Elsevier

Physics Reports

Volume 356, Issues 1–2, January 2002, Pages 1-117
Physics Reports

Modulation calorimetry and related techniques

https://doi.org/10.1016/S0370-1573(01)00031-XGet rights and content

Abstract

Modulation techniques for measuring specific heat, thermal expansivity, temperature derivative of resistance, thermopower, and spectral absorptance are reviewed. Owing to the periodic nature of temperature oscillations, high sensitivity and excellent temperature resolution are peculiar to all these methods. Various methods of modulation and of measuring the temperature oscillations are presented. Some applications of the modulation techniques for studying physical phenomena in solids and liquids are considered (equilibrium point defects, phase transitions, relaxation phenomena in specific heat).

Introduction

Modulation techniques of studying thermophysical properties consist in periodically modulating the power that heats the sample and creating thereby temperature oscillations in the sample around a mean temperature. The amplitude of these oscillations depends on the heat capacity of the sample. Corbino (1910) discovered this principle at the beginning of the last century. However, at that time such measurements could not be precise.

In modulation calorimetry, it is enough to measure the oscillations of the heating power and of the sample's temperature. The use of periodic temperature changes provides important advantages. When the modulation frequency is sufficiently high, corrections for heat losses from the sample are negligible even at the highest temperatures. In this respect, the method is comparable to adiabatic calorimetry. Against the pulse method, the modulation technique has the advantage that selective amplifiers and lock-in detectors measure the harmonic temperature oscillations. This feature becomes very important when good temperature resolution is required as, e.g., in studies of phase transitions.

The modulation methods provide a unique possibility to perform measurements with temperature oscillations in the range 0.1–10mK and even smaller if necessary. Modifications of the method differ in the ways of modulating the heating power (heating by an electric current, radiation or electron-bombardment heating, induction heating, use of separate heaters, Peltier heating) and by the methods of detecting the temperature oscillations (by the resistance of the sample or radiation from it, and by the use of thermocouples, resistance thermometers, or pyroelectric sensors).

The modulation techniques allow one to perform measurements in a wide temperature range, from fractions of a kelvin up to melting points of refractory metals, and with high sensitivity. In many cases, it is possible to assemble compensation schemes whose balance does not depend on the amplitude of the oscillations in the applied power and to automatically record quantities to be measured. The measurements can be controlled by a data-acquisition system and fully automated. All these features have made the method very attractive and widely used. In treating the data, the mean temperature and the amplitude of the temperature oscillations are considered constant throughout the sample. In this respect, the modulation techniques differ from the method of temperature waves. As a rule, the measurements are carried out in a regime where the amplitude of the temperature oscillations in the sample is inversely proportional to its heat capacity.

Corbino (1910) developed the theory of modulation calorimetry and carried out the first modulation measurements of specific heat. He used the oscillations of the sample's resistance to determine the temperature oscillations. They were detected by passing through the sample a supplementary AC current of a frequency equal to that of the temperature oscillations or by the third-harmonic technique (Corbino, 1911). Fermi (1937) highly praised this work in a paper in memory of Corbino entitled “Un Maestro”. In some measurements, the temperature oscillations were detected by the oscillations of the thermionic current from the sample (Smith and Bigler, 1922; Bockstahler, 1925). Zwikker (1928) measured the specific heat of tungsten up to 2600K.

Considerable progress in modulation calorimetry has been achieved in the 1960s due to advances in experimental techniques. At the first stage, the method was used exclusively at high temperatures. Its most important feature was the smallness of the correction for heat losses. The samples, in the form of a wire or a rod, were heated by an electric current passing through them. The temperature oscillations were deduced from oscillations in the resistance of the samples or radiation from them. By the use of this method, the high-temperature specific heat of refractory metals has been determined. Later, the method was employed in studies of phase transitions, where the main requirement became good temperature resolution. In these experiments, the samples were also heated by an electric current.

At the second stage, the modulation technique was applied to measurements at low and middle temperatures and for studying nonconducting materials. Even at low and middle temperatures, the traditional domain of adiabatic calorimetry, modulation calorimetry ensures better temperature resolution and higher sensitivity. In addition, small dimensions of the samples often are of importance. Absolute values obtainable by the modulation technique are less accurate than those obtained by adiabatic calorimetry. Even under very favorable conditions, the accuracy of modulation measurements is about 1–2%. Nevertheless, the method is preferable in studies of phase transitions due to its excellent resolution. It is applicable under high pressures.

Filippov (1960) was the first to employ the third-harmonic method for studying thermal properties of a liquid surrounding a probe. Temperature oscillations in a heater immersed in a liquid under study depend on the product of the specific heat and thermal conductivity of the liquid. This method was rediscovered by Birge and Nagel (1985) and used in their well-known studies of supercooled liquids near the glass transition. The authors founded a new branch in modulation calorimetry, specific-heat spectroscopy.

For a long time, there was no special term for modulation calorimetry. The term ‘modulation method for measuring specific heat’ was proposed in the paper describing the equivalent-impedance method for determining the specific heat of wire samples (Kraftmakher, 1962). However, most investigators have become acquainted with modulation calorimetry only from the famous papers of Sullivan and Seidel 1966, Sullivan and Seidel 1967, Sullivan and Seidel 1968. These authors considered thermal coupling in a system including a sample, a heater, and a thermometer, and performed measurements at low temperatures. Sullivan and Seidel (1968) stressed the significant advantages of this technique as follows. “(1) The sample may be coupled thermally to a bath. (2) The method is a steady-state measurement. (3) Changes in heat capacity with some experimentally variable parameter may be recorded directly. (4) Extremely small heat capacities may be measured with accuracy. (5) The method possesses a precision an order of magnitude better than existing techniques.” The term ‘AC calorimetry’ introduced by the authors is now generally accepted. A new term recently appeared, ‘the temperature-modulated calorimetry’ (Gmelin, 1997).

The excellent resolution peculiar to modulation calorimetry appeared very useful in studies of high-temperature superconductors. The aim was to investigate contributions that amount only to a few percent of the total specific heat. Small samples are generally used in modulation calorimetry, but a challenge has arisen when it was necessary to perform measurements on a microgram sample.

Modulation calorimetry provides unique opportunities: temperatures down to 0.1K (Feng et al., 1988; Steinmetz et al., 1989); magnetic fields up to 30T (Yu et al., 1988; Fortune et al., 1990); pressures up to 3.5GPa (Eichler and Gey, 1979; Eichler 1980, Eichler 1981); samples as small as 1μg (Fominaya 1997a, Fominaya 1997b, Fominaya 1999a, Fominaya 1999b); a resolution of the order of 0.01%. Temperature oscillations necessary for calorimetric measurements are of the order of 1K at high temperatures, of 1mK at room temperatures, and of 1μK at liquid helium temperatures (Mehta and Gasparini 1997, Mehta and Gasparini 1998; Mehta et al., 1999). One can measure specific heat as a function of an external parameter, e.g., magnetic field or pressure. Sullivan and Seidel 1967, Sullivan and Seidel 1968 have proposed and confirmed this approach. One of the recent achievements is the noncontact calorimetry successfully employed in space during a mission of the shuttle Columbia (Wunderlich et al., 1997). Modulation calorimetry became such a necessary technique that a commercial instrument has already appeared (calorimeter ACC-1, Sinku-Riko, Inc.).

The modulation principle is also powerful for studying some other thermophysical properties. Measurements of oscillations in the sample's length permit a direct determination of thermal expansivity. This technique enables one to avoid the main drawback of high-temperature dilatometry caused by the creep of the samples and to significantly improve the temperature resolution. The temperature derivative of resistance is available by registering oscillations of the sample's resistance caused by the temperature oscillations. Direct measurements of this quantity more reliably reveal the behavior of electrical resistivity, e.g., near phase transitions of the second order. Measuring temperature oscillations by two thermocouples provides a direct comparison of their thermopowers. The modulation method was applied also to measurements of the spectral absorptance.

High sensitivity and unique temperature resolution are peculiar to all modulation techniques. Both features are due to the periodic nature of the temperature changes. Employment of selective amplifiers and lock-in detectors reduces any influence of noise and interference. The theory of the measurements is simple and quite adequate to experimental conditions. In contrast to pulse or dynamic calorimetry, the modulation technique is a steady-state method: the amplitude and the phase of the temperature oscillations in the sample do not depend on time. Electronic equipment necessary for modulation measurements is widely used in fundamental and applied studies and is quite accessible. It is therefore possible to perform the measurements using common scientific instruments, as well as data-acquisition systems and computers for controlling the measurements and processing the data.

Until today, only modulation calorimetry gained recognition by the scientific community. Other modulation methods are still waiting for a wider practical use. Some modulation techniques were reinvented several times. This fact is a convincing confirmation of their usefulness. On the other hand, the principle of modulation was discovered long ago, and its application to studying various physical properties seems to be natural and even ordinary.

The long history of the modulation techniques is presented in Table 1.1. Many review papers and book chapters are devoted to the modulation calorimetry (Filippov 1966, Filippov 1967, Filippov 1984; Gmelin, 1997; Kraftmakher 1973a, Kraftmakher 1984, Kraftmakher 1988, Kraftmakher 1992a). Specific items of modulation calorimetry were considered by Hatta and Ikushima (1981), Garland (1985), Huang and Stoebe (1993), Finotello and Iannacchione (1995), Birge et al. (1997), Finotello et al. (1997), Hatta 1997a, Hatta 1997b, Jeong (1997), Minakov (1997), Hatta and Nakayama (1998), Hatta and Minakov (1999), Kraftmakher 1992b, Kraftmakher 1994a, Kraftmakher 1996a, and Wunderlich (2000). The author has reviewed modulation dilatometry and other modulation techniques (Kraftmakher 1973b, Kraftmakher 1978a, Kraftmakher 1989).

Section snippets

Basic equation of modulation calorimetry

The basic theory of modulation calorimetry is very simple (Corbino, 1910). The power heating the sample is modulated by a sine wave and thus equals p0+psinωt. The sample's temperature therefore oscillates around a mean value T0. For a short time interval Δt, during which the quantities involved remain constant, the heat-balance equation takes the form(p0+psinωt)Δt=mcΔT+P(T)Δt.Here m,c and T are the mass, specific heat and temperature of the sample, P(T) is the power of the heat losses from the

Modulation of heating power

The choice of a method to periodically heat the sample depends on its shape and electrical conductivity and on the temperature range of the measurements. At high temperatures, direct electrical heating or electron bombardment is preferable. In studies of nonconducting samples, separate heaters or modulated-light heating are employed. Various methods of heating provide different accuracy of data. For example, the modulated-light heating is usable where there is no need to accurately determine

Measurement of temperature oscillations

Various methods are applicable to detect temperature oscillations. First, they were determined from oscillations in the resistance of the sample, thermionic current, or radiation from it. Later, separate temperature sensors, thermocouples and resistance thermometers, served for this purpose. Oscillations of the thermionic current are usable only in special cases, and this technique is now out of use. A choice of an adequate method to measure the temperature oscillations is very important. It

Methods of calorimetry, a brief review

Calorimetric measurements seem, at first glance, to be simple and straightforward. By definition, one has to supply some heat to the sample and to measure the corresponding increment in its temperature. However, no simple solution for this problem exists in a wide temperature range. First, the accuracy of temperature measurements in various temperature ranges is very different. Second, it is impossible to completely avoid uncontrollable heat exchange between the sample and its surroundings when

Methods of dilatometry, a brief review

At present, methods for measuring the dilatation of solids provide a sensitivity of the order of 10−10m and even better. However, difficulties of high-temperature dilatometry are caused by the poor stability of the samples rather than by the lack of sensitivity. This is why one had to accept data on thermal expansivity averaged within wide temperature intervals. Important improvements in this field have been made in the last decades. Along with a significant progress in traditional dilatometry,

Temperature derivative of resistance

The modulation technique allows one to directly measure the temperature derivative of electrical resistance. The method consists in oscillating the sample temperature around a mean value and measuring the oscillations in the sample's resistance along with the temperature oscillations (Kraftmakher, 1967a). A nickel sample was heated by a DC current from a stabilized source and by a modulated AC current. A thermocouple measured the temperature oscillations in the sample. Simultaneously,

Principles of noise thermometry

Thin wire samples heated by an electric current passing through them are well suited for modulation measurements at high temperatures. However, an accurate determination of their temperature poses a serious problem. It is solvable by measurements of the thermal noise of the samples. As was theoretically shown by Nyquist, the mean squared thermal-noise voltage generated by a resistor in a narrow frequency band Δf equalsΔU2〉=4kBRTΔf,where R is the resistance, kB is Boltzmann's constant, and T is

Electronic instrumentation for modulation measurements

Modulation techniques require numerous electronic instruments. However, the necessary equipment is now readily available. Specific requirements for this equipment are quite moderate. A list of electronic instruments and their features important for modulation measurements is given below. This equipment can be used to assemble modulation setups and to further develop the modulation techniques.

DC sources are necessary for heating the samples to a desired mean temperature. Their main features are

Accuracy of modulation measurements

As in other cases, errors of modulation measurements fall into two categories: errors arising from differences between the theoretical model and experimental conditions, and instrumental errors arising from the inaccuracy of measuring instruments. The theoretical model of the modulation calorimetry includes the following assumptions.

(1) The mean temperature is the same over the calorimetric cell, as well as the amplitude and phase of the temperature oscillations. This means that when a separate

Applications of modulation techniques

Studies employing modulation techniques are very numerous and cannot be presented in this review completely. Examples picked here illustrate the most important features of the modulation techniques: (i) the capability of studying metals at extremely high temperatures; (ii) the unique temperature resolution, from about 10−6K at liquid helium temperatures to 1K at high temperatures; (iii) the high precision, of the order of 10−4; (iv) the capability of measuring the specific heat as a function of

Conclusion

In conclusion, it is worth attracting attention to some points presented in the review.

(1) Measurements of high-temperature specific heat employing reference samples, such as tungsten or platinum. When provided with blackbody models, such samples provide temperature oscillations of definite amplitude. These oscillations can be compared with those in other samples with a blackbody model (Section 4.2).

(2) Application of direct measurements of the temperature coefficient of the specific heat (

Acknowledgements

Many thanks to the editor, Professor A.A. Maradudin, for his work for improving the review. The support of the Ministry of Science and Technology of Israel is gratefully acknowledged.

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