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Measurement of thermal expansion coefficient of
rock minerals using XRD and its implications to
thermal damage mechanism
To cite this article: Zhaoyang Ma et al 2021 IOP Conf. Ser.: Earth Environ. Sci. 861 072021
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11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
Measurement of thermal expansion coefficient of rock minerals
using XRD and its implications to thermal damage mechanism
Zhaoyang Ma1; Guanglei Zhang2; Ranjith Pathegama Gamage1*; Chengpeng Zhang3*
Deep Earth Energy Laboratory, Building 60, Monash University, Melbourne, Victoria 3800,
Australia.
1
2
Department of Earth Science and Engineering, Imperial College London, London SW7 2BP,
UK
3
State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University,
Chongqing, 400044, China.
Abstract. The measurement of thermal expansion coefficient (TEC) of rocks is of vital
importance to various geotechnical engineering applications, such as high-level radioactive
waste disposal, enhanced geothermal energy exploitation, ultra-deep borehole stability, liquid
nitrogen storage, etc. However, previous researchers mainly focused on the thermal expansion
coefficient of various rocks, instead of the constituent minerals. In this manuscript, we measured
the both temperature-dependent volume and linear thermal expansion coefficients of two rockforming minerals (quartz and albite) using both high-temperature X-ray diffraction (XRD) as
well as low-temperature XRD. It is found that there exist strong anisotropic linear thermal
expansion coefficients of the two minerals, which can provide strong reference for the relevant
engineering applications and numerical simulations.
1. Introduction
Thermal stresses induced during the heating or cooling of rocks cause fracture and fragmentation, which
is of vital importance to various geotechnical engineering applications, such as high-level radioactive
waste disposal, enhanced geothermal energy exploitation, ultra-deep borehole stability, etc. Rocks are
inherently heterogeneous and they are polycrystalline with various anisotropic mineral aggregates.
Generally, thermal stresses can be generated due to the following reasons: (1) thermal expansion
(contraction) coefficient mismatch between different phases (grains) (2) anisotropic thermal expansion
(contraction) coefficient of individual phase (3) thermal gradient (heating or cooling rate) (4) phase
change and/or structural chemical reactions (5) temperature-dependent elastic modulus of each
individual phase (6) composition, distribution and orientation of individual grains (7) presence of crack
or fracture (8) previous maximum temperature (9) presence of water (moisture content) (10) pressure
[1] (11) thermal cycling (temperature history). Take granite for instance, it consists of three major
mineral constituents, namely, quartz, feldspar and mica. The thermal expansion coefficient of these
three dominant minerals is of great difference. For each single phase, there exists a distinct anisotropic
linear thermal expansion coefficient, which is relatively less studied. Thermal expansion coefficient (𝛼𝛼)
indicates the percentage of material expands or contracts with temperature changes. Under a constant
pressure, volumetric thermal expansion coefficient (𝛼𝛼𝑉𝑉 or 𝛽𝛽) is defined by the variations of volume by
a differential temperature change (∆𝑇𝑇) and it is normally described as the following relationship:
𝛼𝛼𝑉𝑉 =
1 𝜕𝜕𝜕𝜕
( )
𝑉𝑉 𝜕𝜕𝜕𝜕 𝑃𝑃
where 𝑉𝑉 is the volume at a temperature 𝑇𝑇 under a constant pressure 𝑃𝑃.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
Similarly, the linear thermal expansion coefficient is defined as:
𝛼𝛼𝑙𝑙 =
1 𝜕𝜕𝜕𝜕
( )
𝑙𝑙 𝜕𝜕𝜕𝜕 𝑃𝑃
Accurate thermal expansion coefficients of rock-forming minerals are of great essence to the
clarification of the equation of state of the earth’s interior [2] and it is emphasized that most serious
error in the calculation of thermodynamic functions arises from the uncertainty of thermal expansivity
at high temperature [3]. In most cases, rocks are assumed to be homogeneous and isotropic for the
convenience of analysis. Only a few studies focused on the thermal expansion coefficient measurement
of rock minerals. For instance, Jay [4] measured the lattice parameters of quartz from 18 to 730 ℃ using
the high-temperature Debye-Scherrer method and it was found that thermal expansion coefficients in
both axial direction of the hexagonal unit cell increased from 18 to 580 ℃ and both axes apparently
decreased in length from 580 to 730 ℃. Ackermann and Sorrell [5] measured the thermal expansion
coefficient of quartz using high-temperature X-ray diffractometry from 22 to 1400 ℃. It was found that
the expansion was zero from 574 to 1000 ℃ and decreased from 1000 to 1400 ℃. The maximum
volume thermal expansion coefficient was 100±20 × 10−5 /degree between 572 and 574 ℃. In addition,
they [6] measured the thermal expansion of both single-crystal quartz and quartz powder using the
highly sensitive dilatometer apparatus. It was found that the expansion coefficient characteristics were
identical with X-ray diffractometer measurement below 560 ℃ . Single-crystal quartz reached a
maximum volume thermal expansion coefficient 250× 10−5 /degree between 571 and 573 ℃ while the
quartz powder reached a maximum volume thermal expansion coefficient 150× 10−5 /degree between
571 and 576 ℃.
Richter and Simmons [7] measured the thermal expansion coefficient of several igneous rocks from 25
to 550 ℃ and they found that the measured thermal expansion coefficient is a function of crack porosity,
heating rate, previous maximum temperature as well as mineralogical composition and preferred crystal
orientation. In their study, they only measured the linear thermal expansion coefficient of rock core
along the long dimension of the sample only, and the volume expansion was assumed to be three times
the linear expansion coefficient. In addition, the anisotropy of expansion was not investigated. The
experimental volume expansion coefficients of the rocks and calculated value using the Turner’s
formula and the input value for the Turner’s calculation is from [7]. Alvaro, Angel et al. [8, 9] measured
the unit cell volume of quartz using the high-temperature single-crystal X-ray diffraction from 294K to
872K.
2. Temperature Calibration
In this manuscript, the thermocouple measurements were calibrated using the lattice parameter changes
for magnesium oxide (MgO) as a function of the temperature. We measured the XRD pattern of MgO
with the change of temperature, as shown in figure 1. It can be found that there is an obvious peak shift
of MgO in the temperature range between room temperature and 1200 ℃. Figure 2 demonstrated the
calibrated temperature of MgO with the set temperature and it can be found that the two curves fit very
well, which indicates the accuracy of temperature.
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11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
Intensity (A.U.)
1200 °C
1100 °C
1000 °C
900 °C
800 °C
700 °C
600 °C
500 °C
400 °C
300 °C
200 °C
100 °C
30 °C
41
42
43
2θ (degree)
44
45
Figure 1. MgO-high-temperature from room temperature to 1200 ℃
Calibrated temperature (οC)
1200
HTK Calibrated temperature using MgO
Linear fitting of tempeture
1000
800
600
400
y = a + b*x
-10.55255 ± 3.67413
1.02643 ± 0.0052
0.99986
0.99972
0.99969
Equation
Intercept
Slope
Pearson's r
R-Square (COD)
Adj. R-Square
200
0
0
200
400
600
800
Set temperature (oC)
1000
1200
Figure 2. The HTK calibrated temperature using MgO
3. XRD experimental results for quartz and albite
3.1 Thermal expansion coefficient and unit cell parameters of quartz
Figure 3 demonstrates the XRD pattern of quartz from room temperature to 1200 ℃ and figure 4 shows
the magnification of XRD pattern of quartz from room temperature to 1200 ℃. Linear and volume
expansion coefficients are calculated using the following equation:
𝛼𝛼𝑖𝑖 =
1 ∆𝑌𝑌𝑖𝑖
1 + 𝑌𝑌𝑖𝑖 ∆𝑇𝑇
where the suffix 𝑖𝑖 represents a, b, c (each crystallographic axis) and 𝑣𝑣 (volume). The unit cell
parameters of quartz under various temperatures were gained using the Rietveld method and the results
are shown in Table 1.
3
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
1200°C
1000°C
600°C
Quar
tz
Intensity (A.U.)
800°C
30°C
20
25
30
35
40
45
2θ (degree)
50
55
60
Figure 3. XRD pattern of quartz from room temperature to 1200
1200°C
Intensity (A.U.)
1000°C
800°C
600°C
30°C
25
26
27
28
2θ (degree)
29
30
Figure 4. Magnification of XRD pattern of quartz from room temperature to 1200 ℃
Table 1. Relationship of unit cell parameters of quartz with various temperatures
Set
Temperature
(℃)
30
60
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
Calibrated
temperature
(℃)
30.432
60.864
101.44
152.16
202.88
253.6
304.32
355.04
405.76
456.48
507.2
557.92
608.64
659.36
710.08
760.8
811.52
Unit cell
Volume
(Å3)
112.997
113.13
113.317
113.569
113.841
114.133
114.455
114.803
115.194
115.645
116.212
117.203
118.032
118.027
118.012
117.992
117.973
Unit Cell
Volume
error
0.0012
0.001
0.0009
0.0009
0.001
0.0013
0.001
0.0011
0.0014
0.0014
0.0013
0.0019
0.0018
0.0018
0.0011
0.0016
0.0013
4
a
(Å)
Error of a
c
(Å)
Error of c
4.9134
4.915654
4.918826
4.923111
4.927676
4.932559
4.93795
4.943732
4.950275
4.957828
4.967182
4.983595
4.997388
4.997536
4.99748
4.997404
4.997257
0.00002299
0.00001813
0.00001666
0.00001594
0.000018
0.00002422
0.00001722
0.00001938
0.00002614
0.00002224
0.00002298
0.00003551
0.00003069
0.00002897
0.00001614
0.00002594
0.00002155
5.404704
5.406116
5.408062
5.410651
5.413549
5.416709
5.420121
5.423896
5.428004
5.432687
5.438776
5.44908
5.457351
5.456792
5.456229
5.455496
5.454927
0.00002789
0.00002369
0.00002655
0.00002269
0.00002417
0.00002825
0.00003026
0.00003268
0.00003047
0.00004227
0.00003027
0.0000433
0.00004623
0.00005055
0.00003419
0.00004672
0.00003838
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
850
900
950
1000
1050
1100
1150
1200
862.24
912.96
963.68
1014.4
1065.12
1115.84
1166.56
1217.28
117.947
117.921
117.899
117.876
117.856
117.835
117.817
117.798
0.0012
0.0017
0.0021
0.0015
0.0013
0.0017
0.0014
0.0016
4.997015
4.996836
4.996665
4.996482
4.996339
4.996121
4.99601
4.995864
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
0.0000193
0.00003286
0.00003852
0.00002703
0.00002134
0.00002793
0.00002658
0.00002707
5.46
Unit Cell-c (Å)
5.00
Unit Cell-a (Å)
0.00003698
0.00003534
0.00004706
0.00003817
0.00003759
0.00004893
0.00002752
0.00004141
Unit Cell a change with temperature 5.47
Unit Cell c change with temperature
5.02
5.45
4.98
5.44
4.96
5.43
4.94
5.42
4.92
4.90
5.45425
5.453431
5.452801
5.452133
5.451528
5.451025
5.450411
5.449864
5.41
0
200
400
600
800
Temperature (°C)
1000
1200
5.40
Figure 5. Unit cell parameters change of quartz from room temperature to 1200 ℃
Figure 5 shows the unit cell parameters change of two directions of quartz (namely, a and c). There
exists an obvious variation of unit cell parameters near the phase change temperature (namely, 573 ℃).
The quartz 𝛼𝛼 − 𝛽𝛽 transition is marked by a kink in the unit cell parameters, as shown in the above
figures. From the above curves, we can find that quartz has a steep, nonlinear expansion curve until is
transition temperature and a slight contraction exists beyond the transition temperature.
3.2 Thermal expansion coefficient and unit cell parameters of albite
Similarly, figure 6 demonstrates the XRD pattern of albite from room temperature to 1200 ℃ and figure
7 shows the magnification of XRD pattern of albite from room temperature to 1200 ℃. The unit cell
parameters of albite under various temperatures were gained using the Rietveld method and the results
are shown in Table 2. Figure 8 shows the unit cell parameters change of three directions of albite and
figure 9 demonstrates the unit cell angle variation of albite with temperature. Figure 10 shows the both
the volume and linear thermal expansion coefficient of albite with the calibrated temperature and it can
be seen that there exists a strong anisotropic thermal expansion coefficient of albite and both the volume
and linear thermal expansion coefficient of albite increases with the temperature.
5
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
Intensity (A.U.)
500 °C
300 °C
A l bi te
950 °C
700 °C
30 °C
20
40
60
80
2θ (degree)
100
Figure 6. XRD pattern of albite from room temperature to 950 ℃
950 °C
Intensity (A.U.)
700 °C
500 °C
300 °C
30 °C
27.0
27.5
28.0
2θ (degree)
28.5
29.0
Figure 7. Magnification of XRD pattern of albite from room temperature to 950 ℃
Table 2. Relationship of unit cell parameters of albite with various temperatures
Set
temperature
(℃)
30
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
Calibrated
temperature
(℃)
30
101.008
151.728
202.448
253.168
303.888
354.608
405.328
456.048
506.768
557.488
608.208
658.928
709.648
760.368
811.088
861.808
912.528
963.248
Volume
(𝐴𝐴̇3 )
664.938
666.167
667.03
668.001
668.877
669.882
670.88
671.912
673.02
673.905
675.013
675.949
677.007
678.091
679.136
680.172
681.371
682.629
683.925
a (Å)
b (Å)
c (Å)
8.143125
8.151428
8.157127
8.163595
8.169331
8.175948
8.182209
8.188472
8.195641
8.201536
8.208406
8.214505
8.221115
8.22804
8.23472
8.241806
8.250021
8.258159
8.266808
12.79024
12.79433
12.79744
12.80085
12.80423
12.80755
12.81216
12.8164
12.82044
12.82384
12.82798
12.83135
12.83528
12.83933
12.84418
12.84739
12.85182
12.85653
12.86114
7.160918
7.161221
7.161281
7.161787
7.162122
7.163084
7.163663
7.1647
7.165916
7.166494
7.167861
7.168712
7.169874
7.171152
7.171764
7.17293
7.174199
7.175586
7.177362
6
α (°)
94.24636
94.2226
94.19119
94.16269
94.13197
94.09352
94.05622
94.00687
93.96167
93.90932
93.85375
93.79725
93.74332
93.6799
93.61521
93.55089
93.47926
93.40101
93.31534
β (°)
116.6172
116.567
116.5319
116.4983
116.4682
116.4387
116.4107
116.383
116.3553
116.3329
116.3056
116.2819
116.254
116.2304
116.2066
116.1839
116.1627
116.1356
116.1153
γ (°)
87.74094
87.73409
87.72876
87.71893
87.70934
87.69948
87.68577
87.68124
87.67019
87.663
87.65198
87.63967
87.62926
87.61681
87.60656
87.59337
87.58058
87.57001
87.56183
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
8.28
8.24
7.178
12.86
a
b
c
8.26
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
7.176
12.85
7.174
12.84
8.22
8.20
8.18
7.172
12.83
7.170
12.82
7.168
7.166
12.81
8.16
12.80
8.14
12.79
0
200
400
600
800
1000
7.164
7.162
7.160
1200
Calibrated temperature (°C)
Figure 8. Variation of unit cell parameter of albite (length) with the calibrated temperature
94.4
94.2
116.6
94.0
116.5
93.8
116.4
α
β
γ
93.6
93.4
93.2
87.76
116.7
200
87.72
87.70
87.68
87.66
87.64
87.62
116.3
87.60
116.2
116.1
0
87.74
400
600
800
Calibrated temperature (°C)
1000
87.58
87.56
87.54
1200
Figure 9. Variation of unit cell parameter of albite (angle) with the calibrated temperature
3.1
1.65 6.0
Volume thermal expansion*10^-5
Linear thermal expansion a*10^-5
Linear thermal expansion b*10^-6
Linear thermal expansion c*10^-7
3.0
1.60
2.9
5.6
25
20
1.55 5.4
5.2 15
2.8
1.50
2.7
5.0
1.45 4.8
10
4.6 5
2.6
1.40
2.5
5.8
0
200
400
600
800
Calibrated temperature (°C)
1000
4.4
0
1200
Figure 10. Volume and linear thermal expansion coefficients of albite with the calibrated temperature
4. Conclusions:
In this study, the relationship between the rock mineral thermal expansion coefficient and temperature
was established. The unit cell parameters (a, b, c, 𝛼𝛼, 𝛽𝛽, 𝛾𝛾 and V) were refined using the Rietveld method.
For quartz, there exists an obvious variation of unit cell parameters near the phase change temperature
(namely, 573 ℃) and quartz has a steep, nonlinear expansion curve until is transition temperature and
7
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 072021
IOP Publishing
doi:10.1088/1755-1315/861/7/072021
a slight contraction exists beyond the transition temperature. For albite, there exists a strong anisotropic
thermal expansion coefficient and both the volume and linear thermal expansion coefficient of albite
increases with the temperature.
This study if of vital importance to the relevant geotechnical engineering, such as high-level radioactive
waste disposal, enhanced geothermal energy extraction, etc. In addition, the accurate measurement of
thermal expansion coefficients of these rock minerals can provide baselines for the numerical
simulation.
5. Acknowledgement
We appreciate the help from Jisheng Ma to carry out the high-temperature XRD tests and its
postprocessing. The first author would like to acknowledge the financial support provided by the China
Scholarship Council (No. 201706420055). In addition, this work is funded by various projects, namely,
China Postdoctoral Science Foundation (No. 2020 M673142), Chongqing Science and Technology
Bureau Foundation (No. cstc2020jcyj-bsh0059), Chongqing Human Resources and Social Security
Bureau Foundation (No. cx2019100), Chongqing Science and Technology Bureau Foundation (No.
cstc2020jcyj-zdxm0122) and National Natural Science Foundation of China (No. U19B2009).
6. References
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Sweet, J.N., Pressure effects on thermal conductivity and expansion of geologic materials.
1979, Sandia Labs., Albuquerque, NM (USA).
[2].
Suzuki, I., Thermal expansion of periclase and olivine, and their anharmonic properties.
Journal of Physics of the Earth, 1975. 23(2): p. 145-159.
[3].
Jay, A., The thermal expansion of quartz by X-ray measurements. Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character,
1933. 142(846): p. 237-247.
[4].
Thirumalai, K. and S. Demou, Effect of reduced pressure on thermal‐expansion behavior of
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p. 5147-5151.
[5].
Ackermann, R. and C.A. Sorrell, Thermal expansion and the high–low transformation in
quartz. I. High-temperature X-ray studies. Journal of Applied Crystallography, 1974. 7(5): p.
461-467.
[6].
Richter, D. and G. Simmons. Thermal expansion behavior of igneous rocks. in International
Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts. 1974. Elsevier.
[7].
Clark, S.P., Handbook of physical constants. Vol. 97. 1966: Geological Society of America.
[8].
Johnson, B., A. Gangi, and J. Handin. Thermal cracking of rock subjected to slow, uniform
temperature changes. in 19th US Symposium on Rock Mechanics (USRMS). 1978. American
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[9].
Friedman, M. and B. Johnson. Thermal cracks in unconfined Sioux quartzite. in 19th US
Symposium on Rock Mechanics (USRMS). 1978. American Rock Mechanics Association.
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