High accuracy, self-calibrating photopyroelectric device for the absolute determinationof thermal conductivity and thermal effusivity of liquidsPreethy Chirukandath Menon, Ravindran Nair Rajesh, and Christ Glorieux

Citation: Review of Scientific Instruments 80, 054904 (2009); doi: 10.1063/1.3131625 View online: http://dx.doi.org/10.1063/1.3131625 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/80/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Self consistently calibrated photopyroelectric calorimeter for the high resolution simultaneous absolutemeasurement of the specific heat and of the thermal conductivity AIP Advances 2, 012135 (2012); 10.1063/1.3684962 Assessment of calibration procedures for accurate determination of thermal parameters of liquids and theirtemperature dependence using the photopyroelectric method Rev. Sci. Instrum. 73, 2773 (2002); 10.1063/1.1488147 Strip inverse-configuration photopyroelectric technique to measure the thermal conductivity of bulk samples Appl. Phys. Lett. 78, 469 (2001); 10.1063/1.1342782 Photopyroelectric thermal wave setup for the absolute measurement of the thermal conductivity of low densitygases Rev. Sci. Instrum. 71, 3506 (2000); 10.1063/1.1288253 Absolute values of specific heat capacity and thermal conductivity of liquids from different modes of operation ofa simple inverse photopyroelectric setup AIP Conf. Proc. 463, 306 (1999); 10.1063/1.58200

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High accuracy, self-calibrating photopyroelectric device for the absolutedetermination of thermal conductivity and thermal effusivityof liquids

Preethy Chirukandath Menon, Ravindran Nair Rajesh, and Christ Glorieuxa�

Departement voor Natuurkunde en Sterrenkunde, Laboratorium voor Akoestiek en Thermische Fysica,Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

�Received 20 February 2009; accepted 18 April 2009; published online 29 May 2009�

An improved photopyroelectric approach to simultaneously determine the thermal conductivity andthermal effusivity of minute quantities of liquids, which is based on a combined scan of themodulation frequency and the piston-sensor distance, is presented. A thorough sensitivity analysisand statistical analysis of the fitting uncertainties show that the method is a very accurate tool for thesimultaneous determination of the thermal conductivity and thermal effusivity of liquid samples.© 2009 American Institute of Physics. �DOI: 10.1063/1.3131625�

I. INTRODUCTION

Photothermal �PT� techniques have been extensively andsuccessfully used for many years for the nondestructiveevaluation of different materials. Among the various distinctPT techniques, the photopyroelectric �PPE� technique hasproven to be a very useful tool for measuring the thermalproperties of liquid samples.1–3 The PPE technique enablesthe study of the dynamic thermal parameters such asthermal conductivity �� �W m−1 K−1��, thermal diffusivity�� �m2 s−1��, and thermal effusivity �e �J m−2 K−2 s−1/2��, inaddition to the specific heat capacity �c �J kg−1 K−1��. Thethermal diffusivity of a material is defined as �=� / ��c� andthe thermal effusivity as e=��c�, with � �kg m−3� the den-sity. The PPE method also has an important advantage ofhaving good signal to noise ratio, which allows to detectsmaller temperature variations than similar ac-calorimetrictechniques that are based on electrical heating and thermo-couple detection of the temperature oscillations. These favor-able aspects have led to its successful application on com-plex liquids such as liquid crystals and mixtures of liquidsfor high resolution monitoring and analysis of thermal prop-erties in the neighborhood of temperature driven phasetransitions.4–6

In general, PPE measurements are based on the use of apyroelectric transducer to detect the amplitude and phase ofthe temperature variations caused by light induced periodicheating �at frequency f� of a quasi-one-dimentional �1D� lay-ered sample-transducer assembly. The conversion of the av-erage temperature variations in the transducer to an ac elec-tric voltage or ac current signal happens via pyroelectricallyinduced changes in the dielectric polarization charge over thetransducer electrodes. The PPE signal depends in a rathercomplex way on the optical and thermal parameters of thelayers and on the electronic frequency response of the detec-tion circuitry.7,8 However, with suitable calibration and a fewapproximations, one can obtain analytical expressions in

which the amplitude and phase of the signal depend only onone or a few of the sample’s thermal parameters.

One of the most usual detection schemes, the “back de-tection” PPE configuration, is widely used to determine thevalues of the thermal diffusivity. This configuration requiresa quasithermally thick and optically opaque sample layer,which absorbs a large fraction of the modulated light on itsfront side, thus generating a periodical heat flux through thesample. A pyroelectric sensor, situated in good thermal con-tact with the sample’s rear side, detects the fraction of theheat flux variations that has diffused through the sample. Theamplitude of the heat flux reaching the sensor depends expo-nentially and the phase linearly on the thickness and thermaldiffusivity of the sample and on the square root of the modu-lation frequency. As a result, in such a configuration, thethermal diffusivity can be determined by performing a fre-quency scan and fitting the experimental curve for the depen-dence of the signal amplitude and phase on frequency. Therecently developed “thermal wave resonant cavity” approachproposed usage of the sample’s thickness as a control param-eter, allowing a powerful and highly accurate alternative tothe classical frequency scan schemes.9–15 Due to the accu-racy of the results, this particular PPE approach was usedto detect molecular associations in liquid mixtures and thecorresponding concentration dependence of the thermalparameters.13,15

An alternative PPE configuration is the so-called “front”or “inverse” PPE �IPPE� geometry in which the sensor’sfront surface is irradiated and the sample is placed in goodthermal contact with its rear surface. The IPPE signal de-pends in the first place on the thermal effusivity and specificheat capacity of the sensor itself. However, for frequencieswhere the sensor is “thermally thin” so that the heat fluxoscillations generated at the front surface partially reach thesample at the sensor’s rear side, the PPE signal also containsinformation about the thermal effusivity of the sample.16

In a variant of this configuration, the sample is sand-wiched between the illuminated sensor and a �high effusiv-ity� metal piston, which acts as heat sink with quasi-invarianta�Electronic mail: christ.glorieux@fys.kuleuven.be.

REVIEW OF SCIENTIFIC INSTRUMENTS 80, 054904 �2009�

0034-6748/2009/80�5�/054904/9/$25.00 © 2009 American Institute of Physics80, 054904-1

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temperature. When the frequency is chosen sufficiently lowso that the sample layer is “quasithermally thin,” similar toclassical stationary methods for determining the thermal re-sistance of materials,17 then the sample acts as a thermalresistance between the sensor and the heat sink. The tem-perature oscillation pattern in the sample follows a quasilin-ear temperature gradient that is inversely proportional to itsthermal conductivity and quasi-independent of its specificheat capacity.18 This configuration allows the precise deter-mination of changes in the thermal conductivity of a samplewithout influence of possible changes in the sample’s spe-cific heat capacity.

In most of the reported PPE implementations, with theexception of those determining the thermal effusivity ofsemi-infinite samples, the thickness of the sample layer inthe one dimensional stacked PPE sandwich configurationwas determined by a calibration measurement on a referenceliquid with known thermal properties, usually water. In thisapproach, the frequency dependences of the PPE signal ob-tained from the calibration run and the one from the actualmeasurement are compared. In order to extract the thermalparameters, both measurements have to be made in exactlythe same circumstances. Changes in location and size of theheated area and the intensity of the incident radiation, as wellas electrical parameters of the setup, quite often can causedramatic changes in the signal, deteriorating the accuracy ofthe method. In particular small changes in the sensor-pistondistance, induced by mechanical manipulations during thefilling and emptying of the measurement cell with referenceand sample liquid, have a severe negative impact on the re-liability of the derived thermal conductivity.

In this article, inspired by the elegant approach sug-gested by Caerels et al.,19 Balderas-López and Mandelis,20

and Delenclos et al.,15 we report on a new, robust, and user-friendly PPE measurement approach that combines the ad-vantages of thickness and frequency scans to determine withvery high accuracy and simultaneously the thermal conduc-tivity and thermal effusivity. The achieved improvements in-volve the combined least-squares fit of frequency and thick-ness scans to extract the thermal parameters. Thoroughstatistical analysis is performed to verify the degeneracy ofthe fitting values for thermal conductivity, thermal effusivity,and offset sample thickness, ensuring the precision and ac-curacy of the reported value.

In Sec. I, we summarize the theoretical principlesbehind the PPE method and describe the advantages of acombined thickness and frequency scan approach to extractthe thermal conductivity and thermal effusivity of liquidsfrom the PPE signal dependence. Details of the ex-perimental setup are given in Sec. II. Thermal propertyvalues extracted from thickness and frequency scans forwater, paraffin oil, ethanol, and the ionic liquid choline-bis�trifluoromethylsulfonyl�imide �choline bistriflimide� arepresented, and their reliability is discussed in Sec. III. Con-clusions are given in Sec. IV.

II. METHODOLOGY

In a PPE configuration, the pyroelectric response S�t� ofthe detector due to the absorption of periodically �with radial

frequency �=2�f� modulated incident light �with intensitygiven by the real part of I�t�= IDC+ I0 exp�i�t�� by one of theinterfaces or layers in a typically quasi-1D stacked configu-ration �containing a sensor and sample layer� is given by21

S�t� =pI0Ap

2Lp�pcp����E���exp�i�t� , �1�

with �p and cp being the density and specific heat capacity ofthe sensor. I0 �W m−2� is the radiation intensity, p�C m−2 K−1� is the pyroelectric coefficient, and Ap �m2� andLp �m� are the illuminated sensor area and its thickness. E���represents the global frequency response of the used electriccircuitry that converts the oscillating temperature inducedpolarization charges into the ac voltage or current signal thatis synchronously detected by the lock-in analyzer. ���� is adimensionless response factor containing relevant informa-tion about the thermal properties and thickness of the differ-ent layers, among which is the sample layer of interest.

In the typical schematic IPPE geometry that is depictedin Fig. 1, the sample is put on top of a pyroelectric trans-ducer, whose opposite side is irradiated with modulated lightof intensity Io. In reality, the illumination of the sensor by atypically Gaussian laser beam is not laterally uniform. How-ever, the configuration can be considered one dimensionalsince the thermal diffusion lengths �= �2� /��1/2, i.e., thedistances over which heat flux oscillations can diffuse at therate f =� / �2��, with � representing the thermal diffusivityof the sample and the other layers, are small compared to alllateral dimensions. Moreover, the expression for the laterallyintegrated �integration over the sensor surface� oscillatingtemperature field can be reduced to the one of a hypothetical1D configuration where the excitation and layers are uniform

FIG. 1. Schematic geometry of IPPE setup. The sensor is illuminated fromthe bottom by periodically modulated light. The liquid sample is sandwichedbetween the top side of the sensor and a metal backing. In our case thebacking is mounted on a movable piston giving the possibility to vary thesample thickness.

054904-2 Menon, Rajesh, and Glorieux Rev. Sci. Instrum. 80, 054904 �2009�

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in the lateral direction. In order to show this, we first writethe PPE signal as a volume integral and express the lateraltemperature oscillation amplitude T�x ,y ,z ,�� in the inte-

grand in terms of the double Fourier transform in the xyplane parallel with the sensor surface of the correspondingquantity in Fourier domain T�p ,q ,z ,��,

SPPE��� � �0

Lp

dz�−

+

dx�−

+

dyT�x,y,z,�� = �0

Lp

dz�−

+

dx�−

+

dy�−

+

dp�−

+

dqT�p,q,z,��exp�ipx�exp�iqy�

= �0

Lp

dz�−

+

dp�−

+

dqT�p,q,z,���−

+

dx exp�ipx��−

+

dy exp�iqy� = �0

Lp

dz�−

+

dp�−

+

dqT�p,q,z,���p

= 0��q = 0� = �0

Lp

dzT�p = 0, q = 0,z,�� .

In the derivation above we have assumed that the tem-perature oscillations outside of the sensor surface are negli-gible with respect to the ones in the middle so that the inte-grals in the lateral directions �x and y� over the finite sensorsurface can be extended to infinity. The obtained expressionfor SPPE��� only contains the quantity T�z , p=0, q=0,��,which corresponds to the temperature oscillation amplitudeone would obtain with laterally uniform heat source, sample,and temperature field.

In the configuration presented here, the thin liquidsample is sandwiched between the upper interface of the sen-sor �typically 200–300 �m thick� and a thermally thickbacking layer ���Lb with Lb several millimeters� of a mov-able metal piston, which acts as a heat sink with high thermaleffusivity �eb�es ,ep�. The sensor is illuminated from belowthrough the air, where the optically opaque metallic �in our

case Ni–Cr� electrode at the interface between the gas andthe sensor absorbs the modulated �in our case near infrared�light radiation, thus serving as a modulated heat source.

In the following, we consider two particular situations,namely, the case of a thermally thin sample ��s Ls� and ofa thermally very thick sample ��s�Ls�. These cases, whichcan be achieved by choosing an appropriate combination ofsample thickness and modulation frequency, lead to a sim-plification of the factor ����, with a resulting specific depen-dence on, respectively, the thermal conductivity and thermaleffusivity of the sample.

In the so-called “thermal conductivity mode,” a ther-mally very thin sample layer is considered ��s Ls�. For thiscase, which is achieved by using a sufficiently low modula-tion frequency, the sample properties in the factor �,22 i.e.,

���� =�bbp + 1 + �sLsbbs��exp��pLp� − 1� + �bbp − 1 − �sLsbbs��exp�− �pLp� − 1�

�bbp + 1 + �sLsbbs�exp��pLp� + �bbp − 1 − �sLsbbs�exp�− �pLp�, �2�

with bbp=eb /ep and bbs=eb /es appear only via the quantity

bbs�sLs =eb�1 + i�Ls

es

��f

�s=

eb�1 + i�Ls��f

�s, �3�

where �s,p= �1+ i���f /�s,p= �1+ i� /�s,p, the indices s, p, andb refer to sample, sensor, and backing material, respectively,and �s and �p are the thermal diffusivities in the sample andsensor layer, respectively. In terms of sample properties, thepyroelectric signal is thus only sensitive to the thermal resis-tance Ls /�s of the sample and is independent of its heatcapacity. Note that for the typical case where eb�es ,ep, theexpression for ���� simplifies to

���� =�1 + �sLsbps��exp��pLp� − 1� + �1 − �sLsbps��exp�− �pLp� − 1�

�1 + �sLsbps�exp��pLp� + �1 − �sLsbps�exp�− �pLp�

�3a�

so that the signal is independent on the details of the backingmaterial and only the thermal conductivity and thickness ofthe sample appear via the quantity

bps�sLs =ep�1 + i�Ls

es

��f

�s=

ep�1 + i�Ls��f

�s. �3b�

With increasing frequency, one reaches the so-called “ef-fusivity mode,” where the sample is thermally thick ��s

�Ls�. In this case, the so-called “thermal wave,” i.e., thedamped wavelike oscillating heat flux diffusion, does not

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reach the opposite side of the sample-pyroelectric assembly,so there is no longer a “reflection” term in the signal fromthe backing material. In this case,

���� =�bsp + 1�exp��pLp� + �bsp − 1�exp�− �pLp� − 2bsp

�bsp + 1�exp��pLp� + �bsp − 1�exp�− �pLp�, �3c�

and only the effusivity of the sample is of influence in thesignal through the ratio bsp=es /ep.

In intermediate cases ���� depends on both the thermalconductivity and thermal effusivity of the sample materialand on the thickness of the sample layer. The frequency de-pendence of the sensitivity of the PPE signal to the thermalconductivity of the sample, expressed as the procentual am-plitude change and change in degrees of the signal phase fora 1% change in thermal conductivity while keeping the spe-cific heat capacity fixed, is depicted in Fig. 2 for typicalvalues for the thermal parameters of a liquid sample �Ks

=0.25 W m−1; es=650 J m−2 K−1 s−0.5�, a sample layerthickness of Ls=100 �m, and a lead zirconate titanate �PZT�sensor of thickness of 260 �m �Table I�. Also the sensitivityto effusivity changes �1%� of the sample, which was ob-tained in an analogous way while keeping the thermal diffu-sivity of the sample fixed, is shown. Though the sensitivityvaries nonmonotonically with frequency, some general ten-dencies can be observed. At low frequencies, the sensitivityto the thermal conductivity of the sample becomes more im-portant, while the sensitivity to the thermal effusivity de-creases to zero. In this case the sample tends to act as athermal resistance for the thermal wave traveling from thetransducer toward the piston, similar to the situation in a

stationary thermal conductivity measurement setup. For verylow frequencies, the effect of the heat capacity of the ther-mally thin sample is overwhelmed by the effect of the heatcapacity of the piston material. Because of the fact that lessthermal wave reach the piston at higher frequencies, the in-fluence of the heat capacity and thus also of thermal effusiv-ity is initially growing. In the limit for very high frequencies,the sensitivity of the signal to the thermal effusivity of thesample is again decreasing because in that case, the thermalwaves generated at the air-transducer interface decreasinglyreach the backside �sensor-sample interface� of the sensorand thus become insensitive to the sample properties. In or-der to select between the thermal conductivity mode, theintermediate mode, or the effusivity mode, one has to varythe crucial quantity �s /Ls either via a change in the samplethickness or via the modulation frequency since �s is in-versely proportional to �f .

The procedure of fitting experimentally obtained datacurves for ���� in a broad frequency range to determinethermal properties of liquids has been convincingly illus-trated in several reports. It is intrinsic to all methods that theextraction of �s requires the knowledge of the sample thick-ness Ls. Also in the case of the PPE method �s only appearsin ���� in combination with Ls via the thermal resistanceLs /�s. Therefore, till now, the PPE measurement procedureto determine �s required a separate determination of sensor-piston distance, i.e., the sample thickness Ls by means of acalibration measurement on a liquid with known thermalconductivity. Unfortunately, the manipulations to fill the cellwith the reference liquid followed by emptying and cleaningthe cell and refilling it with the sample liquid make thisprocedure quite tedious and subject to errors due to smallchanges in Ls.

In the following, we propose an alternative procedure todetermine the value of Ls without the need of a referenceliquid. We thereby exploit the fact that our measuring cell isequipped with a computer controlled stepping-motor thick-ness scanning option. Though this device does not allow toset an absolute value for Ls, it does yield a high precision onrelative thickness variations �Ls. It has been shown in Ref.15 for a PPE configuration in “diffusivity mode” that usingthickness scans with precisely controlled thickness steps cir-cumvents the requirement of knowing the absolute offsetthickness. Indeed, in the thermally quasithermally thick limitof the thermal diffusivity configuration, it is easy to showthat the complex ratio of a pair of PPE signals obtained attwo sample thicknesses that differ by an amount �Ls is pro-portional to the quantity R=exp�−�1+ i���Ls /�s��. R is inde-pendent of the individual absolute thicknesses Ls1 and Ls2 sothat given only the value of their difference �Ls, their indi-

TABLE I. Properties of the sensor and backing used for fitting.

Thickness��m�

Thermal diffusivity�10−7 m2 s−1�

Thermal effusivity�J m−2 K−1 s−1/2�

Thermal conductivity�W m−1 K−1�

PZT 260 3.51�0.15 �4.3%� 1733�42 �2.4%� 1.03�0.04 �3.9%�Piston �steel� 12 000 ¯ 5441�33 �0.6%� ¯

FIG. 2. Frequency dependence of the sensitivity of the PPE signal amplitudeexpressed in percent change �upper plot� and phase change �lower plot�in degrees per 1% change in thermal conductivity �dashed lines, specificheat capacity is kept fixed� and per 1% change in thermal effusivity�dotted line, thermal diffusivity is kept fixed�. The simulation is done aroundtypical values for the thermal parameters of a liquid sample �Ks

=0.25 W m−1, es=650 J m−2 K−1 s−0.5�, a sample layer thickness of Ls

=100 �m, and a PZT sensor of thickness of 260 �m.

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vidual values are not necessary to determine �s from theamplitude and phase of R, and from �s=��s / ��f�, the ther-mal diffusivity �s of the sample of interest.

The typical spectra depicted in Fig. 3�a� show that fordifferent sample layer thicknesses �50, 100, and 150 �m� inan air-sensor-sample-piston configuration in thermal conduc-tivity mode, the dependence of the amplitude and phase ofthe normalized PPE signal spectra on Ls /�s is more compli-cated, quite nonlinear, and even nonmonotonic. As a result,as illustrated in Fig. 3�b�, the complex ratio R of 2 PPEsignals obtained at thicknesses Ls0 and Ls1=Ls0+�Ls �in Fig.3�b�, R is obtained for the thickness pairs �50 and 100 �m�and �100 and 150 �m�� cannot be expressed in a simple wayas a function of �Ls /�s but remains dependent on the un-known Ls0. Therefore, we have tried an alternative strategyto extract �s, namely, by collecting the frequency depen-dence of PPE signals Sj��� for NL different sample thick-nesses Lsj =Ls0+ j�Ls, with j=0. . .NL−1, Ls0 unknown, but�Ls well controlled by adjusting the micrometer. It turns outthat each individual spectrum Sj��� depends in a differentand nonlinear way on �s and on the offset thickness Ls0.

When choosing the frequency range wide enough, the �highfrequency parts of the� spectra also contain information ones. These observations indicate that different signals togetheryield nondegenerate information on �s, Ls, and es. Therefore,in order to extract the two parameters of interest, �s and es,experimentally, here we perform a least-squares fit with thethree fitting parameters �s, Ls0, and es, minimizing the globalcost function

�2 =1

N�j=1

NL−1

�k=1

Nf

�Sj,exp��k� − Sj,calc��k,�s,Ls0 + j�Ls,es��2,

�4�

with N= �NL−1�Nf, the number of summed quadratic differ-ences between experimental and calculated data; �k �k

FIG. 3. �a� Variation in the normalized PPE signal amplitude �upper plot�and phase �lower plot� spectrum for a sample thickness of 50 �m �dottedline�, 100 �m �full line�, and 150 �m �dashed line�. The normalization wasdone with respect to the configuration air-sensor �semi-infinite sample�. Foreach case curves are shown for �s=0.1 �inflection point toward lower fre-quency� and �s=0.60 W m −1 K−1 �inflection point toward higher fre-quency� �b�. Relative amplitude ratio �upper plot� and phase difference�lower plot� between the 50 and 100 �m spectra �full line� and between the100 and 150 �m spectra �dotted line� in �a� for the case of �s

=0.60 W m −1 K−1. The simulations are done for fixed heat capacity of �s

=1000 kg m−3, cs=1000 J kg−1 K−1, and a PZT sensor of thickness of260 �m. Although the sample thickness difference is the same for the twocases, the relative amplitude and phase spectra of the normalized signals aredifferent.

FIG. 4. �Color online� Photograph of the PPE device. Illumination of thebottom sensor electrode is done by a built-in near IR LED of 50 mW/Sr�Vishay® TSHA5201� maximum light power. The liquid sample is pouredinto a cup and mounted on the sensor. The piston backing, whose surface iskept parallel to the upper sensor surface, is immersed in the liquid. Thepiston-sensor distance, which determines the sample thickness, can be con-trolled and scanned by a stepping motor. The cylindrical structure allows toslide a metal cylinder, provided with a heater and a thermistor, over themeasurement chamber, allowing high precision temperature control.

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=1. . .Nf�, the experimentally available frequencies;Sj,exp��k�, the normalized complex PPE signals obtainedfor sample thicknesses Ls0+ j�Ls; and Sj,calc��k , �s , Ls0

+ j�Ls , es�, the corresponding calculated PPE signals for agiven trial set of �s, Ls0, and es. The normalization of allsignals, which is necessary to remove influences of the non-flat frequency response of the electrical circuitry, was doneby performing a complex division of the frequency spectrumof each signal, with the one of a reference signal obtained inthe thermally thick sample regime, where the piston wasmoved around 3 mm away from the sensor.

III. RESULTS AND DISCUSSION

All results were obtained with a PPE cell �Fig. 4� con-taining a 260 �m thick PZT sensor of 20 mm in diameterand a chromium coated steel piston of 8 mm in diameter,which was mounted on a stepping motor �computer con-trolled with motion precision of 0.025 �m� with its flat edgeparallel �within about 10 �m� with the upper side of thesensor. The bottom side was illuminated by sinusoidallymodulated light from a �50 mW/Sr maximum angular inten-sity and 875 nm peak wavelength� near infrared light emit-ting device �Vishay® TSHA5201�. The current signal, col-

lected from the Ni–Cr electrodes of the sensor, was amplifiedand analyzed by an SR830 �Stanford Research Systems®�lock-in amplifier.

A typical result for water is shown in Fig. 5. Frequencyscans were performed for NL=4 piston-sensor distances.Clearly, a set of �s, Ls0, of es is found in a very good fitbetween the experimental signals and theoretical curves. Wehave also calculated the uncertainties on the fitted values byanalyzing the variation of the cost function when varyingeach of the three parameters �s, Ls0, and es around theirrespective best fitting value while keeping the other twofixed. These uncertainties �Table II� are quite small. How-ever, this procedure does not guarantee that the found set offitting values is unique. In order to check for possible degen-eracy of the pairwise dependences on �s, Ls0, and es, thecombined pairwise dependences of the cost function are plot-ted in Fig. 6�a�. The result confirms our expectations: theminimum of �2��s ,Ls0� is point symmetric; there is only oneunique pair of parameters minimizing the cost function. Fig-ure 6�c� shows that this is not the case when �2 is analyzedfor only one piston-sensor distance �NL=1�. In this case, thepairwise dependence of �2 on �s and Ls shows a valley ofequivalent minima characterized by a constant Ls /�s. On theother hand, even from a single frequency scan, there is nodegeneracy between the fitted thermal conductivity and effu-sivity values.

The sharpness of the �2 minima, which is inversely pro-portional to the fitting uncertainty, decreases with decreasingnumber of piston-sensor distances. This is illustrated bycomparing Figs. 6�a� and 6�b� for, respectively, NL=4 andNL=2. In particular the removal of frequency scans obtainedat the smallest sample thickness deteriorates the reliability ofthe fit.

It is impossible to visualize in a two-dimensional �2D�plot a possible threefold degeneracy between �s, Ls0, and es.However, multifold degeneracy can be verified by perform-ing a so-called “most-squares” fitting analysis.23 In this ap-proach, the dependence of the cost function around the bestfitting parameter set on a given parameter, e.g., �s, is deter-mined by optimizing for each chosen �s the two other pa-rameters �es and Ls0� until �2 is minimized. This is in con-trast with the classical “least-squares analysis,” where es andLs0 are kept fixed while varying �s.

Figure 7 shows that using multiple piston-sensor dis-tances Lsj, the most-squares sum of quadratic differences be-tween experimental and calculated data remains parabolic for

FIG. 5. Experimental result �symbols� and best fit �lines� for the frequencydependence of the normalized PPE signals with a water sample for 4 differ-ent sample thicknesses differing by �Ls=40 �m. During the fit, both thethermal conductivity �s and the thermal effusivity es of the sample werevaried, as well as the offset thickness Ls0. The assumed properties of thesensor and the backing are summarized in Table I. In order to be indepen-dent on the frequency response of LED excitation and signal detection, allthe curves have been normalized with the curve obtained for Ls 3 mm forwhich the sample can be considered thermally very thick over the wholefrequency range �i.e., the configuration is in the effusivity mode�.

TABLE II. Thermal properties of water at 25 °C with frequency and thickness scan approach.

Frequency scans Thickness scans

Values fromliterature Unit

Amplitudefitting

Complexfitting

Phasefitting

Amplitudefitting

Complexfitting

Phasefitting

Thermal conductivity 0.602�0.002 0.600�0.002 0.597�0.002 0.55�0.03 0.54�0.04 0.54�0.04 0.598,a 0.66�0.07 c W m−1 K−1

Thermal effusivity 1588�1 1589�3 1589�2 1538�29 1541�42 1547�33 1589,a 1582,b 1555c J m−2 K−1 s−1/2

Offset thickness 45.3�0.3 44.3�0.7 44.0�0.4 143�8 131�6 111�3 �m

aDirect or derived properties according to tabulated data in Ref. 25 at T=25 °C.bAccording to Ref. 26 at room temperature.cAccording to Ref. 19 at room temperature.

054904-6 Menon, Rajesh, and Glorieux Rev. Sci. Instrum. 80, 054904 �2009�

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all three fitting parameters, confirming the absence of fittingdegeneracy. The uncertainties �p on each parameter P=�s,Ls0, and es mentioned in the result Tables II–V are based onthe expression24

�p =� 2

N�2

�P2 ��2/�min2 �

, �5�

with �min2 the value of �2 for the best fitting parameter set.

Overall, even for moderately small NL, the proposed mea-surement procedure results in very small statistical fittinguncertainties on the thermal parameters of the sample. Theflat curve for NL=1 confirms the occurrence of threefold fit-ting degeneracy when only one thickness is considered.

In Tables II–V, the fitting results are given for water.Also results for ethanol, paraffin oil, and choline bistriflimideare summarized. For the materials where literature data areavailable, the correspondence with our results is verygood.25–28 Interestingly, the procedure works not only byconsidering �2 based on the vectorial differences in the com-plex plane between the experimental and theoretical data butconsistent results are also obtained by considering only theamplitudes or only the phases.

It also worthy to mention that for a given frequency

TABLE III. Thermal properties of paraffin oil at 25 °C with frequency and thickness scan approach.

Frequency scans Thickness scans

Values fromliterature Unit

Amplitudefitting

Complexfitting

Phasefitting

Amplitudefitting

Complexfitting

Phasefitting

Thermal conductivity 0.166�0.003 0.162�0.004 0.161�0.003 0.18�0.03 0.19�0.03 0.20�0.05 0.15a W m−1 K−1

Thermal effusivity 513�4 512�3 509�3 528�16 569�22 589�31 506a J m−2 K−1 s−1/2

Offset thickness 71.4�0.5 70.8�0.5 70.5�0.2 237�5 255�6 270�8 �m

aDirect or derived quantities according to Ref. 27 at room temperature.

TABLE IV. Thermal properties of ethanol at 25 °C with frequency and thickness scan approach.

Frequency scans Thickness scans

Values fromliterature Unit

Amplitudefitting

Complexfitting

Phasefitting

Amplitudefitting

Complexfitting

Phasefitting

Thermal conductivity 0.237�0.005 0.237�0.003 0.236�0.004 0.29�0.03 0.27�0.03 0.26�0.05 0.256a W m−1 K−1

Thermal effusivity 804�1 803�2 802�2 817�14 821�13 824�29 J m−2 K−1 s−1/2

Offset thickness 73�1 73�1 74�2 208�5 209�6 212�8 �m

aAccording to Ref. 28 at room temperature.

TABLE V. Thermal properties of choline pms at 25 °C with frequency and thickness scan approach.

Frequency scans Thickness scans

UnitAmplitude fitting Complex fitting Phase fitting Amplitude fitting Complex fitting Phase fitting

Thermal conductivity 0.26�0.01 0.26�0.01 0.26�0.01 0.29�0.03 0.28�0.03 0.20�0.05 W m−1 K−1

Thermal effusivity 672�3 663�3 660�3 641�26 648�23 653�19 J m−2 K−1 s−1/2

Offset thickness 57�1 57�1 56.7�0.4 104�5 108�6 110�8 �m

FIG. 7. Least-square �full line� and most-square sum of quadratic differ-ences between the fitting function and the experimental data �dashed line�for �s, es, and Ls0 for the frequency scans at �a� NL=4 and �b� NL=1 thick-nesses. The sharpness of the parabolic minima is an indicator for the statis-tical uncertainty on the best fitting values. The most-squares minima ofevery considered parameter are broader than the least-squares minima, be-cause they take into account covariance with the other fitting parameters. Allmost-squares curves, except for the ones based on experimental data ob-tained with only one sample thickness, show a quite sharp minimum, con-firming the absence of fitting degeneracy and the high intrinsic accuracy ofthe method.

054904-7 Menon, Rajesh, and Glorieux Rev. Sci. Instrum. 80, 054904 �2009�

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range �in this case 150 logarithmically spaced frequenciesbetween 0.08 and 1000 Hz�, the thickness range can be op-timized by sufficiently covering the three regimes of interest,namely, Ls /�s�1, Ls /�s1, and Ls /�s 1. In particular,the first regime, which requires quite small sample thick-nesses, is of importance to ensure sufficient sensitivity of thedata to the thermal conductivity. As mentioned before, thecomparison between Figs. 6�a� and 6�b� illustrates that re-moving frequency scans obtained at small thicknesses fromthe data used in the fitting procedure drastically deterioratethe accuracy on the extracted thermal conductivity. In prac-tice, the optimization of the thickness range can be doneiteratively by successively improving the thickness rangebased on fitted intermediate values for Ls0, �s, and es.

Figure 8 illustrates that combining variations in fre-quency and thickness, necessary to lift the degeneracy be-tween �s, Ls0, and es, can also be achieved by performing aset of thickness scans at different frequencies. This scheme istypically faster to perform in practice. For most cases, thefitting values �Tables II–V� are consistent with the ones offrequency scans at different thicknesses. The uncertainties�Fig. 9� are somehow larger due to the typically larger thick-ness offset value.

The analysis described above is based on the assumptionthat the thermal parameters of the sensor and backing mate-rials, which appear together with the sample properties in Eq.�3�, are accurately known or that possible errors on the as-sumed values have limited impact on the extracted sampleproperties. In order to verify this, we have performed thefollowing actions a priori to use the setup for liquid charac-terization. Before using the setup for sample characteriza-tion, we have performed a set of measurements versus fre-

quency and thickness with water as a sample with accuratelyknown thermal parameter values taken from literature.25 Thisdata set was fitted with the theoretical model, with the effu-sivity of the steel backing as the only fitting parameter. Avalue of 5441�13 J m−2 K−1 s−1/2 was found, which is con-sistent with literature value.29,30

In addition, we have also fitted a posteriori on the basisof the same data set the thermal properties of water under thefollowing artificial conditions. First, fits were performed byassuming piston effusivity values between 5386 and5495 J m−2 K−1 s−1/2. It turns out that a deviation of 1%from the true value of the piston effusivity leads to an errorof 0.8% on the fitted thermal conductivity value and 0.9% onthe fitted thermal effusivity value of water. In a similar waywe have determined the relation between an error on thethermal effusivity, the diffusivity, and the thickness of thesensor on one hand and the fitted values for the thermalconductivity and effusivity of water on the other hand. Thecorrelations are summarized in Table VI. Given our accurateknowledge of the piston and sensor values �Table I�, and bysome simple error propagation calculus, we can concludethat overall, possible impacts of deviations in those values onthe fitted sample properties are smaller than 1% for the ther-mal conductivity and 1% for the thermal effusivity.

IV. CONCLUSION

While previous work showed that PPE techniques arequite effective to determine thermal properties of liquids, thePPE methods to determine the thermal conductivity of liq-uids required the use of a reference liquid to calibrate thesample thickness, a procedure that has quite some disadvan-tages. In this paper, we propose an improved PPE approachthat is based on a combined scan of the modulation fre-quency and the piston-sensor distance and therefore is self-calibrating without need for a calibration measurement on aknown reference liquid.

TABLE VI. Change in thermal parameters with 1% change in different fitting parameters with water as sample.

Effusivity of steel Effusivity of PZT Diffusivity of PZT Thickness of PZT Unit

Thermal conductivity 0.597�0.005 �0.9%� 0.597�0.001 �0.2%� 0.597�0.001 �0.2%� 0.597�0.001 �0.2%� W m−1 K−1

Thermal effusivity 1589�15 �0.9%� 1589.3�0.5 �0.03%� 1589.3�0.5 �0.03%� 1589.3�0.5 �0.03%� J m−2 K−1 s−1/2

Offset thickness 44.00�0.02 �0.05%� 44.0�0.02 �0.05%� 44.00�0.47 �1%� 44.0�0.9 �2%� �m

FIG. 8. Thickness dependence of the normalized PPE amplitude �upperplot� and phase �lower plot� with water as a sample at modulation frequen-cies 0.2 Hz �circles�, 0.3 Hz �squares�, 0.4 Hz �stars�, and 0.5 Hz �dia-monds�. The data were normalized with the curve obtained for a largesample thickness of 3 mm, where the sample can be considered thermallythick �Ls��s�. The full lines indicate the best fit. The amplitude behavior isnonmonotonic due to thermal wave interference effects.

FIG. 9. �Color online� Contour plots vs pairs of fitting parameters of the �2

cost function obtained by taking into account thickness scans at NL=4 fre-quencies for the experimental data and that were fit in Fig. 8. For all threeparameter pairs, �i� ��s ,Ls�, �ii� ��s ,es�, and �iii� �es ,Ls�, �2 shows a clearminimum around the best fitting values, confirming the absence of pairwisefitting degeneracy in this case.

054904-8 Menon, Rajesh, and Glorieux Rev. Sci. Instrum. 80, 054904 �2009�

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A thorough sensitivity analysis and statistical analysis ofthe fitting uncertainties show that the method is a very accu-rate tool for the simultaneous determination of the thermalconductivity and thermal effusivity of liquid samples andfrom there the specific heat capacity and thermal diffusivity.In addition, comparison of obtained results for water, etha-nol, and paraffin oil �thermal conductivities ranging from0.16 to 0.60 W m−1 K−1� with values found in literature�consistent within 1%–8%� indicates that the method has ahigh absolute accuracy and very high precision �typicallybetter than 1% statistical uncertainty on fitted values forthe performed frequency scans at different thicknesses�.The high detection sensitivity of PPE ac-thermometry�nanokelvin�, allowing very small temperature gradients overthe sample �millikelvin�, infers an excellent potential for theapplication of the method to thermophysical characterizationin general and studies of thermodynamic phase transitions inparticular.

ACKNOWLEDGMENTS

The authors would like to express their gratitude toFWO-V �Project No. G.0230.07� and K. U. Leuven �OT

Project Nos. 2007/035 and IDO/05/005� for financial sup-port, to the Department of Education and Training of theFlemish Government, Belgium, and Service de Coopérationet d’Action Culturelle, France �Tournesol Project No.T2006.05� for possibilities to exchange know-how with Uni-versité du Littoral, Dunkerque, France, and to Dr. StéphaneLonguemart and Dr. Sylvain Delenclos from Université duLittoral for their very kind and precious advice and helpduring the development of this research.

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FIG. 6. �Color online� �a� Contour plots of the �2 cost function vs pairs offitting parameters obtained by taking into account frequency scans at NL

=4 thicknesses for the experimental data and fit for water as a sample in Fig.5. For all three parameter pairs, �i� ��s ,Ls�, �ii� ��s ,es�, and �iii� �es ,Ls�, �2

shows a clear minimum around the best fitting values, confirming the ab-sence of pairwise fitting degeneracy in this case. �b� Similar contour plot butfor �2 based only on two, rather large sample thicknesses �NL=2�. Due tothe removal of important information present in the data for small thickness,the minima become less sharp along the �s and Ls direction, indicating adeteriorated uncertainty on those fitting parameters. �c� Contour plot of �2

based on only one sample thickness. Due to the degeneracy between �s andLs, their pairwise contour plot shows a stretched valley, indicating that thedata only contain about the combined quantity �s /Ls.

054904-9 Menon, Rajesh, and Glorieux Rev. Sci. Instrum. 80, 054904 �2009�

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