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Combinatorial nanocalorimetry

Patrick J. McCluskey and Joost J. Vlassaka)

School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138

(Received 28 April 2010; accepted 23 June 2010)

The parallel nano-scanning calorimeter (PnSC) is a silicon-based micromachined devicefor calorimetric measurement of nanoscale materials in a high-throughput methodology.The device contains an array of nanocalorimeters. Each nanocalorimeter consists of asilicon nitride membrane and a tungsten heating element that also serves as a temperaturegauge. The small mass of the individual nanocalorimeters enables measurements onsamples as small as a few hundred nanograms at heating rates up to 104 K/s. Thesensitivity of the device is demonstrated through the analysis of the meltingtransformation of a 25-nm indium film. To demonstrate the combinatorial capabilities, thedevice is used to analyze a Ni–Ti–Zr sample library. The as-deposited amorphous samplesare crystallized by local heating in a process that lasts just tens of milliseconds. Themartensite–austenite transformation in the Ni–Ti–Zr shape memory alloy system isanalyzed and the dependence of transformation temperature and specific heat oncomposition is revealed.

I. INTRODUCTION

As materials scientists strive to optimize the perfor-mance of materials, often the best performers are found incomplex materials systems, i.e., materials systems withthree components or more. This class of materials systemsis still largely unexplored. The reason why complex mate-rials systems remain unexplored is twofold: first, the scopeof the problem is vast and second, conventional measure-ment methods are too slow. For example, considering ter-nary and quaternary combinations of elements lead toover four million materials systems of which less than 1%are well known.1 Conventional measurement methodsconsider just one composition of one material system ata time. This approach is insufficient to explore the vastmaterials space in a reasonable amount of time. Thedependence of materials properties on temperature, scale,and processing conditions further increases the scopeof the problem. Clearly, high-throughput techniques canimprove the efficiency of materials discovery and propertyoptimization.

This work introduces an instrument, the parallel nano-scanning calorimeter (PnSC), which combines techniquesfrom combinatorics and nanocalorimetry to create a usefultool for materials research and discovery.2 More specifi-cally, the PnSC allows high-throughput measurementof enthalpy-related materials properties in thin-film sam-ples. The design of the PnSC allows for fast synthesis ofsample libraries by conventional thin-film growth tech-niques. Samples can range from nanometer to micrometer

thickness and libraries can vary by composition, thickness,temperature history, etc. Once created, the entire samplelibrary is measured sequentially with millisecond mea-surement times, allowing fast sample analysis over a widerange of temperatures.Combinatorial materials science is a rapidly growing

field of materials research that pairs traditional measure-ment techniques with high-throughput methods to accel-erate materials discovery.3–6 The combinatorial approachhas been applied to such diverse applications as catalysisof transition metal-containing compounds,7 polymerthin-film dewetting,8 and more recently to the optimiza-tion of shape memory alloy transformation behavior.9,10

This is a very limited sample of the growing field ofcombinatorial material science; nevertheless, theseexamples demonstrate the broadly applicable nature, theefficiency, and the economy of the combinatorialapproach.Calorimetry is an essential tool in the study of materials

that is used to measure transformation temperatures,enthalpies, and heat capacities. It is also used to investigatethe kinetics of phase transformations and reactions.Nanocalorimetry makes use of thin-film and micro-machining technologies to significantly reduce the adden-dum of the calorimeter, enabling ultrasensitive calorimetricmeasurements.11–14 Efremov and colleagues,15 in particular,have demonstrated sensitivities on the order of 10 pJ/Kusing a differential measurement scheme. Nanocalorimetryis a proven method for measuring the thermal energies ofnanoscale quantities of materials. Nanocalorimetry, in gen-eral, possesses characteristics that make it suitable forcombinatorial material science: sample fabrication methodsare generally compatible with combinatorial sample library

a)Address all correspondence to this author.e-mail: [email protected].

DOI: 10.1557/JMR.2010.0286

J. Mater. Res., Vol. 25, No. 11, Nov 2010 © 2010 Materials Research Society2086

fabrication techniques and short measurement timesfacilitate high-throughput measurements.

By combining nanocalorimetry with combinatorialmethods, the PnSC accelerates materials synthesis andanalysis. The sensitivity of the PnSC is first demonstratedby measuring the heat of fusion and melting temperaturefor a 25-nm indium thin film. The combinatorial capabil-ities of the device are then demonstrated by synthesizing asample library of the high-temperature shape-memoryalloy Ni–Ti–Zr with a two-dimensional compositiongradient to reveal the dependence of the martensite trans-formation characteristics on the chemical composition.

II. DEVICE DESCRIPTION AND WORKINGPRINCIPLE

A. Description of device

The PnSC device consists of a substrate with a numberof micromachined thermal sensors. The thermal sensorsare arranged in a 5 ! 5 array to facilitate combinatorialsample preparation (Fig. 1). When a thin-film samplewith an in-plane composition gradient is deposited onthis substrate, the film is essentially discretized at eachthermal sensor, allowing the simultaneous creation of 25samples with unique composition. One can also envisionsystematic variations of other parameters, such as samplethickness, processing conditions, etc.

The design and operation of the thermal sensors is simi-lar to the nanocalorimetric cells developed by Olson et al.13

and Efremov et al.,15 with different materials, fabricationmethod, and geometries. Each nanocalorimetric sensor con-sists of a thin-film thermistor sandwiched between twoelectrically insulating ceramic layers that form a membranesupported by the substrate [Fig. 2(a)]. The thermistor isfabricated from an electrically conductive film and servesto both measure temperature and heat the sample. Samples

to be measured are limited to the thermistor area of eachsensor, and may be deposited on either side of the mem-brane. The membrane design of the sensor thermally insu-lates the sample from the surroundings and ensures that thethermal mass of the sensor, i.e., the addendum, is verysmall.

Referring to the schematic in Fig. 2(b), the widestraight line down the center of the membrane is theheating element; the metal lines connected to the heaterare the voltage probes, and the portion of the heatingelement between the voltage probes is the thermistor.A current passed through the heating element heats thesample and the calorimetric cell. The power dissipated inthe thermistor is determined experimentally from the cur-rent supplied to the thermistor and the potential dropbetween the voltage probes. The local temperature changeis determined from a four-point thermistor resistancemeasurement that has been calibrated to temperature.

Not all power dissipated in the thermistor is used toheat the sample and addendum; some power is lost to theenvironment. At moderate temperatures, heat transferfrom the thermistor to the membrane dominates this heatloss. As the temperature increases, radiation from thethermistor and the membrane becomes important and

FIG. 1. Photograph of the parallel nano-scanning calorimeter.

FIG. 2. Layout of the nanocalorimeter cell: (a) cross-section schematic,and (b) plane-view schematic. Heater linewidth is 0.8 mm and voltageprobe linewidth is 0.1 mm.

P.J. McCluskey et al.: Combinatorial nanocalorimetry

J. Mater. Res., Vol. 25, No. 11, Nov 2010 2087

eventually dominates the heat loss. Measurements areperformed in vacuum to eliminate convection losses andto provide a chemically inert testing environment. Con-duction and radiation heat losses can be accounted for bymodeling or through the use of a reference measurementscheme. In this work, results are generally analyzedusing a reference measurement scheme; a thermal modelfor the sensor is given in Appendix A.

The substrate of the PnSC device is made of silicon,which is readily micromachined. The ceramic membraneconsists of silicon nitride, selected because it is a goodelectrical insulator and because it is made easily into thinmembranes. Silicon nitride also has a low thermaleffusivity, which reduces the heat loss into the mem-brane. The thermistor is made of tungsten, because of itslarge temperature coefficient of resistance and its smallresistivity, both of which are beneficial to measurementsensitivity.13 The high melting temperature of tungstenalso results in excellent thermal stability of the thermis-tor. The electrical leads and contact pads on the substrateare made of copper to reduce the resistance of the signallines on the substrate and to facilitate contact to the PnSCdevice. Specific dimensions of the components of thePnSC device can be found in Table I.

B. Operating principle

The power dissipated in the thermistor can be parsedinto stored power and power lost to the surroundings. Atconstant pressure, the stored power results in a change ofthe enthalpy of the sample and calorimeter addendum. Ifwe define a control volume (CV) that comprises thesample and the calorimeter addendum, then

P " _H # Q ; $1%

where P is the total power dissipated in the thermistor,_H is the time rate of change of the enthalpy within theCV, and Q is the heat loss through the boundaries of theCV. The rate of change of the enthalpy can be written as

_H " dH

dT

dT

dt; $2%

where T is the temperature of the thermistor. SubstitutingEq. (2) into Eq. (1) and rearranging results in

P_T" dH

dT# Q

_T; $3%

where _T is the heating rate of the thermistor. The left sideof Eq. (3) can be directly calculated from measuredquantities and is defined as the calorimetric signal fromthe sensor. If Q is known or if its contribution to Eq. (3)is negligible (e.g., in the case of large heating rates), thechange in enthalpy with temperature, dH/dT, can bedetermined directly from the calorimetric signal. If nophase transformations or reactions take place, dH/dT isequal to the heat capacity CP of the control volume;during a phase transformation this term also includes thelatent heat of transformation HL.To reduce the effect of the calorimeter addendum

and/or heat loss on the measurement, it is often conve-nient to perform a reference measurement. Equation (3)can then be rewritten to define the differential calorimet-ric signal as

DP_T" D

dH

dT# D

Q_T

; $4%

where D represents the difference between a sensor witha sample and a sensor that is either empty or contains areference sample. Comparing measurements in this man-ner eliminates the contribution of the addendum and re-duces heat loss contributions to the signal (see AppendixA). If the heating rate of both sensors is identical, thenthe heat conducted into the membrane is the same forboth sensors and the heat loss term in Eq. (4) vanishes.The differential calorimetric signal is the quantity that isanalyzed to determine transformation temperatures andlatent heats throughout this study. In addition toperforming differential measurements, the effect of heatlosses can also be reduced by increasing the heating rateof the calorimeter.In deriving Eq. (3) temperature uniformity in the CV is

implicitly assumed. Any nonuniformity of the tempera-ture results in a broadening of features in the calorimetricsignal curve. At short times this effect is small andit increases with time. An approximate analysis of thetemperature nonuniformity of the CV is presented inAppendix B.

III. EXPERIMENTAL METHOD

A. Device fabrication

The fabrication process starts with (100)-oriented Siwafers, 200 mm in diameter and polished on one side.These wafers are delivered with a coating of &80 nm ofSi3N4 grown on both sides using a low-pressure chemicalvapor deposition process [Fig. 3(a)]. Special care is takenthroughout the fabrication process to protect the Si3N4 onthe polished side of the wafer. This film will eventuallyform the base membrane layer of the PnSC; even shallowscratches may result in ruptured membranes.Each Si wafer is cleaved into seven 55 mm ! 55 mm

square substrates. The substrates are rinsed in deionized

TABLE I. Dimensions of PnSC components.

Length (x) Width (y) Thickness (z)

Substrate (Si) 55 mm 55 mm 0.7 mmMembrane (Si3N4/SiNx) 5 mm 2.5 mm 80 nm/100 nm

Thermistor (W) 3.6 mm 800 mm 125 nm

Note: thermal sensor center-to-center spacing is 8 mm in both the x- andy-directions.


J. Mater. Res., Vol. 25, No. 11, Nov 20102088

water and blown with nitrogen to remove any particles.Next, 125 nm of tungsten and then 1.2 mm of copperare deposited on the polished side of a square substrateusing direct current (dc) magnetron sputtering. Substraterotation produces uniform film thickness [Fig. 3(b)].Immediately prior to film deposition, the substrates aresputter-cleaned using an Ar plasma to remove any con-tamination and to improve adhesion of the sputteredcoatings. The sputter chamber used for the fabrication ofthe PnSC device has a base pressure of &8 ! 10'8 Torr;other processing conditions are shown in Table II.

After deposition of the metal coatings, the wafer isbaked at 150 (C for 5 min. Shipley 1805 photoresist(S1805, Dow Chemical Co., Midland, MI) is then spin-

coated and patterned on both sides of the wafer. Thefront side of the substrate is exposed to ultraviolet lightthrough a mask with the metallization artwork, and thebackside is exposed through a mask with the cavity win-dow artwork. Both sides of the substrate are developedsimultaneously in Microposit CD-30 (MicroChem Corp.,Newton, MA) for 1 min [Fig. 3(c)]. The Si3N4 on thebackside of the wafer is reactively etched in CF4 to createrectangular openings in the silicon nitride layer. Copperis etched in an aqueous solution of phosphoric, nitric, andacetic acid at 50 (C. The Cu etch exposes the underlyingtungsten, which is then etched in 30% H2O2 at 50 (C[Fig. 3(d)]. Both etch steps take &3 min. After the wet-etch processes, the remaining resist is exposed andremoved. Next, S1805 photoresist is reapplied to themetallization side and patterned with the rectangular cav-ity artwork [Fig. 3(e)]. Copper is then etched from themembrane area, leaving only tungsten within the areathat will form the membrane. After patterning the metal-lization layers, the device is coated with &100-nmplasma-enhanced chemical vapor deposition (PECVD)silicon nitride (SiNx) [Fig. 3(f)] using a Nexx Systems(Billerica, MA) Cirrus 150 with 265 W microwavepower, 40 sccm 3% SiH4 in Ar, 5.8 sccm N2, 20 sccmAr gas flow rates, and a working pressure of 100 mTorr.Using the previously described lithography and etch pro-cedure, the SiNx is removed from the Cu contact pad area.At this point, the device is annealed at 450 (C for 8 h in avacuum furnace with a base pressure of 10'7 Torr tostabilize the SiNx and tungsten thermistor.

In a last step, freestanding membranes are created byan anisotropic etch of the Si in a solution of 15 g KOH in50 mL H2O at 85 (C for &9 h [Fig. 3(g)]. The patternedSi3N4 coating on the backside of the substrate serves as ahard mask for this step. During this procedure, the metal-lization is protected by a sample holder that exposes thebackside of the device to the KOH solution, while isolat-ing the front side from the KOH solution. If a membranehappens to break as this etch step is completing, thePECVD silicon nitride layer protects the metallization.

B. Sample preparation

It is possible to deposit samples on the PnSC usingseveral different techniques. In this study, indium sam-ples were deposited by thermal evaporation and Ni–Ti–Zr samples were deposited by magnetron sputtering

FIG. 3. Cross-section schematic of a nanocalorimeter cell during steps(a)–(g) in the PnSC fabrication process, as detailed in the text.

TABLE II. Magnetron fabrication processing conditions.

Pressure(mTorr)

Power(W)

Time(min)

Thickness(mm)

Sputter clean 20 25 5 NATungsten 6 151 15 0.125

Copper 5 200 60 1.2



(Fig. 4). In each case, samples were deposited through ashadow mask micromachined from a silicon wafercoated with LPCVD Si3N4. The deposition shadow maskis formed so that extrusions in the shadow mask fit intothe recesses of the PnSC and align an opening in theshadow mask with each thermistor on the PnSC. Eachopening of the shadow mask has nominally the samearea, although small variations in the dimensions causedby the shadow mask fabrication process require directmeasurements of the opening areas for accurate samplevolume determination. The areas of the openings in theshadow mask were measured by optical transmissionmicroscopy using a Nikon Eclipse ME600L microscope(Tokyo, Japan) equipped with a charge-coupled devicecamera. Digital images were then processed to determinethe opening areas with a pixel counting scheme. Thevolumes of the samples are determined by depositingreference samples on a dummy substrate immediatelybefore depositing the calorimetry samples and by mea-suring the reference sample thickness with a VeecoDetak 6M profilometer (Plainview, NY) to determinethe deposition rates. The deposition rate at each sensorlocation and the area of the corresponding shadow maskopening determine the flux of sputtered material; there-fore, controlling the deposition time controls the volumeof material deposited.

Indium samples were deposited to a thickness of 25 )1 nm in a thermal evaporator with a base pressure in the10'7 Torr range. Ni–Ti–Zr samples were sputter depos-ited in a chamber with a base pressure in the 10'8 Torrrange to a thickness of about 290 nm at a deposition rateof &11 nm/min. The Ni–Ti–Zr samples were depositedfrom confocal sputtering guns using three elementaltargets, each with a 25.4 mm diameter (Fig. 4). The

sputtering guns were tilted toward the concentric pointso that the chimneys were close to touching. This posi-tioned the guns directly under the device and reduced thepossibility of any secondary shadowing from the openingof the shadow mask. By this same reasoning, the workingdistance was set to the maximum for the sputtering sys-tem, i.e., &120 mm. The dc power to each gun wasdetermined from an iterative calibration process to obtainthe desired composition range, resulting in 72, 150, and60 W to the Ni, Ti, and Zr guns, respectively. Samplefilm compositions were measured by energy dispersivex-ray spectroscopy (EDS) using an EDAX systeminstalled on a Zeiss Ultra55 field-emission scanning elec-tron microscope (FE-SEM; Oberkochen, Germany). Thecompositions were also measured by wavelength disper-sive x-ray spectroscopy (WDS) using a JEOL JXA-8200Superprobe (Tokyo, Japan). Both instruments were cali-brated using pure element standards for the Ni–K and Ti–Klines at 15 kV accelerating voltage. For Zr, the L-line wasused for EDS and the M-line was used for WDS. Ni–Ti–Zr deposition reference samples were relatively thick,&1 mm, for accurate quantitative composition analysis.The reference samples were deposited through a shadowmask with openings defined by the same dimensions asthe PnSC membranes (Table I). Composition measure-ments were made at the center of each reference sample.The composition variation within the samples was deter-mined by measuring the composition along the centerlinesof a reference sample. Measurements were performed atfive locations in each direction, spaced 0.83 mm alongthe length and 0.48 mm across the width of the sample.

C. Experimental setup and procedures

PnSC measurements are controlled and recorded witha personal computer and a National Instruments PCI-6221 data acquisition card (DAQ; Austin, TX) (Fig. 5).The DAQ is used to send a control voltage to a voltage-to-current converter, with a linear mapping of 1 V to10 mA. The current source consists of a precision opera-tional amplifier (OPA227, Texas Instruments, Austin,TX), a power operational amplifier (OPA549, TexasInstruments), and a differential amplifier (INA133,Texas Instruments) arranged in a modified Howland

FIG. 4. Sputter deposition schematic of Ni–Ti–Zr samples. FIG. 5. Measurement setup schematic.



configuration. It is powered by a Protek 3030D dual dcpower supply (Englewood, NJ) running in series mode,providing a constant 30 V controlled power. Excludinginternal losses, the current supply is limited by &20 Vof compliance and can supply a maximum current of100 mA.

The output of the current source, I, is monitored withthe DAQ by measuring the voltage drop VI across a 100 Oprecision resistor RI. The DAQ also reads the voltage dropV across the thermistor. This is shown in schematic formin Fig. 5 and a typical result is shown in Fig. 6. All signalsare recorded at a sampling rate of 100 kHz and with aresolution of 16 bits. In this setup, the voltage range of theDAQ is adjusted on the basis of the maximum expectedvalue for each measurement to maximize the precision ofthe 16-bit analog to digital conversion.

With reference to Fig. 5, the voltage signals are used todetermine the resistance of the thermistor by R " RIV/VI

and the electrical power dissipated in the thermistor byP " VIV/RI. The temperature of the thermistor is thencalculated from

T " T0 #R' R0

lR0; $5%

where l is the temperature coefficient of resistance of theheating element, T0 is the ambient temperature, and R0 isthe resistance at T0. The heating rate is calculated from

_T " R

lR0

_V

V'

_VI

VI

! "; $6%

where _V and _V1 are the voltage rates calculated from amoving linear least squares fit to the respective datahistories. With the power input and heating rate known,the calorimetric signal can be expressed explicitly interms of measurable quantities as

P_T" l R0 I2

_V

V'

_VI

VI

! "'1

: $7%

This calorimetric signal still contains the contributionsfrom heat loss to the environment and from the calorim-eter addendum. These contributions were eliminatedthrough use of reference measurements: for the indiummeasurements, two cells were used—a cell with anindium sample and a bare neighboring cell for a refer-ence measurement. To reduce random noise, 100 temper-ature cycles were averaged for the sample measurementand 10 cycles were averaged for the reference measure-ment. In the case of the Ni–Ti–Zr samples, both thesample measurements and the reference measurementswere made using the same cell in a scheme where calori-metric measurements on nontransforming phases wereused as reference measurements for transforming phases;the as-deposited amorphous films were fully crystallizedin one cycle. Subsequent cycles showed no indication offurther crystallization; 10 of these cycles served as thereference measurement for the crystallization process.The as-deposited amorphous samples cycled 100 timesin a low temperature range served as the reference for themartensite–austenite measurements on the crystallinesamples. An advantage of using a cell plus sample as itsown reference, as opposed to a bare reference cell, is thatits mass is exactly the same. Consequently, any deviationbetween calorimetric signals must be related to a changein the enthalpy of the sample.

All calorimetric measurements were conducted in avacuum chamber with a vacuum level of &10'5 Torr toeliminate convection losses and side reactions. Appliedcurrent amplitudes and durations for each measurementcan be found in Table III. Prior to performing calorimet-ric measurements, the thermistors need to be calibrated.The temperature coefficient of resistance was measuredby placing the PnSC substrate in an oven and steppingthe oven through a temperature range. During this cali-bration process the temperature of the substrate was mea-sured with a thermocouple. The resistance of a thermistorwas recorded at each temperature step by applying a1 mA monitoring current for a period of 20 ms. Becausethis measurement was performed under atmospheric con-ditions, the 1 mA current pulse caused a negligibleamount of Joule heating. The initial resistance R0 andthe temperature coefficient of resistance l were deter-mined from a linear least squares fit of the resistance dataas a function of temperature (Fig. 7). The value of l wascalculated using Eq. (5), yielding a value of (1.50 )0.04) ! 10'3 K'1 for the tungsten thermistors on the

FIG. 6. Typical voltage response for a PnSC cell with Ni–Ti–Zr sam-ple to an 85-mA current pulse lasting 25 ms.

TABLE III. Current pulse amplitude and duration for calorimetricmeasurements.

Measurement Amplitude (mA) Duration (ms)

Indium 28 20Ni–Ti–Zr crystallization 85–90 60

Ni–Ti–Zr martensite 85–90 22–25



device with the Ni–Ti–Zr samples, and a similar valuefor the device with the indium sample. The value of lwas determined for one cell on each substrate, whereasthe value of R0 was measured for each cell on a substrate.

IV. RESULTS AND DISCUSSION

A. Indium measurements

The calorimetric signals from a PnSC cell with anindium sample and a cell without indium sample areshown in Fig. 8(a). The small offset between the twosignals is caused by the presence of the indium sample;the slope is primarily the result of heat loss to the ambi-ent. Because these cells were heated at the same rate(&14 ! 103 K/s), the heat loss is the same for both cellsand the difference in calorimetric signals [Fig. 8(b)] rep-resents the calorimetric trace for the indium sample.

A linear baseline is used to determine the meltingtemperature TM, the heat of fusion Hf, and the heatcapacity CP of the indium sample. The melting temper-ature is taken as the temperature at which the calorimet-ric signal peaks, and is found to be 157 (C, in excellentagreement with the reported literature value of156.6 (C.16 Unlike traditional calorimetry, the peak ofthe melting signal is the most appropriate point to definethe melting temperature, because the very small thermalresistance between the sensor and sample produces anegligible temperature difference between them (seeAppendix B). The shape of the melting peak is deter-mined mainly by the temperature distribution within theheating element. Thus, the peak temperature representsthe average heater temperature at which most of thesample melts. This temperature is a better approxima-tion for the melting temperature than the onset of thepeak used traditionally. This approach is confirmed withheating rate experiments, which do not show an increasein the peak temperature with increased heating rates aswith traditional calorimeters.

The specific heat of fusion is obtained by numericallyintegrating the peak area between the differential calori-metric signal DP= _T and the baseline, and normalizingthis result by the mass of the sample. The mass of thesample is calculated using the measured sample volume,VIn " (27.5 ) 3) ! 103 mm3, and the handbook value forthe density, rIn " 7.31 g/cm3,16 and is equal to (201 )20) ng. This procedure results in hf " (23 ) 2) J/g, whichis less than the literature value of 28.7 J/g for bulkindium.16 Similar reductions in hf have been demon-strated for nanostructures by Lai et al.,17 Efremovet al.,18 and Zhang et al.19

The fact that the specific heat of fusion is reduced,while the melting temperature remains unchanged com-pared to bulk indium, is explained by the observation that

FIG. 7. Calibration fit of the temperature coefficient of resistance.

FIG. 8. Indium melting results. (a) Indium melting curve and referencecurve and (b) differential calorimetric signal with baseline showing themelting transformation of In nanostructures (inset).



the indium film breaks up into droplets when it is firstmelted. When the film cools, these droplets solidify intoparticles with a broad range of sizes as illustrated in theSEMmicrograph in Fig. 8(b). The smallest particles havea reduced hf and TM; the largest particles with diametersin excess of 100 nm have values that are the same as forbulk indium.18,19 From the position of the low-temperatureshoulder, the small indium particles begin to melt at&125 (C. According to Ref. 19 this corresponds to a parti-cle size on the order of 7 nm, which is consistent withSEM observations. Because the mass of the small particlesrepresents only a small fraction of the total sample mass,the small particles give rise to a low-temperature shoulderon the melting peak in Fig. 8(b), leaving the peak temper-ature unaltered from its bulk value. The specific heat offusion, by contrast, is determined by integrating over theentire peak, including the low-temperature shoulder, and isreduced as a result.

The room-temperature specific heat of the indiumsample is found by evaluating the baseline at T0 andnormalizing that value by the sample mass. This proce-dure results in CP " (260 ) 60) J/kg K, which is compa-rable to the literature value of 230 J/kg K.16 Most of theerror in the specific heat measurement is associated withtaking the difference between the sample and referencemeasurements, which were performed on two differentcells with slightly different addendum. As the samplemakes up only 8% of the total heat capacity of the sam-ple cell, a small difference in addendum between the twocells can result in a much larger error in the heat capacityof the sample. The accuracy of the CP measurementcould be improved considerably by performing the refer-ence measurement on the same cell as the sample mea-surement before the sample is deposited.

B. Composition of the Ni–Ti–Zr samples

The compositions of the Ni–Ti–Zr samples are listedin Table IV, arranged by sample location on the device.The concentration of a particular element is greatestclosest to the respective target. The sample library coversan atomic composition range of 44.6–52.7 at.% Ni, 27.7–38.5 at.% Ti, and 15.1–22.6 at.% Zr. Chemical composi-tion results measured by EDS are within 0.5 at.% of thecomposition results obtained by WDS, good agreementconsidering an uncertainty in the measurements of&0.5 and 0.7 at.% for EDS and WDS, respectively. Com-position variations within individual samples are readilyestimated from the dimensions of the cells and the com-position spread across the substrate. The maximum vari-ation in Ni content within a sample is 0.7%, which isapproximately the same as the uncertainty in the Ni mea-surements. The concentration variations for Ti and Zr aresignificantly smaller than for Ni. These observationsare borne out by the experimental measurements of the

within-sample composition variation, which do not showany trends and are all within experimental error.

C. Crystallization of the as-deposited Ni–Ti–Zrsamples

X-ray diffraction (XRD) showed that the as-depositedNi–Ti–Zr samples were amorphous. An 85–90 mA cur-rent pulse lasting 60 ms was used to crystallize the sam-ples. Figure 9(a) shows typical temperature responses ofa calorimetric cell with a sample in the as-depositedamorphous state and of the same cell with the sample inthe crystalline state. The exothermic crystallization ofthe amorphous sample creates a step in the temperaturehistory of the as-deposited sample.

The crystallization peak for the as-deposited sample isevident in the calorimetric signal [Fig. 9(b)]; the lack oftransformation is apparent in the crystallized sample. Thenonzero slope in the calorimetric signal at low tempera-tures is caused primarily by the temperature dependenceof the specific heat and to a lesser extent by the conduc-tive heat loss into the membrane. The steep rise in thesignal at elevated temperatures is the result of radiativeheat loss to the environment.

The differential calorimetric signal is plotted as afunction of temperature in Fig. 9(c). Plotting the differ-ential signal further amplifies the crystallization peakand reveals the onset of the glass transformation (475 )2 (C) immediately before crystallization. The crystal-lization peak temperature (733 ) 3 (C) is significantly

TABLE IV. Calorimetric cell numbering scheme with correspondingsample chemical composition from EDS analysis (at.%). Measurementuncertainty can be estimated as 0.6, 0.4, and 0.4 at.% for Ni, Ti, andZr, respectively.

Cell no. 1 Cell no. 2 Cell no. 3 Cell no. 4 Cell no. 5

52.6% Ni 52.7% Ni 52.7% Ni Broken 52.7% Ni

32.3% Ti 31.3% Ti 30.3% Ti cell 27.7% Ti15.1% Zr 16.0% Zr 17.0% Zr 19.6% Zr


0.4% Ni 50.6% Ni 50.8% Ni 50.6% Ni 50.5% Ni

34.0% Ti 32.8% Ti 31.5% Ti 30.3% Ti 29.1% Ti

15.6% Zr 16.6% Zr 17.7% Zr 19.1% Zr 20.4% Zr

Cell no.11 Cell no. 12 Cell no. 13 Cell no. 14 Cell no. 15

48.2% Ni 48.6% Ni 48.9% Ni 48.6% Ni 48.3% Ni35.8% Ti 34.3% Ti 32.8% Ti 31.6% Ti 30.5% Ti

16.0% Zr 17.1% Zr 18.3% Zr 19.8% Zr 21.2% Zr


46.4% Ni 46.6% Ni 46.8% Ni 46.6% Ni 46.4% Ni

37.1% Ti 35.7% Ti 34.3% Ti 33.0% Ti 31.7% Ti

16.5% Zr 17.7% Zr 18.9% Zr 20.4% Zr 21.9% Zr


44.6% Ni 44.7% Ni 44.7% Ni 44.7% Ni 44.6% Ni

38.5% Ti 37.1% Ti 35.7% Ti 34.3% Ti 32.8% Ti

16.9% Zr 18.2% Zr 19.6% Zr 21.0% Zr 22.6% Zr



higher than previously reported values (&500 (C) forthin films of similar composition.20 This difference isdue to the kinetics of the crystallization process and thedifference in heating rates between the studies. In thisstudy the heating rate is nominally 2.0 ! 104 K/s whileKim and colleagues20 reported using a heating rate of&0.17 K/s. All Ni–Ti–Zr samples were crystallized in asimilar fashion. The width of the crystallization peak inthis study can be attributed to the kinetics of the crystalli-zation process, the formation of multiple phases, and thetemperature nonuniformity in the thermistor (Appendix B).A detailed analysis of the crystallization process as a func-tion of film composition and the resulting microstructure isthe subject of a future publication.

D. Martensite–austenite transformation incrystalline samples

Typical low-temperature responses of a cell with asample in the amorphous state and of the same cell withthe sample in the crystalline state are shown in Fig. 10(a).The martensitic transformation in the crystalline sampleis not immediately obvious from the temperature histo-

ries because the martensite–austenite latent heat is sig-nificantly smaller than the crystallization latent heat, butthe transformation shows up clearly in the calorimetricsignal [Fig. 10(b)]. The differential calorimetric signal isshown in Fig. 10(c). The peak transformation tempera-ture TM–A and the enthalpy of transformation HM–A arereadily obtained from the curve: HM–A is given by thearea between the transformation peak and a linear base-line fit to the calorimetric signal outside the transformingregion; TM–A is defined as the temperature where thecalorimetric signal during the transformation is furthestfrom the baseline. For this particular sample TM–A "(183 ) 2) (C and HM–A " (32.0 ) 0.5) mJ. Also apparentfrom Fig. 10(c) is the negative slope of the differentialcalorimetric signal at temperatures above TM–A. Thisnegative slope is observed for all samples, independentof composition, and is a clear indication that the specificheat capacity of the amorphous samples increases fasterwith temperature than that of the crystalline samples.Repeating the previous analysis on the remaining sam-

ples reveals that only the rows with the two lowest Niconcentrations demonstrate a measurable transformationsignal (Fig. 11). The analysis results are summarized for

FIG. 9. Typical Ni–Ti–Zr crystallization results. (a) Temperature response of as-deposited and crystallized Ni–Ti–Zr sample to an 85-mA currentpulse lasting 60 ms. (b) Calorimetric signal of as-deposited and crystallized Ni–Ti–Zr sample plotted as a function of as-deposited sampletemperature. (c) Differential calorimetric signal for crystallizing Ni–Ti–Zr sample.



the transforming samples in Table V, where HM–A hasbeen normalized by the sample mass to produce the spe-cific latent heat of transformation hM–A. The mass mS ofeach sample was determined from the measured samplevolume VS, the measured sample composition, the volumeof the martensite unit cell VUC for Ni49.5Ti50.5–xZrx withx " {5, 10, 15, 20},21 the molar mass Mi, and the factthat Ni–Ti–Zr martensite has four atoms per unit cell22,23

mS " 4VSfNiMNi # fTi MTi # fZrMZr

NA VUC

! "; $8%

where f represents atomic fraction and NA is Avogadro’snumber. The sample volume measurements wereobtained from as-deposited amorphous samples and theunit cell volume is for the crystalline martensite phase.This discrepancy does not cause a significant error, asInoue has reported previously that the density change oncrystallization is small (<1%).24

The visual representation of the martensite–austenitetransformation temperature results can be found inFig. 12, along with results compiled from Hsieh et al.25,26

The transformation temperature varies linearly with Zrcontent in the range of 16% to 23%. This result confirmsthe trend previously reported by Hsieh et al. The Nicontent, by contrast, does not seem to affect the transfor-

mation temperature, at least for the samples in whichtransformations were observed. This behavior is alsoobserved for binary NixTi50–x alloys with x < 50%.27 Ifx > 50%, however, the transformation temperature candecrease by as much as 200 (C for a 1% increase in Niconcentration. This behavior, which also occurs inNi–Ti–Zr,28 may provide an explanation as to why notransformations are observed in the Ni-rich samples—thetransformations take place at temperatures well below themeasured range. The temperature dependence of the heatcapacity provides indirect evidence that the “nontrans-forming” samples are in the high-temperature phase: theslope of the calorimetric signal of the “nontransforming”samples is the same as the slope for the austenitic phasein the transforming samples. The slope of the marten-sitic phase, by contrast, is closer to that of the amorphoussamples (Fig. 11). It is interesting to note that the transitionbetween transforming samples and “nontransforming” sam-ples takes place at a Ni concentration between 46.8 and48.2 at.%, which is smaller than for the bulk NiTi system.

It is apparent from Fig. 12 that the thin-film samplesshow a marked depression in TM–A compared to the resultsfor bulk Ni–Ti–Zr. This depression of the transformationtemperature is caused by the fine microstructure of thesamples. SEM observations show that the films in this

FIG. 10. Typical Ni–Ti–Zr martensite transformation results. (a) Temperature response of the as-deposited and crystallized sample to an 85-mAcurrent pulse lasting 22 ms, with the crystallized sample transforming martensitically. (b) Calorimetric signals of a sample in the amorphous andthe crystallized phase during heating. (c) Differential calorimetric signal for the martensite–austenite transformation during heating.



study have a nanoscale grain structure (&10–50 nm di-ameter) as a result of the fast heating and cooling rates. Asimilar depression of the transformation temperature hasbeen demonstrated for NiTi samples with a nanoscalemicrostructure29,30 and is attributed to a decrease in thestability of the martensite, related to the energy-cost offorming twin-boundary interfaces.The dependence of hM–A on chemical composition

(Table V) is somewhat more complicated than the trans-formation temperature. The majority of samples (cells 16,20, 22–25) transform with a similar specific enthalpy,averaging !hM'A " 10) 1 J/g, which is less than thereported bulk value of 21.4 J/g, but comparable to thevalue of 9.8 J/g reported for melt-spun ribbons withmicron-scale thickness.28 A few samples (cells 17–19)transform with a reduced enthalpy, which may be an indi-cation that the R phase is formed instead of the B190

martensitic phase. In the binary NiTi system, a reducedenthalpy of transformation is indeed associated with theR phase.31 The R phase has a reduced lattice distortionrelative to the austenite phase, which decreases theentropy of transformation and therefore the specific latentheat.32 In NiTi, the R phase has been shown to stabilize atreduced length scales33 and with increasing Ni concentra-tion.34 It is reasonable to expect that the reduced specificlatent heats in cells 17–19 are the result of an R-phasetransformation. However, a competing hypothesis couldbe that the martensite volume fractions in these samplesare simply reduced,33 which would also reduce the effec-tive specific latent heats of transformation. Structuralanalysis is required to draw an unambiguous conclusion.

V. CONCLUSION

A device has been developed, the parallel nano-scan-ning calorimeter, which combines nanocalorimetry andcombinatorial methods to create a powerful instrumentfor materials screening and analysis. The device consistsof a substrate with an array of micromachinednanocalorimeter sensors, each one of which is individu-ally addressable. The nanocalorimeter sensors are verysensitive, with a resolution of &10 nJ/K, allowing ther-mal analysis of small quantities of material. The smallmass of the sensors makes it possible to achieve heatingrates as fast as 104 K/s, thus making the device a usefultool for exploring the kinetics of phase transformationsand reactions. Measurements can be performed over atemperature range from room temperature to &900 (C.By depositing samples with a composition gradient ontothe PnSC, combinatorial studies can be performed muchfaster than by conventional techniques. In short, thePnSC enables the thermal study of complex materialssystems at the nano- and the microscale. The capabilityof the PnSC is demonstrated by applying it to a 25-nmcoating of indium and to a library of thin-film Ni–Ti–Zr

FIG. 11. Differential calorimetric signal (from '150 to 300 nJ/K) ofthe martensite transformation versus temperature (from 30 to 350 (C)for each sample during heating. Results are arranged as positionedon the PnSC. See Table IV for composition results and Table V forquantitative transformation results.

TABLE V. Martensite-austenite transformation results for composi-tions demonstrating a transformation in the calorimetric signal. Chemicalcomposition uncertainty can be estimated as 0.6, 0.4, and 0.4 at.% forNi, Ti, and Zr, respectively.

Cell no. Composition (at.%) TM–A ((C) hM–A (J/g)

16 Ni46.4Ti37.1Zr16.5 136 ) 2 8.4 ) 0.2

17 Ni46.6Ti35.7Zr17.7 149 ) 3 4.8 ) 0.1

18 Ni46.8Ti34.3Zr18.9 180 ) 3 0.80 ) 0.03

19 Ni46.6Ti33.0Zr20.4 229 ) 3 1.14 ) 0.0320 Ni46.4Ti31.7Zr21.9 250 ) 1 10.3 ) 0.2

21 Ni44.6Ti38.5Zr16.9 100 ) 1 5.6 ) 0.1

22 Ni44.7Ti37.1Zr18.2 183 ) 2 9.8 ) 0.2

23 Ni44.7Ti35.7Zr19.6 209 ) 1 10.4 ) 0.224 Ni44.7Ti34.3Zr21.0 252 ) 3 11.2 ) 0.2

25 Ni44.6Ti32.8Zr22.6 277 ) 1 10.6 ) 0.3

FIG. 12. Martensite–austenite peak transformation temperatures as afunction of Zr content for two Ni poorest rows. The bulk sampleresults of Hsieh et al.25,26 are shown for comparison. Trend lines aredrawn as guides to the eye.



samples. The PnSC readily detects the melting point ofthe indium coating. After the first heating cycle, theindium breaks up into islands with a wide size distribu-tion. The peak melting temperature, which correspondsto the melting of the larger islands, is in good agreementwith the bulk value. The average specific heat of fusion,however, is depressed below the bulk value because ofthe size dispersion of the indium islands. The characteri-zation of the Ni–Ti–Zr sample library illustrates the useof the PnSC device in combinatorial studies. As-depositedNi–Ti–Zr samples are amorphous and are crystallizedduring the first temperature scan. The PnSC readilyreveals the crystallization reaction, as well as the glasstransition temperature of the samples in their amorphousstate. After crystallization, the martensite transformationis detected in a subset of the samples depending on chem-ical composition.

ACKNOWLEDGMENTS

The authors would like to thank Jim MacArthur fordesigning the current source and Ofer Kfir for his helpwith the Ni–Ti–Zr measurements. The work presented inthis paper was supported by the Air Force Office ofScientific Research (AFOSR) under Grant No. FA9550-08-1-0374 and by the Materials Research Science andEngineering Center (MRSEC) at Harvard University. Itwas performed in part at the Center for Nanoscale Sys-tems (CNS), a member of the National NanotechnologyInfrastructure Network (NNIN), which is supported bythe National Science Foundation under NSF Award No.ECS-0335765. CNS is part of the Faculty of Arts andSciences at Harvard University.

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APPENDIX A: HEAT LOSS MODEL

The power balance of the control volume CV isdefined by Eq. (1) in the main body of the work. Thisappendix defines the model that describes the heat lossfrom the thermistor Q. The heat loss is decomposed intocomponents as

Q " Qx # Qy # Qz ; $A1%

where x, y, and z represent the coordinate axes asshown in Fig. 2. Physically, Qx represents the conduc-tion losses at the ends of the thermistor, Qy refers tothe conduction losses into the membrane, and Qz repre-sents the radiation losses from the top and bottom freesurface of the CV.

The temperature in the thermistor is assumed to beuniform (see Appendix B for a justification). From thisassumption, the Qx term vanishes while Qy and Qz aregiven by,

Qz " 2Az eCVs f t* + # T0$ %4 ' T40

# $; $A2%

and

Qy " '2AMkM @TM=@y : $A3%

In Eq. (A2), Az is the emissive area, eCV is the effec-tive emissivity of the CV, s is the Stefan–Boltzmannconstant, T0 is the temperature of the surroundings, andf(t) represents the average temperature change of theheater, f t* + " TAve ' T0, as a function of time. InEq. (A3), the subscript M denotes membrane parameterswith AM as the cross-sectional area of the membrane, andkM as the thermal conductivity of the membrane. Thetemperature gradient in the membrane, @TM=@y, is eval-uated at the CV-membrane boundary. Substituting theexpressions for Qy and Qz into Eq. (A1) yields

Q " '2AMkM@TM@y

# 2Az eCVs f t* + # T0$ %4 ' T40

# $:

$A4%

Equation (A4) is not yet explicit in terms of the rele-vant thermal parameters because the temperature gradi-ent in the membrane cannot be measured directly. Thetemperature gradient depends on kM, on the volumetricheat capacity of the membrane (r cP)M, on the membraneemissivity, and on the temperature history of the CV. Todetermine the temperature gradient, we solve the one-dimensional (1D) thermal diffusion equation for the tem-perature profile in the membrane,

$r cP%M@TM@t

" kM@2TM@y2

' 2eM shM

$T4M ' T4

0% : $A5%

Here eM and hM are the emissivity and the thickness ofthe membrane, respectively. The factor of two in the radi-ation term arises because the membrane radiates fromthe top and bottom surfaces. If we let t " TM ' T0,and approximate the radiation term with a linear Taylorexpansion about t " 0, the radiation term then becomes8 eMsT3

0=hM% &

t. By letting a " kM= r cp% &

M, and b "

eMsT30=hM r cp

% &M, Eq. (A5) reduces to

@t@t

" a@2t@y2

' bt ; $A6a%

with initial and boundary conditions,

t*y; 0+ " 0; t*0; t+ " f t* +; t 1; t$ % " 0 : $A6b%

Here, the temperature in the membrane is assumed tobe initially uniform. The temperature at the left boundary(y " 0, i.e., at the CV-membrane boundary) evolves as afunction of time, while the right boundary (y " 1)remains fixed at the initial value. Comparing these con-ditions to the physical thermal cell, the initial conditionis satisfied by letting the sensor equilibrate with its sur-roundings for an appropriate length of time (approxi-mately one second for these sensors). The left boundary



condition is given by the experimentally measured tem-perature history, f [t], of the thermistor. The right bound-ary condition remains valid as long as the thermaldiffusion length is smaller than the distance to the edgeof the membrane. The linearization of the radiation termin the membrane is valid as long as the temperaturedifference between the membrane and its surroundingsremains appropriately small.

To solve Eq. (A6), we follow Sneddon’s example forthe solution to Eq. (6) without the radiation term.35 Thecomplete solution can be found in Ref. 36. For the sakeof brevity we reproduce the solution here without proof

t y; t* + "Z t

0

f t0* +yExp 'y2

4a t't0$ % ' b t' t0$ %h i

2''''''pa

pt' t0$ %3=2

dt0 : $A7%

Equation (A7) represents the temperature profile in themembrane for a given temperature history of the heatingelement. The first factor inside the integral is the forcingfunction, while the second factor is the Green’s functionof the problem.

The above integral [Eq. (A7)] can be solved explicitlywhen the temperature history f [t0] is represented by apolynomial. For the sake of simplicity, we assume alinear temperature history, f *t0+ " _TCnstt0, where _TCnst

represents a constant heating rate. Solving Eq. (A7) withthe linear temperature history assumption and taking thederivative with respect to y at y " 0 results in

@t@y

((((y"0

"' _TCnst

'''1

a

s

! t1=2e'b t'''p

p # 1

2b'1=2 # b1=2t

!

Erf b t* +

!

:

$A8%

Since @t=@y " @TM=@y, we substitute Eq. (A8) intoEq. (A4) and find,

Q" 2AM_TCnstjM t1=2

e'b t'''p

p # 1

2b'1=2#b1=2t

!

Erf b t* +

!

#2Az eCVs$$ _TCnstt#T0%4'T4

0% : $A9%

Here, the thermal effusivity is defined as f "'''''''''''k r cp

p.

Equation (A9) establishes the heat loss model for a nano-calorimetric cell. Once a nanocalorimetric cell has beencalibrated, Eq. (A9) provides a simple means for calculatingthe heat loss from the cell. Reference 36 describes a detailedprocedure for determining the thermal parameters of thecell. If the temperature history of the cell is nonlinear,higher-order polynomials can be used to represent the tem-perature history in Eq. (A7). Solutions of the integral forhigher-order polynomials can be found in Ref. 36.

The thermal model shows that the heat loss is a func-tion of the calorimeter geometry, thermophysical mate-rials properties, and temperature history. Because thereference measurement is performed on the same or asimilar calorimetric cell as the sample measurement, thegeometry and thermophysical materials properties aresimilar and the heat loss should be nearly identical. If areference measurement does not exist, then the thermalmodel can be used to account for the heat loss. In somecases the reference measurement will have a slightlydifferent heating rate due to a change in heat capacity orlatent heat. In such situations the thermal model can beused to correct the differential calorimetric signal on thebasis of the difference in temperature histories.

APPENDIX B: TEMPERATURE UNIFORMITY

The assumption of temperature uniformity of thecontrol volume CV can be addressed by consideringthe temperature gradients in the x-, y-, and z-directions(Fig. 2). Starting with the z-direction, we can take thetemperature in this direction to be uniform because atthe time scale of the measurements the thermal diffu-sion length is generally much longer than the sensorthickness. The relationship between the diffusionlength and the length scale relevant to a problemis commonly represented by the Fourier number,Fo " a t/h2, where a is the thermal diffusivity, t isthe relevant timescale for the problem, and h is thelength scale. Even the worst-case scenario of a 1-mmsample of silicon nitride and an experimental samplingrate of 100 kHz yields a Fourier number in excess of10. For comparison, a simple 1D model shows that thetemperature deviation in the sample is less than 1% ofthe measured temperature if the Fourier number isgreater than two. This example uses the shortest rele-vant timescale for the measurements and a relativelythick and poorly conducting sample, yet the Fouriernumber is still approximately five times the value thatwould produce a 1% temperature deviation. This anal-ysis demonstrates that the temperature through thethickness can be considered uniform, i.e., the tempera-ture measured at the thermistor is also the temperatureof the sample. Similar findings have been demon-strated using finite elements.36

The temperature distribution in the y-direction can beestimated by assuming the shape of the steady-statesolution (a worst-case assumption) to the 1D diffusionproblems with uniform heat generation, i.e., a parabolictemperature distribution. The temperature distributionis defined by considering the symmetry of the heatingelement, which causes the temperature gradient atthe center of the heater to be zero, dT=dy center " 0j ,and because the temperature and heat flow mustbe continuous across the CV membrane transition,



kCVACV dT=dy$ %CV " kMAM dT=dy$ %M, where the sub-scripts CV and M denote control volume and membranevalues, respectively. By assuming a constant heatingrate of the thermistor and neglecting radiation fromthe membrane, the temperature gradient in the mem-brane at the edge of the thermistor can be defined as,

dT=dy$ %M " '2''''''''''''''t=$pa%

p_TCnst. The temperature profile

can then be written as

T " T0 # _TCnst t' 2'''p

p AMkMACVkCV

a'1=2 y2

w' w

12

! " ''t

p! ";

$A10%

with the origin of y at the center of the heater. Equation(A10) can be used to calculate the standard deviation ofthe temperature sT, which is a measure of the temperaturenonuniformity in the heater. For example, using litera-ture materials properties (Table AI) and a typical heating

rate ( _TCnst " 15 ! 103 K/s) yields sT " 0.4% for anempty sensor at 400 (C. For a sensor at 900 (C and thesame heating rate, we find sT " 0.3%. The addition of asample increases the difference in conductance betweenthe membrane and the CV, reducing the gradient at theedge of the thermistor and improving the temperatureuniformity.

The temperature nonuniformity in the x-direction canbe estimated by solving for the approximate 1D tem-perature distribution along the length of the heater,governed by

VH$rcP%H@T

@t" VHkH

@2T

@x2' 2'''

pp AMfM

''t

p @T

@t

'2Az eHs$T4 ' T40%#PH ; $A11a%

with boundary and initial conditions all equal to theinitial temperature T0,

T*x; 0+ " T*0; t+ " T*l; 0+ " T0 t , 0 : $A11b%

In Eq. (A11a), VH is the volume of the heater includ-ing the silicon nitride in direct contact with the heaterand any sample. As the sample is discontinuous alongthe length of the heater, the volume and thermal prop-erties along the heater are discontinuous as well. Withthis is mind, (r cP)H is the effective volumetric heatcapacity and kH is the effective thermal conductivityalong the heater. The second term on the right siderepresents the heat loss into the membrane for a con-stant heating rate, with AM as the cross-section area ofthe membrane perpendicular to the heater, and fM isthe thermal effusivity of the membrane where fM "(kM (r cP)M)

1/2. The third term represents the radiativeheat loss from the heater, and PH is the power dissi-pated in the heater. Equation (A11) was solved for acell with a sample by the finite difference method usingliterature values for materials properties (Table AI) andtypical experimental parameters (see Secs. II and III).For the region inside the voltage probes, the standarddeviation of the temperature is better than 1% at mod-erately elevated temperatures (T < 450 (C) and betterthan 6% at elevated temperatures (T < 950 (C). Thisresult shows that for typical experiments the tempera-ture distribution in the CV is quite uniform. Conse-quently, the assumption, Qx " 0, in the thermalanalysis is valid (Appendix A), and peak broadening isminimal for short measurement times.

TABLE AI. Thermal properties of PnSC materials.16,37–39

k (W/m K) r (kg/m3) cP (J/kg K)

Thermistor (W) 174 19,300 132Membrane (SiNx) 3.2 3000 700

Sample (NiTi) 5.5 6500 500



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