High-temperature specific heat by a pulse-heating method

~ pulse~heating .method. for measuring specific heat and electrical resistivity to high temperatures 1s descnbed. This techruque can be used for electrically conductive materials from room temperature to near 1300 K. As an example of this method, measurements of zirconium are presented.


INTRODUCTiON
The measurement of specific heat at high temperatures is important in the understanding ofthe thennodynamic properties of materials.In particular, one can study phase stability and the energetics of phase transformations at these high temperatures.In many cases, a comparison can also be made of materials parameters measured at low temperatures and their YalUCS measured at high temperatures (e.g., y and 8 D ).
Several ofthe usual probiems that arise in the measurement of propenies at high temperatures can be circumvented by using transient pulse heating.1.zIn this technique, the probJem of the sample reacting with imperfect vacuum is greatly reduced because the sample only stays at elevated temperatures for a sbort time.The correction for radiation losses is also greatly reduced because of the short duration of the experiment.Another advantage of the pulse-heating technique is that it can be used to study metastable materials for which the cooling rate of the sample is sufficient to retain the metastable phase.Lastly, the pulse-heating method, by its very nature, is well suited for studying the effects ofheating rates on phase transformations.
We describe here a puJse-heating method for measuring specific heat and electrical resistivity of metallic samples at high temperatures.This method differs from that of Ref. l because we use thermocouples rather than optical pyrometry to measure the sample temperature.This allows us to measure from room temperature, but limits our highest temperatures of measurement.The method described in Ref. 2 resembles ours, but has somewhat slower heating rates that allow (and require) a measurement of the sample's emissivity, but require longer samples.Thus, our technique extends the method's applicability to brittle intermetallic compounds where long samples are difficult to make.

l. S?ECl:MEN PREPARATION
The specimens to be measured by this technique must be electrically conductive and formed into the shape of a long thin rod or wire.The specimen may be produced in this shape by a variety of methods including drawing, swaging, and vacuum casting. 3We have used all of these methods with satisfactory results.In principle, specimens may also be formed by spark cutting from a !arge piece of the material to be studied.Since the specimen is resistively self-heated, it is important that it have uniform cross section along its length as weil as being macroscopically homogeneous and free of voids.
The length of specimen necessary is determined by several parameters.The temperature profile for a resistively self-heated wire (clamped at both ends) is initially flat topped, but eventually becomes parabolic in the steady state. 1 Therefore, the important criterion that must be satisfied is that the thermal gradient end effects must not propagate into the center region of the sample, between the voltage leads, during the current pulse.To estimate this propagation time for a particular material, the temperature profile is calculated as a function of time by numerically solving the onedimensiona) heat equation. 4Measured, estimated, or literature values of the electrical resistivity, thermal conductivity, and specific heat at room temperature are used as input parameters for this calculation.An example of a temperature profile for UA1 2 is shown in Fig. 1. lt is clear from this model  that for voltage leads with a separation less than 0.8 cm, and a pulse length less than 400 ms, thermal gradient end etfects do not affect the measurement.

U. APPARATUS DESIGN
The specimen holder is built with two high-current leads, two voltage leads, and three thermocouple leads.The specimen is mounted into the sample holder by clamping it between the two current leads.The clamps holding the specimen are either copper or molybdenum strips held together with 0-80 (1.5-mm o.d.) stainless-steel screws.Because the measurements are done in a high vacuum ( l 00 µPa), all of the components ofthe sample holder must be high vacuum compatible.The voltage leads are 50.8-µm(2 mil) wire chosen of a material to minimize reaction with the sample at high temperatures.The thermocouples used were either 12. 7 µm (0.5 mil) Chromel-Alumel or Chromel-Constantan.Since the response time of a thermocouple is proportional to its crosssectional area, it is important to have the thermocouple wires as smaU as possible.Because it is not possible to attach two thermocouple leads in exactly the same position along the sample, a three-wire technique is used.In this technique, the voltage drop due to current flowing in the specimen can be adjusted to zero with an external potentiometer (p in Fig. 2).
The electronics used in this measurement are shown schematically in Fig. 2. The current flowing through the sample is controlled by the pulser which is triggered by a TTL edge from the data-acquisition unit.This current pulse is variable from 0-1 s and 0-300 A. The amount of current flowing during a pulse is held constant by a current controller which has six 50-A current drivers in parallel.The power source for the !arge current is a standard automotivetype 12-V battery.
The data-acquisition system we used was fabricated at UCSD, but commercial systems are now available that are adequate for this application.The minimum requirements for the data-acquisition unit are: I2-bit resolution, three simultaneous input channels, 5000 readings/s in each channel, variable data-acquisition rate, computer interfaceable, and capable of taking 1000 datapoints/channel in each pulse.
Each channel going into our data-acquisition unit goes through an amplifier to bring all of the signals to the same level.These amplifiers must have truly differential input with a high common mode rejection ratio and be fast enough to not change the signal shape (de to 30-kHz frequency response is adequate). 5The computer used for the data acquisition and analysis was a HP9825 desktop computer and the data were taken in real time in direct memory access mode.The data-acquisition rate was established with an external frequency generator and measured with a frequency counter accurate to 1 Hz.

m. MEASUREMENT PROCEDURE
For the measurement the sample is mounted in the sample holder and the voltage and tbermocouple leads are attached by spot welding.The thennocouple wires, e.g" two Chrome!and one Alumel, are connected to the sample making sure that the Alumel wire is between the two Chromel wires and that all three wires are as close together as possible.The sample holder is then placed into a high-vacuum chamber and pumped to -1001.1.Pa.
The potcntiometer for the thennocouple, p in Fig. 2, is coarsely adjusted by passing a small ac current through the sample and adjusting the potentiometer to give a null or minimum ac signal on the output of the amplifier that reads the thermocouple.Fine adjustments are made during preliminary pulses by minimizing the offset tbat occurs when the current pulse stops.
The sample is then "pulsed up" to temperature by sending pulses of successively !arger energy through the sample.The temperature that a given sample can be "pulsed up" to is usually determined by the temperature at which the leads react with the sample or the sample melts (or breaks).During each pulse, voltage, current, and temperature are monitored as a function of time and this data set is recorded.In the "pulse up " process, the gains of the amplifiers are adjusted to keep the signal levels between 33% and 100% of full scale.Successivc pulses are not started until the sample has retumed to room temperature.

tV. DATA REDUCT~ON AND RESUL TS
The measured quantities T (t ), V (t ), and I (t ) are then related to the specific heat (per formula weight) of a sample of circular cross section by where r is the radius of the sample, I is the distance between the voltage leads, € is the emissivity of the sample, q is the Stefan-Boltzmann constant, T 0 is room temperature during the measurement, M is the formula weight ofthe sample, and p is the mass per unit length of the sample.The second term in Eq. ( 1) is a radiation correction, where a value for € is usually estimated. 6To reduce the relative size of the radi-Speclflc heat 1224 .ation correction, we keep the pulse length short and the volume-to-surface ratio of the specimen !arge.For pulse lengths of0.5 s this term usually amounts to a 2% correction at 1300 K and, therefore, it is oflittle importance to know e weil.The assumption is made here that the temperature ofthe surface of the sample (as measured by the thermocouple) is the same as at the center ofthe sample.Since the radiation correction is small and the power is uniformly dissipated throughout the sample, we feel that this assumption is reasonablc.The derivative in Eq. ( 1) was taken numerically by a ratio of ditferences method and is thus the major source of random error in the CP calculations.To reduce the noise introduced by taking this derivative, we have averaged the data over 5-20 points.An example of the specific heat of Zr measured by this method is shown in Fig. 3.The pea.k in these data at -1130 K is the a-+ß phase transformation.We see the scatter in the data using this averaging is ± 1 %.
In a similar manner, the enthalpy can be calculated from the data.The enthalpy is given by extracting the latent heat of a transformation.Shown in Fig. 4 is the enthalpy of Zr near the a-+ß transformation.An extrapolation to the midpoint of the transformation gives La-ß = 4.0 kJ/g-at.and T = 1125 K for Zr in good agreement with the accepted values of 3.9 and 1136. 7Since the enthalpy does not contain the derivative ofthe temperature, it does not have the noise problems of the specific heat.The integral in Eq. ( 2) was approximated as a sum with step size equal to the time between data points.The high-temperature resistivity is given by 1TrV p= --.

II
(3) In this case the resistivity is a ratio oftwo raw data channels normalized by the proper geometrical factor.Shown in Fig. 5 is the resistivity of Zr.The drop in the resistivity at high temperature is caused by the a-ß transformation.

F1G. 3 .
Specific heat of Zr measurcd by a pulse-heating technique.These data were averaged over 18 datapoints.The error bar represents a ± 1 % crror.

FIG. 4 .ZircooiumF10. 5 .
FIG.4.Enthalpy ofZr near thea-ß transfonnation.These data were averaged over six datapoints.We believe that the overshoot at the beginning of thc transforrnation is due to a slight superheating of the transfonnation.
F1c. 2. Block diagram ofthe pulse-heating electronics.The outside thermocouple wires are of the same type with the center wire being its complement.