3 NEW THEORETICAL MODEL AND VALIDATION BY LOCK-IN MICRO-THERMOGRAPHIC TEMPERATURE MAPPING

5th European Thermal-Sciences Conference, The Netherlands, 2008
3ω SCANNING THERMAL MICROSCOPY:
NEW THEORETICAL MODEL AND VALIDATION
BY LOCK-IN MICRO-THERMOGRAPHIC
TEMPERATURE MAPPING
M. Chirtoc, J.F. Henry
Thermophysics Lab., UTAP, Univ. Reims, BP 1039, 51687 Reims Cedex 2, France
Abstract
We developed a general theory of ac hot wire method formulated in terms of dimensionless
parameters. Simple analytical solutions are found for special cases of experimental relevance in
scanning thermal microscopy (SThM) with Wollaston resistive thermal probe (ThP). Contrary to
the ac or dc SThM electrical signal which is related to the spatial average temperature of the probe
wire, the use of infrared micro-thermography permits to address directly the local parameters
involved in the model and to confront the experimental results with the theoretical predictions. In
particular, we investigated by dc and lock-in thermography the temperature distribution in the ThP
as a function of modulation frequency, and the temperature profile along the heated wire on upscaled, self-made thermal probes, with the probe in air and in contact with solid and liquid samples.
At last, we exploit the amplitude and phase thermograms of a thin composite sample out-of-contact
and in-contact with the Wollaston ThP in order to evaluate the heat flux transferred to the sample
through the air and by the physical contact. The latter amounts to 8% from the total electrical power
and is frequency-independent. The separation of the two contributions is achieved by vector
subtraction.
1 Introduction
Resistive thermal probes (ThP) with Wollaston wire have been employed in conjunction with
atomic force microscopes (AFM) for micro-thermal analysis [Pollock and Hammiche (2001)] and
for scanning thermal microscopy (SThM) [Gmelin et al. (1998)]. The ThP consists of a thin PtRh
wire bent at sharp angle. With modulated excitation current, its temperature-dependent resistance
oscillates at the second harmonic while the voltage across the wire has a component at the third
harmonic, proportional to the temperature amplitude averaged along the wire (3ω method) [Altes et
al. (2004), Lefèvre and Volz (2005)].
In SThM, one of the problems in validating theoretical models by the experiment resides in the fact
that the primary parameters which intervene in the models, such as the temperature profile along the
wire, the local temperature at the ThP apex, the sample thermal conductance due to solid-solid
contact only, the fraction of heat loss to ambient air, etc. are not directly accessible by the dc or ac
electrical response of the ThP. This is because this signal is generated by the average wire
temperature, in the sense of a spatial integral over length l, leading to spatial and temporal (i.e., high
frequency) resolution loss. The motivation for the use of IR micro-thermography method in this
study was to address directly the mentioned parameters and to confront the experimental results
with the predictions of our model.
2 Theoretical background
5th European Thermal-Sciences Conference, The Netherlands, 2008
We developed a general theory of ac hot wire method accounting for distributed as well as for
localized heat exchange mechanisms [Chirtoc and Henry (2007)]. The former depends on the
exchange coefficient h with the ambient while the latter is relevant for SThM operation. The salient
features of this model are:
- dimensionless formulation in terms of normalized thermal admittances;
- modular structure i.e., the heat balance in the wire is separated from the modeling of heat
transfer parameters;
- simple analytical solutions for local and average temperature field in the wire;
- it accounts for possible non-linear effects.
For thermal conductivity k mapping, the apex of the ThP is brought in contact with a solid sample.
The resulting heat transfer through the contact surface is quantified by a dimensionless sample
thermal admittance y. The solution of the differential fin equation is expressed as a form factor
F(x,f)=θ(x,f)/θ0 representing the dimensionless temperature profile along the wire half-length l (with
f the thermal modulation frequency corresponding to 2ω):
F ( x) =
[
1
(ml )
2
)]
(
[
(
)]
⎧
ml + y 1 − e −ml e mx + ml − y 1 − e ml e −mx ⎫
⎬
⎨1 −
ml(1 + yM ) e ml + e −ml
⎭
⎩
(
)
(1)
In particular, at the two ends F(l)=0 since the thick silver prongs supporting the wire act as heat
sinks, while at the apex:
F (0) =
MM 1 / 2
2(1 + yM )
(2)
with M(ml) = [tanh(ml)]/(ml) and M1/2 = M(ml/2). The argument ml is the fin parameter depending
on geometrical and thermal properties of the wire, on h coefficient and on modulation frequency f.
The 3ω voltage signal is proportional to F(x,f) averaged over the length l:
V3ω ∝< F(x, f ) >= F( f )=
1 ⎛
1+ yM1/ 2 ⎞
⎜1− M
⎟
2⎜
1+ yM ⎟⎠
(ml) ⎝
(3)
The equation (3) can be approached by a low pass transfer function [Chirtoc et al. (2004), (2006)]:
F( f ) =
F ( f << f c )
1 + j f / fc
(4)
with fc=3α/2πl2 the cross over frequency and α the thermal diffusivity of wire material. In general,
fc(y)>fc0(y=0). Equation (4) has two limiting cases (operation modes) in the frequency domain:
isothermal (I) for f<fc and adiabatic (A) for f>fc. Measurements of thermophysical properties are
possible in (I) mode in which F is independent of frequency. Then F(y) varies between 1/3 and 1/12
when 0<y<∞, yielding a theoretical dynamic range DR=4 in signal amplitude. A maximum phase
variation of 35o is found around fc.
The parameter y is related to the constriction thermal conductance [Carslaw and Jaeger (1959)],
proportional to rck (rc is the radius of tip-sample contact area). Moreover, y is subjected also to large
variability due to contact characteristics (area, force, humidity, surface roughness, hardness) and to
ThP characteristics. We showed that most of these unwanted influences can be efficiently cancelled
5th European Thermal-Sciences Conference, The Netherlands, 2008
by normalization techniques [Chirtoc et al. (2006)]. Two limiting situations can be used for
reference measurements at low frequency (I mode). With the tip in air (y→0), F(y)=1/3 and
neglecting the heat loss to ambient, the temperature profile of equation (1) reduces to equation (5a).
With the tip in perfect contact with an ideal heat sink (y→∞), F(y)=1/12 and instead of equation
(5a) one obtains equation (5b). Both represent a parabolic temperature profile along the wire:
2
1⎡ ⎛x⎞ ⎤
F ( x) = ⎢1 − ⎜ ⎟ ⎥
2⎣ ⎝ l ⎠ ⎦
(5a)
2
1 ⎡x ⎛ x⎞ ⎤
F ( x) = ⎢ − ⎜ ⎟ ⎥
2⎣l ⎝ l ⎠ ⎦
(5b)
When a thin sample suspended in air is in contact with the ThP, it is equivalent to an insulated and
thermally thin (d<<μ) radial fin with axial symmetry. The radial temperature field θ(r,f) can be
approximated by [Carslaw and Jaeger (1959)]:
θ (r , f ) = −
P/d
2π k
π⎞
⎛ 1.26 r
+ i ⎟⎟
⎜⎜ ln
μ
4⎠
⎝
(6)
with μ=(α/πf)1/2 the thermal diffusion length and P the heat flux amplitude leaving the ThP apex.
From the real part of equation (6) one obtains:
P
− Re[θ (r , f )]
=
2π kd ln(1.26 r / μ )
(7)
Based on the model presented above, a number of figures of merit for the 3ω SThM can be defined
[Chirtoc et al. (2004), Chirtoc and Henry (2007)]:
- the mixing efficiency;
- the temperature calibration factor;
- the crossover frequency between isothermal and adiabatic mode of operation;
- the dynamic range of amplitude and phase signals.
3 Materials and methods
The Wollaston ThP produced by Veeco Co. (figure 1a), comprises a sensing element made of
Pt0.9Rh0.1 wire (on the left in the figure) and is 2l=200 μm long and 5 μm in diameter. The thick
prongs with coaxial structure (Wollaston wire) are 75 μm in diameter Ag sheaths. The epoxy glue
ball is for mechanical rigidity. Behind it one can note the small mirror plate which reflects a diode
laser beam in order to achieve the z-feedback control in the SThM.
We used an IR camera type CEDIP JADE IRC 320-4 Long-Wave (LW) with 3:1 or 1:1 imaging
objectives, which acquires 320x240 pixels/frame with 10x10 μm2 pixel size. Amplitude and phase
thermograms in the range 0.4-90 Hz were obtained by lock-in processing and accumulation of video
frames using the software developed in our laboratory [Pron et al. (2000), Pron and Bissieux
(2004)]. The W (2l=3 mm and 100 μm in diameter) and Ni wires used for the home-made ThP were
covered by lampblack to increase their IR emissivity and were imaged by at least two pixels across
the diameter. The composite specimen studied in section 4.3 was a 2x2 cm2 typewriter carbon paper
consisting of 20 μm thick paper substrate facing the ThP and 10 μm thick carbon-powder based
black layer facing the IR camera.
5th European Thermal-Sciences Conference, The Netherlands, 2008
a)
c)
b)
d)
Figure 1: Wollaston thermal probe for SThM. a) Optical microscopy photograph, b) dc IR
thermogram (red corresponds to strong IR radiation), c) amplitude and d) phase lock-in
thermograms at 24 Hz frequency of excitation current.
4 Results and discussion
There are three types of heat exchanges in a thermal microscopy experiment: the thermal balance
within the thermal probe, the heat exchanged between the ThP tip and the sample following several
heat flow channels, and the heat diffusion within the sample in relation with its structure and
geometry.
4.1 Investigation of the Wollaston thermal probe
The dc thermogram of figure 1b shows that with dc current excitation the whole body of the
Wollaston ThP is heated above ambient by the heat developed mainly in the thin PtRh wire at the
left end of the probe. The latter does not appear to be very bright due to its low emissivity typical
for metals, but in fact it has the highest temperature in the image. On the lock-in thermograms of
figures 1c and 1d one can see that with ac excitation the ac temperature field is well localized at the
tip already at a frequency of 24 Hz. It follows that with the 3ω technique we may restrict the
analysis of heat exchanges to the thin PtRh wire.
As shown in section 2, the value of fc parameter scales with l-2. In order to verify this statement, we
studied by thermography the temperature distribution along a tungsten (W) wire welded at each end
to brass blocks [Chirtoc et al. (2004), (2005)]. The calculated temperature oscillation was 5 K for
1.4 A excitation current amplitude. Lock-in amplitude and phase profiles are shown in figure 2.
Numerical values extracted from amplitude thermograms are shown in figure 3. The representation
in figure 3a is based on dimensionless parameters f/fc and x/2l. The lines were computed for the
Wollaston ThP with a finite element model assuming only conductive heat loss along the wire to the
end-supports, and are identical to equation (1). The good agreement demonstrates the validity of the
scaling law and the assumption of the predominance of heat loss by conduction through the wire to
the supports. The saturation for f/fc<<1 (I mode) is clearly visible. By contrast, at high frequency
f/fc>10 (A mode), the profile tends to be flat along the wire.
5th European Thermal-Sciences Conference, The Netherlands, 2008
0,4 Hz
200 μm
4 Hz
30 Hz
Figure 2: Amplitude (left) and phase (right) temperature profiles along a linear, ac heated W wire,
obtained by lock- in thermography. The wire is 3 mm long and 100 μm in diameter.
The average temperature data of figure 3b are proportional to F(f) factor of equation 4. For this upscaled system, the cross over frequency between (I) mode and (A) mode is situated at fc=5.19 Hz
corresponding to an effective length of 2l=5 mm (αW=67.4x10-6 m2s-1). This is longer than the free
length of the wire and can be explained by residual thermal resistance at the welding sites.
4.2 Effect of the tip-sample contact
We investigated the thermal balance of a self-made ThP with Ni wire as a function of various
contact situations [Henry and Chirtoc (2005)]. We used a well-conducting liquid (Hg) in order to
determine the DR of the ThP for large y in absence of thermal contact resistance, figure 4. At 0.5 Hz
(I mode), the ratio between the integral over the half-length l of the experimental profile with the tip
in air (red triangles) and the one with the tip in contact with the Hg droplet (red circles), yields
DR=2.9. The maximum theoretical value saturates at DR=4 for y→∞, in agreement with equations
(3) and (4). The same result is obtained by using the respective curves in figure 4c, calculated with
finite element software or with equations (5a) and (5b). By contrast, at the same frequency, the DR
of the local temperature at the contact is DR=9, and in theory it tends to infinity for a perfect
contact with an ideal heat sink (from equation (2)). These findings show that the measurement of
local contact temperature rather than the average one over the wire length (such as with electrical
signals) might increase the performance of SThM.
NORMALISED AMPL.
1
0,1
(b)
fT
0,01
0,1
1
10
100
f (Hz)
Figure 3: Temperature profiles of half-length wire (a) and integrated temperature along the wire
(b), for the ac heated W wire of Figure 2. Points are data obtained from the thermograms. Lines
are finite element computations.
5th European Thermal-Sciences Conference, The Netherlands, 2008
1,0
In air 0,5 Hz
In air 40 Hz
Perfect contact 0,5 Hz
Perfect contact 40 Hz
In air 0,5 Hz
Contact Hg 0,5 Hz
Contact Hg 40 Hz
(a)
(b)
Normalized temperature amplitude
0,9
0,8
0,7
0,6
(c)
0,5
0,4
0,3
0,2
0,1
0,0
-1,0
-0,9
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0,0
Normalized distance x / L
Figure 4: dc thermograms of a thermal probe made of Ni wire (2l=3 mm and 40 μm in diameter)
in the proximity (a) and in contact (b) with a Hg droplet, at 0.5 Hz. c) Temperature amplitude
profiles of half-length wire in two extreme situations: lock-in thermograms (points) and finite
element computations or with equations (5a,b) (lines).
We measured also temperature profiles of the wire in case of contact with solid samples, figure 5.
The ThP was attached to a micrometric z-translation stage and the sample was placed on a balance.
By lowering the height of the ThP further beyond contact, the balance displayed the contact force in
units of mass with a resolution of 0.1 mg. In addition to the sample conductivity, the figure 5c
shows that the local temperature is strongly dependent on the contact force which modifies the size
of contact surface. As expected, this dependence is less pronounced with hard materials like glass.
For Cu, at 20 mgf, the local DR is 3.3, which is again larger than DR=1.5 of the spatial average.
The theoretical values of the DR with solids cannot be estimated in absence of contact radii data. In
the thermograms of figures 4a,b and 5a,b, the reflection of the bright (hot) wire at the Hg and Cu
surface, respectively, allow to control the limit between in-contact and out-of-contact.
1.0
(a)
(b)
Normalized temperature
.
0.9
0.8
0.7
0.6
(c)
0.5
0.4
0.3
0.2
0.1
Glass
Steel
Th
Aluminium
Copper
f = 1 Hz
0.0
0
5
10
15
20
25
30
35
Force ( mgf)
Figure 5: a) and b) same as Figures 4 a) and b), for polished Cu surface, at 1 Hz.
c) Temperature amplitude at wire apex during contact, normalized to temperature out of contact,
vs. contact force.
5th European Thermal-Sciences Conference, The Netherlands, 2008
4.3 Heat diffusion in a thin sample
We measured in back-detection configuration the temperature distribution over a thin composite
paper sample in the region of contact with the ThP (figure 6) [Henry et al. (2007)]. Experimental
data fit with equations (6) and (7) allowed for determination of sample in-plane thermal diffusivity
as α = 2.93x10-6 m2/s ±7% which results in k = 4.32 Wm-1K-1 the equivalent thermal conductivity of
the composite sample (ρc = 1.47x106 kg m-3).
The direct heat transfer via the contact was determined by vector subtraction of out-of-contact data
from in-contact data, for different modulation frequencies. In figure 7a, P decreases with increasing
frequency due to thermal wave attenuation by conduction across the air layer. The maximum power
is reached at r ≈ 0.2 mm which represents the (surprisingly large) equivalent heat source radius in
absence of contact. From the total ac heat flux, 6% to 23% are transferred by air conduction for f =
40 ... 0.2 Hz, respectively. On the other hand, from figure 7b a fraction of only about 8% is
transferred directly to the sample via the contact. It is almost frequency- and radius-independent, in
agreement with the point-heat source and radial-fin model employed.
(a)
(b)
Figure 6: Radial distribution of temperature amplitudes (a) and phases (b) relative to the
center, when theWollaston thermal probe is in contact with the specimen, at 0.2...40 Hz, from
top to bottom. Insets show typical amplitude and phase lock-in thermograms.
(a)
(b)
Figure 7: ac power (according to equation (7)) transferred from the thermal probe to the
sample, vs. distance from the center; a) out-of-contact; b) difference between
in-contact and out-of-contact.
5th European Thermal-Sciences Conference, The Netherlands, 2008
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