A general expression (master equation, ME) is derived for the thermal impedance in photothermal experiments in a multilayer system, based on one-dimensional (1D) periodic heat diffusion. The ME in a compact form benefits from newly defined generalized, higher-order thermal reflection and transmission coefficients. The modeled system comprises seven layers among which a semitransparent sample and a transducer that integrates the temperature field within it (e.g., a pyroelectric sensor). The ME can be adapted to various experimental cell structures used in photopyroelectric, photoacoustic, photothermal radiometry, or thermoreflectance methods using volume- or surface-temperature detection, in view of spectroscopic applications or thermophysical properties determination. The derivation of special cases is facilitated by applying simple contraction rules to dimensionless quantities. Modeling multiple heat sources in the system is done by superposition of individual solutions. The possible extension of the 1D model to 2D geometry is demonstrated, in general, and practical criteria are discussed.
REFERENCES
1.
A. C.
Tam
, “Applications of photoacoustic sensing techniques
,” Rev. Mod. Phys.
58
, 381
(1986
). 2.
M. V.
Iravani
and M.
Nikoonahad
, “Photothermal waves in anisotropic media
,” J. Appl. Phys.
62
, 4065
(1987
). 3.
Z.
Bozóki
, A.
Miklós
, and D.
Bicanic
, “Photothermoelastic transfer matrix
,” Appl. Phys. Lett.
64
, 1362
(1994
). 4.
D. P.
Almond
and P. M.
Patel
, “Physics and its applications
,” in Photothermal Science and Techniques
, edited by E. R.
Dobbs
and S. B.
Palmer
(Chapman and Hall
, London
, 1996
), Vol. 10.5.
P.
Grosse
and R.
Wynands
, “Simulation of photoacoustic IR spectra of multilayer structures
,” Appl. Phys. B
48
, 59
(1989
). 6.
J. F.
Power
, J.
Karanicolas
, and S. W.
Fu
, “Method of Images direct mapping of one-dimensional heat flow patterns in an arbitrarily layered solid
,” Appl. Phys. B
71
, 57
(2000
). 7.
J.
Karanicolas
, S. W.
Fu
, and J. F.
Power
, “Depth profiling the thermal reflection coefficient of an opaque solid via an inverse thermal wave scattering theory based on the Method of Images
,” Appl. Phys. B
71
, 217
(2000
). 8.
D.
Maillet
, S.
André
, J. C.
Batsale
, A.
Degiovanni
, and C.
Moyne
, Thermal Quadrupoles Solving the Heat Equation Through Integral Transforms
(Wiley
, New-York
, 2000
).9.
A.
Mandelis
, Diffusion-Wave Fields: Mathematical Methods and Green Functions
(Springer
, New York
, 2001
).10.
P.
Grossel
and F.
Depasse
, “Thermophysical continuous profiles and their discretization
,” Int. J. Therm. Sci.
50
, 2078
(2011
). 11.
A.
Chauchois
, E.
Antczak
, D.
Defer
, and O.
Carpentier
, “Formalism of thermal waves applied to the characterization of materials thermal effusivity
,” Rev. Sci. Instrum.
82
, 074902
(2011
). 12.
P. M.
Patel
, D. P.
Almond
, and H.
Reiter
, “Thermal-wave detection and characterisation of sub-surface defects
,” Appl. Phys. B
43
, 9
(1987
). 13.
M. A.
Karam
, “A thermal wave approach for heat transfer in a nonuniform soil
,” Soil Sci. Soc. Am. J.
64
, 1219
(2000
). 14.
A.
Rosencwaig
and A.
Gersho
, “Theory of the photoacoustic effect with solids
,” J. Appl. Phys.
47
, 64
(1976
). 15.
A.
Mandelis
and M. M.
Zver
, “Theory of photopyroelectric spectroscopy of solids
,” J. Appl. Phys.
57
, 4421
(1985
). 16.
M.
Chirtoc
and G.
Mihailescu
, “Theory of the photopyroelectric method for investigation of optical and thermal materials properties
,” Phys. Rev. B
40
, 9606
(1989
). 17.
A.
Seas
and C.
Christofides
, “Photopyroelectric spectroscopy in the presence of an air gap transparent phase-shifter
,” Appl. Phys. B
62
, 563
–574
(1996
). 18.
H.
Coufal
, “Photothermal spectroscopy using a pyroelectric thin-film detector
,” Appl. Phys. Lett.
44
, 59
(1984
). 19.
A.
Mandelis
, “Frequency domain photopyroelectric spectroscopy of condensed phases: A New, simple and powerful spectroscopic technique
,” Chem. Phys. Lett.
108
, 388
(1984
). 20.
D.
Dadarlat
, M.
Chirtoc
, R. M.
Candea
, and I.
Bratu
, “Direct pyroelectric detection of optical absorption in non-transparent materials
,” Infrared Phys.
24
, 469
(1984
). 21.
M.
Chirtoc
, “Investigation of layered systems by photothermal methods with periodic excitation
,” in Thermal Wave Physics and Related Photothermal Techniques: Basic Principles and Recent Developments
, edited by E.
Marín
(Transworld Research Network
, Kerala
, 2009
), Chap. 2, pp. 29
–63
.22.
M.
Chirtoc
, R. M.
Candea
, and J.
Pelzl
, “Photopyroelectric (PPE) spectroscopy of radiation conversion efficiency in fluorescent dye ink layers
,” Appl. Spectrosc.
47
, 1071
(1993
). 23.
M.
Chirtoc
and I.
Chirtoc
, “A model for inverted spectral peaks in reflection mode photopyroelectric spectroscopy
,” Infrared Phys.
29
, 847
(1989
). 24.
M.
Chirtoc
, C.
Glorieux
, and J.
Thoen
, “Thermophysical properties and critical phenomena studied by the photopyroelectric (PPE) method
,” in Thermal Wave Physics and Related Photothermal Techniques: Basic Principles and Recent Developments
, edited by E.
Marín
(Transworld Research Network
, Kerala
, 2009
), Chap. 5, pp. 125
–158
.25.
J. A.
Balderas-López
, “Generalized 1D photopyroelectric technique for optical and thermal characterization of liquids
,” Meas. Sci. Technol.
23
, 065501
(2012
). 26.
P. K.
John
, L. C. M.
Miranda
, and A. C.
Rastogi
, “Thermal diffusivity measurement using the photopyroelectric effect
,” Phys. Rev. B
34
, 4342
(1986
). 27.
M.
Marinelli
, F.
Murtas
, M. G.
Mecozzi
, U.
Zammit
, R.
Pizzoferrato
, F.
Scudieri
, S.
Martellucci
, and M.
Marinelli
, “Simultaneous determination of specific heat, thermal conductivity and thermal diffusivity at low temperature via the photopyroelectric technique
,” Appl. Phys. A
51
, 387
(1990
). 28.
S.
Delenclos
, M.
Chirtoc
, A. H.
Sahraoui
, C.
Kolinsky
, and J. M.
Buisine
, “Assessment of calibration procedures for accurate determination of thermal parameters of liquids and their temperature dependence using the photopyroelectric method
,” Rev. Sci. Instrum.
73
, 2773
(2002
). 29.
C.
Christofides
and A.
Seas
, “Theory of photopyroelectric spectroscopy of a two-layer sample
,” Ferroelectrics
165
, 55
(1995
).30.
J. A.
Balderas-López
and A.
Mandelis
, “Photopyroelectric spectroscopy of pure fluids and liquid mixtures: Foundations and state-of-the-art applications
,” Int. J. Thermophys.
41
, 78
(2020
). 31.
A.
Mandelis
, J.
Vanniasinkam
, S.
Budhudu
, A.
Othonos
, and M.
Kokta
, “Absolute nonradiative energy-conversion-efficiency spectra in Ti3+:Al2O3 crystals measured by noncontact quadrature photopyroelectric spectroscopy
,” Phys. Rev. B
48
, 6808
(1993
). 32.
H.
Coufal
, “Photothermal spectroscopy of weakly absorbing samples using a thermal wave phase shifter
,” Appl. Phys. Lett.
45
, 516
(1984
). 33.
J. S.
Antoniow
, M.
Chirtoc
, and M.
Egee
, “A photopyroelectric method with air as the thermal coupling fluid
,” J. Phys. D: Appl. Phys.
30
, 1934
(1997
). 34.
J.
Shen
and A.
Mandelis
, “Thermal-wave resonator cavity
,” Rev. Sci. Instrum.
66
, 4999
(1995
). 35.
E.
Marin
and H.
Vargas
, “Recent developments in thermal wave interferometry for gas analysis
,” in Thermal Wave Physics and Related Photothermal Techniques: Basic Principles and Recent Developments
, edited by E.
Marín
(Transworld Research Network
, Kerala
, 2009
), Chap. 4, pp. 99
–123
.36.
J. S.
Antoniow
, M.
Chirtoc
, M.
Egee
, and B.
Burlet
, “Device for photopyroelectric analysis
,” French patent FR2715226A1 (18 January 1994
).37.
J. S.
Antoniow
, M.
Chirtoc
, and M.
Egée
, “Characterization of sedimentation processes using photopyroelectric monitoring
,” Ferroelectrics
165
, 215
(1995
). 38.
A.
Matvienko
and A.
Mandelis
, “Quantitative one-dimensional thermal-wave cavity measurements of fluid thermophysical properties through equivalence studies with three-dimensional geometries
,” Rev. Sci. Instrum.
77
, 064906
(2006
). 39.
J. A. P.
Lima
, E.
Marin
, M. G.
da Silva
, M. S.
Sthel
, S. L.
Cardoso
, D. F.
Takeuti
, C.
Gatts
, H.
Vargas
, C. E.
Rezende
, and L. C. M.
Miranda
, “On the use of the thermal wave resonator cavity sensor for monitoring hydrocarbon vapors
,” Rev. Sci. Instrum.
71
, 2928
(2000
). 40.
J. A. P.
Lima
, E.
Marin
, O.
Correa
, M. G.
da Silva
, S. L.
Cardoso
, C.
Gatts
, C. E.
Rezende
, H.
Vargas
, and L. C. M.
Miranda
, “Measurement of the thermal properties of liquids using a thermal wave interferometer
,” Meas. Sci. Technol.
11
, 1522
(2000
). 41.
C. A.
Bennett
and R. R.
Patty
, “Thermal wave interferometry: A potential application of the photoacoustic effect
,” Appl. Opt.
21
, 49
(1982
). 42.
M.
Chirtoc
, J. S.
Antoniow
, and M.
Egee
, “The effective thermal thickness: a new concept for photothermal investigation of layered systems,”
AIP Conf. Proc.
463
, 84
(1999
). 43.
J. A.
Balderas-Lopez
, A.
Mandelis
, and J. A.
Garcia
, “Thermal-wave resonator cavity design and measurements of the thermal diffusivity of liquids
,” Rev. Sci. Instrum.
71
, 2933
(2000
). 44.
D.
Dadarlat
and C.
Neamtu
, “Recent developments of photopyroelectric calorimetry of liquids
,” in Thermal Wave Physics and Related Photothermal Techniques: Basic Principles and Recent Developments
, edited by E.
Marín
(Transworld Research Network
, Kerala
, 2009
), Chap. 3, pp. 65
–97
.45.
D.
Dadarlat
, M.
Chirtoc
, C.
Neamtu
, R. M.
Candea
, and D.
Bicanic
, “Inverse photopyroelectric detection method
,” Phys. Status Solidi A
121
, K231
(1990
). 46.
M.
Chirtoc
, D.
Bicanic
, and V.
Toşa
, “A versatile inverse photopyroelectric (IPPE) technique and instrument for real time observation of the condensation of water vapor in the atmosphere
,” Rev. Sci. Instrum.
62
, 2257
(1991
). 47.
D.
Dadarlat
and A.
Frandas
, “Inverse photopyroelectric detection of phase transitions
,” Appl. Phys. A
57
, 235
(1993
). 48.
J.
Caerels
, C.
Glorieux
, and J.
Thoen
, “Photopyroelectric thermal wave setup for the absolute measurement of the thermal conductivity of low density gases
,” Rev. Sci. Instrum.
71
, 3506
(2000
). 49.
E. H.
Bentefour
, C.
Glorieux
, M.
Chirtoc
, and J.
Thoen
, “Broadband photopyroelectric thermal spectroscopy of a supercooled liquid near the glass transition
,” J. Appl. Phys.
93
, 9610
(2003
). 50.
S.
Pittois
, M.
Chirtoc
, C.
Glorieux
, and J.
Thoen
, “Direct determination of thermal conductivity of solids and liquids at very low frequencies using the photopyroelectric method
,” Anal. Sci. (Japan)
17
, s110
(2002
). 51.
S.
Pittois
, B.
Van Roie
, C.
Glorieux
, and J.
Thoen
, “Thermal conductivity, thermal effusivity, and specific heat capacity near the lower critical point of the binary liquid mixture n-butoxyethanol-water
,” J. Chem. Phys.
121
, 1866
(2004
). 52.
M.
Chirtoc
, I.
Chirtoc
, S.
Pittois
, C.
Glorieux
, and J.
Thoen
, “Thermal resistance of adhesive tapes measured by the photopyroelectric method
,” Rev. Sci. Instrum.
74
, 632
(2003
). 53.
U.
Zammit
, F.
Mercuri
, S.
Paoloni
, M.
Marinelli
, and R.
Pizzoferrato
, “Simultaneous absolute measurements of the thermal diffusivity and the thermal effusivity in solids and liquids using photopyroelectric calorimetry
,” J. Appl. Phys.
117
, 105104
(2015
). 54.
H.
Vargas
and L. C. M.
Miranda
, “Photoacoustic and related photothermal techniques
,” Phys. Rep.
161
, 43
(1988
). 55.
B. K.
Bein
and J.
Pelzl
, “Analysis of surfaces exposed to plasmas by nondestructive photoacoustic and photothermal techniques
,” in Plasma Diagnostics Surface Analysis and Interactions
, edited by O.
Auciello
and D. L.
Flamm
(Academic Press
, 1989
), Vol. 2
, Chap. 6, pp. 211
–326
.56.
M.
Chirtoc
, N.
Horny
, I.
Tavman
, and A.
Turgut
, “Photothermal spectroscopy of polymer nanocomposites
,” in Spectroscopy of Polymer Nanocomposites
, edited by S.
Thomas
, D.
Rouxel
, and D.
Ponnamma
(Elsevier
, Amsterdam
, 2016
), Chap. 11, pp. 312
–361
.57.
Y.
Fujii
, A.
Moritani
, and J.
Nakai
, “Photoacoustic spectroscopy theory for multi-layered samples and interference effect
,” Jpn. J. Appl. Phys.
20
, 361
(1981
). 58.
59.
P.
Helander
, “Signal processing in optothermal spectroscopy
,” J. Appl. Phys.
59
, 3339
(1986
). 60.
M.
Chirtoc
, D.
Bicanic
, I.
Chirtoc
, M.
Lubbers
, B.
Arnscheidt
, and J.
Pelzl
, “Laser photoacoustic spectroscopy of liquids using a large dynamic range optothermal window cell; application to ammonium in water, fatty acids and lubricating oils
,” J. Mol. Struct.
348
, 469
(1995
). 61.
M.
Chirtoc
, D.
Bicanic
, I.
Chirtoc
, and M.
Lubbers
, “Processing of weak photothermal signals from the optothermal window (OW) cell; Application to the determination of ammonium ion in water
,” Instrum. Sci. Technol.
26
, 281
(1998
). 62.
P.-E.
Nordal
and S. O.
Kanstad
, “Photothermal radiometry
,” Phys. Scr.
20
, 659
(1979
). 63.
R.
Santos
and L. C. M.
Miranda
, “Theory of the photothermal radiometry with solids
,” J. Appl. Phys.
52
, 4194
(1981
). 64.
R. D.
Tom
, E. P.
O’Hara
, and D.
Benin
, “A generalized model of photothermal radiometry
,” J. Appl. Phys.
53
, 5392
(1982
). 65.
R.
Fuente
, E.
Apinaniz
, A.
Mendioroz
, and A.
Salazar
, “Simultaneous measurement of thermal diffusivity and optical absorption coefficient using photothermal radiometry. I. Homogeneous solids
,” J. Appl. Phys.
110
, 033515
(2011
). 66.
A.
Salazar
, R.
Fuente
, E.
Apinaniz
, A.
Mendioroz
, and R.
Celorrio
, “Simultaneous measurement of thermal diffusivity and optical absorption coefficient using photothermal radiometry.: II. Multilayered solids
,” J. Appl. Phys.
110
, 033516
(2011
). 67.
M.
Chirtoc
, A.
Fleming
, N.
Horny
, and H.
Ban
, “Nonlinear heterodyne photothermal radiometry for emissivity-free pyrometry
,” J. Appl. Phys.
128
, 153101
(2020
). 68.
P. E.
Hopkins
, “Thermal transport across solid interfaces with nanoscale imperfections: Effects of roughness, disorder, dislocations, and bonding on thermal boundary conductance
,” ISRN Mech. Eng.
2013
, 1
(2013
). 69.
K. T.
Regner
, S.
Majumdar
, and J. A.
Malen
, “Instrumentation of broadband frequency domain thermoreflectance for measuring thermal conductivity accumulation functions
,” Rev. Sci. Instrum.
84
, 064901
(2013
). 70.
N.
Horny
, M.
Chirtoc
, A.
Fleming
, G.
Hamaoui
, and H.
Ban
, “Kapitza thermal resistance studied by high-frequency photothermal radiometry
,” Appl. Phys. Lett.
109
, 033103
(2016
). 71.
M.
Chirtoc
, J.
Gibkes
, H. G.
Walter
, A.
Christ
, J. S.
Antoniow
, D.
Bicanic
, Z.
Bozoki
, G.
Szabo
, B. K.
Bein
, J.
Pelzl
, M.
Kleebauer
, H.
Bader
, and M.
Marinelli
, “Comparative study of coating thickness determination in packaging composite materials using PTR, PA and PPE methods
,” Anal. Sci. (Japan)
17
, S185
(2001
). 72.
M.
Chirtoc
, “Investigation of layered systems and temperature-dependent thermophysical characterization by photothermal methods with periodic excitation
,” J. Phys.: Conf. Series
214
, 012005
(2010
). 73.
M.
Chirtoc
, E. H.
Bentefour
, J. S.
Antoniow
, C.
Glorieux
, J.
Thoen
, S.
Delenclos
, A. H.
Sahraoui
, S.
Longuemart
, C.
Kolinsky
, and J. M.
Buisine
, “Current mode versus voltage mode measurement of signals from pyroelectric sensors
,” Rev. Sci. Instrum.
74
, 648
(2003
). 74.
E. W.
Weisstein
, See http://mathworld.wolfram.com/HankelTransform.html for Hankel Transform, from MathWorld - A Wolfram Web Resource.75.
S.
Pittois
, Ph.D. thesis
, Catholic University
, Leuven, Belgium
, 2004
(in Dutch).76.
G.
Hamaoui
, E.
Villarreal
, H.
Ban
, M.
Chirtoc
, and N.
Horny
, “Spatially localized measurement of isotropic and anisotropic thermophysical properties by photothermal radiometry
,” J. Appl. Phys.
128
, 175104
(2020
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
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