Direct observation of a transverse vibrational mechanism for negative thermal expansion in Zn(CN)2: An atomic pair distribution function analysis

Article abstract of DOI:10.1021/ja055197f

The instantaneous structure of the cyanide-bridged negative thermal expansion (NTE) material Zn(CN)₂ has been probed using atomic pair distribution function (PDF) analysis of high energy X-ray scattering data (100-400 K). The temperature dependence of the atomic separations extracted from the PDFs indicates an increase of the average transverse displacement of the cyanide bridge from the line connecting the Zn^II centers with increasing temperature. This allows the contraction of non-nearest-neighbor Zn-Zn′ and Zn-C/N distances despite the observed expansion of the individual direct Zn-C/N and C-N bonds. Thus, this analysis provides definitive structural confirmation that an increase in the average displacement of bridging atoms is the origin of the NTE behavior. The lattice parameters reveal a slight reduction in the NTE behavior at high temperature from a minimum coefficient of thermal expansion (α = dl/ldJ) of -19.8 × 10^-6 K ^-1 below 180 K, which is attributed to interaction between the doubly interpenetrated frameworks that comprise the structure.

Full text of DOI:10.1021/ja055197f

Published on Web 10/18/2005
Direct Observation of a Transverse Vibrational Mechanism for
2
Negative Thermal Expansion in Zn(CN) : An Atomic Pair
Distribution Function Analysis
Karena W. Chapman, Peter J. Chupas, and Cameron J. Kepert*
†
‡
,†
Contribution from the School of Chemistry, UniVersity of Sydney, Sydney, NSW 2006 Australia;
AdVanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439
Received July 31, 2005; E-mail: c.kepert@chem.usyd.edu.au
Abstract: The instantaneous structure of the cyanide-bridged negative thermal expansion (NTE) material
Zn(CN) has been probed using atomic pair distribution function (PDF) analysis of high energy X-ray
2
scattering data (100-400 K). The temperature dependence of the atomic separations extracted from the
PDFs indicates an increase of the average transverse displacement of the cyanide bridge from the line
II
connecting the Zn centers with increasing temperature. This allows the contraction of non-nearest-neighbor
Zn‚‚‚Zn′ and Zn‚‚‚C/N distances despite the observed expansion of the individual direct Zn-C/N and C-N
bonds. Thus, this analysis provides definitive structural confirmation that an increase in the average
displacement of bridging atoms is the origin of the NTE behavior. The lattice parameters reveal a slight
reduction in the NTE behavior at high temperature from a minimum coefficient of thermal expansion
-
6
-1
(
R ) dl/ldT) of -19.8 × 10
K
below 180 K, which is attributed to interaction between the doubly
interpenetrated frameworks that comprise the structure.
diatomic bridges such as the cyanide anion.1
4-16
In both families
Introduction
Negative thermal expansion (NTE) behavior is an exotic
of NTE materials, the increase in average transverse displace-
ment of the bridging atoms, through thermal population of low
energy transverse vibrational modes, with negligible expansion
of the nearest-neighbor distances, is generally considered to
induce a contraction of the next-nearest-neighbor separations
1-3
material property with attractive potential applications, most
notably in moderating the predominantly positive thermal
expansion (PTE) behavior of materials, particularly those in high
precision applications where instability associated with tem-
perature fluctuation often reduces performance. This behavior
has been identified in a range of oxide-based materials contain-
ing strong single-atom bridges such as the AM2O8, AM2O7,
A2M3O12, and NASICON families. These include some tung-
5
,15,17,18
and, accordingly, the lattice parameter.
The typical exis-
1
9
tence of rigid unit modes (RUMs) in these systems, which
model the possible lattice vibrations as the librational motion
of corner sharing or bridged rigid polyhedra, is largely consistent
with the proposed population of transverse vibrational
4
-7
states (notably, ZrW2O8 and Sc2W3O12),
molybdates (e.g.,
ZrMo2O8),8 phosphates (e.g., NbZr(PO4)3 and ZrP2O7),
vanadates (e.g., ZrV2O7) and many zeolites (e.g., ALPO-17
9,10
15,20,21
modes.
10
The direct experimental confirmation of the transverse
vibrational mechanism for NTE has remained elusive and,
indeed, this mechanism has been challenged in recent local
structural studies which include variable temperature EXAFS
studies on ZrW2O8 and in studies modeling the pair distribution
function of the zeolite chabazite using a reverse Monte Carlo
and chabazite).1
1-13
More recently, NTE behavior has been
recognized in molecular framework materials containing linear
†
University of Sydney.
Argonne National Laboratory.
‡
22
(
(
(
(
(
1) Evans, J. S. O. J. Chem. Soc., Dalton Trans. 1999, 3317-3326.
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23
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15630
9
J. AM. CHEM. SOC. 2005, 127, 15630-15636
10.1021/ja055197f CCC: $30.25 © 2005 American Chemical Society

Negative Thermal Expansion in Zn(CN)
2
A R T I C L E S
Figure 2. Representation of the cubic unit cell of the Zn(CN)2-type
framework indicating the tetrahedral Zn(C/N)4 coordination (a) and an
II
illustration of the network interpenetration (b). The Zn ions lie on special
Figure 1. Representation of a single (a) and double-atom (b) bridged moiety
in which a transverse vibration is excited. The ‘apparent’ bond length, which
is inferred from the distance between the average positions of two atoms
obtained from Bragg crystallographic analysis, and the ‘true’ bond length,
which is the average distance between a pair of atoms, are indicated.
positions and the cyanide bridge is disordered (C/N-N/C) along the
Zn‚‚‚Zn′ axis of symmetry.
from the average long-range structure, providing insight into
the instantaneous structure. As such, peaks in the PDF cor-
respond to the average distance between pairs of atoms, that is,
the ‘true’ bond lengths, which can be used to directly determine
the magnitude of any displacement of bridging atoms from the
bonding direction.
Here we report the direct observation of an increase in average
transverse displacement with increasing temperature of the
bridging atoms in Zn(CN)2 through PDF analysis of variable
temperature high energy X-ray scattering data. The advantage
of this approach is that the structural information is extracted
directly from the PDFs and, therefore, the interpretation of the
means that the instantaneous structure deviates considerably
from the average structure measured by conventional crystal-
lographic techniques reliant on analysis of Bragg intensities
(Figure 1). Consequently, the application of Bragg analysis to
NTE materials typically reveals an unphysical contraction, with
increasing temperature, of the ‘apparent’ nearest-neighbor bond
lengths inferred from the distance between the average positions
of two atoms, and an elongation of the displacement parameters
of the bridging atoms perpendicular to the bonding direction;
features which are generally consistent with a transverse
vibrational mechanism.24-26
data is not subject to artifacts that can arise from structural
22,23
modeling
including from under-constrained (or over-
In ZrW2O8, the oxide-based material exhibiting the most
pronounced isotropic NTE, the contraction of the Zr-O-W
linkages between the oxo-bridged ZrO6 octahedra and WO4
tetrahedra and, correspondingly, the lattice parameter, is ac-
companied by a contraction of the apparent Zr-O bond length
29
constrained) models fitted through RMC methods. The rich
structural and compositional diversity of cyanide-bridged mo-
lecular framework materials is expected to yield a vastly
expanded range of NTE materials with enhanced NTE properties
arising from the additional flexibility associated with the double-
atom bridge and, as such, the study of the mechanism underlying
the NTE in the cyanide-bridged Zn(CN)2 structure is of
particular relevance.
The cubic Zn(CN)2 structure (Pn 3h m) consists of tetrahedral
Zn centers bridged linearly by the cyanide anion to form a
doubly interpenetrated expanded â-cristobalite-like framework
Figure 2). In the Bragg analysis, the C/N bridging atoms are
constrained to lie on the Zn‚‚‚Zn′ 3-fold axis with the Zn
centers in turn confined to special positions, highlighting the
need for a total scattering approach to probe any transverse
displacement. The observed increase in transverse displacement
of the cyanide bridge corresponds to a contraction in non-
nearest-neighbor distances which counteracts the observed
expansion of the bond length for directly bonded atom pairs,
confirming the population of transverse vibrational modes as
the mechanism underlying the NTE behavior. This analysis also
reveals the presence of longer range correlated lattice vibrations
which can contribute to the NTE.
5
as determined by Bragg techniques. In the cyanide-bridged
Zn(CN)2 frameworks, which exhibit isotropic coefficients of
thermal expansion (CTEs, R ) dl/ldT) up to more than double
that of ZrW2O8, the increase in the average transverse vibrational
amplitude of the cyanide bridge is suggested by the larger
relative increase in the component of the C/N displacement
parameter perpendicular to the Zn‚‚‚Zn′ axis.15 However, it is
not clear that the harmonic approximation adequately describes
any transverse vibrational motion and hence the magnitude of
the displacement cannot be extracted reliably.
II
14
(
II
Total scattering approaches, which include atomic pair
distribution function (PDF) analysis, have been applied suc-
cessfully to probe the structure of systems containing dynamic
2
7,28
disorder.
The PDF is a radial distribution function related
to the probability of finding two atoms at a distance r (i.e.,
atom-atom correlations), including those which may deviate
(
(
(
23) Martinez-Inesta, M. M.; Lobo, R. F. J. Phys. Chem. B 2005, 109, 9389-
396.
24) David, W. I. F.; Evans, J. S. O.; Sleight, A. W. Europhys. Lett. 1999, 46,
61-666.
25) Li, J.; Yokochi, A.; Amos, T. G.; Sleight, A. W. Chem. Mater. 2002, 14,
9
6
Experimental Section
2
602-2606.
Sample Preparation. Distilled water was carefully layered above
2
O (0.22 g, 5 mL, 0.1 M) and
-Zn(CN) (0.12 g, 5 mL, 0.1 M) contained within small glass vials
(
26) Forster, P. M.; Sleight, A. W. Int. J. Inorg. Mater. 1999, 1, 123-127.
aqueous solutions of Zn(NO
3
)
2
‚6H
(
27) Tucker, M. G.; Dove, M. T.; Keen, D. A. J. Phys.: Condens. Matter 2000,
K
2
4
1
2, L425-L430.
(
28) Chupas, P. J.; Chaudhuri, S.; Hanson, J. C.; Qiu, X.; Lee, P. L.; Shastri, S.
D.; Billinge, S. J. L.; Grey, C. P. J. Am. Chem. Soc. 2004, 126, 4756-
(29) Tucker, M. G.; Dove, M. T.; Keen, D. A. J. Appl. Crystallogr. 2001, 34,
630-638.
4
757.
J. AM. CHEM. SOC.
9
VOL. 127, NO. 44, 2005 15631

A R T I C L E S
Chapman et al.
and the vials placed inside a larger water filled glass beaker (total
volume ca. 100 mL). Slow diffusion of the reagents resulted in
crystalline samples of Zn(CN) within 2-3 weeks. The samples were
2
filtered, washed in water and ethanol, and dried in air.
Variable Temperature Powder X-ray Diffraction. The high energy
X-rays (90.48 keV, λ ) 0.13702 Å) available at the 11-ID-B beamline
at the APS at Argonne National Laboratory were used in combination
with a MAR-345 imaging plate (IP) detector to record diffraction
-
1
patterns to high momentum transfers (Q ≈ 21 Å ) for the sample
3
0
housed in a polyimide capillary. The sample was cooled to nominal
temperature of 85 K using an Oxford Cryosystems Cryostream 700
and data were collected in 3 min exposures with continuous heating at
-
1
a rate of 100 Khr to a maximum temperature of 400 K. This
corresponds to the collection of diffraction images at 7 K intervals. A
temperature correction (see Supporting Information) was applied to the
nominal temperature of the cryostream obtained from a calibration run
with an Omega K-type thermocouple added at the sample position,
such that the correct temperature range of the experiment was ca.
Figure 3. Temperature-dependent lattice parameters refined for Zn(CN)2
relative to the lattice dimensions forecast at 100 K(]) from Rietveld
refinement of X-ray powder diffraction data (11-ID-B, APS). Error bars
are given as (1 esd. The corresponding isotropic coefficients of thermal
expansion (×) are shown.
100-400 K. The sample-to-detector distance was increased for images
used in Rietveld analysis to improve Q resolution. The raw images
3
1,32
were processed using Fit-2D.
tilt of the IP relative to the beam were refined using a LaB
PDF Analysis. The PDFs, G(r) ) 4πr[F(r) - F ] where F(r) and F
are the instantaneous and average densities, were extracted using
The sample-to-detector distance and
6
calibrant.
temperature. The static disorder of the cyanide ligand was modeled
with 50% C/N occupancy, with the C/N atoms constrained to have the
same position and atomic displacement parameters. The CTEs,
R ) dl/ldT, quoted for specific temperatures use the dl/dT fitted to 5
points in a ( 14 K range.
o
o
33
PDFgetX2, subtracting the contributions from the sample environment
and background to the measured diffraction intensities. Corrections for
multiple scattering, X-ray polarization, sample absorption, and Compton
scattering were then applied to obtain the structure function S(Q). Direct
Results
Fourier transform of the reduced structure function F(Q)
)
Bragg Analysis. The refinement of the lattice parameters
showed an isotropic contraction of the cubic lattice parameter
in the temperature range 100-400 K (Figure 3).
-
1
Q[S(Q) - 1] up to Qmax ≈ 21 Å gave G(r), the pair distribution
function. Distances of interest were extracted directly from the G(r)’s
by direct fitting of a Gaussian function to the peaks at ∼2 Å (dZn-C/N
)
)
While the NTE behavior appears approximately linear, with
and ∼3.2 Å (dZn‚‚‚C/N) and the two peaks at ∼5 Å (dZn‚‚‚Zn and dZn‚‚‚C/N′
-
6
-1
3
4
an average CTE of -16.0(2) × 10
K
(100-400 K), slight
within KUPLOT. The distances reported are the average of duplicate
variable temperature experiments. Although the termination ripples can
influence the apparent peak positions, an effect which is most
pronounced at low r, the rigorous treatment of this effect can be
complicated by the fact that the low r-region also contains the most
significant contribution from systematic errors in the data. Thus,
improved fits are typically not obtained when termination ripples are
considered, and a common approach is to fit only a Gaussian to the
nonlinearities in the thermal expansion are evident. The CTE,
-
6
-1
which attains a minimum of -19.8 × 10
increases slightly at high temperature, approaching -14 ×
10 at ca. 400 K.
Atomic Pair Distribution Function Analysis. The PDFs
K
below 180 K,
-
6
-1
K
were obtained from the high energy X-ray scattering data by
direct Fourier transform of the reduced structure functions, F(Q),
3
5,36
peak position.
To minimize any effect of the termination ripples,
-1
up to Q ≈ 21 Å (Figure 4a). The PDFs contain well-defined
Fourier transforms were performed over a constant Q-range to ensure
a consistent effect and to eliminate any influence on the observed change
in the bond lengths and distances. Refinement of a model based on the
Rietveld refined structure, against G(r) and generation of partial PDFs
peaks to high r, as is consistent with the long range order
requisite for Bragg crystallographic analysis (Figure 4b). The
form of G(r) evolves with increasing temperature, with general
peak broadening and shifting of some peaks to lower r. The
greatest changes, as a function of temperature, are evident at
high r (> 6 Å). By contrast, the correlations that define the
3
7
were performed within PDFFIT. A more complex starting configu-
ration with the bridging atoms disordered over 6 positions displaced
equidistant from the Zn‚‚‚Zn′ axis was also refined.
II
Bragg Analysis. Structural analysis of the Bragg intensities within
Zn coordination sphere (∼2 Å) do not change visibly.
GSAS used the Rietveld method (2θ ) 1.2-10.6°; at 108 K wR
p
)
)
Refinement of a model based on the crystallographic structure
yielded a moderately good fit to the experimental PDF at
0
0
.0287, R
.0211; RF2 ) 0.1062).
p
) 0.0222; RF2 ) 0.0487; at 396 K wR
p
) 0.0275, R
p
3
8,39
Complete parameters from the Rietveld
1
00 K (R ) 36.8%, Figure 5). Refinement of a more complex
structural refinement are included in the Supporting Information. The
lattice parameter, isotropic atomic displacement parameters, C/N
position and peak shape parameters were refined independently at each
model in which the cyanide bridge is disordered over six
equivalent positions displaced from the Zn‚‚‚Zn′ axis yielded a
slightly improved fit (R ) 34.9%). However, both models are
unable to completely describe some of the long-range correla-
tions with periodicities greater than a single unit cell, for
example at r ≈ 12.5, 17 Å. This is highly unusual for a
crystalline system, for which the PDF fits usually improve at
higher r, where the long-range structure approaches the average
(
30) Chupas, P. J.; Qiu, X.; Hanson, J. C.; Lee, P. L.; Grey, C. P.; Billinge, S.
J. L. J. Appl. Crystallogr. 2003, 36, 1342-1347.
(
31) Hammersley, A. P.; Svensson, S. O.; Hanfland, M.; Fitch, A. N.;
H a¨ usermann, D. High-Pressure Res. 1996, 14, 235-248.
(32) Hammersley, A. P., ESRF Internal Report 1997, ESRF97HA02T.
33) Qiu, X.; Thompson, J. W.; Billinge, S. J. L. J. Appl. Crystallogr. 2004,
(
3
7, 678.
(
34) Proffen, T.; Neder, R. B. J. Appl. Crystallogr. 1997, 30, 171-175.
35
structure. This indicates that there exists a greater extent of
(
35) Qiu, X. Y.; Proffen, T.; Mitchell, J. F.; Billinge, S. J. L. Phys. ReV. Lett.
2
005, 94, 177203.
(
36) Peterson, P. F.; Bozin, E. S.; Proffen, T.; Billinge, S. J. L. J. Appl.
Crystallogr. 2003, 36, 53-64.
(38) Toby, B. H. J. Appl. Crystallogr. 2001, 34, 210-213.
(39) Larson, A. C.; Von Dreele, R. B. General Structure Analysis System (GSAS)
2000, Los Alamos National Laboratory Report, LAUR 86-748.
(
37) Proffen, T.; Billinge, S. J. L. J. Appl. Crystallogr. 1999, 32, 572-575.
15632 J. AM. CHEM. SOC.
9
VOL. 127, NO. 44, 2005

Negative Thermal Expansion in Zn(CN)
2
A R T I C L E S
Figure 4. Reduced structure function, F(Q) ) Q[S(Q) - 1], at ca. 100 K
a) and the PDFs, G(r), at representative temperatures (b) from high energy
(
X-ray diffraction of Zn(CN)2 (11-ID-B, APS).
Figure 5. PDF generated for the crystallographic structural model (red)
refined against the experimental PDF (gray). The residual (blue) is given
below.
Figure 6. Calculated total PDF (a) and the partial PDFs corresponding to
Zn-Zn (b), Zn-C/N (c), and C/N-C/N (d) correlations, calculated for the
structural model based on the crystalline Bragg starting configuration for
Zn(CN)2 at 100 K. No peak sharpening, arising from correlated atomic
motion, has been included. The experimental PDF (gray dashed line) has
been overlaid for comparison.
vibrational distortions than can be accounted for within a single
unit cell. Thus, the PDF refinement suggests the existence of
vibrational correlations with periodicities over several unit cells.
Reported analyses of vibrations involving in-phase atomic
motion indicate that such vibrations are associated with a
sharpening of correlations in the PDF.40 Application of such
sharpening for the first coordination shell, where there is obvious
in-phase atomic motion, was attempted. However, comparison
of the experimental and calculated PDF (Figure 5) shows that
the peaks up to ca. 6 Å appear sharpened relative to the peaks
at high r, suggesting correlation of longer range atomic motion.
Analysis of this sharpening is beyond the scope of this paper
and is currently being pursued in conjunction with refinement
of a larger configuration model of multiple unit cells using a
RMC modeling approach.41
The peaks in the PDFs correspond principally to Zn-Zn and
Zn-C/N correlations (Figure 6). The low intensity correlations
between the carbon and nitrogen atoms, which have low
scattering power relative to zinc, are of the same order as the
termination ripples arising from the direct Fourier transform of
F(Q), which are most pronounced in the low r region, and, as
such, cannot be extracted accurately.
The individual peaks below r ≈ 6 Å, corresponding to
Zn-Zn and Zn-C/N atomic correlations, were identified and
fit with a Gaussian function to extract accurate peak positions,
that is, bond lengths and distances, as a function of temperature
3
2
(
Figure 7).
The bond lengths and distances from the Bragg and PDF
analyses, and their corresponding temperature dependence, show
(
40) Jeong, I. K.; Proffen, T.; Mohiuddin-Jacobs, F.; Billinge, S. J. L. J. Phys.
Chem. A 1999, 103, 921-924.
(
41) Nield, V.; Keen, D. A. Diffuse Neutron Scattering from Crystalline
Materials; Oxford/Clarendon Press: New York, 2001.
J. AM. CHEM. SOC.
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VOL. 127, NO. 44, 2005 15633

A R T I C L E S
Chapman et al.
where dZn-C/N, dZn‚‚‚C/N and dZn‚‚‚Zn′ are the Zn-C/N, Zn‚‚‚C/
N, and Zn‚‚‚Zn′ distances extracted from the PDF (Figure 9).
Figure 7. Peaks in the PDF below r ≈ 6 Å (a) relate to the shortest local
atomic correlations corresponding to the direct C-N and Zn-C/N distances,
non-nearest-neighbor Zn‚‚‚C/N and Zn‚‚‚Zn′ intra-framework distances and
Zn‚‚‚C/N′ inter-framework distance.
Table 1. Selected Distances and the Corresponding CTEs from
Bragg and PDF Analysis
Bragg analysis
PDF analysis
d(100K)/ d(400K)/
d(100K)/ d(400K)/ CTE/
×
CTE/ ×
6 1
-6
-
1
-
-
Å
Å
10
K
Å
Å
10 K
Figure 9. Average displacement of the C/N bridging atoms from the line
connecting the Zn centers. The straight line is the linear least-squares fit
to the data.
C-N
Zn-C/N
Zn‚‚‚C/N
1.200 1.194 -19(5)
1.986 1.977 -15.1(16) 2.015 2.021 +10.2(10)
3.186 3.171 -16.4(8) 3.209 3.203 -6.4(10)
II
Zn‚‚‚C/N′other net 4.883 4.860 -15.8(2)
Zn‚‚‚Zn′ 5.172 5.147 -15.9(2)
4.871 4.837 -23(4)
5.191 5.172 -12.42(12)
The average transverse displacement increases with increasing
temperature by 0.04 Å per 100 K from 0.46 Å at 100 K to
0
.58 Å at 400 K.
The range of the direct C-N bond length can also be derived
significant differences (Table 1). The apparent bond lengths and
distances inferred from the average atomic positions (Bragg)
are contracted relative to the ‘true’ (PDF) distances, a feature
that is consistent with a difference in the average and instan-
taneous structures. The Zn-C/N bond length from the Bragg
analysis appears to contract with increasing temperature (Figure
from the PDF distances based on the two extremes of the
displacement modes with the two bridging atoms to the same
side of the Zn‚‚‚Zn′ axis and to opposite sides of the Zn‚‚‚Zn′
axis (Figure 10). These both show the correct PTE behavior.
Discussion
The PDF analysis provides direct evidence of the increase in
the average displacement of the C/N bridging atoms from the
Zn‚‚‚Zn′ direction at higher temperatures. This induces the
contraction of the non-nearest-neighbor Zn‚‚‚C/N and Zn‚‚‚Zn′
distances, despite the observed expansion of the nearest-neighbor
Zn-C/N and C-N bond lengths which is associated with the
inherent anharmonicity of the interatomic potential for directly
bonded atom pairs. The Bragg analysis shows the opposite,
unphysical trend for the apparent direct Zn-C/N and C-N bond
lengths, confirming the necessity of a total scattering approach
to accurately determine the temperature dependence of atomic
correlations in the dynamic structure. A similar discrepancy in
the high-temperature bond lengths from Bragg and total
scattering approaches has been demonstrated by Keen and co-
8
), contradicting known behavior for directly bonded atoms for
which thermal expansion occurs due to the inherent anharmo-
nicity of the interatomic potential. In contrast, the true
Zn-C/N bond length from the PDF analysis expands with
increasing temperature. Again, this feature is consistent with
an increase in the deviation of the average and instantaneous
structures at higher temperatures.
The average displacement (D) of the C/N bridging atoms from
II
to line connecting the Zn centers can be calculated directly
from the PDF results according to the geometric relation
2
2
2 2
(
dZn-C/N) + (d
) - (dZn‚‚‚C/N)
2
Zn‚‚‚Zn
D )
(dZn-C/N) -
(
)
x
2dZn‚‚‚Zn
27
workers for the related â-cristobalite (SiO2) structure. Thus,
Figure 8. Temperature-dependent Zn-C/N bond length (b) and next-nearest-neighbor Zn‚‚‚C/N distance (9), as inferred from the average atomic positions
from the Bragg analysis (a) and as extracted from the PDFs (b). Error bars are given as (1 esd.42 The straight lines are the least-squares fit to the data.
15634 J. AM. CHEM. SOC.
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VOL. 127, NO. 44, 2005

Negative Thermal Expansion in Zn(CN)
2
A R T I C L E S
vibrations, such as through steric interaction of the interpen-
etrated frameworks. The closest inter-framework interaction
between the interpenetrated frameworks is between the mid-
points of the cyanide bridges, an interaction that will be
influenced significantly by the amplitude of the transverse
displacement of these bridges. Disregarding any correlation in
this transverse displacement between frameworks, the PDF
results indicate that the average distance between bridging atoms
if displaced directly toward each other decreases from 3.29 Å
at 100 K to 3.04 Å at 400 K, thus bringing the inter-framework
contacts within the van der Waals radii. The unfavorable nature
of such an interaction suggests that a degree of correlation may
exist between the vibrational modes in the interpenetrated
frameworks to avoid this steric interaction. Thus, increasing
inter-framework correlations may be responsible for the subtle
reduction in the NTE at higher temperatures (>180 K). The
Zn(CN)2 family are the only NTE materials yet reported with
an interpenetrated structure and hence this effect has not been
observed previously, although the thermal expansion of unre-
Figure 10. Expansion of the direct C-N bond lengths calculated from
the PDF distances assuming displacement of bridging atoms in opposite
(]) and identical (4) directions. The straight lines are linear least-squares
fits to the data.
the PDF analysis confirms the increase in the transverse
vibrational motion of the bridging atoms as the origin of the
observed NTE behavior.
44
lated guests is known to influence the NTE in porous zeolites.
II
Both the correlated vibrations of the Zn centers and the
interpenetrated framework are associated with periodicities
beyond a single unit cell and may therefore be responsible for
the poor fit of the crystallographic model to the PDF at high r.
The PDF analysis indicates a PTE contribution from the
individual bond lengths that to some extent counteracts the NTE
influence of the transverse vibrations of the cyanide bridge.
Although the generally stated mechanism for NTE in such
systems assumes that bond length PTE effects are minor, the
present analysis clearly indicates a large PTE behavior for the
Conclusion
The application of PDF methods to probe the structure of
the NTE material Zn(CN)2 provides unique insights into the
dynamic structure that cannot be obtained with traditional Bragg
crystallography. The PDF analysis provides definitive evidence
of the increasing average transverse displacement of the cyanide
bridge with increasing temperature, confirming the population
of transverse vibrational modes of the bridging atoms as the
mechanism underlying the NTE behavior. The data further show
that this effect is sufficient to counteract the PTE contribution
from the significant thermal expansion of the individual
Zn-C/N bond lengths. Modeling of the PDF suggests vibra-
tional correlations that extend beyond a single unit cell, such
-
6
-1
Zn-C/N bond length with a CTE of +10.2(10) × 10 K ,
considerably larger than the Si-O bond expansion coefficient
-
6
-1 27
(
+2.2 × 10 K ) such as in â-cristobalite and greater in
-6 -1 5
magnitude than the bulk NTE in ZrW2O8 (-9.1 × 10 K ).
Previous analyses of the atomic displacement parameters, which
assume harmonic motion, were in apparent agreement with
15
thermal expansion data, however, such analyses assume
negligible expansion of the individual bond lengths,4 an effect
which the present study demonstrates to be more significant
that previously postulated. It is noteworthy that the more
significant NTE contribution from the transverse vibrations of
the cyanide bridge in this highly flexible material15 is sufficient
to produce a large overall NTE property.
3
II
as involving the correlated displacements of the Zn centers
from the special positions and interaction between the pairs of
interpenetrated frameworks. We propose that the latter may
cause a slight reduction in the NTE at high temperature.
Application of a PDF approach can be readily envisioned to
study thermal expansion in a variety of materials including other
cyanide-bridged frameworks such as the Prussian blue ana-
The PDF analysis shows a significantly less rapid contraction
of the intra-framework Zn‚‚‚Zn′ distance than for the Bragg
distance or lattice parameter. This suggests a contribution from
II
lattice vibrations with displacement of the Zn ions from the
special positions to which they were confined in the Bragg
analysis, such that the true Zn‚‚‚Zn′ distance is greater than the
Bragg distance. Correlated vibrations moving atoms off special
positions are also evident in other types of dynamic structures,
including perovskites. For example, in the high-temperature
cubic form of AlF3, which exhibits the ReO3 structure, transverse
16
logues, oxide-based NTE materials, and systems that display
45-47
anomalous thermal expansion due to other mechanisms.
Acknowledgment. This work was supported by the Australian
Synchrotron Research Program, which is funded by the Com-
(
42) The esd’s for the bond length extracted from the PDF are taken as the esd
for the Gaussian fit to the peak. Systematic errors in the measurement of
S(Q) tend to occur as long wavelength modulations in S(Q). Thus, the
systematic errors in the PDF are most pronounced at low r, where actual
error in the values contains a component associated with the systematic
errors. See Toby, B. H.; Billinge, S. J. L. Acta Crystallogr., Sect. A:
Fundam. Crystallogr. 2004, 60, 315-317.
2
8
vibrational motion has been identified using the PDF method
although for AlF3 this effect is not sufficient to counteract the
(
significant PTE of the individual Al-F bonds). In the dynamic
structure, the rigid AlF6 octahedra are involved in correlated
motions around their 3-fold symmetry axis in which the Al
(
43) Busing, W. R.; Levy, H. A. Acta Crystallogr. 1964, 17, 142-146.
44) Jardim, P. M.; Marinkovic, B. A.; Saavedra, A.; Lau, L. Y.; Baehtz, C.;
Rizzo, F. Micropor. Mesopor. Mater. 2004, 76, 23-28.
(
atoms move off the special positions. The same effect is
expected to be true in Zn(CN)2 with displacement of the ZnII
(45) Salvador, J. R.; Guo, F.; Hogan, T.; Kanatzidis, M. G. Nature 2003, 425,
702-705.
ions from the special positions.
The observed reduction in the NTE behavior at higher
temperatures may be due to dampening of high amplitude
(46) Jayarama, A.; Bucher, E.; Dernier, P. D.; Longinot, L. Phys. ReV. Lett.
1
973, 31, 700-703.
(
47) Arvanitidis, J.; Papagelis, K.; Margadonna, S.; Prassides, K.; Fitch, A. N.
Nature 2003, 425, 599-602.
J. AM. CHEM. SOC.
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VOL. 127, NO. 44, 2005 15635

A R T I C L E S
Chapman et al.
monwealth of Australia under the Major National Research
Facilities. Work performed at the A.P.S. was supported by the
U.S. D.o.E., Office of Science, Basic Energy Sciences, under
Contract No. W-31-109-Eng-38. C.J.K. thanks A. L. Goodwin
for valuable discussions. We thank C. Kurtz, G. Jennings, and
N. Leyarovska for experimental support.
Supporting Information Available: Details of PDF and
Bragg analysis (PDF). This material is available free of charge
via the Internet at http://pubs.acs.org.
JA055197F
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