Similar to muscovite, we also
determined values at ρPEX=0P in addition to calculating LDA and
P=0
GGA results at ρPLD=0A and ρGG
A. Comparison of our calculated
elastic constants with those of Sato et al. (2005) reveals sizable
discrepancies that are much larger than one would expect from
Table 2. Elastic→constants
of muscovite model 1 in the conventional set→
ting Z ||c calculated with GGA and LDA at different densities
LDA
LDA
LDA
LDA GGA GGA Experiment
PLDA = 0
(a,b,c,β)EXP V = VEXP V = VEXP PGGA = 0
3
V (Å ) 904.5 917.2
921.9
921.9 921.9
984.3
921.9(3)
a (Å) 5.140 5.150 5.1579 5.154 5.187
5.244
5.1579(9)
b (Å) 8.925 8.945 8.9505 8.952 9.012
9.101
8.9505(8)
c (Å) 19.806 19.998 20.071 20.073 19.818 20.713
20.071(5)
β (°)
95.48 95.44
95.75
95.43 95.61
95.39
95.75(2)
C11
187.5 182.7
180.3
180.9 194.3
170.1 181.0 ± 1.2
C22
178.1 172.3
169.9
170.0 188.0
162.1 178.4 ± 1.3
C33
71.5
62.7
60.1
60.3
91.1
55.6
58.6 ± 0.6
C44
22.1
19.9
19.1
18.4
25.2
14.4
16.5 ± 0.6
C55
28.1
25.0
22.4
23.8
30.5
17.8
19.5 ± 0.5
C66
71.7
71.1
70.6
70.5
71.3
67.6
72.0 ± 0.7
C12
59.8
55.1
53.3
53.4
68.1
47.4
48.8 ± 2.5
C13
34.8
29.1
27.1
27.2
43.2
20.6
25.6 ± 1.5
C15
–15.4 –14.7
–14.3
–14.7 –14.3
–14.1 –14.2 ± 0.8
C23
31.0
25.6
23.6
23.5
39.5
18.2
21.2 ± 1.8
C25
1.8
1.5
2.3
1.4
1.7
–0.8
1.1 ± 3.7
C35
–0.7
–1.0
0.6
–1.0
1.1
0.0
1.0 ± 0.6
C46
–0.7
–1.9
–1.8
–1.8
–0.6
–3.3
–5.2 ± 0.9
Note: Those calculated with LDA at the experimental unit-cell volume in column
4 (bold) are in best agreement with the experimental values of Vaughan and
Guggenheim (1986) in the last column.
Table 3. Elastic
constants of illite-smectite in the conventional set→ →
ting Z ||c
LDA PLDA = 0
LDA
ρ (g/cm3)
2.923
2.8247 a (Å)
5.130
5.156
b (Å)
8.861
8.905
c (Å)
19.738
20.205
α (°)
89.99
89.71
β (°)
106.08
105.94
γ (°)
89.95
89.94
C11
165.5
153.9
C22
197.8
188.5
C33
67.9
27.2
C44
18.6
10.4
C55
35.9
24.8
C66
56.3
55.4
C12
31.3
25.1
C13
25.8
13.2
C15
–23.5
–30.3
C23
18.8
5.2
C25
–6.9
–8.2
C35
–3.2
–5.4
C46
–15.6
–15.9
Note: Our most reliable results are in bold.
GGA
2.8247
5.197
8.964
19.943
89.92
106.25
89.96
169.1
202.8
82.3
10.5
37.0
56.4
37.4
33.4
–24.4
24.9
–6.9
–0.2
–16.7
GGA PGGA = 0
2.600
5.259
9.066
21.116
90.02
105.74
89.95
150.0
191.2
19.6
6.9
19.1
54.3
32.0
16.9
–36.0
8.3
–9.6
–1.9
–15.0
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
two independent first-principles calculations (Table 4). Sato et
al. (2005) report larger values for C11 and C22, while C33 is less
than half of our value.
Discussion
Comparison of our elastic constants for muscovite from
LDA calculations at ρPEX=0P with those for illite-smectite indicate
that the former is harder than the latter, in qualitative agreement
with a recent nanoindentation investigation of the two phases
by Zhang et al. (2009). The Young’s elastic modulus normal to
the basal plane of muscovite derived from our calculated elastic
constants is 54.7 GPa. This is much lower than the value of 79.3
GPa obtained by Zhang et al. (2009), but in good agreement with
values found in earlier investigations, e.g., 58.6 GPa (Vaughan
and Guggenheim 1986) and 60.9 GPa (McNeil and Grimsditch
1993). For illite-smectite we estimate a value of 26.0 GPa,
which is larger than the value of 18.3 GPa obtained by Zhang
et al. (2009). Zhang et al. (2009) suggest that their indenter tip
geometry could have caused their values to be overestimated,
which would explain their larger value for muscovite. For illitesmetcite, where our value is even larger than that of Zhang et al.
(2009), we attribute the difference to the fact that their experimental sample was hydrated, whereas we calculated values for
an anhydrous system.
Following similar arguments to those of Zhang et al. (2009)
we suggest that the differences in the elastic properties of muscovite and illite-smectite can be related to variations in their crystal
structures. In muscovite, all tetrahedral layers have a net negative
charge and potassium cations are present in each interlayer. The
Table 4. Calculated elastic
constants for ideal kaolinite in the conven→ →
tional setting, Z ||c
Sato et al.
This This This This Experiment
(2004, 2005) work
work
work
work by Neder et
al. (1999)
GGA
GGA
GGA LDA LDA PGGA = 0 ρ = ρEXP ρ = ρEXP PLDA = 0
ρ (g/cm3)
2.544
2.506
2.599
2.599
2.711
2.599
a (Å)
5.1445
5.225
5.184
5.179
5.127
5.154
b (Å)
8.9241
9.071
8.999
8.993
8.900
8.942
c (Å)
7.5873
7.464
7.326
7.325
7.182
7.401
α (°)
91.089
91.44
91.73
91.46
91.79
91.69
β (°)
104.60
104.67 105.04 104.66 105.08
104.61
γ (°)
89.869
89.77
89.81
89.74
89.80
89.82
C11
178.5 ± 8.8
166.0
164.1
169.1
169.2
C22
200.9 ± 12.8
177.8
175.5
179.7
178.4
C33
32.1 ± 2.0
70.1
119.3
81.1
149.8
C44
11.2 ± 5.6
13.4
15.3
17.0
19.8
C55
22.2 ± 1.4
21.7
25.2
26.6
29.7
C66
60.1 ± 3.2
56.7
57.4
57.6
58.4
C12
71.5 ± 7.1
64.8
61.2
66.1
65.4
C13
2.0 ± 5.3
16.0
35.1
15.4
41.5
C14
–0.4 ± 2.1
0.0
–0.6
–0.4
–0.2
C15
–41.7 ± 1.4
–37.2
–30.5
–34.0
–28.4
C16
–2.3 ± 1.7
–7.2
–7.7
–7.8
–8.9
C23
–2.9 ± 5.7
11.2
27.2
10.2
33.0
C24
–2.8 ± 2.7
–3.3
–4.7
–3.4
–5.3
C25
–19.8 ± 0.6
–16.2
–11.6
–16.1
–10.8
C26
1.9 ± 1.5
–0.8
–0.3
–0.1
–0.3
C34
–0.2 ± 1.4
–1.3
–0.8
–2.9
–1.2
C35
1.7 ± 1.8
1.3
14.2
6.7
17.0
C36
3.4 ± 2.2
0.2
0.8
–0.1
0.5
C45
–1.2 ± 1.2
0.1
0.1
–0.7
–0.2
C46
–12.9 ± 2.4
–14.2
–13.9
–12.4
–13.0
C56
0.8 ± 0.7
0.6
0.5
1.1
1.1
Note: The experimental density is 2.599 g/cm3. Our most reliable results are
in bold.
131
coulombic attraction between the negative layers and positive
cations, gives rise to strong interlayer bonding. In illite-smectite,
only half of the tetrahedral layers have a net negative charge and
potassium cations are only present in alternate interlayers, which
leads to weaker interlayer bonding. This explains, for example,
why C33 is 50% smaller for illite-smectite than for muscovite.
Differences in other elastic constants are either related to this or
the different concentration and arrangement of defects.
Our results for muscovite and illite-smectite form a quantitative basis for estimating the elastic moduli of the wide variety of
K-dominated di-octahedral sheet silicate compositions that are
expected to occur naturally. We propose that the elastic moduli
will scale with the mean inter-layer charge: Z = +1 in the case of
muscovite, and Z = +0.5 in the case of illite-smectite. Smectites
have values Z < 0.5 and illites have 0.5 < Z < 1.0. Other potentially
important effects that remain to be investigated include the role of
intra-layer cation radius (e.g., Na substitution for K) and the role
of inter-layer water. The dominant role of inter-layer charge and
local charge balance was also emphasized by Stixrude and Peacor
(2002) in their study of the structure of illite-smectite.
The computed elastic constants for kaolinite, dickite, and
nacrite are summarized in Tables 4–6. These are provided in the
→ → → → →
→ →
→
conventional (Z||c
,Y||c ×a , and X = Y×Z) as well as in the more
→
→ →
→ → → → →
intuitive (X||a ,Z||a ×b, and Y = Z×X) orientation. The latter has the
→
advantage that the Z axis is always oriented perpendicular to the
basal planes and the elastic constants can be compared directly. In
→
the conventional setting, the Z axis changes each time a different
stacking is introduced. Differences in elastic constants between
various structures may, in this setting, be partially due to differTable 5. Calculated elastic constants→for different kaolin materials using
→
the conventional setting, Z ||c
Dickite Dickite Nacrite Nacrite Kaolinite Kaolinite
GGA LDA GGA LDA
model 2M* model 2M*
PGGA = 0 ρ = ρEXP PGGA = 0 ρ = ρEXP GGA PGGA = 0 LDA ρ = ρEXP
ρ (g/cm3) 2.505
2.583
2.494
2.607
2.503
2.599
Exp. ρ (g/cm3) 2.583
2.583
2.607
2.607
2.599
2.599
a (Å)
5.221
5.179
9.020
8.910
5.235
5.188
b (Å)
9.077
9.010
5.221
5.165
9.054
8.973
c (Å)
14.545 14.323 14.881 14.571
14.582
14.297
α (°)
90 90
90
90
96.73
96.81
β (°)
96.72
96.62 101.11 101.24
93.32
93.31
γ (°)
90
90
90
90
89.88
89.86
C11
181.1
184.2
147.6
131.8
181.9
184.7
C22
178.6
178.8
160.8
157.9
176.9
185.0
C33
78.6
67.5
64.8
75.0
70.3
81.4
C44
16.9
15.8
7.8
13.2
12.6
16.7
C55
15.5
17.1
8.8
17.7
13.0
16.7
C66
60.6
60.4
62.4
62.2
57.0
57.0
C12
67.7
69.1
45.5
41.5
63.6
65.6
C13
10.8
6.0
5.7
10.2
5.9
6.6
C14
0.0
0.0
0.0
0.6
–7.5
–7.0
C15
–20.3
–17.8
4.4
–16.0
–6.5
-6.9
C16
0.0
0.0
0.0
0.1
–4.6
–4.5
C23
8.0
2.5
12.0
14.0
8.6
9.3
C24
0.0
0.0
0.0
0.4
–20.1
–17.9
C25
–8.6
–6.3
2.8
–3.8
–2.3
–2.9
C26
0.0
0.0
0.0
0.1
–1.8
–0.9
C34
0.0
0.0
0.0
0.5
3.7
3.2
C35
–2.0
5.8
0.2
9.3
4.8
4.6
C36
0.0
0.0
0.0
0.1
1.2
1.3
C45
0.0
–0.0
0.0
–0.5
2.4
0.8
C46
–6.2
–4.8
0.6
–9.4
–3.7
–3.2
C56
0.0
–0.0
0.0
–0.1
–5.4
–5.1
Notes: The densities used in the calculations and the experimental values are
listed. Our most reliable results are in bold.
132
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
ences in orientation. Instead of comparing elastic constants, it is
preferable to study pole figures, where a change in orientation only
leads to a rotation of the contours. When compared in the same
setting (Table 6), the diagonal elastic constants of ideal kaolinite
agree very well with those of dickite and our 2M* structure. Only
C33 shows some deviation. Kaolinite deviates from muscovite
and illite-smectite because it is triclinic which leads to additional
non-zero constants, however, most of them are very small with
the exception of C16 = −7.8 GPa. Nacrite deviates slightly by
exhibiting a softer response, smaller elastic constants, which is
a result of the weaker interlayer hydrogen bonding.
The measured and calculated elastic constants of muscovite
lead to very similar predictions for seismic velocities, as the
spherical P wave velocity surfaces in Figure 5 show. The differences are analyzed in more detail in Figure 6 where P- and
S-wave speeds are plotted for different propagation directions.
→
→
→
The direction is changed continuously from Z to X to Y and back
→
to Z using equal angular increments. vP varies between 4.5 km/s
in the slow direction perpendicular to the sheets to about 8.3 km/s
within the sheets.
The deviations between experiment and model 1 are no larger
than 0.5 km/s, with experimental velocities slightly lower than the
calculated one. The results also agree with recent experimental
studies of muscovite elasticity based on Brillouin scattering (McNeil and Grimsditch 1993) and nano-indentation (Zhang et al.
2009). Just as one expects, splitting of the shear waves is greatest
→
→
in the X and Y directions, where the polarization is within and
→
perpendicular to the layers, respectively and lowest in the Z direction, where both polarizations lie within the plane of the layers.
vP varies between 4.5 km/s in the slow direction perpendicular
to the sheets to about 8.3 km/s within the sheets.
P-wave velocity surfaces for illite-smectite model B from LDA
and GGA are shown in Figure 5. The wave speeds parallel to the
sheets are very similar to those of muscovite but perpendicular
P-velocities are much lower than those for muscovite. The maxi→
mum in P-velocity of about 8.2 km/s (LDA) occurs along the b
direction. The minimum of 2.9 km/s occurs along an axis that is
tilted by 15° away from vertical.
While Sato et al. do not clearly specify the orientation of their
reported elastic constants, Figure 5 confirms that there is genuine
disagreement because the velocity isosurfaces differ substantially
in shape while a difference in the crystal orientation would only
lead to a rotation of the isosurfaces. The results of Sato et al.
(2005) predict the structure to be much more anisotropic, with
vP varying between 3.5 and 9.0 km/s, while we predict variations
between 4.9 and 8.8 km/s (Fig. 7). Both sets of calculations in
Table 4 are based on the Perdew-Berke-Ernzerhof (Perdew et al.
1996) generalized gradient approximation, but use different codes
and pseudopotentials. While Sato et al. used the CASTEP program
and ultrasoft pseudopotentials with a 340 eV plane wave energy
cut-off, we used the VASP code and the projector augmentedwave method with a 586 eV cut-off or higher. The latter is more
reliable than ultrasoft pseudopotentials and is expected to yield
results that are closer to all-electron calculations. We also notice
some differences in the reported structural parameters given in
Table 4 along with the experimental values by Neder et al. (1999).
The angles from both calculations agree well with experiments.
On the other hand, lattice parameters a and b from Sato et al.
(2005) are shorter and agree better with experiment than those of
the present work, while the opposite is true for the c parameter.
This means that our calculations predict a smaller sheet separation, but larger lattice parameters in the planes. Since slowest
wave propagation occurs nearly perpendicular to the planes, the
differences in the structural properties are consistent with Sato
et al. (2005) predicting a higher elastic anisotropy.
One feature in the pole figures in Figure 5 distinguishes
kaolin minerals from the others studied here is that the slowest
propagation does not occur in the direction perpendicular to the
layers. Overall, in-plane propagation is still about 50% faster than
vertical propagation, but there is noticeable asymmetry in the pole
figures. The slowest propagation direction deviates from the plane
vector by 30 ± 1% for ideal kaolinite, dickite and nacrite, as well
as the 2M* structure. It is caused by a sizeable contribution from
off-diagonal elastic constants. There is significant shear stress if
the sample is strained vertically. The shear forces are a result of
the hydrogen bonds that are not aligned vertically.
The results for kaolin minerals in Figure 7 demonstrate shear
bands crossing that gives rise to the ring of S wave degeneracy
in Figure 8. Inside the ring we find a ring of increased splitting
magnitude and another minimum inside of that. Conversely muscovite, which does not have any hydrogen bonds, does not exhibit
this unusual shear band signature. Figures 5 and 6 show very little
shear wave splitting around the vertical direction. This shear band
signature makes kaolinite unique among the shale components
studied here. Figure 7 also shows that the elastic constants of Sato
et al. (2004) imply that the P and both S bands are entangled but
we have no data to support such findings.
While the properties of hydrogen bonds have been analyzed
in different experimental (e.g., Neder et al. 1999; Dera et al.
2003) and theoretical (e.g., Sato et al. 2004; White et al. 2009)
studies, we show here that they create a distinct signature in
kaolin minerals that can be measured in laboratory experiments
and seismic observations.
Table 6. Elastic constants from Tables 4 and 5 computed
with LDA→at
→ → → → →
the
experimental density in the setting X ||a ,Z ||a ×b, and
Y=
→ →
→
Z→×X
with the exception
of→nacrite
where we exchanged X and
→
→ → → → →
→
Y (Y ||a , Z ||a ×b, and X = Y ×Z ) to align it in the same way as
kaolinite and dickite where the principle
axis of the hexagonal
→
lattice in the sheets is parallel to the X direction
C11
C22
C33
C44
C55
C66
C12
C13
C14
C15
C16
C23
C24
C25
C26
C34
C35
C36
C45
C46
C56
Ideal kaolinite
187.4
179.8
83.6
13.7
16.0
61.1
70.5
4.8
–0.5
0.2
–7.5
6.0
0.4
–0.3
0.8
–4.3
–2.6
0.7
–0.2
–0.9
0.2
Dickite
188.4
178.8
69.3
15.2
14.1
60.9
69.7
3.0
0.0
–0.1
0.0
2.0
0.0
1.5
0.0
0.0
1.9
0.0
0.0
0.4
0.0
Nacrite
136.9
157.9
79.3
11.5
13.0
63.9
42.0
5.5
0.8
3.6
–0.0
13.6
0.5
1.7
–0.0
0.4
1.1
–0.0
–0.5
0.6
0.0
Kaolinite model 2M*
185.3
189.3
82.5
14.2
15.3
58.0
66.6
5.0
–0.0
1.1
–3.7
6.8
0.0
0.5
0.3
–2.1
2.0
0.1
–0.4
–0.9
–0.4
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
133
8
7.5
7
6.5
6
5.5
5
4.5
km/s
Elastic wave speed (km/s)
Figure 5. P wave velocity surfaces derived from elastic constants in Tables 2–5: Muscovite computed using model 1 and experimental results
from Vaughan and Guggenheim (1986), illite-smectite comparison of LDA and GGA for ρ = 2.825 g/cm3, ideal kaolinite from Sato et al. and this
→
work, and finally our results for dickite, nacrite and our 2M* stacking model, all calculated at ρPEX=0P using LDA. Equal area projections along the Z
→ →
= [001] direction in the conventional setting, Z||c , is used throughout.
8
7
Experiment
Model 1
vp
6
5
vs1
4
vs2
3
2
Z
X
Y
Propagation direction
Z
Figure 6. Elastic wave velocities for muscovite disorder model 1
(Fig. 2) from Table 2 column 4 and experimental results from Vaughan
and Guggenheim (1986) as function of propagation direction using the
→ →
conventional setting, Z||c .
Application to shale anisotropy
An important application of improved single-crystal elastic
properties of clay minerals is to polycrystal averages for shales.
Shales are complex rocks, rich in sheet silicates. To estimate the
contribution of the matrix to elastic properties, and particularly
elastic anisotropy, we need to know volume fractions of constituent minerals, their orientation distributions and single-crystal
elastic properties. It has previously been observed that measured
acoustic wave velocities are much lower than those computed
from matrix averaging. Is this difference due to reduced elastic
properties of sheet silicates, such as illite-smectite, or is it caused
by the pore structure that is not taken into account for the averaging? This was the motivation behind this study.
To demonstrate the application of the computed single-crystal
properties to natural shales, we have chosen a well-characterized
shale, composed mainly of detrital and authigenic illite, kaolinite,
and quartz, with minor plagioclase, pyrite, and chlorite. The shale
is of Kimmeridgian age (156–151 Ma) and from a drill core in
the North Sea. Elastic properties were determined as a function
of pressure with ultrasonic methods (Hornby 1998). The same
sample was recently reinvestigated with synchrotron X-ray diffraction to quantify orientation distributions of component phases
(Wenk et al. 2010). The Kimmeridge Clay shale shows strong
preferred orientation of kaolinite, illite, and illite-smectite, and
a nearly random orientation distribution for quartz. The pole
figures in Figure 9 display nearly axial symmetry of orientation
distributions. Therefore, strict axial symmetry was imposed in the
following property calculations (transverse isotropy), reducing
the number of independent components of the aggregate elastic
tensor to five: C11 = C22, C33, C12 = C11 − 2C66, C13 = C23, C14 =
C55 and all others are zero.
To estimate the elastic properties of the shale, first the elastic
properties of contributing mineral phases were calculated by
averaging their single-crystal values over the mineral orientation
distributions, using a geometric mean (Matthies and Humbert
1993) in the software Beartex (Wenk et al. 1998). For singlecrystal properties we used the LDA values at ρPL=0
DA for illite and
illite-smectite, and for kaolinite the LDA values for kaolinite. For
quartz we used the experimental data of Heyliger et al. (2002).
It should be mentioned that in these texture-related calculations
Elastic wave speed (km/s)
134
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
9
8
7
6
5
4
3
2
Ideal koalinite
This work
Sato et al.
1
0
Z
X
Nacrite
Y
Z
Propagation direction
Figure 7. Elastic wave speeds compared for ideal kaolinite and
nacrite. The upper branch shows vP and the lower two show vS.
a consistent approach must be used for phases with monoclinic
crystal symmetry. Beartex uses the first setting ([001] twofold
axis) and corresponding transformations of lattice parameters,
pole figure indices, and elastic constants are required (Matthies
and Wenk 2009). In this paper we represent data in second
setting ([010] twofold axis), even though all calculations were
done in the first setting. For the bulk aggregate elastic constants
of the contributing phases were averaged, taking into account
corresponding volume fractions.
Table 7 lists aggregate elastic constants of the contributing
phases, as well as the bulk average and experimental data from
Hornby (1998) measured at the maximum confining pressure of
80 MPa. In the bulk average, the porosity, estimated at 2.5%, was
not taken into account. From aggregate elastic properties, P- and
S-wave velocities were calculated and Figure 10 shows P-velocities and shearwave splitting as function of the angle to the bedding plane. Illite and illite-smectite are strongly anisotropic (32
and 30%, respectively), combining strong preferred orientation
and strong single-crystal anisotropy. The anisotropy of kaolinite
is smaller because of reduced single-crystal anisotropy (16%) and
the contribution of quartz to anisotropy is negligible.
Note that using the new single-crystal elastic constants, the
calculated values for the shale aggregate are still considerably
higher than experimental results in Table 7 but the differences are
less than in previous comparisons (Voltolini et al. 2009; Wenk et
al. 2008). Particularly, using the illite-smectite results instead of
muscovite reduces the value of C33 from 85.0 to 70.5 GPa, which
is still much higher than the measured value of 36 GPa.
The discrepancy between measured and computed aggregate
elastic moduli is likely due to the pore/fracture structure of the
rock. There is an extensive literature that discusses the influence
of pores (e.g., Bayuk et al. 2007; Berryman et al. 2002; Mukerji
et al. 1995), fracture distribution (Sayers 1998) and saturation
(Pham et al. 2002) on the elastic properties of shales. The approach of the models is generally empirical since very little is
known about the details of the pore structure and distribution and
does not consider anisotropy. This may change in the future as
micro- and nano-tomography techniques are becoming available
to map the 3D pore structure (e.g., Bleuet et al. 2008; Herman
2009; Kanitpanyacharoen et al. 2011). Recently Matthies (2010)
proposed a self-consistent model based on the theory of Eshelby
(1957) to include penny- or rod-shaped pores in a heterogeneous
anisotropic medium and this could prove valuable for further
Z=[001]
Figure 8. Comparison of shear wave splitting in muscovite and ideal kaolinite. The latter exhibits a ring of nearly perfect degeneracy in shear
wave velocity. The dashes indicate the direction of polarization of the fast shear wave.
135
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
Table 7. Elastic constants for Hornby Kimmeridge shale (in GPa)
Phase
vol%
Illite-mica
26.7
Illite-smectite 45.5
Kaolinite
5.1
Quartz
22.7
Average
Experiment
a
C11
118.2
83.6
121.7
97.3
97.9
56.2
C33
81.2
49
87.3
97.1
70.5
36.4
C44
30.6
22.6
29.1
43.9
29.9
10.3
C12
36.8
20.6
43.6
8.1
23.3
18.4
C13 P velocity anisotropy (%)
33.3
18.7
19.3
26.7
34.5
16.6
7.8
0
21.2
16.5
20.5
21.6
P velocities (km/s)
7
6
5
Quartz
lllite
l/S
Kaolinite
Average
Exp.
4
3
Figure 9. (001) pole figures for detrital illite, illite-smectite,
kaolinite, and quartz in equal-area projections on the bedding plane.
Contours are in multiples of a random distribution.
10
20
30
40
50
60
70
80
90
Angle from bedding normal (deg.)
b
700
600
ds velocities (m/s)
quantifying elastic properties of shales. The important role of
porosity in the elasticity of the experimental sample is further
emphasized by its variation with confining pressure, which reduces the anisotropy substantially due to closing of pores and/
or cracks (Hornby 1998).
Even without a quantitative model the comparison of
measured and calculated elastic properties reveals important
information about the pore structure in Kimmeridge Clay shale.
Since the relative difference between computed and measured
aggregate moduli is much greater for C33 than for C11, pores must
have very anisotropic shapes and alignment. The pattern suggests
pores that are flattened and aligned in the bedding plane. The
difference between computed and measured Cij could be used in
principle to invert for the properties of a simple pore geometry
model, such as one based on penny-shaped cracks, although this
is beyond the scope of the present study. In the future we plan to
investigate this shale by micro-tomography and use this information to average over both pores and matrix minerals. Combined
with our new values of single-crystal elastic moduli we hope to
obtain more reliable estimates of shale elasticity.
In summary, we have calculated the single-crystal elastic
properties of the layered silicates muscovite, illite-smectite,
kaolinite, dickite, and nacrite, from density functional theory.
Our results suggest that, for dioctahedral sheet silicates, cation
disorder has little effect on the elastic properties of individual
aluminosilicate layers. For muscovite calculated elastic properties are in excellent agreement with experiments. The elastic
properties of naturally occurring clays depend on the degree and
type of isomorphic substitution and interlayer cations, since it
controls the strength of interlayer bonding. Those with higher
mean inter-layer charge (e.g., illites) are expected to be stiffer
then those with lower inter-layer charge (e.g., illite-smectites).
In a similar manner, in the case of kaolin minerals, stacking
arrangements that result in strong interlayer hydrogen bonding
0
500
400
300
Quartz
lllite
l/S
Kaolinite
Average
Exp.
200
100
0
0
10
20
30
40
50
60
70
80
90
Angle from bedding normal (deg.)
Figure 10. P-velocities (a) and shear wave splitting (b) vs. angle to
bedding plane normal. The computed results for the components were
averaged to compare with the experimental values for the aggregate
(Hornby 1998).
(dickite) are expected to be stiffer than those where it is weak
(nacrite). The calculated elastic properties of sheet silicate
minerals provide a quantitative basis on which to discuss the
elasticity of shales, which should be of great benefit for seismic
prospecting of hydrocarbon deposits.
Acknowledgments
B.M. and S.S. acknowledge support from NSF (CMG 0530282) and U.C.
Berkeley’s lab fee grant program. Teragrid and NCCS computers were used.
H.R.W. acknowledges support from DOE-BES (DE-FG02-05ER15637), NSF
(EAR-0337006), and the Esper Larsen Fund. We acknowledge access to the
facilities of beamline 11-ID-C at APS ANL for texture measurements and Y. Ren
for assistance with the experiments. Brian Hornby (BP) kindly provided us with a
sample of Kimmeridge Clay shale. We appreciate constructive reviews by Nico de
Koker and Manuel Sintubin that helped us improve the manuscript.
136
Militzer et al.: Ab initio Calculation of Elastic Moduli of Sheet Silicates
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Manuscript received March 16, 2010
Manuscript accepted August 3, 2010
Manuscript handled by Florian Heidelbach