estimation of heat loss in thermal wave experimentsii. estimation of the heat loss coefficient in a...

Estimation of heat loss in thermal wave experiments

Hilmar Straubea) and Otwin BreitensteinMax Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle, Germany

(Received 3 November 2010; accepted 18 December 2010; published online 28 March 2011)

The theoretical description of thermal wave experiments assumes harmonic heat introduction

(sinusoidally modulated heating and cooling) and an adiabatic (thermally insulated from the

environment) sample. In most experimental setups, however, only pulsed heat is introduced, and

the mean sample temperature stabilizes at a value above ambient determined by the strength of the

heat loss to the surroundings. The question arises as to what conditions have to be satisfied such

that the assumption of an adiabatic sample is applicable to the experimental situation. In this

contribution we present a treatment of heat-loss mechanisms (radiation, conduction/convection to

air, and conduction to the sample holder) together with their implications for signal shape and

temperature control of the sample. The treatment leads to a general mathematical criterion to

decide whether heat conduction inside the sample can be considered to be adiabatic or not,

depending on excitation frequency and spatial frequency content of the heating power distribution.

The theoretical considerations are applied experimentally to typical situations of thermally

thin samples (where heat loss to the sample holder is critical) and thermally thick

samples (where temperature control can be a problem). VC 2011 American Institute of Physics.[doi:10.1063/1.3549734]

I. INTRODUCTION

Heat transfer from the sample to the environment can

occur via radiation, conduction/convection to air, or through

thermal contact resistance to a sample holder. In most exper-

imental works on thermal wave based measurements, the

effect of heat loss on the measurements is tacitly neglected.

Estimations of the influence of heat loss for several ex-

perimental situations can be found in the literature: Salazar

et al.1 and Zhang and Imhof2 for the experimental determina-tion of the thermal diffusivity using lock-in thermography

(considering only conductionlike heat loss and radiation),

Cahill and Pohl3 for thermal conductivity measurements

using the 3x-method, McKelvie4 for thermoelastic stressmapping (assuming linear and homogeneous heat loss coeffi-

cients), Rousset et al.5 for the mirage effect, and, for materialfatigue analysis, Meneghetti6 (although the excitation is

modulated, only steady-state heat loss is of interest for the

technique).

To actually predict heat loss behavior is both theoreti-

cally and experimentally challenging. For convection it is

necessary to solve the Navier–Stokes equation in air coupled

to the time-harmonic heating in the medium.5 This can only

be done through involved numerical simulations and has to

be done anew for every heat source distribution. For ther-

mally thin samples, heat conduction to the sample holder is

typically the dominant heat loss mechanism. The corre-

sponding thermal resistance depends critically on the surface

properties and the contact pressure and is thus difficult to

predict. Finally, conductionlike heat transfer to air is a func-

tion of the spatial frequency content of the heating power

distribution.

Therefore it is not practical to include heat loss behavior

in the formulas for experimental evaluation (for treatments

assuming a linear and homogeneous thermal conductivity to

the environment; see Refs. 7–9). Instead, a simple criterion

is needed to assess the relative importance of heat loss and

the effects of experimental measures taken to reach a regime

where it is negligible (quasiadiabatic conditions10 with

respect to the environment). Such measures can include the

following: doing the experiment at a sufficiently high excita-

tion frequency, moving the sample to vacuum, and/or to

apply a highly reflective coating to suppress radiation heat

loss. Reducing the excitation amplitude, however, alleviates

the problem only in the special case that free convection is

the dominant mechanism.

A certain amount of heat loss is necessary to provide

for temperature control of the sample. All the heat intro-

duced in an excitation period has to be led away by some

mechanism; which means there are conflicting experimen-

tal design goals. On the one hand a low heat loss to the

environment is desirable to have a defined, easily interpret-

able signal. On the other hand a high thermal conductivity

to some kind of temperature controlled heat sink is needed

for stabilizing the sample temperature. The latter is espe-

cially a problem for accurate measurements on thin film

devices on glass substrates: Temperature control from the

surface is often not possible and not very effective through

the substrate.11,12

In this contribution we give practical estimations for the

importance of heat loss, both for the outcome of the thermal

wave experiment and for temperature control. The results are

not limited to one particular experimental situation, they can

be applied directly to thermally thin, thick, or layered sample

geometries if the adiabatic thermal wave propagation in the

samples is known.a)Electronic mail: [email protected].

0021-8979/2011/109(6)/064515/10/$30.00 VC 2011 American Institute of Physics109, 064515-1

JOURNAL OF APPLIED PHYSICS 109, 064515 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

http://dx.doi.org/10.1063/1.3549734http://dx.doi.org/10.1063/1.3549734

II. ESTIMATION OF THE HEAT LOSS COEFFICIENT

In a thermal wave experiment, heat is introduced peri-

odically into the sample. After switching on the experiment,

a transient (the initial heating-up phase; see Refs. 10 and 13

for details on how to correct its effects on the signal) leads to

a quasisteady-state, where only periodic time dependence

remains. In the case of sinusoidally shaped heating, a tem-

perature distribution

Tðr; tÞ ¼ T0 þ ToffsetðrÞ þ TmðrÞ exp i½xtþ uðrÞ� (1)

is established, where T0 is ambient temperature, Tm the am-plitude of the harmonic response to the excitation, and Toffseta steady temperature offset distribution proportional to the

average amount of heat introduced in the sample at a certain

location. As there is no cooling phase in the excitation of

most thermal wave experiments, the heating can be decom-

posed into constant average heating (frequency zero) plus

the oscillatory component (frequency x). Tm is then the ther-mal response to the oscillatory component and Toffset can beviewed as the response to the excitation at zero frequency.

Especially in electronic device testing, square wave ex-

citation is necessary to have a well-defined operating point

of the device. In this case the higher harmonics of the heating

pulse lead to further terms of the form Tmn(r) exp i[nxtþun(r)] for n¼ 1,3,5,… These higher harmonic com-ponents do not interfere with one another and can be eval-

uated separately by performing a sine/cosine correlation at

frequency nx, which is generally only done for the (strong-est) first harmonic n¼ 1.

To compare the individual heat loss mechanisms, it is

useful to describe them using the same parameter. The heat

loss coefficient h connects the surface-to-ambient tempera-ture difference to the heat flux across the interface:

hðT � T0Þ ¼ _q; (2)

where q̇ denotes the heat flux from the sample into the sur-rounding (W m�2). The defining equation for h is known asNewton’s law of cooling.14 In general, the heat loss coeffi-

cient is a function of location (due to inhomogeneity and ex-

citation spatial frequency content), of temperature (due to

nonlinearity, especially for free convection), and of fre-

quency. Therefore, h affects Toffset and Tm differently.In the following section, relevant literature results for

estimating the heat loss coefficient h of the different mecha-nisms shall be summarized and applied to the special case of

periodic excitation. Their influence on thermal wave experi-

ments will be assessed in Sec. III.

A. Radiation

To actually calculate radiation involves integrating the

Planck distribution weighted with the spectral emissivity

of the sample at every point. An upper limit is given by

the blackbody with a constant emissivity of 1 for all

wavelengths.

The heat loss from the surface of a blackbody (radiation

into the half space, solid angle 2p) can be calculated by

linearizing the Stefan–Boltzmann law around ambient tem-

perature T0 (r: Stefan–Boltzmann constant):14

_qbb � 4rT30 ToffsetðrÞ þ TmðrÞ exp i½xtþ uðrÞ�f g: (3)

To linearize heat loss due to radiation is justified if the tem-

perature variation is small compared to the absolute tempera-

ture of the environment Tm, Toffset� T0; which is given in themajority of thermal wave experiments. As an example con-

sider lock-in thermography, which typically has excitation

amplitudes Tm of a few mK and temperature offsets Toffsetbelow 1 K, done at room temperature T0� 300 K.

According to the definition of h, the maximum (black-body) radiation contribution to the heat loss coefficient is

thus

hrad;bb ¼ 4rT30 ; (4)

giving a value of 6.1 Wm� 2 K� 1 at room temperature

(300 K; Tm, Toffset� T0). Every nonblackbody surface has alower heat loss coefficient hrad< hrad,bb.

If the emissivity is spatially varying, e.g., due to low

emissivity metal contacts on the sample surface, hrad can be

a function of location. It does not depend on the excitation

frequency and thus affects Toffset and Tm equally.

B. Conduction and convection

Heat loss to air, other gases, or liquids (“fluids”) occurs

through heat conduction and convective motion in the heated

fluid. After a temperature difference between sample and

fluid is switched on, it takes a certain time until the steady-

state, free convective flow is fully established. Depending on

whether this time is short or long compared to the excitation

period of the experiment, low- and high-frequency behavior

have to be distinguished for the influence of heat loss on Tm.For Toffset (at x¼ 0) only the low frequency behavior is rele-vant. The interaction between low and high frequency behav-

ior for Tm is described in Sec. II B 3.

1. Low-frequency behavior: The developed freeconvective flow

The magnitude of the effect of free convection can be

estimated by considering convection along horizontal and

vertical sample geometries, where the fluid is heated along a

part of the sample that is kept at a temperature that is raised

with respect to the environment. Both situations are shown

in Fig. 1. Consider a sample with a heated portion of finite

length L at an average temperature above ambient. In thevertical case, Fig. 1(a), the boundary layer flow starts to

form below at the leading edge (at x¼ 0) and detaches fromthe sample at the trailing edge of the isothermal surface.

There (at L), a buoyant plume is formed that is no longer incontact with the sample. The influence of the trailing edge

region on the overall heat transport can be modeled in a gen-

eral way and is found to be minimal.15 It is thus permissible

to approximate a finite vertical heated region in the experi-

ment by an infinite model, neglecting the presence of the

trailing edge region (L!1). In the leading edge region, the

064515-2 H. Straube and O. Breitenstein J. Appl. Phys. 109, 064515 (2011)


thermal boundary layer thickness d(x) is smallest, i.e., thetemperature gradients and thus the heat loss is highest; see

Fig. 1. In the horizontal case (b), a flow will develop from

both left and right edges. In the center region (at L/2) of thefinite horizontal surface, these flows merge and again form a

buoyant plume that carries away the heat.16 Experimentally

well-tested steady-state solutions for both situations are

available in the literature.16–18

Details on how to calculate convection for arbitrary flu-

ids can be found in the Appendix, here we will only give spe-

cialized results for air at atmospheric pressure. In the

laminar regime, the results for our cases are17,18

hconv; a ¼ 1:05x�1=4ðT � T0Þ1=4; (5)

for a vertical sample (a) and

hconv; b ¼ 0:397x�2=5ðT � T0Þ1=5: (6)

for a horizontal sample (b). Results are given with all units

dropped, [x]¼m, [T]¼K, [h]¼Wm�2 K�1. Both expres-sions diverge for x ! 0 due to the assumption of an idealstep function in temperature at the leading edge. There, the

thermal boundary layer d(x) is very thin and the increasedtemperature gradient leads to stronger heat conduction from

the surface into the moving fluid. Note also that due to nonli-

nearity in the temperature, no linear approximation for low

temperature offsets is possible.

Figure 2 shows the spatially varying convection heat

loss coefficients for several deviations from the ambient tem-

perature T�T0. For moderate temperature deviations, con-vection heat loss exceeds radiation only close to the leading

edge.

As already mentioned at the beginning of this section,

the system needs a certain time to build up a fully developed,

free convective flow. A characteristic time is the time needed

for flow originating from the leading edge to arrive at a posi-

tion x. For the vertical flow of case (a), this leading edgeeffect was investigated by Goldstein and Briggs19 for several

boundary conditions. For our case (interpolated for air with a

Prandtl number of Pr¼ 0.7 from values19 for Pr¼ 0.1 and1.0; all units dropped, [t]¼ s, [x]¼m, [T]¼K):

t ¼ 16:5ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x

T � T0

r: (7)

Thus, for an oscillating source with, e.g., Tm¼ 10 mK ofcharacteristic extension 10 mm, it takes about 16.5 s to build

up convective effects.

2. High-frequency behavior: Conductionlike heat loss

For time scales much shorter than those estimated by

Eq. (7), an initial stage of convective motion is formed.19

For reasons of symmetry, the convective term drops from the

heat equation. That means the convective motion in this ini-

tial stage does not contribute to heat loss; the fluid—although

it is moving macroscopically for all but perfectly horizontal

surfaces—follows the heat equation of a rigid body.

For a homogeneous heat flux q̇ into the (semi-infinite) fluid, the solution of the heat equation is known.20

Evaluating the temperature modulation amplitude in the fluid

and close to the sample surface (z ! 0) gives the value forthe heat loss coefficient hcond, by its definition in Eq. (2):

T � T0 ¼ _q1

kaffiffiffiffiffiffiffiffiffiffiffiffiffiix=Da

p (8)

FIG. 1. Geometrical situations considered to assess the influence of free

convection effects for thermal wave experiments. The fluid is heated along a

part of the sample that is kept at a raised temperature. The leading edge of

the heated portion of the sample (white) is assumed at x¼ z¼ 0 (and x¼L,z¼ 0 for the horizontal case). In (a) the flow is driven by the buoyancy forceof the heated fluid along the sample, whereas in (b) the flow along the sam-

ple is driven by a pressure difference, which in turn is caused by buoyancy.

The thermal boundary layer thickness d(x) is indicated in dark gray.

FIG. 2. Heat loss coefficient hconv for air at atmospheric pressure, ambient temperature T0¼ 300 K, temperature offset from 0.1 to 10 K.



) hcond ¼ kaffiffiffiffiffiffiffiffiffiffiffiffiffiix=Da

p; (9)

where k is the thermal conductivity and D¼ k/cp. the ther-mal diffusivity; the index a refers to the fluid (air). The com-plex nature of hcond indicates that heat transport due toconduction is phase-shifted to its excitation by an angle of

45�. When estimating the relative impact of heat conduction,however, only the absolute value of h is of interest.

The heat transfer coefficient hcond is proportional to thethermal effusivity

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikaðcp.Þa

p, which describes the ability of

the material to exchange heat with its environment via con-

duction. Note also the proportionality, hcond /ffiffiffiffixp

. The

physical reason for this proportionality is that the tempera-

ture gradient is confined mostly to a distance Ka ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Da=x

p(the thermal diffusion length) from the interface into the

fluid. The heat flux �karT is proportional to the gradient ofthe temperature profile in the fluid for a given temperature

difference. Thus, it increases with rising excitation fre-

quency. The relation (8) is not valid anymore when the oscil-

lation is so slow that the transient to convection begins (see

above). In particular, Toffset (with x¼ 0) is not influenced byconductionlike behavior.

The above discussion was restricted to homogeneous

sources q̇. Inhomogeneous sources can be treated most easilyin Fourier space, i.e., after a 2D Fourier transform in the sur-

face plane of the sample, F 2[q̇(x, y)]¼ q̇(kx, ky). The result is(for details see Sec. IIIA and Ref. 21)

Tðkx; kyÞ � T0 ¼ _qðkx; kyÞ1

kaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiix=Da þ k2

p (10)) hcondðkÞ ¼ ka

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiix=Da þ k2

p(11)

with the absolute value of the spatial frequency vector

k2¼ k2xþ k2y. That means that the average heat loss factordue to conduction is a function of the heat source distribution

and increases for high spatial frequency content of q̇. Thewavelength s¼ 2p/k is a measure of the characteristic dimen-sions of the corresponding details. Close to thin heating

points or lines, very high spatial frequencies occur and,

according to Eq. (9), these have a higher heat loss coefficient

than homogeneous sources. This can be understood using the

thermal diffusion length in air Ka ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Da=x

pas a character-

istic length of heat diffusion into the fluid:

hcondðkÞ ¼ kaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2i=K2a þ ð2p=sÞ

2q

: (12)

For details smaller than this characteristic length Ka, the heattransport occurs not only perpendicular to the surface but

also parallel to it. This additional heat transport increases

with decreasing s.The dependence of the heat loss coefficient hcond on

the excitation frequency f and wavelength s is shown inFig. 3. It can be seen that even for homogeneous heating,

heat conduction to air exceeds thermal radiation for fre-

quencies above 0.2 Hz. Independent of the excitation fre-

quency, heat conduction exceeds thermal radiation for

details smaller than� 3 cm as long as the high-frequencyregime is given.

3. Interaction of conduction and convection for Tmand Toffset

Convection and conductionlike behavior can occur

simultaneously in a given experimental situation. Let us start

our analysis with the case of purely stationary heating

(x¼ 0). If convection is the strongest heat loss mechanism,then an equilibrium is approached between Toffset and theconvective flow caused by Toffset. If another heat loss mecha-nism is stronger, then Toffset is largely determined by thisother mechanism and a corresponding convective flow

develops.

The build-up time for convective effects [see Eq. (7)] is

often much longer than the excitation period. This does not

mean that the oscillatory component Tm is uninfluenced bythe (stationary) convective flow. The maximum Tm of a heatsource affected by a fast convective stream (e.g., close to the

leading edge) will be somewhat lower than the Tm of anothersource of equal strength but under quiescent fluid.

For low excitation frequencies when the characteristic

time for establishing convective effects is smaller than the

excitation period, the convective flow is modulated. The

influence of the modulated temperature Tm on the convectiveflow can, however, often be neglected in the common situa-

tion that Tm� Toffset.Typically, both convection (due to Toffset) and conduc-

tion (due to the oscillation of Tm) occur simultaneously;then, the two effects superimpose. The superposition is

expected to be linear as long as Tm� Toffset holds.

4. Forced convection

Temperature control of the active layer can be a serious

problem for thin film samples on thick substrates. In this

case, higher heat loss coefficients at the active front surface

can be achieved by forced convection; that is, by having an

external flow over the sample.14 The increase in heat loss

coefficient compared to free convection is caused by the

thinner thermal boundary layer. For moderate velocities

(u1� 10 ms�1), the external flow causes an increase of

FIG. 3. Heat loss coefficient hcond for air at atmospheric pressure, 300 K,for a range of excitation frequencies and spatial frequencies k¼ 2p/s. Notethat although the heat loss parameter diverges for high excitation frequen-

cies, the overall heat flux to the environment q̇cond¼ hcondTm approacheszero.



approximately a factor of 10 compared to the free convection

case with h / ffiffiffiffiffiffiu1p (see the Appendix for details). For air atatmospheric pressure (all units dropped; [x]¼m, [u]¼m/s,[h]¼Wm�2 K�1):14

hfc ¼ 1:94x�1=2u1=21 : (13)

Forced convection has a similar dependence on the distance

to the leading edge of the sample as free convection. In con-

trast to free convection, it is not driven by buoyancy but by

external flow with velocity u1 and does not depend on thetemperature difference of the sample surface to the environ-

ment. Thus, hfc is not a function of temperature for forcedconvection.

C. Heat conduction to sample holder

In many setups, the sample is mounted on some kind of

sample holder. The heat capacity of this sample holder typi-

cally exceeds that of the sample by orders of magnitude, in

which case it may be considered a heat sink at room temper-

ature (or thermostatted at any other temperature).

The heat flux q̇ from the sample back surface to the sam-ple holder causes a temperature drop between any location on

the back surface of the sample and the (homogeneous) tem-

perature of the sample holder, according to Eq. (2). It turns

out that the thermal contact conductance hco is a difficult prop-erty to predict theoretically. There is a marked dependence on

the pressure, p, with which the surfaces are pressed to-gether,22,23 that follows roughly h ! p2/3. Correlations can befound in Refs. 22 and 23. Typical values for metal–metal con-

tact surfaces at low pressures (< 20 kPa) range from 103 to104 Wm�2 K�1. For softer sample materials on a metal sam-

ple holder the values are even larger. These values exceed all

other mechanisms by orders of magnitude.

For a thermally thick sample, the back surface is (by

definition) of no consequence to the thermal wave pattern.

Thus, Tm on the sample surface is not influenced by heat lossto the sample holder. For zero frequency (corresponding to

Toffset) no sample is thermally thick (the thermal diffusionlength K¼1 for x¼ 0). The overall heat loss coefficientfor Toffset is then given through the sum of the thermal resist-ance of the thick substrate (l/k) and through the interface re-sistance (1/hco)

hthick;x¼0 ¼ 1=hco þ l=kð Þ�1: (14)

For a thermally thin sample the temperature field in the sam-

ple does not depend on the depth in the sample. Heat loss to

the sample holder thus influences Tm and Toffset equally. Inthe extreme case of an ideal thermal contact hco!1, thereis no difference between sample temperature and sample

holder temperature; i.e., the whole sample is at the sample

holder temperature and no signal is registered in thermal

wave experiments. When thermostatting a thermally thin

sample, the contact conductance, therefore, should not be too

high (such that a controlled signal can develop, ideally unin-

fluenced by the sample holder’s hco) but also not too low(such that the sample holder temperature is at least approxi-

mately the mean temperature of the sample).

If the heat conduction to the sample holder is found

experimentally to be too high for a specific application, it

can be reduced using sample holders that are intentionally

roughened or covered by a metal net.10

III. IMPACT OF HEAT LOSS ON THE RESULTS OFTHERMAL WAVE EXPERIMENTS

A. General treatment

The previous section dealt with practical estimations of

the coefficient h, which unifies the treatment of heat lossmechanisms. We now give a general approach on how to test

whether this heat loss alters the form of the thermal wave

appreciably or not. It is assumed that the sample is laterally

homogeneous and has two parallel surfaces (for a thermally

thick substrate, the back surface does not alter the tempera-

ture distribution in the sample). Only the oscillating signal

Tm is registered, therefore we need only consider the heatloss coefficient for an excitation frequency x.

The idea to estimate the relative influence of heat loss is

to first calculate the temperature field Tm0 resulting from apower distribution P0, disregarding all heat losses, and tolook at the temperature field variations at the upper and

lower surfaces of the sample. The resulting temperature vari-

ation Tm0(r)expi[nxtþun(r)] at every point of the surfacesgives rise to new time-harmonic sources P1 (Wm

�2) on the

upper (at z¼ 0) and lower (at z¼ l) surfaces:

P1ðrÞjz¼0; l ¼ �hTm0ðrÞjz¼0; l: (15)

This additional heat source distribution in turn alters the ther-

mal wave field to a certain extent. In principle, this proce-

dure could be iterated to give the temperature field. More

importantly, it gives a condition for neglecting the influence

of heat loss. Heat loss (radiation, conduction, convection) is

negligible if the first iteration temperature does not differ

appreciably (according to the exigencies of the experiment)

from the temperature variation calculated when neglecting

heat losses.

This condition can be evaluated quantitatively for arbi-

trary heat sources. Any planar heat source can be decom-

posed into spatially sinusoidally varying contributions

through a 2D Fourier transform. It is therefore sufficient to

look at the spatial frequency components to understand the

response of the system. Because of the linearity of the sys-

tem the temperature response, as well as the new heat sour-

ces due to the temperature field at the surface will be

spatially sinusoidally varying. (This approach implicitly

requires a homogeneous and linear h; for the purpose of esti-mation it should be admissible to use a maximum value of has a constant.) For a single Fourier component of the signal,

the thermal response can be described by a single complex

number with the response-to-excitation amplitude ratio

(given in Km2W�1) and a corresponding phase shift. This

complex ratio as a function of the spatial frequency k is iden-tical to the 2D Fourier transform of the point spread function

and may be called the thermal transfer function, in analogy

to the optical transfer function.24 It can be given analytically

for layered systems; see Ref. 21 for details. The transfer



function for a source at depth z0 that is sinusoidally varyingwith spatial frequency k, and whose temperature field isobserved at depth z, is denoted by Fk(z / z0).

The surface temperature values neglecting heat loss are

then given by (l: sample thickness, p: excitation amplitude):

Tm0ð0Þ ¼ pFkð0 z0Þeikx;Tm0ðlÞ ¼ pFkðl z0Þeikx:

The additional surface heating terms corresponding to these

temperature variations at the surfaces are

P1ð0Þ ¼ � hfrontTm0ð0Þ;P1ðlÞ ¼ � hbackTm0ðlÞ:

These lead to the first iteration temperature distribution eval-

uated at z:

Tm1ðzÞ ¼ pFkðz z0Þeikx

� phfrontFkðz 0ÞFkð0 z0Þeikx

� phbackFkðz lÞFkðl z0Þeikx:

The thermal wave field without heat loss and the additional

contributions from both surfaces are again spatially sinusoi-

dally varying functions. The relative influence of these addi-

tional terms is given by the ratio of amplitudes:

vfront ¼hfrontFkð0 z0ÞFkðz 0Þ

Fkðz z0Þ

��; (16)

vback ¼hbackFkðl z0ÞFkðz lÞ

Fkðz z0Þ

��: (17)

The physical interpretation of the relative heat loss v is a roughestimate of the percentage of error introduced by neglecting

heat loss for a given spatial frequency component, k.In the common setup that the heat source and the obser-

vation plane are at the front surface (z0 ¼ 0, z¼ 0), the frontsurface expression simplifies to

vfront ¼ jhfrontjjFkð0 0Þj: (18)

For a thermally thin sample the heat conduction does not

depend on the depth in the sample and thus Fk(l / 0)¼Fk(0 / 0), which gives

vback; thin ¼ jhbackjjFkð0 0Þj: (19)

For a thermally thick sample Fk(l / 0) is zero by defi-nition (the sample is so thick that thermal waves do not pene-

trate) and thus

vback; thick ¼ 0: (20)

The formulas (18) to (20) are now surprisingly simple and

can be used directly to quantify excitation frequency depend-

ent heat loss behavior for a given estimate of the heat loss

coefficient (Sec. II) and the known transfer function.

Some general conclusions can also be drawn: Relative

heat losses are not only proportional to the heat loss coeffi-

cient but also to the transfer function, i.e., the ratio of K

temperature variation per Wm�2 for a given spatial fre-

quency. They do not depend on the excitation amplitude p ifh is not a function of temperature (i.e., if free convection isnegligible). The latter is especially important. It means that

the disturbing influence of heat loss cannot be reduced by

choosing a low excitation amplitude unless free convection

is the dominant mechanism.

B. Application to thermally thin and thermally thicksamples

1. Thermally thin

If a thermally thin sample is suspended in a fluid, con-

duction heat loss will be the dominant mechanism for a wide

range of excitation frequencies; see Fig. 3. For a thermally

thin sample Fk is:21

Fk ¼1

lksðix=Ds þ k2Þ; (21)

where D: thermal diffusivity, k: thermal conductivity,l: thickness, s: index of the sample, a: index of the fluid.Both faces will contribute equally to the heat loss, hence the

factor of 2, with Eq. (9):

vcondðkÞ ¼ 2jhcondjjFkj ¼ 2ka

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=D2a þ k44

plks

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=D2s þ k4

p ; (22)

which is highest for homogeneous excitation. High-spatial-

frequency components are less affected by heat loss

An interpretation of this formula in the homogeneous

case (k¼ 0) is found using the thermal diffusion lengthKa ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Da=x

pas a measure of penetration depth into the

fluid:

vcond ¼ffiffiffi2p KaðcpqÞa

lðcpqÞs: (23)

Apart from the factorffiffiffi2p

this is the ratio of heat capacities

influenced by the periodic temperature change. In conse-

quence, if the heat stored in the fluid in one excitation period

is small compared to that stored in the sample, the thermal

wave pattern is not disturbed by heat loss due to conduction.

If the back surface is mounted on a sample holder, the

factor of 2 in Eq. (22) vanishes, as only one surface contrib-

utes to conduction heat loss. On the other surface, the coeffi-

cient of heat loss to the sample holder is independent of the

excitation frequency, x, and spatial frequency, k, and is typi-cally hco> 1000 Wm

�2 K�1 (see Sec. II C). The back sur-

face heat loss is thus

vcoðkÞ ¼ hco1

lksffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=D2s þ k4

p ; (24)

which is again highest for homogeneous excitation (k¼ 0)There is an interesting physical interpretation of Eq.

(24) for k¼ 0. Using the (areal) heat resistance Rco¼ 1/hcoand the (areal) heat capacity C¼ l(cpq)s¼ lks/Ds, the termRcoC is found in the ratio. In analogy to an electrical resis-tor–capacitor circuit, RcoC can be interpreted as the



characteristic time of heat exchange between a thermally

thin sample and the sample holder. Rewriting the ratio with

the excitation period tp¼ 2p/x instead of the angular fre-quency x,

vcoð0Þ ¼tp

2pRcoC; (25)

it becomes apparent that the relative heat loss v is the ratioof the excitation period tp to the characteristic time RcoCof the asymptotic initial heating-up of the sample

(apart from a constant factor 2p). The assumption of a qua-siadiabatic sample is wrong if a sample approaches thermal

equilibrium with the sample holder during the lock-in

period.10

The relative heat loss v for radiation, conduction, andconduction to the sample holder is shown in Fig. 4 for the

example of a silicon solar cell lightly pressed to a metal sam-

ple holder. The dominant heat loss mechanism is (by far)

heat conduction to the sample holder. For the high thermal

conductivity between thermally thin sample and metal sam-

ple holder assumed here (hco¼ 1250 Wm�2 K�1), experi-ments with excitation frequencies below 10 Hz cannot be

interpreted quantitatively without taking heat loss into

account. If the sample thickness is reduced from 200 to

100 lm, the frequency needs to be doubled for the same heatloss ratio of approximately 5%.

2. Thermally thick

For a thermally thick sample the heat loss over the back

side of the sample (be it in contact with some kind of sample

holder or not) does not influence the oscillatory component.

The dominant heat loss mechanism for a wide range of fre-

quencies is thus conduction.

For a thermally thick sample Fk is:21

Fk ¼1

ksffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiix=Ds þ k2

p ; (26)

This gives, for the relative heat loss due to conduction to the

fluid, using Eq. (11):

vðkÞ ¼ kaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=D2a þ k44

pks

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=D2s þ k44

p : (27)

For the limit of low spatial frequencies (homogeneous exita-

tion, k¼ 0) and using the thermal diffusion length for thefluid Ka and the sample Ks, this can be written as

vð0Þ ¼ kaffiffiffiffiffiffiffiffiffiffiffi1=Da

pks

ffiffiffiffiffiffiffiffiffiffi1=Ds

p ¼ KaðcpqÞaKsðcpqÞs

: (28)

The influence of heat loss due to conduction to the fluid is

thus negligible if the heat capacity of the fluid, which is

affected by periodic temperature change, is small compared

to the affected heat capacity of the sample. This ratio is

about v¼ 0.0007 for glass exposed to air (the dependence onthe excitation frequency cancels out). The influence of heat

loss on the pattern of Tm is therefore negligibly small.For very high spatial frequencies, the terms in k domi-

nate over those in x, giving

vðkÞ ¼ kaks: (29)

This ratio is valid for highly detailed parts of the heating

power distribution and is v¼ 0.024 for glass exposed to air.Better thermal conductors than glass result in a lower relative

heat loss v.

IV. EXPERIMENTAL VERIFICATION

A. Thermally thin: Solar cell on a sample holder

One of the main applications for lock-in thermography

(LIT) is the evaluation of current flow patterns in crystalline

silicon solar cells (shunting, edge currents, diffusion cur-

rents, series resistance).10 The silicon solar cell is also a pro-

totypical case for thermally thin heat propagation.

To reliably contact the cell and for good temperature

control, almost all experiments are carried out with the sam-

ple lightly pressed to a metal sample holder. In this section

we will investigate temperature transients, and the impact of

heat loss in this example situation, experimentally. We will

see that this direct contact between metallic back surface and

sample holder is critical and will become even more so if the

trend to thinner cells continues.

1. Initial heating-up phase

When a lock-in thermography experiment is started, the

cell has the same temperature as its sample holder. Also, this

excitation is positive only; it does not have a cooling phase.

The heating is counteracted by heat loss to the sample

holder, all other heat loss mechanisms are negligible. Thus

in quasisteady-state

hcoToffset ¼ �P (30)

FIG. 4. (Color online) Relative heat loss for a thermally thin sample (values

for a 200 lm silicon solar cell) on a sample holder for homogeneous excita-tion. The contributions from conduction and radiation from the front surface

are negligible compared to that from the heat loss through the back contact

for all frequencies. Convection (not shown) replaces conductionlike behav-

ior at very low frequencies.



holds (with the average heating power density �P). The timeneeded to reach this quasisteady-state is called the initialheating-up phase.10

In the thermally thin case, this phase can be described

theoretically by the thermal capacitance (C¼ cpql, heatcapacity per area) and the thermal resistance to the heat sink

(the sample holder; Rco) with the exponential time depend-ence of a charging capacitor:

Toffset; transientðtÞ ¼ Toffset 1� exp �t

RcoC

� �� : (31)

The resulting transient was measured in the central part of a

monocrystalline silicon solar cell using an InSb thermoca-

mera (bare surface, locally corrected for emissivity). To

avoid further complications, the excitation was not pulsed

(as in an actual LIT experiment) and the whole transient was

recorded (10 s). The averaged signal (15 repetitions) is

shown in Fig. 5.

Evaluation of these transients gives a heat loss coeffi-

cient of hco¼ (1250 6 150) Wm�2 K�1 and a characteristictime-constant of RcoC¼ l(cpq)s/hco¼ (0.36 6 0.07) s. Thederived area heat capacity C is the same as for a slab of sili-con with a thickness of 269 lm. This is plausible for a fin-ished silicon solar cell including its aluminum back contact.

Quasiadiabatic conditions are met if the lock-in period

is much shorter than the characteristic time of heat exchange

with the sample holder RcoC; see Sec. III B1 and Fig. 4. Forthe given sample holder, this is the case only for frequencies

well above 10 Hz.

2. Influence on amplitude and phase of the LIT image

The temperature transient measurements presented

above suggest that heat conduction to the sample holder

(h� 1250 Wm�2 K�1) is an issue for frequencies below10 Hz. A simple test of this assertion is to reduce the heat

loss in one part of the sample and look if the signals from the

two parts of the sample are equal. These should only look

different if heat loss is in fact influencing the thermograms

appreciably.

Figure 6 shows amplitude and phase images of LIT on a

monocrystalline silicon solar cell directly mounted on top of

a copper sample holder with a piece of paper placed between

sample and sample holder in the upper right corner. It can be

seen that for 3 Hz there is a significant influence on the phase

(about 10�). That means that a thermography experiment atsuch a low frequency gives results already flawed by a sys-

tematic error due to heat loss. The influence of heat loss on

the phase of the complex signal is more marked than on the

amplitude because of the RC character of the system. Asexpected, this influence vanishes for higher frequencies. At

25 Hz the experiment is not sensitive to thermal conductivity

variations at the sample/sample holder interface, i.e., the

heat propagation inside the sample is quasiadiabatic.

In the fabrication of these cells, there is a strong trend to

reduce the cell thickness. A possible cell with only half the

thickness of our example cell (100 lm) also has only half itsheat capacity. According to (25), this means that, in order to

reach quasiadiabatic conditions, it is necessary to either dou-

ble the thermal resistance (roughened surface, copper net) or

the lock-in frequency.

B. Thermally thick: Thin film crystalline silicon onglass solar module

As an example system of a sample on a thermally thick

substrate, consider a thin film silicon solar cell on a 3 mm

glass substrate mounted on a temperature controlled sample

holder. The glass substrate is thermally thick for frequencies

above 0.5 Hz.

FIG. 5. Determination of the characteristic time constant and thermal con-

ductivity hco¼ 1/Rco to the sample holder from the transient temperaturebehavior for a monocrystalline silicon solar cell on a copper sample holder.

The power density of P¼ (8.7 6 0.7)mW cm� 2 is given from the power fedinto the cell and a geometry factor of 0.70 6 0.05 due to inhomogeneouscurrent flow. The geometry factor is estimated using a LIT image of the

whole cell at the same bias (0.6 V).

FIG. 6. (Color online) LIT on a monocrystalline silicon solar cell on a cop-

per sample holder (h� 1250 Wm�2 K�1), 0.75 V, 5.0 A at 3 and 25 Hzlock-in frequency. A piece of paper is placed between sample and sample

holder in the upper right corner, reducing the heat conduction to the sample

holder significantly. At 3 Hz the amplitude and particularly the phase of the

lock-in signal are significantly altered by heat conduction to the sample

holder. In accordance to Fig. 4, this effect becomes small for lock-in

frequencies> 10 Hz.



Our estimations in Sec. III B 2 have shown that heat

conduction is the dominant mechanism of heat loss with a

constantly low relative heat loss of v¼ 0.0007 to 0.02(depending on the spatial frequency component) for air at

atmospheric pressure. This holds for frequencies down to 0.2

Hz where radiation and free convection are the dominant

heat loss mechanisms and the heat loss ratio starts to

increase. The Toffset component (corresponding to zero fre-quency excitation), in contrast, can be very high. We meas-

ured the temperature increase in the case that the glass back

surface was pressed lightly to the copper sample holder for

heating with �P¼ 12 mW cm�2 (electrically in the siliconstructure on the surface). Fig. 7 shows the results compared

to a 1D simulation of the structure.

The theoretical curve was calculated by numerically

solving the heat equation for homogeneous excitation:

0 ¼ D @2Tðz; tÞ@z2

� ix @Tðz; tÞ@t

;

� k @Tðz; tÞ@z

jz¼0 ¼ �P;

� k @Tðz; tÞ@z

jz¼l ¼ hco Tðl; tÞ � Tambient½ �;

Tðz; 0Þ ¼ Tambient; (32)

with the power density _P, glass thickness l, and the glass tosample holder contact conductivity hco.

For an infinitely thick glass sample the expected behavior

is a square-root shaped increase of temperature (see Ref. 20,

§ 2.9). This behavior is present in the sample only in the very

first seconds, until the back surface influences the propagation

of the heat wave front in the sample. The amount of heat con-

duction through glass and back contact determines the shape

of the temperature transient; a higher contact conductivity

results in a shorter transient. From our estimations (Sec. II) it

follows that the amount of heat transported from the surface is

much smaller than the heat conducted through the glass to the

sample holder (about 5% of the overall heat dissipation). The

heat resistance of the glass is also known, hglass¼ k/l¼ 360Wm�2 K�1. This leaves one free parameter, hco, to be deter-mined by fitting the shape of the numerical curve to the

experimental curve, hco¼ k/l¼ (300 6 20) Wm�2K�1. Thiscoefficient for the glass/copper interface is significantly smaller

than that of the silicon/copper interface above (Sec. III B 1)

due to the high surface roughness and stiffness of the glass.

In summary, heat loss at the excitation frequency is

mostly due to conduction to air and is negligible. This is typ-

ical behavior for thermally thick samples. Heat loss at zero

frequency, which determines the shape of the initial heating-

up phase and is important for temperature control, occurs

mainly through the glass and the sample holder. For our

model system, we determined a zero frequency heat loss

coefficient of h ¼ ðh�1glass þ h�1co Þ�1 � 170 Wm�2K�1. The

transient behavior is more complicated than in the thermally

thin case and occurs at a time scale determined by the thick-

ness of the glass and the quality of the thermal contact

between sample and sample holder.

Note that for glass suspended in air, the temperature rise

is much more pronounced, as the overall zero frequency heat

loss is only hsurfþ hback� 18 Wm�2 K�1. Accordingly, the(unwanted) steady-state temperature increase with respect to

the environment for the same power density is higher by a

factor of� 10 if no sample holder is used with this thermallythick sample.

V. CONCLUSIONS

We have presented estimations of the heat loss coeffi-

cient h for the modulated signal and its zero frequency com-ponent. The mechanisms considered are: radiation, heat

conduction and convection to air, and heat conduction to a

sample holder. A general measure of the influence of modu-

lated heat loss on the signal is proposed. The heat loss ratio

[Eq. (18)] allows a simple answer to the question whether

heat conduction in a sample can be considered to be adia-

batic with respect to the surrounding or not.

It is found that for modulation frequencies above

0.2 Hz, heat conduction to air is generally a more efficient

heat loss than radiation at 300 K. Only for structures larger

than 3 cm, frequencies below 0.2 Hz, and high IR emissivity,

does radiation become significant. For moderate temperature

differences (some K) free convection becomes more impor-

tant than conduction to air and than radiation only for modu-

lation frequencies below� 0.1 Hz, and is more effective atleading edges of heated regions in the sample. It is also

found that the influence of heat loss can only be reduced by

choosing a lower excitation amplitude in the special case

that free convection is the dominant mechanism.

These theoretical findings are applied to cases com-

monly encountered in practice and confirmed experimentally

using a thin film solar module on a 3 mm thick glass sub-

strate and a standard crystalline solar cell. In the latter case it

was found that excitation frequencies below 10 Hz are criti-

cal if the sample is mounted directly on its sample holder.

When the thickness of wafer-based cells is further reduced,

FIG. 7. (Color online) Determination of the characteristic time constant and

thermal conductivity to the sample holder from the transient temperature for

a thin film silicon solar module on a 3 mm glass substrate pressed to a cop-

per sample holder. Two contributions need to be taken into account for heat

loss at zero frequency, radiation and convection at the surface (hsurf, about5%), conduction through the glass (hglass) and from the glass to the sampleholder (hco, about 95%).



the issue of heat loss is expected to become more severe. For

the film-on-substrate sample, no influence of heat loss on the

oscillating part of the signal is expected. Temperature con-

trol for these samples, however, is a problem. In our example

situation an increase of about 0.5 K was observed when

using a temperature controlled sample holder. If the sample

is suspended in air, the unwanted temperature increase rises

to about 5 K.

ACKNOWLEDGMENTS

The authors wish to acknowledge valuable discussions

with J.-M. Wagner on the concept of the thermal transfer

function. This work was supported by the BMBF project

SiThinSolar (contract no. 03/P607).

APPENDIX: DETAILS ON CONVECTION

A. Free convection

Solving the equations of fluid mechanics is mathemati-

cally challenging. In order to reduce the parameter space, the

scalability in time and space (i.e., the independence of the

solution from the system of units used) is exploited by intro-

ducing dimensionless parameters for all relevant physical

processes and variables. The definitions of the dimensionless

numbers used in the solutions of Ostrach17 for the vertical

problem and of Pera and Gebhart18 for the horizontal prob-

lem are the following [note that x is vertical for situation (a)and horizontal for situation (b); see Fig. 1]:

Nux ¼hx

k; (A1)

Grx ¼gbðT � T0Þx3

m2; (A2)

Pr ¼ mD: (A3)

Where h: heat transfer coefficient, k: thermal conductivity,g: gravitational acceleration, b: coefficient of thermal expan-sion, .: density, l: dynamic viscosity, v: kinematic viscosity,and D: thermal diffusivity. The Nusselt number Nu is thedimensionless expression for the heat transfer coefficient h,the Prandtl number Pr describes the ratio of momentum diffu-sivity v¼ l/. and thermal diffusivity D¼ k/cp . in the fluid,and finally the Grashof number Gr describes buoyancy forces.Using the Grashof number implies the Boussinesq approxima-

tion, which requires that the variable density of the fluid is

only important in the terms describing buoyancy forces.

Because bouyancy is the driving force in free convection, the

Grashof number plays a role similar to the Reynolds number

in forced convection. It also describes the transition to turbu-

lent behavior at Grx� 108. For details, see, e.g., Holman.14In the laminar regime17,18

Nux; a ¼ffiffiffiffiffiffiffiffiGrx4

4

r0:718Pr1=2

ð0:952þ PrÞ1=4; (A4)

for a vertical sample (a) and

Nux; b ¼ 0:394Gr1=5x Pr1=4 (A5)

for a horizontal sample (b).

B. Forced convection

The dimensionless number describing the flow velocity

far away from the sample surface u1 is the Reynolds number

Rex ¼u1x

m; (A6)

where x is again the distance from the leading edge of theheated portion of the sample; the orientation of the sample is

not important in forced convection. The Reynolds number

Rex also controls the transition to turbulent behavior atapproximately Rex > 5� 105, depending on the surface qual-ity of the sample.14

If the leading edge of the sample (x¼ 0) and the begin-ning of the heated portion (x¼ x0) do not coincide, the Nus-selt number for forced convection along a heated plane is14

Nux ¼ 0:332Pr1=3Re1=2x 1�x0x

� 3=4� ��1=3; (A7)

that is, just like free convection, a function of location x onthe sample.

1A. Salazar, A. Mendioroz, and R. Fuente, Appl. Phys. Lett. 95, 121905(2009).

2B. Zhang and R. E. Imhof, Appl. Phys. A: Mater. Sci. Process. 62, 323(1996).

3D. G. Cahill and R. O. Pohl, Phys. Rev. B 35, 4067 (1987).4J. McKelvie, Proc. SPIE 731, 45 (1987).5G. Rousset, F. Charbonnier, and L. Lepoutre, J. Appl. Phys. 56, 2093(1984).

6G. Meneghetti, Int. J. Fatigue 29, 81 (2007).7A. Mandelis, Diffusion-Wave Fields (Springer, New York, NY, 2001).8K. D. Cole, J. Heat Transfer 128, 709 (2006).9K. D. Cole, J. V. Beck, A. Haji-Sheikh, and B. Litkouhi, Heat conductionusing Green’s functions, 2nd ed. (CRC Press, Boca Raton, FL, 2010).

10O. Breitenstein, W. Warta, and M. Langenkamp, Lock-in thermography:Basics and Use for Evaluating Electronic Devices and Materials, 2nd ed.(Springer, Berlin, 2010).

11K. Emery and T. Moriarty, Proc. SPIE, 7052, 70520D (2008).12H. Seifert, J. Hohl-Ebinger, U. Würfel, B. Zimmermann, and W. Warta, in

24th European Photovoltaic Solar Energy Conference Exhibition, 21–25

September 2009 (2009).13R. Gupta and O. Breitenstein, Quant. Infrared Tomogr J. 4, 85 (2007).14J. P. Holman, Heat Transfer, 8th ed. (McGraw-Hill, New York, 1997).15R. Yang and L. S. Yao, J. Heat Transfer 109, 413 (1987).16B. Gebhart, Y. Jaluria, R. L. Mahajan, and B. Sammakia, Buoyancy

Induced Flows and Transport (Springer, Berlin, 1988).17S. Ostrach, NASA Tech. Rep. 1111 (1953).18L. Pera and B. Gebhart, Int. J. Heat Mass Transfer 16, 1131 (1973).19R. J. Goldstein and D. G. Briggs, J. Heat Transfer 86, 490 (1964).20H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed.

(Clarendon, Oxford, 1959).21H. Straube, O. Breitenstein, and J.-M. Wagner, “Thermal wave propaga-

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(McGraw- Hill, New York, 1985), pp. 4-1–4-187.23T. N. Veziroglu, Prog. Astronaut. Aeronaut. 20, 879 (1967).24R. C. Jones, J. Opt. Soc. Am. 48, 934 (1958).



http://dx.doi.org/10.1063/1.3236782http://dx.doi.org/10.1007/BF01594230http://dx.doi.org/10.1103/PhysRevB.35.4067http://dx.doi.org/10.1117/12.799606http://dx.doi.org/10.1063/1.334206http://dx.doi.org/10.1016/j.ijfatigue.2006.02.043http://dx.doi.org/10.1115/1.2194040http://dx.doi.org/10.1117/12.799606http://dx.doi.org/10.3166/qirt.4.85-105http://dx.doi.org/10.1115/1.3248097http://dx.doi.org/10.1016/0017-9310(73)90126-9http://dx.doi.org/10.1115/1.3248097http://dx.doi.org/10.1364/JOSA.48.000934

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estimation of heat loss in thermal wave experimentsii. estimation of the heat loss coefficient in a...

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