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**European Journal of Mineralogy**
An international journal on mineralogy, petrology, geochemistry, and related sciences

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Short summary

The knowledge of physical properties of quartz as an abundant rock-forming mineral in the Earth’s crust allows for a better understanding of its dynamic processes. The thermal transport properties of single-crystal quartz are studied between –120 °C and 800 °C using a laser flash method. First, low-temperature data as well as the role of the low-to-high quartz phase transition (e.g. a transition-related non-ballistic radiative transfer) and size effects on thermal diffusivity are discussed.

The knowledge of physical properties of quartz as an abundant rock-forming mineral in the...

**Research article**
01 Feb 2021

**Research article** | 01 Feb 2021

Anisotropic thermal transport properties of quartz: from −120 °C through the *α*–*β* phase transition

- Institute of Applied Geosciences, Karlsruhe Institute of Technology (KIT), Adenauerring 20b, 76131 Karlsruhe, Germany

- Institute of Applied Geosciences, Karlsruhe Institute of Technology (KIT), Adenauerring 20b, 76131 Karlsruhe, Germany

**Correspondence**: Simon Breuer (simon.breuer@kit.edu)

**Correspondence**: Simon Breuer (simon.breuer@kit.edu)

Abstract

Thermal diffusivities of synthetic quartz single crystals have been
measured between −120 and 800 ^{∘}C using a laser flash method.
At −120 ^{∘}C, the lattice thermal diffusivities are *D*_{[001]}=15.7(8) mm^{2} s^{−1}
and *D*_{[100]}=8.0(4) mm^{2} s^{−1} in the [001] and [100]
directions, respectively. Between −80 and 560 ^{∘}C, the temperature
dependence is well approximated by a $D\left(T\right)=\mathrm{1}/{T}^{n}$ dependency (with *n*=1.824(29) and *n*=1.590(21) for the [001] and [100]
directions), whereas for lower temperatures measured thermal diffusivities
show smaller values. The anisotropy of the thermal diffusivity
*D*_{[001]}∕*D*_{[100]} decreases linearly over *T* in *α*-
and *β*-quartz, with a discontinuity at the *α*–*β*
phase transition at ${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C. In the measured signal–time curves
of *α*-quartz, an unusual radiative heat transfer is observed,
which can be linked to the phase transition. However, the effect is
already observed far below the actual transition temperature. The standard
evaluation procedure insufficiently describes the behaviour and leads to an underestimation of the thermal diffusivity
of ≥20 %. Applying
a new semi-empirical model of radiation absorption and re-emission
reproduces well the observed radiative heat transfer originating in
the phase transition. In the *β*-quartz region, the radiative heat transfer is not
influenced by the phase transition effect observed in *α*-quartz
and for the thermal diffusivity evaluation common models for (semi)transparent
samples can be used.

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How to cite.

Breuer, S. and Schilling, F. R.: Anisotropic thermal transport properties of quartz: from −120 °C through the *α*–*β* phase transition, Eur. J. Mineral., 33, 23–38, https://doi.org/10.5194/ejm-33-23-2021, 2021.

1 Introduction

Thermal transport properties such as thermal conductivity *λ*
and thermal diffusivity *D* play an important role in understanding
Earth's crust and mantle dynamics and their underlying processes (Čermák, 1982; Lenardic and Kaula, 1995).
Furthermore, growing attention is given to a precise knowledge of
heat transport in material sciences as well as in engineering (Taylor and Kelsic, 1986; Turner and Taylor, 1991).
According to Fourier's law, the heat flux *q* is equal to the thermal
conductivity *λ* of a material multiplied by the negative temperature
gradient $\partial T/\partial x$ in one
dimension:

$$\begin{array}{}\text{(1)}& q=-\mathit{\lambda}{\displaystyle \frac{\partial T}{\partial x}}.\end{array}$$

The thermal conductivity quantifies the heat flow through a
material. Comparable to chemical concentration equilibration through diffusion,
thermal diffusivity *D* quantifies the temperature equilibration capability.

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{\partial T}{\partial t}}=D{\displaystyle \frac{{\partial}^{\mathrm{2}}T}{\partial {x}^{\mathrm{2}}}}\end{array}$$

Thermal conductivity and thermal diffusivity
are interrelated through the isobaric specific heat capacity *c*_{p}
and density *ρ*:

$$\begin{array}{}\text{(3)}& \mathit{\lambda}=D\mathit{\rho}{c}_{p}.\end{array}$$

Comparable to the concept of diffusion in the kinetic gas theory, phononic (lattice) heat diffusion in solids is often approximated by (Debye, 1914; Berman, 1976; Kittel, 2005)

$$\begin{array}{}\text{(4)}& D={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}vl,\end{array}$$

with the mean velocity of phonons *v* and their mean free path length
*l*. Regarding insulated solids, phononic heat transport dominates
at low temperatures (e.g. Shankland et al., 1979; Schilling, 1999; Gibert et al., 2005).
In earlier studies, heat transfer was often solely related to acoustic
phonons (Slack, 1965; Slack and Oliver, 1971; Roufosse and Klemens, 1974),
whereas recent studies show that longitudinal and transverse optical
phonons need to be taken into account (Hofmeister, 1999, 2006; Esfarjani et al., 2011).
With increasing temperature and number of phonons, the phonon–phonon
interactions, e.g. three-phonon Umklapp processes, increase (Peierls, 1929; Ross et al., 1984; Kittel, 2005),
resulting in a temperature dependence of the thermal diffusivity frequently
being approximated by (e.g. Eucken, 1911; Zoth and Haenel, 1988; Seipold, 1992; Clauser and Huenges, 1995)

$$\begin{array}{}\text{(5)}& D\propto {\displaystyle \frac{\mathrm{1}}{T}}.\end{array}$$

Even though Fourier's law presumes length independence, heat transport can vary with specimen thicknesses, e.g. due to radiative transport mechanism for optically non-thick conditions (Hofmeister, 2019) and mode mixing effects (Hofmeister et al., 2007; Hofmeister, 2007). Thermal conductivity and thermal diffusivity of crystalline solids are directional, i.e. anisotropic, material properties (except for the cubic system) and can be described by a symmetric tensor of second rank (Nye, 1985).

Kanamori et al. (1968)Beck et al. (1978)Höfer and Schilling (2002)Branlund and Hofmeister (2007)^{a} Cited by Touloukian et al. (1973).
^{b} Only direction ⟂ [001] measured over the *α*–*β*
phase transition up to 700 ^{∘}C.
^{c} Carslaw and Jaeger (1959).
^{d} Extrapolated from 1.95 GPa (multi-anvil press), converted to
*D*.
^{e} Schilling (1999).
^{f} Parker et al. (1961).

Quartz is one of the most abundant rock-forming minerals in the Earth's
crust (Ronov and Yaroshevsky, 1969; Yaroshevsky and Bulakh, 1994). High geothermal
gradients in the uppermost tenths of kilometres (Birch, 1955)
demand a detailed knowledge about the thermal transport properties
of its constituents (Buntebarth, 1984; Siegesmund, 1996).
Thus, quartz plays an important role in developing and testing thermal
transport models (Höfer and Schilling, 2002), and detailed information
on the single-crystal thermal transport properties is required (Touloukian et al., 1973).
In terms of its significance, also for technical applications, single-crystal thermal diffusivity data
on quartz are still scarce and restricted to temperatures above 0 ^{∘}C
(Table 1). For higher temperatures, formerly
reported thermal transport properties (Höfer and Schilling, 2002; Kanamori et al., 1968)
contain both contact losses and spurious radiative gains as has been
shown by Branlund and Hofmeister (2007).

At ambient pressure, *α*-quartz (P3_{2}21) undergoes a displacive
phase transition to *β*-quartz (P6_{2}22) at ${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C (e.g. Le Chatelier, 1890; Bragg and Gibbs, 1925; Dolino, 1990).
In the vicinity of the phase transition, an incommensurate phase is observed
(Dolino, 1990). Already above ∼540 ^{∘}C,
i.e. >30 K below the phase transition, apparent changes
of anelastic (thermal expansion) and elastic properties (e.g. elastic constants and therewith acoustic wave velocities) are observed
(Raz et al., 2002; Klumbach and Schilling, 2014).

To the best of the authors' knowledge, no thermal diffusivity data
of quartz below 0 ^{∘}C have been published so far. Measurements in the [001]
and [100] directions crossing the *α*–*β* phase transition
were reported by different authors: Kanamori et al. (1968), Höfer and Schilling (2002)
and Branlund and Hofmeister (2007). The trend of the thermal diffusivity
around the *α*–*β* phase transition remains uncertain,
as a crossover in the direction of maximum thermal diffusivity was
reported by Höfer and Schilling (2002). Branlund and Hofmeister (2007)
considered this behaviour to be possibly caused by specimen size (thickness)
effects. The only data set not being potentially associated with known
shortcomings of contact methods, e.g. thermal contact resistance
effects (Lee and Hasselman, 1985; Hofmeister, 2007; Abdulagatov et al., 2015),
is the one by Branlund and Hofmeister (2007). But with respect to that,
Branlund and Hofmeister (2007) noted that the used “model of Mehling et al. (1998)
fails to fit the measured signal–time curves near the transition”.
The thermal diffusivity behaviour around the *α*–*β* phase
transition therefore remains unclear.

Phononic heat transport decreases with increasing temperature, approximated by Eq. (5) for the case that all phonon modes are excited. However, this assumption is only valid for high temperatures. At low temperatures, phonon modes freeze out and affect the heat transport properties of the solid (Kittel, 2005). Thus, low-temperature data are of fundamental interest.

The aims of this study are to (1) present low-temperature
behaviour of the heat transport in quartz; (2) better understand
the thermal diffusivity behaviour through the *α*–*β* phase
transition (including a possible crossover occurrence, Höfer and Schilling, 2002,
and size effects, Hofmeister, 2019) – this requires a new
approach to approximate the signal–time data for *α*-quartz measurements
below the *α*–*β* phase transition compared to the model used by
Branlund and Hofmeister (2007); and (3) review a proposed (Höfer and Schilling, 2002) strong correlation
of the phonon velocity and the temperature dependence of the thermal
diffusivity. Therefore, thermal diffusivity laser flash data on synthetic
quartz single crystals have been collected between −120 and 800 ^{∘}C in
different crystallographic directions and at different sample thicknesses (2–10 mm, 10×10 mm cross section).
The measurements furthermore allow us to better distinguish between phononic
and radiative heat transfer.

2 Experiment

The thermal diffusivity is measured by laser flash method (Parker et al., 1961)
using a Netzsch-Gerätebau MicroFlash laser flash apparatus (LFA) 457. A short (∼0.3 ms) laser pulse
(12.7 mm diameter, approximately ∼1 J power output, 1064 nm, IR Sources Inc.) heats up one side of
the specimen with thickness *d*, and the resulting (relative) temperature
history is recorded on the opposite (rear) side using an infrared
(IR) detector. For low-temperature measurements ($T=-\mathrm{120}$–400 ^{∘}C),
a HgCdTe (mercury–cadmium–telluride; MCT) IR detector is used; for high *T* measurements
between room temperature (RT) and 800 ^{∘}C, an indium antimonide (InSb) detector
is installed. Furnace temperatures are measured close to the specimen
using a type-K thermocouple (class 1; DIN EN/IEC 60584-1, 2013).
The chamber is continuously flushed with gaseous N_{2}
(99.99 %) with a flow rate of 50 mL min^{−1}. The temperature
increase Δ*T* of the specimen rear side during one of the measurements
is ≤1 K, and thermophysical properties ($D,\phantom{\rule{0.125em}{0ex}}\mathit{\lambda},\phantom{\rule{0.125em}{0ex}}{c}_{p}$)
and density *ρ* are assumed to be constant within Δ*T*
for each diffusivity determination. The used laser flash setup presumes the IR detector voltage outputs
to be linear to the temperature increase during one measurement.

In laser flash (LFA) experiments on (semi)transparent insulating solids, three major processes influence the detected signal–time curve: adiabatic lattice (phononic) heat transport, fast radiative transport and heat loss to the surroundings (1–3 in Fig. 1). To isolate the phononic contribution to thermal transport, the signal–time curves (black line; Fig. 1) are approximated by a combination of the adiabatic model given by Parker et al. (1961) (dashed blue line; Fig. 1):

$$\begin{array}{}\text{(6)}& T(d,t)={\displaystyle \frac{Q}{\mathit{\rho}{c}_{p}d}}\left[\mathrm{1}+\mathrm{2}\sum _{n=\mathrm{1}}^{\mathrm{\infty}}(-\mathrm{1}{)}^{n}\mathrm{exp}\left({\displaystyle \frac{-{n}^{\mathrm{2}}{\mathit{\pi}}^{\mathrm{2}}\mathrm{D}t}{{d}^{\mathrm{2}}}}\right)\right],\end{array}$$

(*Q* absorbed heat pulse energy per unit area) with a custom heat loss model and
a fast radiative portion (dotted orange and dash-dotted red line;
Fig. 1). Heat loss from the surfaces of
the specimen is approximated by subtracting a proportional temperature
difference d*T*_{i, heat loss} from the temperature *T*_{i}
for each time interval Δ*t* (Schilling, 1998; Höfer and Schilling, 2002):

$$\begin{array}{}\text{(7)}& \mathrm{d}{T}_{i,\phantom{\rule{0.125em}{0ex}}\mathrm{heat}\phantom{\rule{0.125em}{0ex}}\mathrm{loss}}=h{T}_{i}\mathrm{\Delta}t,\end{array}$$

with fitting constant *h* as the only free parameter. This approximation assumes
1-D heat flow and hence mainly heat losses from the front and rear surfaces. This
approximation is well fulfilled for thin platelets. For thicker samples, 2-D effects
may occur, e.g. if heat losses perpendicular to the assumed 1-D heat flow influence the observed signal.

Fitting fast direct radiative heat transfer is different for
measurements at temperatures below and above the *α*–*β*
phase transition (see Fig. 2c, d). For *T*≤573 ^{∘}C,
an absorption/re-emission approximation for semitransparent samples
is applied. For *T*>573 ^{∘}C, the ballistic (boundary-to-boundary)
transfer is defined as portion of the fast temperature decrease
of the specimen's front surface instantaneously heated by the
laser. The corresponding temperature distribution is given in Parker et al. (1961)
and in a good approximation represented by $\propto \mathrm{1}/\sqrt{t}$.

$$\begin{array}{}\text{(8)}& {T}_{i,\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}=\left\{\begin{array}{ll}{T}_{\mathrm{0}\phantom{\rule{0.125em}{0ex}}}{\mathit{\tau}}^{i}+b,& T\le \mathrm{573}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\text{C},\phantom{\rule{0.25em}{0ex}}i\in {\mathbb{N}}_{\mathbb{0}}\\ & \\ \frac{{T}_{\mathrm{0}}}{\sqrt{{t}_{i}}}+b,& T>\mathrm{573}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\text{C},\phantom{\rule{0.25em}{0ex}}i\in \mathbb{N}\end{array}\right.\end{array}$$

*t*_{i} indicates the time steps. Constants approximated during the fit process
are *T*_{0}, which characterizes the initial height of the radiative portion
(see inset Fig. 1), *τ*, which controls
the magnitude of the temperature decrease, and *b*, which is an optional shift
of the derived relative *T* signal. For measurements at temperatures
below 150 ^{∘}C, the radiative heat transfer is negligible (Fig. 2b).

Depending on the individual raw data, the signal–time curve is fitted
to 5–$\mathrm{10}\times {t}_{\mathrm{1}/\mathrm{2}}$ (with ${t}_{\mathrm{1}/\mathrm{2}}=\mathrm{0.1388}\phantom{\rule{0.125em}{0ex}}{d}^{\mathrm{2}}/D$;
Parker et al., 1961) by in-house software using the least-squares
Levenberg–Marquardt algorithm (Levenberg, 1944; Marquardt, 1963).
A pulse-time correction according to Breuer and Schilling (2019)
is applied to each approximation. Thickness corrections with increasing *T* are applied using linear thermal
expansion coefficients *α*_{L}(*T*) for the [001] and [100]
directions as given by Klumbach (2015) (based on Raz et al., 2002).
Reported thermal diffusivities are the average of three independent
measurements at the corresponding temperature. The experimental reproducibility
is 2 %–3 %, while the accuracy is 5 %, confirmed by measurements on Pyroceram
9606, Inconel 600 and stainless-steel 310 standard samples. Uncertainties
can be higher in the vicinity of phase transitions, e.g. due to a pronounced distortion of the signal–time curves (overlap of different heat transport processes)
or by using very small (McMasters et al., 2017) and/or thick
sample dimensions (Swank and Windes, 2014).

The temperature dependence of the thermal diffusivity *D*(*T*) of quartz
below 550 ^{∘}C is fitted using the empirical model proposed by Hofmeister et al. (2014)
(*T* in K):

$$\begin{array}{}\text{(9)}& D\left(T\right)=F{T}^{-G}+HT,\end{array}$$

where the *F**T*^{−G} term describes the contribution of lattice phonons
to heat transport and the rear high *T* term (*H**T*) represents a
contribution of bulk infrared phonon polaritons. Fitting constants are $F,G\phantom{\rule{0.125em}{0ex}}(>\mathrm{0})$ and *H* (≥0).

For the thermal diffusivity measurements, six 10×10 mm platelets with various thicknesses
were prepared from one synthetic quartz single crystal in gem quality
supplied by Maicom Quarz (in accordance with DIN EN/IEC 60758, 2016).
The specimens were cut so that the direction perpendicular to the surface corresponds to the
crystallographic [100] (i.e. *a* axis) and [001] (i.e. *c* axis) direction (Table 2), respectively.
The angular deviations of the samples were proven by ultrasonic sound velocity
measurements and are <1^{∘}. The tabulated thicknesses were measured using a micrometer; the
deviation from parallelism is ≤1^{∘}. To reduce radiative heat
transfer in LFA measurements (Hasselman and Merkel, 1989), all
platelets were sputter coated with Au (99.99 %) using a Denton Vacuum
LLC Desk V to ∼0.1 µm (according to the deposition rates given
by the manufacturer). To ensure consistent laser pulse absorption
(Cernuschi et al., 2002; Stryczniewicz et al., 2017) and
to buffer oxygen fugacity at high *T* (Pertermann et al., 2008),
specimens were coated with graphite spray (CRC Industries) of about
∼15 µm thickness per side. Platelet surfaces were slightly
roughened to mitigate possible non-uniformity of the laser pulse absorption and
to increase the adhesion of the coatings (Branlund and Hofmeister, 2007).

3 Results

The as-derived single-crystal thermal diffusivity data of quartz are shown in Fig. 2a as a function of temperature for the main directions ([001] and [100]; data in Tables S1 and S2 in the Supplement).

The data of high- and low-temperature measurements show a distinct
*D*(*T*) evolution nearly proportional to 1∕*T* between −80 and ∼550 ^{∘}C for
both crystallographic orientations as expected, e.g. for three-phonon
Umklapp processes (Peierls, 1929; Kittel, 2005).
The direction of faster temperature equilibration coincides with the
3-fold axis in the [001] direction. In the vicinity of the *α*–*β* phase transition, thermal
diffusivities stop following the trend of $D\propto \mathrm{1}/T$. The data
show a further distinct decrease in lattice thermal diffusivity, starting
at *T*∼560 ^{∘}C and reaching its minimum of *D*_{[100]}∼0.56 mm^{2} s^{−1}
and *D*_{[001]}∼0.60 mm^{2} s^{−1}
at the *α*–*β* phase transition temperature (HT2_{[001]})
or slightly above (${T}_{D,\phantom{\rule{0.125em}{0ex}}\mathrm{min}}=\mathrm{575.6}$ ^{∘}C for HT1_{[001]} and
578.0 ^{∘}C for HT1_{[100]} and HT2_{[100]}). After the phase transition,
*D* values sharply increase and stay more or less constant up to
800 ^{∘}C at *D*_{[001]}∼0.98 mm^{2} s^{−1} and *D*_{[100]}∼0.86 mm^{2} s^{−1}.
Thermal diffusivities in hexagonal *β*-quartz are about 15 % and 21 % higher for the [001] and [100]
directions compared to fitted *D* values of *α*-quartz extrapolated
to ${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C (black lines; Fig. 2a).
In the *β* phase, the *c* axis is still the direction of maximum
thermal diffusivity. Green and blue areas in Fig. 2a
correspond to thermal diffusivity variations (effect of impurities) of quartz single crystals
measured by Branlund and Hofmeister (2007). Data of this study lie close
to the lower end of these variations and slightly below the thermal diffusivities
reported by Kanamori et al. (1968) (leftwards triangles) and Höfer and Schilling (2002)
(stars). Data fits (least-squares minimization) according to Eq. (9)
for $-\mathrm{120}\le T\le \mathrm{550}$ ^{∘}C are presented as black lines for the [001]
and [100] directions in Fig. 2a (drawn up to
${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C for illustration). The resultant fit
parameters are (*D* in mm^{2} s^{−1}; *T* in K)
${D}_{\left[\mathrm{001}\right]}\left(T\right)=\mathrm{1.86}\left(\mathrm{30}\right)\times {\mathrm{10}}^{\mathrm{5}}\phantom{\rule{0.125em}{0ex}}{T}^{-\mathrm{1.824}\left(\mathrm{29}\right)}$ (*R*^{2}=0.990)
for the [001] direction and ${D}_{\left[\mathrm{100}\right]}\left(T\right)=\mathrm{2.71}\left(\mathrm{31}\right)\times {\mathrm{10}}^{\mathrm{4}}\phantom{\rule{0.125em}{0ex}}{T}^{-\mathrm{1.590}\left(\mathrm{21}\right)}+\mathrm{7.8}\left(\mathrm{29}\right)\times {\mathrm{10}}^{-\mathrm{5}}\phantom{\rule{0.125em}{0ex}}T$
(*R*^{2}=0.996) for the [100] direction, respectively. Fitting the [100]
direction, data benefit from including the *H**T* term (see Eq. 9), whereas for *c*-axis data, the fitting
gives $H=\mathrm{0}\phantom{\rule{0.125em}{0ex}}(\pm \mathrm{5}\times {\mathrm{10}}^{-\mathrm{5}})$. Thus, no improvement of the fit
quality is achieved including the high-temperature term for the latter. For $T<-\mathrm{80}$ ^{∘}C,
the measured thermal diffusivities are lower than the data approximations after
Eq. (9), especially for the higher diffusive [001] direction.

The front parts of raw data signal–time curves are exemplarily shown in Fig. 2b–d
for different furnace temperatures. For measurements at low temperatures
(Fig. 2b), direct radiative heat transfer is effectively suppressed by the sputtered gold layer
and data fittings are independent of Eq. (8). Furthermore,
heat losses are marginal within the short time span of low-temperature
LFA experiments. For measurements above *T*∼150–200 ^{∘}C, fast radiative
heat transfer is explicitly visible in all signal–time curves and
increases with temperature (Fig. 2c). The shape
of the radiative heat transfer portion in raw data signal–time curves
shows a different behaviour for *T*≤573 ^{∘}C and *T*>573 ^{∘}C,
which is considered in the applied approximation procedure (Eq. 8)
(thick grey lines in Fig. 2c, d). Within the *α*-quartz
phase, the intensities of the radiative heat transfer in signal–time curves
decrease only slowly and are characterized by an almost-linear decline
with *t* (Fig. 2c). The origin of this radiation is considered in
the quartz sample itself. This type of intrinsic radiation cannot be (effectively)
suppressed by the gold layer sputtered on the back side of the specimen.

In contrast, at temperatures above the phase transition, this radiative contribution vanishes. Instead, in the *β*
phase region, the radiative heat transfer shows a rapid drop in intensity
well described by a $\propto \mathrm{1}/\sqrt{t}$ dependence (Fig. 2d)
in accordance to a standard boundary-to-boundary (ballistic) transfer.
Even though radiative heat transfer is noticeable for temperatures
as low as *T*∼150–200 ^{∘}C, the evaluated lattice (phononic) *α*-quartz
thermal diffusivity is consistently unaffected by radiative transport
and well described by Eq. (9) until a stronger
decrease in thermal diffusivity starts at ∼560 ^{∘}C (Fig. 2a).

The anisotropy of the single-crystal thermal diffusivity of quartz
is shown in Fig. 3 as *D*_{[001]}∕*D*_{[100]}.
The anisotropy ratios are well approximated by linear fits in the *α*-
as well as in the *β*-quartz region (solid and dashed black lines
in Fig. 3). The fit to the ratio of LT specimens
gives (*T* in degrees Celsius) ${D}_{\left[\mathrm{001}\right]}/{D}_{\left[\mathrm{100}\right]}\left(T\right)=-\mathrm{1.060}\left(\mathrm{11}\right)\times {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}T+\mathrm{1.856}\left(\mathrm{2}\right)$
(solid black line; Fig. 3a) and for specimens
HT1 and HT2 ${D}_{\left[\mathrm{001}\right]}/{D}_{\left[\mathrm{100}\right]}\left(T\right)=-\mathrm{0.963}\left(\mathrm{12}\right)\times {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}T+\mathrm{1.788}\left(\mathrm{5}\right)$
(dashed black line; Fig. 3a, b, *T*≥200 ^{∘}C). LT as well as HT1/2 anisotropy ratios fit well to the data
on anisotropy given by Branlund and Hofmeister (2007) on various quartz
samples (grey area; Fig. 3) and data reported
by Kanamori et al. (1968) within $\mathrm{25}\le T\le \mathrm{500}$ ^{∘}C. The data
on *D*_{[001]}∕*D*_{[100]} by Höfer and Schilling (2002) are within
the shown anisotropy range (Branlund and Hofmeister, 2007) but up
to ∼10 % lower than the data reported in this study. At the phase
transition, the anisotropy *D*_{[001]}∕*D*_{[100]} rapidly decreases
by about 7 % but stays above 1.0 at all temperatures. Thus, in the [001]
direction, the highest thermal diffusivity is observed over the whole temperature range and no
crossover is observed for any probed specimen with thicknesses *d*∼2.0–4.7 mm (see Table 2). The
slope of the linear fit to the anisotropy ratio *D*_{[001]}∕*D*_{[100]}
in the *β*-quartz phase is significantly flatter than for the
*α* phase and gives ${D}_{\left[\mathrm{001}\right]}/{D}_{\left[\mathrm{100}\right]}=-\mathrm{0.088}\left(\mathrm{13}\right)\times {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}T+\mathrm{1.200}\left(\mathrm{8}\right)$
(dotted black line; Fig. 3b).

4 Discussion

The presented data for low-temperature *α*-quartz thermal diffusivity
fit well into the trend of already-published data (see Table 1)
at and above room temperature (Fig. 2a). A small
systematic offset can be recognized for the data measured with the
low- and the high-temperature setup. As the preparation was kept the
same for all samples, this offset might be caused by different characteristics
(nonlinearity, sensitivity) of the two IR detectors and thus reflects the accuracy of the experiment.

It is noticeable that for low-temperature measurements ($T<-\mathrm{80}$ ^{∘}C)
the thermal diffusivities are significantly and systematically lower than
the data fits according to Eq. (9) (using all
data between −120 and 550 ^{∘}C), especially for the higher diffusive *c*-axis
direction. To better understand the origin of this deviation, different
considerations are discussed. (a) This behaviour can be caused by the
effect of IR detector nonlinearity leading to an underestimation of
thermal diffusivity (Hasselman and Donaldson, 1990; Baba and Ono, 2001).
Negative impacts assuming a linear detector voltage to temperature
output are high especially at the lower end of the IR detectors' operating
*T* range and significant even for low absolute temperature rises
Δ*T* during each measurement (Hasselman and Merkel, 1989; Hoefler and Taylor, 1990).
This could also explain the deviation between measurements with low- and high-temperature IR detectors as observed in Fig. 2a, where thermal
diffusivities measured with the high *T* InSb detector (at its low-temperature limit) depict lower values.
(b) Specimens measured at low temperatures have a higher Δ*T*
due to smaller heat capacity *c*_{p} of *α*-quartz compared
to measurements performed at high temperatures (assuming similar pulse
energy absorption) (Lord and Morrow, 1957; Akaogi et al., 1995; Chase, 1998).
As a consequence of higher Δ*T*, thermal diffusivity values
of *α*-quartz at very low *T* are quite possibly too low as
they reflect the property at higher temperatures (e.g. see Akoshima et al., 2013). This effect is more pronounced
if a strong temperature dependence of thermal diffusivity *D* is
present, as is the case for *α*-quartz at very low temperatures
(see Fig. 2a). But deviations of measured
and fitted thermal diffusivities of up to 20 % are very unlikely to
be solely caused by detector nonlinearity and effects of an
increase in Δ*T* due to changes in heat capacity at low temperatures.
The used model to fit temperature-dependent thermal diffusivities
(Eq. 9) is originally tested for *T*≥25 ^{∘}C (Hofmeister et al., 2014). But this approximation
seems to be inappropriate at very low temperatures, i.e. at $T<-\mathrm{80}$ ^{∘}C for *α*-quartz.
(c) At low temperatures, where an increased portion of the phonon modes
is frozen out, the temperature dependence of the heat capacity
strongly increases (∝*T*^{3} at low *T*) and likewise the
thermal conductivity increases before it reaches a maximum followed
by a decrease $\propto {e}^{{\mathrm{\Theta}}_{D}/T}$ (Casimir, 1938; Lord and Morrow, 1957; Zeller and Pohl, 1971; Gross and Marx, 2014).
Furthermore, with an increasing mean free path length *l* due to a reduction
in phonon–phonon interactions at low temperatures, lattice imperfections
and geometrical effects become more and more dominant (de Haas and Biermasz, 1935; Blakemore, 1974; Kittel, 2005).
It seems therefore reasonable for the intrinsic *D*(*T*) trend to deviate
from a $D\propto \mathrm{1}/{T}^{G}$ behaviour for measurements at low temperatures.
Hence, we conclude that the observed behaviour is not the result of
experimental uncertainties alone and that the mentioned deviations
to the data fits are in part a manifestation
of the intrinsic behaviour of *α*-quartz as a reduced number of phonon
modes are excited and the scattering of phonons at 1-D and 2-D defects becomes more dominant at lower temperatures.

With the above exception, fits covering low- and high-temperature measurements
up to 550 ^{∘}C are well approximated by the model proposed by Hofmeister et al. (2014)
(see black lines; Fig. 2a). The presented
high-temperature data have the highest conformity to laser flash data
presented by Branlund and Hofmeister (2007) for phonon-dominated heat transfer processes. In contrast, contact methods
used by Höfer and Schilling (2002) and Kanamori et al. (1968)
only have a good agreement with the data of this study in the lower temperature
range (RT to ∼200 ^{∘}C) and show a less pronounced decrease of
*D*(*T*) towards higher temperatures. This is most likely due to direct
radiative heat transfer not (sufficiently) being taken into account
in these data sets (Andre and Degiovanni, 1995; Mehling et al., 1998; Branlund and Hofmeister, 2007).
A mix of radiative and phononic transfer seems to lead to a positive
slope in thermal diffusivities at temperatures above the phase transition,
whereas the presented data on lattice thermal diffusivities in this
study are in accordance with the data by Branlund and Hofmeister (2007),
which indicate *D*(*T*) being roughly constant at *T*>600 ^{∘}C.
The thermal diffusivities for *α*-quartz samples (this study)
are up to ∼12 % smaller than the *D* range of various quartz
samples presented by Branlund and Hofmeister (2007) (see green
and blue areas; Fig. 2a). Above 573 ^{∘}C, the differences
are less distinct and the thermal diffusivities lie at the lower limits
of published *β*-quartz laser flash data. A small but noticeable
negative influence, i.e. lowering thermal diffusivity, of the
applied graphite coating on the specimens is conceivable (e.g. Albers et al., 2001; Lim et al., 2009) but unable to
explain the observed differences alone. Therefore, low(er) thermal diffusivity data of quartz measurements in this study might be
caused by low amounts of interstitial ions and the occurrence of phonon
scattering at hydroxyls (OH), both decreasing the thermal diffusivity (Branlund and Hofmeister, 2007).
This interpretation is further supported by the low-temperature behaviour ($T<-\mathrm{80}$ ^{∘}C),
where part of the deviation to the applied 1∕*T*^{n} approximation is probably caused by defects.

The two recent publications on thermal diffusivity of single-crystal
*α*–*β* quartz (i.e. studies by Höfer and Schilling, 2002; Branlund and Hofmeister, 2007) show a different behaviour for
the crystallographic direction of the maximum thermal diffusivity
at elevated temperatures. Branlund and Hofmeister (2007) confirmed measurements
by Kanamori et al. (1968) and found that the direction of
maximum thermal diffusivity does not change through the phase transition,
whereas Höfer and Schilling (2002) presented data where a crossover
(i.e. a change in the crystallographic direction of maximum
thermal diffusivity) is described.

In this study, measurements on both
specimens (HT1 and HT2) show no crossover in the *D*(*T*) data
at any temperature (Fig. 2a). After Branlund and Hofmeister (2007),
the crossover observed in measurements by Höfer and Schilling (2002)
can be explained by the specimen dimensions used by the latter. Accordingly,
mixing of longitudinal optic (LO) and transverse optic (TO) modes
can cause an inflation of *D* parallel to [100], and thus the observed
crossover occurs (Hofmeister, 2007; Hofmeister et al., 2007; Hofmeister, 2019).
In order to test this hypothesis of a sample thickness dependence
of the crossover, an additional thick quartz sample has been measured over the
phase transition using the laser flash method with an increased laser power
output of ∼5 J (Fig. 4; data in Table S3).
This cubed sample has an edge length of ∼10 mm (10×10 mm cross section), exceeding the
thinnest specimens measured by Höfer and Schilling (2002).

In general, thermal diffusivities presented in Fig. 4a
show a large variation in *D*(*T*), which is due to different
measurement techniques and specimens used. Relative trends show that
thin specimens measured in this study (e.g. HT1; Fig. 4a)
inhere no crossover in the crystallographic direction of maximum thermal
diffusivity between *α*- and *β*-quartz, similar to the data
presented by Kanamori et al. (1968) and Branlund and Hofmeister (2007)
(not shown in Fig. 4). The derived anisotropy *D*_{[001]}∕*D*_{[100]}
is >1.0 at all temperatures (Fig. 4b). For
specimens with *d*=8.5–20 mm measured by Höfer and Schilling (2002)
and for the quartz cube (*d*∼10 mm, this study; Fig. 4a),
*D*(*T*) data show a crossover around the phase transition with *D*_{[100]}
getting the direction of maximum thermal diffusivity in the *β*
phase. The corresponding anisotropy *D*_{[001]}∕*D*_{[100]} is characterized
by a drop just under 1.0 for the cubic specimen (Fig. 4b).
Thus, increasing the specimen thickness reproduces the crossover formerly
being observed. These results are in accordance with the interpretation
by Branlund and Hofmeister (2007) that TO–LO mixing significantly affects
thermal diffusivity measurements. It should be clarified that for
such thick specimens (∼10 mm) measured by LFA method the reproducibility
and accuracy of the diffusivity determination is much lower compared
to appropriate thin specimens and at the resolution limit of the experiment
for the used setup. This is due to the very small rear-side temperature
increase for thick samples, synonymous with a low signal-to-noise
ratio for the data measured at the IR detector. For the thick samples, modifications of the used simplified
1-D evaluation procedure would be necessary to take 2-D effects into account.

After Hofmeister (2007),
limiting the thickness-to-diameter ratio to *d*∕ø = 0.1 gives
intrinsic thermal diffusivities without or at least negligible influence
of mode mixing. In this study, all specimens exceed this ratio and
have *d*∕ø ∼0.2–0.5 (see Swank and Windes, 2014; ASTM E1461, 2013). As the results in this study
show good agreement with the thermal diffusivity data of single-crystal
quartz presented by, e.g. Branlund and Hofmeister (2007), it is
concluded that a possible influence of mode mixing on thermal diffusivity
LFA measurements is still negligible for quartz specimens at *d*<5.0 mm with the used experimental setup.

Data fits show that the lowest measured thermal diffusivities lie
up to ∼5 K above the actual phase transition temperature of
${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C (except HT2_{[001]}; thermocouple
tolerance is ≈2.3 ^{∘}C at 573 ^{∘}C, DIN EN/IEC 60584-1, 2013).
As the temperature is recorded close to but not directly at the sample surface,
the observed temperature of the thermocouple may record a slightly higher temperature. However, such observation has also been made by
Branlund and Hofmeister (2007). The highest amount of direct radiative heat transfer, linked with
a maximum distortion in signal–time curves (reducing the accuracy of the derived data), can be assigned to measurements
of minimum thermal diffusivity. The observed
shift in the minimum thermal diffusivity could therefore be linked to the formation of
an incommensurate (inc) phase between ∼573 and ∼574.3^{∘}C
(Bachheimer, 1980; Dolino et al., 1983; Dolino, 1990; Klumbach, 2015).
Due to the experimental setup with 2 K temperature steps, thermocouple
uncertainties and most notably the specimen temperature increase Δ*T*
within each measurement, it is not possible to clearly discriminate
such variations and hence processes within this narrow temperature
range. In addition, the phase transition temperature
could be affected by, e.g. impurities and defects in the crystal.

The crossover in *v*_{mean} at the *α*–*β* phase transition
of quartz led Höfer and Schilling (2002) to conclude that the crossover
in the thermal diffusivity observed for the thick samples is the result of a direct
correlation of the thermal diffusivity *D* and the elastic behaviour (i.e. sound velocities). The single-crystal platelets used in this study are much
thinner, better satisfying the hypothesis of 1-D heat transport. Therefore,
the observation made by Höfer and Schilling (2002), that the as-derived
*v*_{mean} is the main factor controlling the *D*(*T*) behaviour at high temperatures, is verified here.

To derive the mean free path lengths *l* of quartz in the [100] and [001] directions, Eq. (4)
is used as a first approximation. For the mean velocities *v*_{mean}, the quadratic mean of
the directional *v*_{p} and both *v*_{s1} and *v*_{s2} sound velocities
are used (${v}_{\mathrm{mean}}=\sqrt{({v}_{p}^{\mathrm{2}}+{v}_{s\mathrm{1}}^{\mathrm{2}}+{v}_{s\mathrm{2}}^{\mathrm{2}})/\mathrm{3}}$) based on the elastic constants of single-crystal quartz
(Kammer et al., 1948; Zubov and Firsova, 1962; Ohno, 1995).
For temperatures between −120 and 20 ^{∘}C, the sound velocities are
extrapolated using an exponential fit which leads to the solid lines
for *v*_{mean} of *α*-quartz shown in Fig. 5.
The crossover in *v*_{mean} at the phase transition temperature
(Fig. 5) is the result of *v*_{p} and *v*_{s1}, *v*_{s2}
velocities in the [100] direction exceeding the sound velocities in the
[001] direction (which in turn expresses a change in elastic constants
with *T*).

The trend of *l*(*T*) (Fig. 5) mainly follows that of
*D*(*T*) (see Fig. 2a). In contrast to the strong variation of the mean
free path length *l*(*T*) from 8.8 nm (in the [001] direction; 4.9 nm in the [100]
direction) to ∼0.5 nm, the mean sound velocities vary only ∼20 % over the entire temperature range (Fig. 5).
Thus, it follows that the influence of *v*_{mean} on *D*(*T*) is much
less distinct than that of *l*(*T*) over the investigated temperature range
(including the behaviour during the *α*–*β* phase transition).
The changes in the sound velocities *v*_{mean} in quartz are
not directly correlated to the mean free path length in thin samples
(this study) as deduced by Höfer and Schilling (2002) for thick samples.
As a consequence, the thermal diffusivity is mainly related to the mean free path length with minor contribution
of the mean phonon sound velocity, assuming 1-D heat transport.

The discussion of Höfer and Schilling (2002) is based on the
assumption that acoustic phonons dominate thermal diffusivity of quartz
and the mean free path length is strongly dependent on the phonon
wavelength. However, the possible size dependence of the measured
thermal diffusivities and the missing direct correlation between the
sound velocities and the mean free path length can be seen as an indication for
2-D heat transport effects and that longitudinal and transverse optical phonons (especially at elevated
temperatures) need to be taken into account to better describe the
observed behaviour. This would be in agreement with independent density functional
theory calculations and observed phonon lifetimes for other oxides
such as magnesium oxide (MgO) (Dekura and Tsuchiya, 2017; Giura et al., 2019). Together with the
deviation of the thermal diffusivity from a simple 1∕*T*^{n} dependence at low
temperatures ($T<-\mathrm{80}$ ^{∘}C) and the comparison to data presented by
Branlund and Hofmeister (2007), one can conclude that the main portion of the
temperature dependence of quartz at elevated temperatures is the result of
phonon–phonon interactions, whereas the absolute thermal diffusivity and the
low-temperature behaviour indicate a significant influence of 1-D/2-D defects
and inclusions, and that the minor variations in the elastic properties have
a subordinate influence on the thermal transport behaviour.

The influence of direct heat transfer by radiation on the raw data
signal–time curve in laser flash experiments of (semi)transparent
materials is well known (e.g. Andre and Degiovanni, 1995; Mehling et al., 1998).
But for materials undergoing phase transitions within the temperature
range of thermal diffusivity measurements, it seems likely that
underlying processes of radiative heat transfer can be different
from those of common ballistic transfer in transparent or semitransparent
media. Branlund and Hofmeister (2007) noticed in their study that direct
radiative heat transfer exceedingly rises close to the *α*–*β*
phase transition, unable to get fitted by Mehling et al. (1998)'s
model. But this finding was not considered in their evaluation routine, and it
is shown here how a separate evaluation procedure can be applied (Figs. 2c–d,
7c). Direct radiative heat declines within a few degrees above the
phase transition. Branlund and Hofmeister (2007) assume that the laser
pulse triggers an *α* to *β* phase transition (in portion)
of the sample at and close to the heated surface already at *T*<573 ^{∘}C.
The following *β*–*α* inversion is coupled with a release
of latent heat observable through a non-ballistic radiative contribution in LFA
measurements. This process lasts longer (Fig. 2c, Eq. 8,
*T*≤573 ^{∘}C) than common ballistic
direct radiative heat transfer (Fig. 2d, Eq. 8, *T*>573 ^{∘}C), which
represents fast cooling of the specimen front surface after
the heating by the laser pulse.

To visualize and better understand the change of heat transfer by radiation in *D*(*T*)
measurements, the ratio of the direct radiative portion *T*_{0, rad}
(Eq. 8; see inset Fig. 1)
to the theoretical adiabatic maximum temperature *T*_{max} on the
specimen rear side (see Fig. 1)
is shown in Fig. 6 in logarithmic scale (data in the Supplement; Tables S4–S9). Scattering
of the data in the low-temperature region is the result of experimental
uncertainties (i.e. approaching resolution limit of the detector,
*T*_{0, rad} very small and close to 0 V), exaggerated by the
logarithmic scaling.

The normalized radiation ratio (${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$) shows a first strong increase
with *T* up to the phase transition at ${T}_{\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}}=\mathrm{573}$ ^{∘}C.
This trend is described by ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}=\mathrm{9.9}\left(\mathrm{4}\right)\times {\mathrm{10}}^{-\mathrm{3}}/(\mathrm{1}-\mathrm{1.7}(\mathrm{1})\times {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}T)\times \mathrm{100}$ %
(solid line; Fig. 6, *T* in degrees Celsius). No difference
can be seen for specimens measured in the [100] (white filling) and
[001] directions (black filling) or for measurements on thinner
(upwards triangles) and thicker specimens (downwards triangles). The ratios
of the data measured with the HgCdTe IR detector for low temperatures
(squares) lie slightly above the data measured by InSb detector for
*T*>150 ^{∘}C, which is likely due to different detector characteristics (e.g. sensitivity, nonlinearity). At the *α*–*β* phase transition, the radiative contribution ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$
reaches its maximum followed by a sharp and distinct decrease within a few Kelvin.
In the *β*-quartz phase, the ratio ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$ again
shows a strong increase with *T* but a dependence on specimen
thicknesses (upwards/downwards triangles) becomes apparent. Thicker HT2 specimens have
a higher onset and stronger increase with *T* than thinner HT1 specimens,
while again no significant differences for different crystallographic
directions ([001] and [100]) are present. To guide the eyes,
the temperature development of ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$ in the *β*
phase is shown by dashed (HT2) and dotted (HT1) lines (Fig. 6) for ∼2 mm and ∼4–4.7 mm thick samples.

The observed strong decrease in the radiation ratio directly after passing
the phase transition at 573 ^{∘}C indicates that at least two different
mechanisms influence the radiative heat transfer in LFA measurements through the phase
transition, also noticeable through the different shapes of the radiative heat transfer portion in the signal–time
data comparing Fig. 2c (*T*<573 ^{∘}C) and Fig. 2d
(*T*>573 ^{∘}C), in the first 200 ms. This is further supported by the fact that
there is no ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$ dependence on sample thickness *d* below
573 ^{∘}C but above.
The radiation below the transition is supposed to be
primarily caused by the phase change itself within the sample and thus non-ballistic
(related to the release of latent heat; see above and Branlund and Hofmeister, 2007).
This affects laser flash measurements as low as *T*∼150–200 ^{∘}C
(see Figs. 2c, 6), which is well conceivable as the absolute temperatures at the front
surface of the specimen initially rise to high values (Parker et al., 1961),
well above the phase transition temperature due to the short duration
of the laser pulse. That the gold layer on the back side of the specimen
cannot suppress this radiative heat generated within the sample below
the phase transition might be explained by a decreased Au reflectance
for the wavelength of this non-ballistic radiation. However, the effectively
suppressed ballistic contribution around 600 ^{∘}C is clear evidence
that the thickness of the gold coating is sufficient and that this non-ballistic
contribution is not the result of the coating. The net effect of radiative heat transfer originating from the phase
transition is approximated by subtracting the ballistic radiative heat
transfer (dashed and dotted line; Fig. 6) from the assumed
non-ballistic portion (dark grey line in Fig. 6).
If this is taken into account (Eq. 8), *D*(*T*) data fits
according to Eq. (9) describe the measured
data well in both crystallographic directions ([001] and [100]) up
to *T*∼560 ^{∘}C (see black lines in Fig. 2a).
This indicates that the used semi-empirical approach to consider the radiative
heat transfer for *T*<573 ^{∘}C (Eq. 8) is well
suited to describe the measured data while taking latent heat effects into account. In contrast, at
higher temperatures (*T*>573 ^{∘}C), where ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$ shows
a sample thickness dependence (Fig. 6), the
radiation is inferred to be the common boundary-to-boundary (ballistic)
radiative heat transfer. This is also supported by the fact that standard
evaluation routines accounting for ballistic radiative transfer
(Eq. 8 for *T*>573 ^{∘}C and Mehling et al., 1998)
fit the observed signal–time data well (see Figs. 2d,
7d). This direct ballistic radiative heat
transfer increasingly inheres the signal–time data towards higher
temperatures (around the *α*–*β* phase transition and above;
dashed and dotted lines in Fig. 6), which is
explained by a reduced reflectance of the sputtered Au layer at high
temperatures (Loebich, 1972). As the thermal diffusivity
*D*(*T*) in the *β* phase shows no specimen thickness dependence but the ratio ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$
does, this supports the fact that the model chosen here is well
suited to approximate the signal–time curves for thermal diffusivity determination at temperatures above 573 ^{∘}C (see
Figs. 2d, 7b, d). A further
increase in fit quality may be achieved by mixing the two radiative
heat transfer models (Eq. 8) in the temperature
range ∼400–573 ^{∘}C as indicated by ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$ in
Fig. 6.

Against the background of having two different effects contributing
to the radiative heat transfer portion in the signal–time curves,
measured HT1 data have been re-evaluated using the most frequently
used standard fit procedure for (semi)transparent specimens after
Mehling et al. (1998). This way, systematic errors for not
taking into account the non-ballistic radiative transfer in the data
evaluation routine below 573 ^{∘}C can be estimated.
The derived differences Δ*D*
between the two model approaches are given as percentages in Fig. 7a, b. For measurements up to ∼200–250 ^{∘}C, the data fits show no significant differences independent
of the used fit routine and $\mathrm{\Delta}D\le \pm \mathrm{2}$ % (Fig. 7a).
At higher temperatures up to 573 ^{∘}C (coinciding with the first rise of radiation
ratio ${T}_{\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{rad}}/{T}_{\mathrm{max}}$; Fig. 6a), differences
in the fitted thermal diffusivities are increasing, exceeding −5 %
at *T*∼500 ^{∘}C, and the maximum is reached at the phase transition
with $\mathrm{\Delta}D\ge -\mathrm{20}$ % (Fig. 7b). Negative differences mean that the data approximations
of this study (accounting for non-ballistic and ballistic transfer) yield higher thermal diffusivities than fits using the
model by Mehling et al. (1998), which only account for standard ballistic radiative transfer. Exemplary signal–time data fits
for the two different evaluation routines are presented in Fig. 7c
for *T*=550 ^{∘}C. It shows that the front part of fast radiative heat
transfer (inset) is well approximated by the non-ballistic absorption/re-emission
model proposed in this study (thick grey line in Fig. 7c)
and that the standard model for (semi)transparent specimens improperly
approximates the observed behaviour within the first few hundred microseconds
(dashed red line; Fig. 7c). The resulting derived
thermal diffusivities are in this example *D*=0.89(3) mm^{2} s^{−1}
(this study) and *D*=0.82(5) mm^{2} s^{−1} (fit after Mehling et al., 1998),
and thus $\mathrm{\Delta}D\sim -\mathrm{8}$ %. Above the phase transition temperature,
the derived thermal diffusivity becomes independent of the used fitting procedure for both crystallographic directions. At
*T*=700 ^{∘}C, the fitted signal–time data shown in Fig. 7d
reveal that both approximations accounting for ballistic radiative heat transport
(Eq. 8 for *T*>573 ^{∘}C and Mehling et al., 1998) are comparably well with a minor exception
for the very first few tens of milliseconds of the model by Mehling et al. (1998).
Consequently, there is no significant difference in the deduced diffusivity values
independently of the used fitting routine with *D*=0.98(3) mm^{2} s^{−1}
(this study) and *D*=0.97(4) mm^{2} s^{−1} (after Mehling et al., 1998),
respectively.

In view of the direct radiative heat transfer, it is concluded that laser flash measurements of (semi)transparent specimens can be significantly influenced by phase transitions already far away from the phase transition temperature (i.e. several hundred degrees Celsius) in flash technique measurements. In this case, direct radiative heat transfer in signal–time data has its origin (in parts) in the phase transition itself and is thus supposed to be non-ballistic (e.g. release of latent heat due to inversion of phase transition, Branlund and Hofmeister, 2007, strong variation in heat capacity). This process results in radiative heat transfer signal–time data differently shaped compared to regular boundary-to-boundary (ballistic) radiative heat transfer. Hence, standard fit algorithms are inappropriate and signal–time data affected by phase transitions need to be specifically evaluated using appropriate fit procedures for (semi)transparent specimens.

5 Conclusions

Laser flash thermal diffusivity measurements were performed on quartz
synthetic single crystals between −120 and 800 ^{∘}C. The temperature dependence
of the thermal diffusivity *D* between −80 and 560 ^{∘}C can be approximated
by the model of bulk phonon polaritons given by Hofmeister et al. (2014).
Low-temperature data show that an extrapolation to lower temperatures of
previously measured thermal diffusivity values (at and above room temperature)
leads to an overestimation of thermal diffusivity values, especially for
temperatures below −80 ^{∘}C. The [001] direction (*c* axis) shows the
maximum thermal diffusivity at all temperatures for samples with
thicknesses *d**≲*5 mm (cross section 10×10 mm). The anisotropy of the thermal diffusivity *D*_{[001]}∕*D*_{[100]}
decreases linearly over the measured temperature range with a discontinuity
around the *α*–*β* phase transition at 573 ^{∘}C. The latter
is linked to a change in the crystallographic direction of maximum
averaged sound velocities (crossover), reflecting changes in elastic
constants over the phase transition.

A sample thickness dependency of the thermal diffusivity is observed
above the phase transition. No crossover in the direction of the maximum thermal diffusivity is
observed for sample thicknesses *d**≲*5 mm in laser
flash measurements. However, while using a centimetre-sized sample (cross section 10×10 mm),
a crossover in thermal diffusivity over the *α*–*β* phase
transition is observed but at the limit of the experimental resolution.

Direct radiative heat transfer in signal–time curves is fundamentally different
in flash measurements for *T*≤573 ^{∘}C and *T*>573 ^{∘}C.
It was shown that the phase transition has (1) a significant influence
on signal–time curves, (2) already at temperatures far below the actual
phase change. This is not only supported by the difference in shape
of the direct radiation transfer in the raw data but also by a sharp
drop of radiative heat transfer intensities just above the phase transition.
In the *β* phase, the intensities of direct radiative heat transfer
increase and can be described by common ballistic (boundary-to-boundary) transfer.
A comparison of data fits shows that a frequently used model only accounting for
boundary-to-boundary radiative heat transport in (semi)transparent
specimens is insufficient to adequately consider the supposed non-ballistic
radiative heat transfer for *T*≤573 ^{∘}C originating in the phase
transition. An absorption/re-emission approach introduced approximates
the observed fast radiative heat transfer well in signal–time curves at
temperatures below 573 ^{∘}C.

These findings that radiative heat transport in quartz flash technique raw data is caused by different effects give rise to the assumption that flash method thermal diffusivity measurements on (semi)transparent matter undergoing phase changes are affected in a similar manner even though temperatures are several hundred degrees away from the actual phase transition. This can result in a strong underestimation of the thermal diffusivities when using standard fit routines that cannot separate the different radiative heat transport mechanisms effectively.

Data availability

Thermal diffusivity signal–time data for low- and high-temperature laser flash measurements are available at https://doi.org/10.5445/IR/1000119716 (Breuer and Schilling, 2020).

Supplement

The supplement related to this article is available online at: https://doi.org/10.5194/ejm-33-23-2021-supplement.

Author contributions

SB was responsible for data curation, software codes and implementation, visualizations and writing the manuscript. FRS was in charge of funding, providing resources, supervision of the study, and reviewing and editing of the manuscript. SB and FRS together worked on the conceptualization of the study, on the formal analysis of the data, and on the creation of models and methodology, as well as on the validation of research outputs.

Competing interests

The authors declare that they have no conflict of interest.

Acknowledgements

This work was made possible by a foundation of Martin Herrenknecht. The authors thank Jana Späthe, Ruben Stemmle and Damien Guth for their help in performing the laser flash measurements, and Gemma de la Flor for comments on an early version of the manuscript. The authors gratefully acknowledge the unknown reviewers for their in-depth, constructive and helpful comments on an earlier version of the manuscript. The authors further acknowledge the support by the KIT-Publication Fund of the Karlsruhe Institute of Technology.

Financial support

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

Review statement

This paper was edited by Etienne Balan and reviewed by two anonymous referees.

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Short summary

The knowledge of physical properties of quartz as an abundant rock-forming mineral in the Earth’s crust allows for a better understanding of its dynamic processes. The thermal transport properties of single-crystal quartz are studied between –120 °C and 800 °C using a laser flash method. First, low-temperature data as well as the role of the low-to-high quartz phase transition (e.g. a transition-related non-ballistic radiative transfer) and size effects on thermal diffusivity are discussed.

The knowledge of physical properties of quartz as an abundant rock-forming mineral in the...

European Journal of Mineralogy

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