The 33rd International Symposium on Transport Phenomena
24-27 September 2023, Kumamoto, JAPAN.

ISTP33ANALYSIS OF CURRENT FLOW PATH AND TEMPERATURE DISTRIBUTION OF
Ag-NANOWIRE NETWORK USING THERMOREFLECTANCE IMAGING
Y. SUGIHARA1, R. UEMURA1, K. TAMAI1, R. Kuriyama1,2 and K. TATSUMI1
1.

Department of Mechanical Engineering and Science, Kyoto University
Kyotodaigakukatsu, Nishikyo-ku, Kyoto 615-8540, Japan
2.

Japan Science and Technology Agency, PRESTO Researcher* Corresponding Author: e-mail address (tatsumi@me.kyoto-u.ac.jp)
Keywords: Ag nano-wire, Network, Contact resistance, Joule heating, Thermo-reflectance Imaging
ABSTRACT
A sheet consisted of Ag nano-wire (Ag-NW) which is
randomly dispersed and forming a network can be applied toelectronic devices and wearable devices as flexible andtransparent conductive film material and heater. To evaluate
the performance and reliability of the sheet and device, it isnecessary to investigate and understand the characteristics ofthe current and temperature distribution of the Ag-NW
network. The current path is based on the dispersed patternof the Ag-NWs and the electric and thermal resistances of the
Ag-NWs and contact point of the Ag-NW junction. These
resistances change depending on the properties of thepolymer coating of the Ag-NW and contact conditions at thejunction. Therefore, it is important to analyze thecharacteristics of the current and temperature distributions in
the Ag-NW network, and evaluate the degree of the contact
resistance variation and their effect on the current andtemperature distribution. In this study, we measured thetemperature distribution of Joule-heated Ag-NW network
using thermoreflectance imaging (TRI). Numericalcomputation was conducted for the electric current flow andheat transfer of the Ag-NW network. The contact electric
resistance was obtained by solving the optimization problem
to minimize the difference between the temperature
distributions of the measurement and computation. The
results showed that the contact electric resistance varies over
the Ag-NW network at certain magnitude. This variation
significantly affected the temperature distribution bydefining the current path and producing a non-uniform
current distribution.
INTRODUCTION
A sheet consisted of Ag nano-wires (Ag-NWs) gives highelectric conductivity, and furthermore, it can become flexible
and transparent if the layer thickness and concentration of the
NWs are small(1). The sheet can also be used as flexible film
heater(2). Therefore, Ag-NW sheet is considered to be a
promising material which can be applied to electronic
devices and wearable devices.
High conductivity and heating performance can beobtained when the current flows uniformly over the NWs.
However, limitation of the paths of the current are observedin the actual NW network and the current shows non-uniform
distribution. This is attributed to the connecting pattern of
NWs and the variation of the resistance at NWs and contact
points. The non-uniform current distribution affects the Joule

Figure 1. (a) Photograph of the Ag-NW network dispersed
on glass substrate between Au electrodes, and (c) opticalsystem and (b) timing chart of the TRI measurement.
heating and generates high temperature regions (hot-spots)
which can lead to thermal fatigue of the NW network.
Therefore, it is important to understand the characteristics ofthe current path and temperature distribution of the NW
network, and evaluate the effect of the contact resistancevariation on these distributions to provide insight into thedesign of high performance and reliable NW network.
The contact electric resistance changes depending on theproperties and thickness of the polymer coating of the AgNW which is generally applied to the Ag-NWs to keep theproperties and strength of the Ag-NW stable. The contact
condition of the Ag-NWs can affect the resistance also.
However, although contact electric resistance at a single
point of two Ag-NWs have been investigated(3), the variationof the resistance in the Ag-NW network has not be elucidated.
Further, the current distribution - current path – in the AgNW network has not been measured. This is due to thedifficulty in the measurement of the current in twodimensional form and sub-micrometer scale, which isdifficult in the temperature measurement as well.

The 33rd International Symposium on Transport Phenomena
24-27 September 2023, Kumamoto, JAPAN.

ISTP33In this study, we measured the two-dimensionaltemperature distribution of the current-applied-Ag-NW
network using the thermo-reflectance imaging (TRI)
technique. TRI can measure the two-dimensionaltemperature of the material surface with the accuracy of submicrometer in space, and sub-microsecond in time.
We further conducted numerical computation for theelectric current and heat transfer of the NWs and estimatedthe contact electric resistance (contact electric conductance)at the NW junction by matching the temperature distributionof the Ag-NW network to the measurement. Discussion ismade on the path and non-uniformity of the current and its
resulting temperature distribution. We evaluated the degreeof the contact electric resistance variation by comparing thecurrent and temperature distributions with the case of
uniform contact electric resistance, and how the contact
electric resistance contributes to the generation of the current
path and temperature distribution.
METHODS
Measurement sample
Fig. 1 (a) shows the photograph of the Ag-NW-network.
Ag-NWs with diameter of 120-150 nm and length of 20-50
m (Sigma-Aldrich: #739448) were dispersed on the glass
substrate in the area located between two parallel Au
electrodes using vacuum filtration method. Au parallel
electrodes were patterned on the glass substrate using thelithography and electron beam vapor deposition methods.
The distance between the electrode was 100 m. In thevacuum filtration, a thin layer of Ag-NWs was deposit on the
porous filter. Ag-NWs located outside the area of the deigned
pattern were first removed, and the remaining Ag-NWs weretransferred to the glass substrate at the area between theelectrodes. The width of the area where the Ag-NWs wastransferred was 100 m, therefore the measured area wassquare 100 m on a side. The concentration of the Ag-NWs
represented by the total length of the Ag-NWs per unit area
was 8.3105 m-1. The electric resistance of Ag-NW network(the area between the electrodes) was 18.7 .
Thermo-reflectance imaging
The TRI measures the two-dimensional distribution of
the surface temperature utilizing the phenomenon that thereflectance of the material changes with temperature(thermo-reflectance). The surface temperature can beobtained by measuring the reflectance and applying it to thefollowing relationship.
∆𝑅 =

𝜕𝑅
∆𝑇
𝜕𝑇

(1)

R and T are the reflectance and temperature, respectively.
Having the intensity of the reflected light given as I, we get
the relationship between the variation of I and T as follows.
∆𝐼
1 𝜕𝑅
=(
) ∆𝑇 = 𝐶TR ∆𝑇
𝐼
𝑅 𝜕𝑇

(2)

In Eq. (2), the part enclosed in parentheses is called thermoreflectance coefficient CTR. The value of CTR is a physicalproperty of the material and depends on the wavelength ofthe light.

Table 1. Properties and parameters of the computation
Parameter
ValuesContact thermal conductance (W/K)
1.010-7
Ag-NW
Diameter (nm)135
Density (kg/m3)
1.05104
234
Specific heat capacity (J/(Kkg))
Electric resistivity (m)
1.5810-8
420
Thermal conductivity (W/(mK))
Glass substrate
Density (kg/m3)
2.5103
670
Specific heat capacity (J/(Kkg))
0.65
Thermal conductivity (W/(mK))
Fig. 1 (b) shows the schematic of the optical system of
the measurement. LED light was used as the incident light inthe measurement. The light was irradiated to the object
through a half mirror and objective lens (Olympus,
LMPLFLN 100X NA=0.8). The reflection image wasrecorded using a CMOS camera (Hamamatsu Photonics,
ORCA Fusion). The pixel resolution of the system was
65nm/pixel. Autofocus system was installed in themeasurement system to keep obtaining the focused images
during the measurement. Autofocus operated by irradiating
light of bandwidth of 530 nm to the object and measuring theperiodic pattern generated at the surface. Therefore,
bandpass filter was inserted before the camera to cut the lightof the autofocus system and measure the reflected light of the
LED only.
Current was periodically applied to the electrode in aform of square pulse with width of 10 s and period of 250
s. Reflectance was measure by LED flash light with widthof 1.5 s and period of 250 µs. The nominal wavelength of
the LED light was 530 nm. Timing of irradiation was
adjusted to the phases at which the Ag-NW gives the
maximum temperature and the temperature equal to room
temperature, respectively. The former image was referred toas hot-image, and the latter as reference-image. The timing
chart of the measurement is shown in Fig. 1 (c).
Five thousand pairs of the images were measured. The
temperature difference T of each pair was calculated andT was averaged over 5,000 pairs. The reflectance wasconverted to temperature using the TR coefficient CTR of
8.9610-5 K-1. This value was obtained from the calibration
measurement on CTR for Ag conducted by the authors using
the TRI system and Ag plane mirror.
Computation
Numerical computation was conducted to solve thecurrent and temperature distribution of the Ag-NW network.
The conservation equations of electric current and energy(heat transfer) of the NWs was solved in the computation.
The voltage applied to the electrode and the total current
supplied to the Ag-NW were adjusted to those of themeasurement. The contact electric resistance of the NWs wasderived to make the temperature distribution of thecomputation fit with that of the measurement by solving theminimization problem of the loss function for theThe 33rd International Symposium on Transport Phenomena
24-27 September 2023, Kumamoto, JAPAN.

ISTP33temperature difference at each divided segment of the NWs.
In the computation, we used the contact electric conductancewhich is the inverse value of resistance. Therefore, it should
be noted that hereafter contact electric conductance will be
used to explain the numerical procedure and presented insome of the results and discussion.
The heat transfer computation considered the heat
conduction of the NWs, heat transfer at the contact point of
the NW junction, and heat transfer to the glass substrate. The
temperature of the glass substrate was kept constant as theheating time was 10 s for the hot-image, and temperaturechange of the substrate during this time was negligible.
Thermal conductivity at the contact point of the NW junction,and the physical properties of the Ag-NW and glass werereferred to other works. The properties and values of theparameters are summarized in Table 1.

(a) TRI measurement

Numerical procedure
Contact electric conductance k of the NW junction andthe contact thermal conductance between the NW and glass
substrate were calculated by the following procedure.
The value k was calculated by minimizing the loss
function L defined as Eq. (3) representing the temperature in
vector format.
𝐿 = |α𝑻s − 𝑻m |2(3)

Ts is the calculated temperature, Tm is the measured
temperature, and  is the correction coefficient.  wasapplied to correct the difference in the overall temperature
between the measurement and computation. This difference
appeared due to having the contact thermal conductance
between the NWs and glass substrate g initially unknown.
The contact electric conductance and contact thermal
conductance contributes to the current and temperature
distribution of the NW network. On the other hand, thecontact thermal conductance between the NWs and glass
substrate mainly affects the overall temperature as thetemperature of the glass substrate is constant during theheating of the Ag-NWs. The  was calculated as Eq. (4)using the temperature Ts at each moment.
α = (𝑻s ∙ 𝑻m )/𝑻2s(4)

Once  was applied, the value k of each NW junction
which minimize L was calculated. The number of iteration to
calculate k was 3,000, and the variation of L was 3.410-4 %in the last 300 steps. To match the overall electric resistanceof the area to the measurement, we corrected the value of k
by multiplying k of all junctions with constant magnitude.
Every 100 iteration steps of the calculation for k, g was
updated finding the value which minimize the loss function
shown in Eq. (5) using the dichotomous exploration method.
𝐿 = |𝑻s − 𝑻m |2(5)

RESULTS AND DISCUSSION
Temperature distribution
Fig. 2 shows the temperature of the Ag-NWs obtained by
the measurement and computation. The pattern of the Ag-(b) Computation (adjusted contact conductance)(c) Computation (uniform contact conductance)
Figure 2. Distribution of the temperature increase T of
the Ag-NW network obtained by TRI measurement andcomputation. The results of the computation show thecases of contact electric conductance adjusted at eachjunction, and applying equal value to all junction(uniform value).
NWs is the same with the one shown in Fig. 1(a). Fig. 2 (a)shows the temperature of the TRI measurement. Nonuniform distribution of temperature and hot-spots generation
are found showing that the current does not flow uniformly
in the network and specific paths are formed. Several NWs
join at some of the junctions, and the current paths converge
to the junction. This produces high concentration current andlarge Joule heating leading to the generation of hot-spots.
Fig. 2 (b) shows the temperature distribution obtained by
the computation calculating k of the junctions to make thetemperature match with the measurement. Comparing Fig. 2(a) and (b), the results of the computation agree well with theThe 33rd International Symposium on Transport Phenomena
24-27 September 2023, Kumamoto, JAPAN.

ISTP33measurement. The minimum value of the contact electric
resistance was 95.3 . The resistance varied in the range ofapproximately 100~50 × 104 .
Fig. 2 (c) shows the result of the computation in which
the value of k was kept equal for all Ag-NW junctions. The
value was defined so as the overall resistance matched themeasurement. Comparing the results with Fig. 2 (b), thetemperature of the high temperature region, especially the
hot-spots, decreases and a more uniform distribution is
obtained. Further, we identified the junctions whichsignificantly contributed to the temperature pattern by
evaluating the sensitivity of the temperature distribution to
the value k of each junction. The number of these key
junctions was 178 out of total number of junctions 2358. The
evaluation method used here will be mentioned in the next
section.
These results show that the variation of k increases thenon-uniformity of the temperature distribution, and the valueof k of a limited number of junctions dominated the current
and temperature distributions in the case k varies.
The junction with large k obstructed the current flow andlimited the path in the Ag-NW network. This produced highcurrent density regions and was the cause of the hot-spot
generation. This concludes that, it is important to evaluate
not only the dispersion pattern of the NWs but the variationof the contact resistance to understand the temperature
distribution of the Ag-NW network.
Electric resistance and its effects
To evaluate the accuracy of the estimated contact electric
resistance of the analysis, we numerically created an Ag-NW
network consisted of randomly placed Ag-NWs. The contact
electric conductance was applied randomly to the junction.
The current, Joule heating, and temperature distributions
were calculated by computation. We further applied
Gaussian noise with standard deviation of 5K to thetemperature of each segment. Using this data as a reference
temperature distribution, we conducted the computation
which was used in the previous section to calculate thecontact electric conductance of the reference network. The
results giving the estimated values were compared with thoseof the reference data initially applied. The temperature
distribution of the reference data and that obtained by thecomputation are shown in Fig. 3. The two values agree wellindicating the validity of the computation.
The accuracy of the estimated contact electric resistancewas evaluated based on an error parameter r shown in Eq. (6)𝑟=|𝑘e − 𝑘t |𝑘t

(6)ke and kt are the estimated value and the applied one (thereference data) of the contact electric conductance,
respectively. Fig. 4 (a) shows the frequency distribution of r.
Although the temperature distribution agrees with thereference data, r is large and the accuracy is low for many
junctions. As k is estimated from the temperature distribution,
k cannot be defined specifically for the junctions located in
the area where the temperature variation is small due to thelack of information. These areas correspond to the Ag-NWs
with zero or negligibly small currents, and the contribution

Figure 3. Distribution of the temperature increase T of
the Ag-NW network in the case of (a) reference data, and(b) obtained by the computation estimating the contact
electric conductivity by matching the temperature to thereference data (a).

Figure 4. (a) Frequency distribution of error parameter r
for the NW junctions, and (b) relationship between the rand sensitivity of the temperature distribution to thecontact electric conductance of a junction.
of these junctions on the current path and temperature
distribution is believed to be small.
We defined another evaluation function as shown in Eq.
(7) which gives the sensitivity of the network temperature
distribution to the variation of k of a junction.
𝑆i = |𝑻H − 𝑻L |(7)

TH and TL are the vectors representing the network
temperature when k of junction i was change to 10 -1 and10-10 -1, respectively. The relationship between S and r of
all junctions were examined for several reference networks.
The results are shown in Fig. 4 (b).
In Fig. 4 (b), junction with large S gives small r showing
an inverse correlation. Further, the number of junctions withlarge S is small. This indicates that only a limited number ofjunctions contributes to the formation of the current path andtemperature distribution.
This procedure was used to the evaluate the contributionof the contact electric resistance of the junction to thetemperature distribution obtained by the measurement. As
previously mentioned, the number of junctions of S >5.0103 was 178 out of 2358 which is the number of all
junctions. This shows that if the electric resistance of such
key junction changes to some extent, the pattern of thecurrent path and the current density of the NWs can easilychange. This will lead to the change of the temperature
distribution and number of hot-spots.
Figs. 5 and 6 respectively show the current distribution
and the electric conductivity k at the junctions obtained by
the simulation matching the temperature distribution of themeasurement, namely the current distribution which gives
T shown in Fig. 2 (b). In Fig. 6 the values of k only at theThe 33rd International Symposium on Transport Phenomena
24-27 September 2023, Kumamoto, JAPAN.

ISTP33junctions of which the sensitivity parameter S is greater than
1,000 K2 are plotted.
The distribution shown in Fig. 5 is non-uniform andshows several paths with current of large values. The locationof large current corresponds to that of large T shown in Fig.
2 (b). This is reasonable as the temperature increase is caused
by Joule heating. On the other hand, the magnitude of T at
the hot-spot is more conspicuous and the distribution gives
greater contrast compared to the current distribution. The
reason for this is twofold. One is that Joule heating is
proportional to the square of current. Therefore, the variationin the distribution increases in the case of T. The otherreason is the effect of heat transfer. Although the heating time(current applied time) is 10 s, heat transfer occurs during
this period in the Ag-NW network. In the area of small
current, the number of connected Ag-NWs is large and theapparent electric conductivity of the area is large. This AgNW structure affects the heat transfer in a similar way, thatthe apparent heat conduction and thermal diffusivity is large
in the area resulting in decrease of T at points of small T.
In Fig. 6, the junctions of large S is concentration along
the current path, and the value of k shows both large andsmall values. The junctions of small k obstruct the current
flow and the junctions of larger k is the point at which thecurrent flows and concentrates. The combination of these
two generates the current path shown in Fig. 5. These results
represents that the contact electric resistance variation should
be considered carefully to evaluate the current path andtemperature distribution of the Ag-NWs network.
CONCLUSION
Temperature distribution of the current-supplied Ag-NW
network was measured using thermo-reflectance imaging.
The current distribution and the contact electric resistance of
the Ag-NW network was obtained numerically by matching
the temperature distribution with the measurement. The
contact electric resistance varied at each junction, and played
an important role in generating specific current paths andhot-spots in the Ag-NW network. The results showed that theperformance and reliability of conductive sheets and heaters
using Ag-NW network should consider the effect of thevariation and reliability of the contact electric resistance,
which similarly could occur with the contact thermal
resistance.
NOMENCLATURE
CTR: thermo-reflectance coefficient
I: light intensity
k: contact electric conductance
L: loss function
R: reflectance
T: temperature
: correction coefficientg: contact thermal conductance
Subscripts
s: computation
m: measurement

[K]
[-]
[1/]
[K2]
[-]
[K]
[-]
[W/K]

Figure 5. Current distribution of the Ag-NW network
obtained by the computation in the case of matching thetemperature to the measurement: the distribution
corresponds to the results of T shown in Fig. 2 (b).

Figure 6. Contact electric conductivity distribution at thejunction of which the sensitivity parameter S is greaterthan 1,000 K2.
ACKNOWLEDGMENT

This work was supported by JSPS KAKENHI Grant Number
22K18769 and 21H01262.
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