186 Volume 57, Number 2, 2003 APPLIED SPECTROSCOPY0003-7028 / 03 / 5702-0186$2.00 / 0q 2003 Society for Applied Spectroscopy

Effective Normalization Method for Sample-Position-Dependence Effect in Photoacoustic Spectrometry

JUN SHEN,* JIANQIN ZHOU, CHENG HU, and JIANHUA ZHAOInnovation Centre, National Research Council of Canada, 3250 East Mall, Vancouver, British Columbia, Canada V6T 1W5

Sample position dependence effect in photoacoustic (PA) spectrom-etry has been reported by several scientists. This effect must betaken into account in a PA application that requires a quantitativetheoretical treatment. In this work, we experimentally investigatedPA signal magnitude varying with sample-to-window distance in anMTEC Model 300 Photoacoustic Detector, which has a � xed empty(gas) volume in addition to the sample-to-window-distance-depen-dent gas volume. An operative method was introduced to obtain thecoef� cient, which considered the sample-to-window distance and theadditional gas volume. With this coef� cient, the one-dimensional PAmodel, developed by Aamodt, Murphy, and Parker, can be em-ployed to quantitatively process PA experimental data, no matterwhat the sample-to-window distance is. Quantitative measurementsof thermal effusivities of two samples were performed to prove thiseffective normalization method.

Index Headings: Sample position dependence; Photoacoustic spec-trometry; One-dimensional photoacoustic theory; Photoacoustic celleffects.

INTRODUCTION

Since 1970, the photoacoustic (PA) technique has beenwidely employed to study the optical and thermal prop-erties of inorganic, organic, and biological samples.1–5

The sample under investigation is usually put in a sealedcell (photoacoustic cell) containing air (or other gas), inwhich one wall is a light window. It has been reportedthat the position of the sample relative to the window hasa considerable effect on the PA signal, due to cell wallsand window acting as thermal sinks,6–8 which are depen-dent on the geometric dimensions of the cell. For PAapplications that require a quantitative theoretical treat-ment, this sample-position-dependence effect must betaken into account.

The ideal theory is three-dimensional (3D) because areal photoacoustic cell and heat � ow in the cell are three-dimensional. Nevertheless, the most commonly used PAtheories are one-dimensional models,9–11 which are mucheasier to use than three-dimensional ones. It is wellknown that some conditions should be met when one-dimensional (1D) theories are quantitatively applied. Forthe purpose of minimizing lateral heat � ow, the lightbeam radius on the sample surface should be largeenough compared with thermal diffusion length in thesample, which is in contrast with the effort of minimizingthe axial heat � ow where the light beam spot size mustbe � nely focused, such as in the case of thermal lensmeasurements, especially when the sample is thin.12 One-dimensional PA models predict that the photoacousticsignal magnitude is inversely proportional to cell volume

Received 25 August 2002; accepted 8 October 2002.* Author to whom correspondence should be sent.

(i.e., gas volume) if the wall and window effects can beignored. For a real 3D photoacoustic cell, the 1D theoriescan be applied when the lateral dimension is much largerthan the thermal diffusion lengths in the sample and gasto diminish the thermal sink effect of the cell wall.

Aamodt, Murphy, and Parker (AMP) developed a 1Dmodel that realistically considers the thermal sink of thewindow.10 It assumes that the cell volume is directly pro-portional to the sample-surface-to-window distance, L.8

For a thermally and optically thick sample, the PA signalin AMP theory can be expressed as:8,10

S } {b[C 2 C 1 D (S 1 hS )]}PA 2 1 2 1

4 {k k (b 1 k )s s s

22 23 [2C C 2 1 1 «S S (1 2 « 1 BDq )1 2 1 2

1 q (D 1 B)(«S C 1 S C )]} (1)1 2 2 1

Here b is the optical absorption coef� cient of the sample;C1 5 cos(k0L); S1 5 sin(k0L); k0 5 v/C0, where v 5 2p f ;f is the modulation frequency; and C0 is the speed ofsound in the cell gas. For material i 5 g, s, or w, standingfor cell gas, the sample, and window, respectively, k i 5( jv/ai)1/2, where ai 5 ki /(ric i) is the thermal diffusivity;ki is the thermal conductivity; ri the density, and c i is theheat capacity. C2 5 cosh(kgL); S2 5 sinh(kgL); B 5 kgkg /ksks; D 5 kgkg /kwkw; h 5 k0 /kg; « 5 [(g 2 1)h]21, whereg is the heat capacity ratio of the cell gas. q 5 1 2 (g 21)h2. For an MTEC photoacoustic detector (Model 300),which is commonly used in photoacoustic FT-IR mea-surements, there is an additional, � xed empty (gas) vol-ume (0.07 cm 3) that is independent of the sample-to-win-dow distance L.8 To quantitatively apply the AMP modelto this PA cell, some modi� cations should be made. Tak-ing into account the additional volume of this cell, Jonesand McClelland 8 (JM) recently suggested that the PA sig-nal should be converted to a volume-corrected signal SVC

by replacing the inverse dependence on L with an inversedependence on the total empty volume of the cell V:

LS } S (2)VC PAV

To verify Eq. 2, we experimentally investigated thesignal magnitude varying with sample-to-window dis-tance L . In this paper, we report the results of the inves-tigation, in which an empirical equation to quantitativelycalculate SVC was found. With the results, an operativemethod was introduced to normalize the sample positiondependence. This method was experimentally proven byquantitative thermal property measurements.

APPLIED SPECTROSCOPY 187

FIG 1. Schematic diagram of PA experimental set up.

FIG 2. PA signals vs. different L values. From top to bottom, the cal-iper measured L 5 0.2, 0.241, 0.256, 0.296, 0.436, 0.522, and 0.758cm, respectively.

EXPERIMENTAL

The experimental apparatus was simple, as shown inFig. 1. A 5 mW HeNe laser (Melles Griot, 05-LHP-151,632.8 nm) beam traveled though a lens and a mechanicalchopper (EG&G, Model 197) to hit the sample sitting inthe MTEC photoacoustic detector (Model 300). The laserspot was expanded by the lens to about 3.5 mm in di-ameter on the sample surface to diminish the lateral heat� ow. After being ampli� ed by the preampli� er providedby MTEC, the PA signal was fed to a lock-in ampli� er(EG&G, Model 7206), which was connected with a com-puter to collect data. The gain factor of 10 of the pre-ampli� er was used throughout this investigation. Thispreampli� er introduces a phase shift, which is frequencydependent. However, in our experiment the chopper fre-quency is from 30 to 150 Hz, and the phase shift in thisfrequency range is � at.8

With this apparatus, two kinds of experiments wereconducted. The � rst one was the measurement of PA sig-nals vs. sample positions. The sample was glassy carbonprovided by MTEC Photoacoustics, Inc. The sample is10.2 mm in diameter and 2.08 mm in thickness. The sam-ple position was veri� ed by putting spacers under thesample. The spacers were 10.2 mm in diameter and ofdifferent thickness. The overall thickness of the glassycarbon sample and a spacer was measured by a digitalvernier calliper (display accuracy: 10 mm), and the cor-relating LCaliper can be calculated. For an MTEC Model300 photoacoustic detector, the sample holder has a ra-dius of 0.525 cm and a 1.03 cm holder-� oor-windowheight.8 Therefore, LCaliper 5 1.03 cm 2 the overall thick-ness in cm. The second kind of experiment was the mea-surement of thermal effusivities of a rubber sample 10.2mm in diameter and 6.20 mm in height and a graphiteplate sample 10.2 mm in diameter and 3.00 mm in height.In the experiments, modulation frequency was scannedfrom 30 to 150 Hz, and the gas in the cell was air.

RESULTS AND DISCUSSION

Sample Position Dependence. Combining Eqs. 1 and2, the volume-corrected PA signal SCV may be written as:

S 5 {C (L)b[C 2 C 1 D (S 1 hS )]}CV 2 1 2 1

4 {k k (b 1 k )s s s

22 23 [2C C 2 1 1 «S S (1 2 « 1 BDq )1 2 1 2

1 q (D 1 B)(«S C 1 S C )]} (3)1 2 2 1

Here, C(L) is the proportional coef� cient, which is afunction of the sample-to-window distance L and possi-bly the modulation frequency f. With the dimensional in-formation about the MTEC Model 300 photoacoustic de-tector (an additional, � xed empty (gas) volume 5 0.07cm3, and the sample holder with a 0.525 cm radius anda 1.03 cm holder-� oor-window height), we have

L L5 (4)

2V 0.525 pL 1 0.07

Here, cm is the unit for L . According to Eq. 2, C(L) maybe expressed as

LC (L) } (5)

20.525 pL 1 0.07

Figure 2 shows PA signals from the � rst kind of ex-periment. These PA signal curves are not parallel to eachother. With the literature values of the glassy carbon sam-ple (kgc 5 8 W/m·K; rgc 5 1.45 3 103 kg/m3; cgc 5 1.13 103 J/kg·K)8 and the window material, potassium bro-mide (kw 5 2.92 W/m·K; rw 5 2.75 3 103 kg/m3; cw 5435 J/kg·K) ,8 a least-square curve � tting to Eq. 3 wasperformed for each curve to obtain its L and C(L). Wename the value of L thus measured LPA. The value of theabsorption coef� cient b of glassy carbon used in the � t-ting was 105 cm21, and the exact value of b had littleeffect on the calculation results when b . 105 cm21.8 Theresults of curve � tting to Eq. 3 are shown in Fig. 3. InFig. 3a, the PA-measured L values LPA are in excellentagreement with the caliper-measured ones with the cor-relation coef� cient R . 0.99. The best linear � tting re-

188 Volume 57, Number 2, 2003

FIG 3. The measured data and best � tting results: (a) L, and (b) C(L).

FIG 4. PA experimental data and curve � tting results for a rubbersample and a graphite plate sample.

veals that the PA-measured L values LPA in cm can beexpressed as:

LPA 5 1.02LCaliper 2 0.00734 (6)

It is as expected that the slope is close to one and theintercept is almost zero, indicating the validity of Eq. 3for predicting the PA signals for an MTEC Model 300photoacoustic detector in the frequency range of 30 to150 Hz. Figure 3b shows the PA-measured C(L) valuesand the values calculated with Eq. 4. These two groupsof values are not in the same distribution pattern, al-though they are close at larger L values, suggesting thatthe proportional coef� cient between C(L) and L /V in Eq.5 could not be a constant and could be a function of L.It is convenient to � nd the mathematical expression ofthe PA-measured C(L) by simply � tting the PA-measuredC(L) values to a three-order polynomial equation with thecorrelation coef� cient R . 0.998. The equation is:

C(L) 5 20.0513 1 3.73L 2 4.55L 2 1 2.14L3 (7)

With this C(L), it becomes possible for Eq. 3 to quanti-tatively process PA experimental data measured by an

MTEC detector. Here L is in cm. The maximum relativedifference of the C(L) value between experiment and Eq.7 is less than 0.3% for the glassy carbon sample.

Thermal Effusivity Measurements. To prove the pos-sibility of measuring the thermal properties of materialswith this normalization method, the second kind of ex-periment was conducted. Before the experiment, a set oftheoretical data of magnitudes vs. frequency (30 to 150Hz) was generated using Eqs. 3 and 7 and the literaturevalues of glassy carbon. It was found that it is dif� cultto obtain the correct values of C(L), L, kgc, and the prod-uct of rgccgc all at once by � tting the theoretical data toEq. 3. With known C(L) and L, the obtained values ofkgc and rgccgc varied with their initial values for the curve� tting. However, the obtained value of the product ofkgcrgccgc was steady, which means that with experimentaldata, thermal effusivity, Es 5 (ksrscs)1/2, can be obtainedby curve � tting to Eq. 3, providing that C(L) and L arewell known. This may be due to the one-dimensionallytheoretical treatment of the model, which is sensitive tothermal effusivity.13

To � nd C(L) and L in the second kind of experiment,glassy carbon was employed as a standard material. First,in the MTEC Model 300 photoacoustic detector, glassycarbon sat on the sample under investigation, and PAmeasurements were conducted. The experimental data ofglassy carbon thus obtained was used to obtain C(L) andLPA by curve � tting. Then the sample was placed abovethe glassy carbon, and PA signals were again measured.In this way, in these two measurements, the two L valueswere kept the same. With the values of C(L) and LPA

found in the � rst PA measurement, thermal effusivitycould be determined by least-square � tting the experi-mental data of the sample to Eq. 3. The experimental dataand � tting results of a rubber sample and a graphite sam-ple are shown in Fig. 4. Equation 3 � ts the experimentaldata nicely with R . 0.999 for both samples. Thermaleffusivity values found are 1 3 103 W·s1/2·m22·K21 and 53 103 W·s1/2·m22·K21, respectively. They are in goodagreements with the results measured by the photother-

APPLIED SPECTROSCOPY 189

mal beam de� ection method in our laboratory for thesame samples: (1.2 6 0.2) 3 103 W·s1/2·m22·K21 and (5.66 0.2) 3 103 W·s1/2·m22·K21. Recalling that the thick-nesses of the rubber and graphite samples are 6.20 mmand 3.00 mm, respectively, the successful measurementof their thermal effusivity demonstrates that the positioneffect can be eliminated by the method presented above.With the measured C(L), Eq. 3 can be employed to quan-titatively process experimental data from an MTEC Mod-el 300 photoacoustic detector that has an additional emp-ty (gas) volume.

CONCLUSION

With an MTEC Model 300 photoacoustic detector thePA signal magnitude varying with sample-to-windowdistance L has been investigated. A method to measureC(L) using a well-known material has been presented.This C(L) takes into account the sample-to-window dis-tance L and the additional gas volume in the MTEC Mod-el 300 photoacoustic detector. With this C(L) the AMPmodel can be quantitatively applied to PA measurementswith the MTEC Model 300 photoacoustic detector, no

matter what the sample-to-window distance L is, thus ef-fectively normalizing the sample-position-dependence ef-fect.

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