The influence of edge effects is well-detailed in Fang et al. [27, 28] and reported predicted the U values (centre-of-glass and overall) to be 0.26 and 0.65 Wm2 K1 , respectively. These values were reported for a TVG sample size of 500 500 mm with 4 mm (top), 4 mm(middle) and 4 mm (bottom) un-tempered soda lime glass sheets having four layers (1-top, 2-middle and 1-bottom) of low-e coatings (ε of 0.03) with a frame rebate depth of 10 mm and the width of indium-alloy edge seal 6 mm (k of 83.7 Wm1 K1 ). It can be compared, considering all factors, with the results of this paper. It is found that an increase of 0.07 Wm2 K1 (deviation of 26.9%) and 0.4 Wm2 K1 (deviation of 61.54%) of centre of glass and overall U value, respectively. Such deviations, as per FEM calculations, are due to: the sample size influences to a small extent as the TVG fabricated sample size was of the size 300 300 mm, the width of the edge seal influences to a large extent because the TVG fabricated sample width of edge seal used was total 14 mm (10 mm wide Cerasolzer seal and 4 mm wide J-B Weld supportive seal), the use of three SnO2 low-e coatings (ε of 0.15) in the fabricated sample instead of four Ag low-e coatings (ε of 0.03), and there is no frame rebate depth utilised in the fabricated sample of TVG that has an influence to a small extent on the overall U value of the TVG because the purpose of this paper is focused on the hermetic sealing materials thermal conductivities and triple vacuum glazing area thermal performance only. By accounting all these factors and including the thermal conductivity of the sealing materials, the FEM model predictions are in good agreement.

A finite element calculations of the temperature loss due to a wider edge seal are analysed, along the outer surface lines AA and BB (as shown in Figure 5), showing the temperature gradient from the glazing edge to the central area of the glazing as illustrated in Figure 6. An

Figure 6. The temperature loss, due to a wider edge seal, along the glazing surface lines AA and BB showing the temperature gradient from the glazing edge to the central area of the glazing of the one quarter of total sample of

Figure 5. The FEM based isotherms on (a) the indoor (b) the outdoor glass surface showing the temperature variations

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from the edge area towards the central glazing area.

300 300 mm using ASTM boundary conditions.

The simulated isotherms of the triple vacuum glazing for the outdoor and indoor surfaces are shown in Figure 5. The mean glass surface temperatures were simulated to be 12.55 and 6.71C for the outdoor and indoor surfaces of the total glazing area. The mean surface temperatures for the centre of glass area were simulated to be 16.43 and 16.60C for the outdoor and indoor surfaces respectively. It can be seen the temperature discrepancies on the outdoor side are smaller than the indoor side. This is, however, due to the use of 14 mm composite edge seal as compared to the edge seal thickness of 6 mm [28]. Thus, the edge effects need to be reduced by narrowing the width of edge-seal to 9 mm (6 mm and 3 mm for primary and secondary respectively) which reduces the centre-of-glass U-value of 0.043 Wm2 K1 .

Thermal Conductivity Measurement of Vacuum Tight Dual-Edge Seal for the Thermal Performance Analysis… http://dx.doi.org/10.5772/intechopen.74255 141

The internal and external surface heat transfer coefficients were set to 8.3 and 30 Wm2 K1

The finite element 3D modelling results show a centre of glass and overall U-value of 0.33 and

The U value (centre of glass) 0.2 Wm2 K1 [13] was based on the parametric model of triple vacuum glazing without frame focused on the central glazing area. This value was achieved with 6 mm (top), 4 mm (middle) and 6 mm (bottom) thick untampered soda lime glass sheets having four layers (1-top, 2-middle and 1-bottom) of low-e coatings (ε of 0.03). It is compared with the results of this paper, it is found that an increase of U value (centre-of-glass)

paper which is made of 4 mm(top), 4 mm (middle) and 4 mm(bottom) untampered soda lime glass sheets having three layers of low-e SnO2 coatings (ε = 0.15). The reason, to use such dimensions and SnO2 coatings, is the conventional availability of glass sheets from Pilkington

The influence of edge effects is well-detailed in Fang et al. [27, 28] and reported predicted the U

were reported for a TVG sample size of 500 500 mm with 4 mm (top), 4 mm(middle) and 4 mm (bottom) un-tempered soda lime glass sheets having four layers (1-top, 2-middle and 1-bottom) of low-e coatings (ε of 0.03) with a frame rebate depth of 10 mm and the width of

with the results of this paper. It is found that an increase of 0.07 Wm2 K1 (deviation of 26.9%) and 0.4 Wm2 K1 (deviation of 61.54%) of centre of glass and overall U value, respectively. Such deviations, as per FEM calculations, are due to: the sample size influences to a small extent as the TVG fabricated sample size was of the size 300 300 mm, the width of the edge seal influences to a large extent because the TVG fabricated sample width of edge seal used was total 14 mm (10 mm wide Cerasolzer seal and 4 mm wide J-B Weld supportive seal), the use of three SnO2 low-e coatings (ε of 0.15) in the fabricated sample instead of four Ag low-e coatings (ε of 0.03), and there is no frame rebate depth utilised in the fabricated sample of TVG that has an influence to a small extent on the overall U value of the TVG because the purpose of this paper is focused on the hermetic sealing materials thermal conductivities and triple vacuum glazing area thermal performance only. By accounting all these factors and including the thermal conductivity of the

The simulated isotherms of the triple vacuum glazing for the outdoor and indoor surfaces are shown in Figure 5. The mean glass surface temperatures were simulated to be 12.55 and 6.71C for the outdoor and indoor surfaces of the total glazing area. The mean surface temperatures for the centre of glass area were simulated to be 16.43 and 16.60C for the outdoor and indoor surfaces respectively. It can be seen the temperature discrepancies on the outdoor side are smaller than the indoor side. This is, however, due to the use of 14 mm composite edge seal as compared to the edge seal thickness of 6 mm [28]. Thus, the edge effects need to be reduced by narrowing the width of edge-seal to 9 mm (6 mm and 3 mm for primary and secondary

values (centre-of-glass and overall) to be 0.26 and 0.65 Wm2 K1

sealing materials, the FEM model predictions are in good agreement.

respectively) which reduces the centre-of-glass U-value of 0.043 Wm2 K1

, respectively, this is compared to the predictions of [26] i.e. 0.2 Wm2 K1 and

. Such deviation is due to the design of the fabricated sample reported in this

, respectively. These values

). It can be compared, considering all factors,

.

respectively [26].

1.05 Wm2 K1

0.13 Wm2 K1

[27] i.e. 0.26 Wm2 K1 as follows.

140 Impact of Thermal Conductivity on Energy Technologies

Glass and its use in the UK dwelling.

indium-alloy edge seal 6 mm (k of 83.7 Wm1 K1

Figure 5. The FEM based isotherms on (a) the indoor (b) the outdoor glass surface showing the temperature variations from the edge area towards the central glazing area.

A finite element calculations of the temperature loss due to a wider edge seal are analysed, along the outer surface lines AA and BB (as shown in Figure 5), showing the temperature gradient from the glazing edge to the central area of the glazing as illustrated in Figure 6. An

Figure 6. The temperature loss, due to a wider edge seal, along the glazing surface lines AA and BB showing the temperature gradient from the glazing edge to the central area of the glazing of the one quarter of total sample of 300 300 mm using ASTM boundary conditions.

influence of wider edge seal on temperature loss around the edge area has affected the centreof-pane U value. Figure 6 shows there are smaller temperature gradients on the cold surface as compared to the warm surface of the glazing which is due to the periodic shape of the edge sealing area of triple vacuum glazing as shown in Figure 1(b). It can be seen the influence of temperature gradient loss spread to 54 and 84 mm on the cold and warm side respectively. When comparing with the temperature gradient profiles of [26, 27] the results are in good agreement with the results presented in this paper.

Nomenclature

a overall radius of the sensor [m] d thickness of the glass sheet [mm]

k thermal conductivity [Wm�1 K�1

ΔTi initial temperature difference [K]

side of the sensor [K]

U thermal transmittance [Wm�2 K�1

ASTM American Society for Testing and Materials

CIBSE chartered institution of building services engineers

R resistance [ohms]

T temperature [�C]

FEM finite element model TPS transient plane source TVG triple vacuum glazing

o before the sensor is heated at

] α thermal diffusivity of the sample [mm2 s

c t = 0 s characteristic

r density [kgm�3

t time [s]

Abbreviations

Subscripts

Greek letters

ave. average

D dimensionless time dependent variable

Po total output power from the sensor [W]

]

Thermal Conductivity Measurement of Vacuum Tight Dual-Edge Seal for the Thermal Performance Analysis…

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143

]

CALEBRE consumer appealing low energy technologies for building retrofitting

�1 ]

ΔTaveð Þτ average temperature increase of the sample surface on the other

### 4. Conclusions

In this study, a hot disk thermal constants analyser TPS 2500s using transient plane source technique with a sensor element in the shape of double spiral and resistance thermometer is proved to be an adequate in measuring and analysing the thermal conductivity of hermetic edge sealing materials (i.e. Cersolzer CS186 and J-B Weld). The technique was validated by measuring the thermal conductivity for Mild Steel and Indium and comparing these results with those available in literature. This validated technique based on hot disk thermal constants analyser was then used to measure thermal conductivity of Cerasolzer CS186 alloy and J-B Weld epoxy steel resin and found to be 46.49 and 7.47 Wm1 K1 , respectively. It has been shown that an increase in temperature has direct relation with respect to reporting time with highest increase in temperature was recorded for JB-Weld for a given time period, which is in agreement with the experimental data. It has also been shown that the increase in temperature in the epoxy J-B Weld is greater than that of mild steel. This observation is linked to the fact that the heat flow in semi-polymeric materials is less as compared to that of metallic materials. The temperature increase in the sample made from Cerasolzer alloy was higher than that of indium sample. These values were utilised for the numerical prediction of thermal performance of triple vacuum glazing using 3D FE model. The simulated resulted showed that the centre-of-glass and total U-value of 300 300 mm triple vacuum glazing are 0.33 and 1.05 Wm2 K1 , respectively. The thermal transmittance values can be reduced by using soft low emittance coatings and by reducing the width of the hermetic edge seal to 9 mm. An influence of wider edge seal on temperature loss spreading from the edge to the central glazing area was further analysed with the FEM model calculations. In which it is concluded that the wider edge seal has affected the U-value to 0.043 Wm2 K1 because of the temperature gradient loss spread to 54 and 84 mm on the cold and warm side respectively.

#### Acknowledgements

The author acknowledge the valuable advice and guidance received from Prof P. C. Eames during the course of this research work. This work was supported by the EPSRC funded project CALEBRE (Consumer Appealing Low energy Technologies for Building Retrofitting) [EP/G000387/1].

### Nomenclature

influence of wider edge seal on temperature loss around the edge area has affected the centreof-pane U value. Figure 6 shows there are smaller temperature gradients on the cold surface as compared to the warm surface of the glazing which is due to the periodic shape of the edge sealing area of triple vacuum glazing as shown in Figure 1(b). It can be seen the influence of temperature gradient loss spread to 54 and 84 mm on the cold and warm side respectively. When comparing with the temperature gradient profiles of [26, 27] the results are in good

In this study, a hot disk thermal constants analyser TPS 2500s using transient plane source technique with a sensor element in the shape of double spiral and resistance thermometer is proved to be an adequate in measuring and analysing the thermal conductivity of hermetic edge sealing materials (i.e. Cersolzer CS186 and J-B Weld). The technique was validated by measuring the thermal conductivity for Mild Steel and Indium and comparing these results with those available in literature. This validated technique based on hot disk thermal constants analyser was then used to measure thermal conductivity of Cerasolzer CS186 alloy and J-B

shown that an increase in temperature has direct relation with respect to reporting time with highest increase in temperature was recorded for JB-Weld for a given time period, which is in agreement with the experimental data. It has also been shown that the increase in temperature in the epoxy J-B Weld is greater than that of mild steel. This observation is linked to the fact that the heat flow in semi-polymeric materials is less as compared to that of metallic materials. The temperature increase in the sample made from Cerasolzer alloy was higher than that of indium sample. These values were utilised for the numerical prediction of thermal performance of triple vacuum glazing using 3D FE model. The simulated resulted showed that the centre-of-glass and total U-value of 300 300 mm triple vacuum glazing are 0.33 and

low emittance coatings and by reducing the width of the hermetic edge seal to 9 mm. An influence of wider edge seal on temperature loss spreading from the edge to the central glazing area was further analysed with the FEM model calculations. In which it is concluded that the wider edge seal has affected the U-value to 0.043 Wm2 K1 because of the temperature

The author acknowledge the valuable advice and guidance received from Prof P. C. Eames during the course of this research work. This work was supported by the EPSRC funded project CALEBRE (Consumer Appealing Low energy Technologies for Building Retrofitting)

gradient loss spread to 54 and 84 mm on the cold and warm side respectively.

, respectively. The thermal transmittance values can be reduced by using soft

, respectively. It has been

agreement with the results presented in this paper.

142 Impact of Thermal Conductivity on Energy Technologies

Weld epoxy steel resin and found to be 46.49 and 7.47 Wm1 K1

4. Conclusions

1.05 Wm2 K1

Acknowledgements

[EP/G000387/1].

### Abbreviations

#### Subscripts

#### Greek letters

ϕ heat loss [W]

ε emittance

### Author details

Saim Memon

Address all correspondence to<EMAIL_ADDRESS>
Centre for Advanced Materials, Division of Electrical and Electronic Engineering, School of Engineering, London South Bank University, London, UK

[7] Eames PC. Vacuum glazing: Current performance and future prospects. Vacuum. 2008;

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145

[8] Robinson SJ, Collins RE. Evacuated windows—theory and practice. ISES Solar World

[9] Collins RE, Simko TM. Current status of the science and technology of vacuum glazing.

[10] Collins RE, Tang JZ, Merrylands. Design improvements to vacuum glazing. 1999. US

[11] Griffiths PW, Di Leo M, Cartwright P, Eames PC, Yianoulis P, Leftheriotis G, Norton B. Fabrication of evacuated glazing at low temperature. Solar Energy. 1998;63:243-249. DOI:

[12] Zhao JF, Eames PC, Hyde TJ, Fang Y, Wang J. A modified pump-out technique used for fabrication of low temperature metal sealed vacuum glazing. Solar Energy. 2007;81:1072-1077.

[13] Manz H, Brunner S, Wullschleger L. Triple vacuum glazing: Heat transfer and basic mechanical design constraints. Solar Energy. 2006 Dec 31;80(12):1632-1642. DOI: https://

[14] Memon, S. Investigating energy saving performance interdependencies with retrofit triple vacuum glazing for use in UK dwelling with solid walls, Sustainable Development on Building and Environment: Proceedings of the 7th International Conference, Reading,

[15] Memon S, Farukh F, Eames PC, Silberschmidt VV. A new low-temperature hermetic composite edge seal for the fabrication of triple vacuum glazing. Vacuum. 2015;120:73-82. DOI:

[16] Gustavsson M, Karawacki E, Gustafsson SE. Thermal conductivity, thermal diffusivity, and specific heat of thin samples from transient measurements with hot disk sensors. Review of Scientific Instruments. 1994 Dec;65(12):3856-3859. DOI: https://doi.org/10.1063/

[17] Gustafsson SE. Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials. The Review of Scientific Instruments. 1991

[18] Hot Disk Thermal Constants Analyzer. 2013. TPS 2500s Instruction Manual, Mathis Instru-

[19] Bohac V, Gustavsson MK, Kubicar L, Gustafsson SE. Parameter estimations for measurements of thermal transport properties with the hot disk thermal constants analyzer. The Review of Scientific Instruments. 2000 Jun;71(6):2452-2455. DOI: 10.1063/

82:717-722. DOI: https://doi.org/10.1016/j.vacuum.2007.10.017

Congress, International Solar Energy Society, Kobe, Japan; 1989

Patent No. 5891536

10.1016/S0038-092X(98)00019-X

DOI: 10.1016/j.solener.2007.03.006

doi.org/10.1016/j.solener.2005.11.003

UK; 2015. ISBN-13: 978–0993120701

Mar;62(3):797-804. DOI: 10.1063/1.1142087

10.1016/j.vacuum.2015.06.024

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1.1145178

1.1150635

Solar Energy. 1998;62:189-213. DOI: 10.1016/S0038-092X(98)00007-3

### References

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τ dimensionless time dependent function

144 Impact of Thermal Conductivity on Energy Technologies

ϕ heat loss [W]

ε emittance

Author details

Saim Memon

References

ac.uk/2134/14562

2005.04.002

φ temperature coefficient of the resistivity [K1

Address all correspondence to<EMAIL_ADDRESS>
2012. pp. 4636-4643. ISBN:<PHONE_NUMBER>923

Engineering, London South Bank University, London, UK

]

Centre for Advanced Materials, Division of Electrical and Electronic Engineering, School of

[1] Memon S, Eames PC. Heat load and solar gain prediction for solid wall dwellings retrofitted with triple vacuum glazing for selected window to wall area ratios, World Renewable Energy Forum, WREF 2012, Including World Renewable Energy Congress XII and Colorado Renewable Energy Society (CRES) Annual Conference, Colorado, USA. 6,

[2] Memon S, Eames PC. Predicting the solar energy and space-heating energy performance for solid-wall detached house retrofitted with the composite edge-sealed triple vacuum

[3] Memon S. Analysing the potential of retrofitting ultra-low heat loss triple vacuum glazed windows to an existing UK solid wall dwelling. International Journal of Renewable

[4] Memon S. Design, fabrication and performance analysis of vacuum glazing units fabricated with low and high temperature hermetic glass edge sealing materials. [PhD Thesis]. UK: Loughborough University Institutional Repository; 2013. DOI: https://dspace.lboro.

[5] Fang F, Eames PC, Norton B, Hyde TJ. Experimental validation of a numerical model for heat transfer in vacuum glazing. Solar Energy. 2006;80:564-577. DOI: 10.1016/j.solener.

[6] Benson DK, Tracy CE. Evacuated window glazings for energy efficient buildings. Proceedings of SPIE 0562, Optical Materials Technology for Energy Efficiency and Solar

glazing. Energy Procedia. 2017;122:565-570. DOI: 10.1016/j.egypro.2017.07.419

Energy Development. 2014;3(3):161. DOI: 10.14710/ijred.3.3.161-174

Energy Conversion IV. San Diego, CA; 1985. p. 250

[20] Memon S, Eames PC. An investigation on edge sealing materials for the fabrication of vacuum glazing, 3rd School of Electrical, Electronic and Systems Engineering (EESE) Research conference, 2013 Mar 21–22, Loughborough University, UK

**Section 3**

**Thermal Conductivity of Nano Systems**

**Thermal Conductivity of Nano Systems**

[20] Memon S, Eames PC. An investigation on edge sealing materials for the fabrication of vacuum glazing, 3rd School of Electrical, Electronic and Systems Engineering (EESE)

[21] Memon S. Experimental measurement of hermetic edge seal's thermal conductivity for the thermal transmittance prediction of triple vacuum glazing. Case Studies in Thermal

[22] ASTM, Standard procedures for determining the steady state thermal transmittance of fenestration systems, ASTM Standard E 1423-91, in: 1994 Annual Book of ASTM Stan-

[23] Memon S, Farukh F, Eames PC, Silberschmidt VV. A new low-temperature hermetic composite edge seal for the fabrication of triple vacuum glazing. Vacuum. 2015;120:73-82. DOI:

[24] Collins RE, Robinson SJ. Evacuated glazing. Solar Energy. 1991 Jan;47(1):27-38. DOI:

[25] Wang J, Eames PC, Zhao JF, Hyde T, Fang Y. Stresses in vacuum glazing fabricated at low temperature. Solar Energy Materials & Solar Cells. 2007;91(4):290-303. DOI: 10.1016/j.

[26] Fang Y, Hyde T, Hewitt N, Eames PC, Norton B. Comparison of vacuum glazing thermal performance predicted using two-and three-dimensional models and their experimental validation. Solar Energy Materials & Solar Cells. 2009;93(9):1492-1498. DOI: https://doi.

[27] Fang Y, Hyde TJ, Hewitt N. Predicted thermal performance of triple vacuum glazing.

[28] Fang Y, Hyde TJ, Arya F, Hewitt N, Wang R, Dai Y. Enhancing the thermal performance of triple vacuum glazing with low-emittance coatings. Energy and Buildings. 2015;97:186-195.

Solar Energy. 2010;84(12):2132-2139. DOI: 10.1016/j.solener.2010.09.002

Engineering. 2017;10:169-178. ISSN 2214-157X. DOI: 10.1016/j.csite.2017.06.002

dard 04.07, American Society of Testing and Materials; 1991. pp. 1160-1165

10.1016/j.vacuum.2015.06.024

146 Impact of Thermal Conductivity on Energy Technologies

10.1016/0038-092X(91)90060-A

org/10.1016/j.solmat.2009.03.025

DOI: 10.1016/j.enbuild.2015.04.006

solmat.2006.10.007

Research conference, 2013 Mar 21–22, Loughborough University, UK

**Chapter 9**

**Provisional chapter**

**Structural and Thermoelectric Properties**

**Structural and Thermoelectric Properties**

Dedi, Indah Primadona, Ping-Chung Lee, Chi-Hua Chien and Yang-Yuan Chen

Dedi, Indah Primadona, Ping-Chung Lee, Chi-Hua Chien and Yang-Yuan Chen

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

yielded high accuracy for size-dependent studies.

http://dx.doi.org/10.5772/intechopen.76635

**Nanowire**

**Abstract**

1.4‒4.3 x 10−3

**1. Introduction**

**Nanowire**

**Characterization of Individual Single-Crystalline**

**Characterization of Individual Single-Crystalline**

DOI: 10.5772/intechopen.76635

Herein, we report a method for structural characterization as well as TE properties measurements of individual single-crystalline Lead telluride (PbTe) NWs by employing a new microchip design. In this work, the single PbTe NW was characterized in four different types of measurement: structural characterization, Seebeck coefficient *S*, electrical conductivity *σ*, and thermal conductivity *κ*. The structural characterization by transmission electron microscope (TEM) revealed that the PbTe NWs were high-quality single crystals with a growth along the [100] direction. The TE properties *S*, *σ*, and *κ* measurement results of individual 75 nm PbTe NW at room temperature were −54.76 µV K−1, 1526.19 S m−1, and 0.96 W m−1 K−1, respectively. Refer to the result of *S*, *σ* and *κ*; the figure of merit *ZT* values of a 75 nm PbTe NW at the temperature range of 300‒350 K were

to previous reported, which indicates that thermal transport through the individual PbTe NWs is limited by boundary scattering of both electrons and phonons. The results show that this new technique measurement provided a reliable *ZT* value of individual NW

**Keywords:** lead telluride, nanowires, thermal conductivity, size-dependent, figure of merit

Recently, nano-engineered thermoelectric (TE) materials used for converting waste heat into electricity have become an interesting research topic. TE energy converters are devices that

. Furthermore, it was observed that the *κ* value is size-dependent compared

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

#### **Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline Nanowire Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline Nanowire**

DOI: 10.5772/intechopen.76635

Dedi, Indah Primadona, Ping-Chung Lee, Chi-Hua Chien and Yang-Yuan Chen Dedi, Indah Primadona, Ping-Chung Lee, Chi-Hua Chien and Yang-Yuan Chen

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.76635

#### **Abstract**

Herein, we report a method for structural characterization as well as TE properties measurements of individual single-crystalline Lead telluride (PbTe) NWs by employing a new microchip design. In this work, the single PbTe NW was characterized in four different types of measurement: structural characterization, Seebeck coefficient *S*, electrical conductivity *σ*, and thermal conductivity *κ*. The structural characterization by transmission electron microscope (TEM) revealed that the PbTe NWs were high-quality single crystals with a growth along the [100] direction. The TE properties *S*, *σ*, and *κ* measurement results of individual 75 nm PbTe NW at room temperature were −54.76 µV K−1, 1526.19 S m−1, and 0.96 W m−1 K−1, respectively. Refer to the result of *S*, *σ* and *κ*; the figure of merit *ZT* values of a 75 nm PbTe NW at the temperature range of 300‒350 K were 1.4‒4.3 x 10−3 . Furthermore, it was observed that the *κ* value is size-dependent compared to previous reported, which indicates that thermal transport through the individual PbTe NWs is limited by boundary scattering of both electrons and phonons. The results show that this new technique measurement provided a reliable *ZT* value of individual NW yielded high accuracy for size-dependent studies.

**Keywords:** lead telluride, nanowires, thermal conductivity, size-dependent, figure of merit

### **1. Introduction**

Recently, nano-engineered thermoelectric (TE) materials used for converting waste heat into electricity have become an interesting research topic. TE energy converters are devices that

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

can harvest renewable energy for power generation and thermal sensing application [1–3]. The efficiency of TE materials is evaluated based on the dimensionless figure of merit *ZT*, which is written as *S2 σT*/(*κe* + *κl* ), where *S*, *σ*, *κe* , and *κl* denotes the thermal power or Seebeck coefficient, the electrical conductivity, the electronic thermal conductivity, and the lattice thermal conductivity, respectively. The quantity of *S2 σ* is defined as the power factor (PF). Theoretically, a reduction in dimensionality from three dimensions to one dimension yields a dramatically increased electronic density of states (DOS) at the energy band edges and a decreased thermal conductivity. As a result, the thermoelectric PF and assuredly *ZT* value enhances [4, 5]. Numerous studies have reported that the enhancement of *ZT* value in nanomaterials is the result of quantum confinement effects and increased surface phonon scattering [3, 6, 7].

intensively explored in the past decades [17–27]. Besides that, there have also been some studies on the TE properties of PbTe-based nanostructures such as PbTe/PbSeTe quantum dot superlattice [28] and PbTe NWs [20] with room temperature *ZT* value 0.75 and 0.0054, respectively.

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline…

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151

In this work, the synthesis of single-crystalline PbTe NWs via a stress-induced method is described elsewhere [27], in a way similar to the on-film formation (ON–OFF) growth of other semiconductor NWs. [29] Briefly, the PbTe was made by mixing elemental Pb (Alfa Aesar, −200 mesh, 99.9%) and Te (Alfa Aesar, −325 mesh, 99.999%) inside a carbon-coated silica tube. After that, the resulting mixtures inside the tube was vacuumed up to 10−6 Torr, sealed, and slowly heated to 1000°C over a period of 12 hours. On the top of the heating process, the temperature was held for 4 hours, and then cooled down to room temperature. The resulted ingot was then cut by a diamond saw into a disc shape with a diameter of 10 mm. Before fabricating the PbTe films, the disc target and substrates were both ultrasonically cleaned in acetone, isopropanol, and then rinsed with deionized water. The PbTe films were prepared by depositing the synthe-

system (LPX Pro 210). The base pressure of the vacuum system was 5.0×10−7 Torr. The excimer laser was applied for 15 min at room temperature with energy and frequency of 140 mJ and 10 Hz, respectively. The substrate rotation speed was approximately 10 rpm. With the all mentioned set up, the total thickness of the PbTe films was about 20 nm. To synthesis PbTe NWs, the PbTe films were sealed in a vacuumed quartz tube below 5×10−6 Torr, annealed at 450°C for 5 days, and then cooled slowly to room temperature. During the annealing process, the NWs grew from the film to release the compressive stress caused by the difference in thermal expan-

As mentioned above, the challenge in working with TE NW was the measurement of thermoelectric properties as well as structural analysis on specific single-crystalline nanowire, to get a high accuracy of *ZT* value. Here, in order to solve that problem, a novel design of microchip was discussed. The preparation processes of the measurement platform are shown in **Figure 1**. First,

by using spin coating method (**Figure 1b**), followed by standard photolithography processes. After that the exposure soluble photoresist can be developed by the developer. The Si wafer was then put into the reactive ion etching system (RIE, Model: ANELA DEM-451T) for dry etching.

sium hydroxide (KOH) solution for wet etching. The silicon which is exposed to KOH will be

membrane for further processes.

/Si (100) substrates in a pulsed laser deposition (PLD)

N4 layer (**Figure 1a**) on both sides was covered with the photoresist

will be etched by the reactive ion. Then, the wafer is immersed into potas-

/Si substrate (0.5×10−6/°C)/

**2. Synthesis and structural characterization of PbTe NWs**

sion coefficients between the PbTe film (19.8×10−6/°C) and the SiO2

**2.1. Synthesis of PbTe NWs**

sized PbTe ingot on single-crystal SiO2

(2.4×10−6/°C).

**2.2. Microchip preparation**

the silicon (Si) wafer with Si3

etched and leaving a thin Si3

N4

N4

The unprotected Si3

Due to the nanometer scale effect, it is believed that in NWs comprising TE materials, such as PbTe, the value of *S* is higher than its bulk counterpart [8, 9]. According to the certain carrierscattering assumptions, the enhancement of *S* value occurs because of the sharp increase in the local DOS around the Fermi level, which also can be interpreted as an increased local DOS effective mass (md\*). However, with the overall benefit of such an improvement in *S*, nanosize will cause a declining in carrier mobility (*μ*) and thus affect to the decreasing the ZT value. This is occurs because the increased local DOS usually leads to a heavier transport-effective mass of carriers. In the most well-known high-temperature TE, the carriers are predominantly scattered by phonons. [10] Increasing the *S* is an obvious goal for obtaining a high efficiency TE materials. Nevertheless, other changes in transport properties often sacrifice the *σ* correlated with an increase in the *S* and thus do not ultimately lead to an improvement in *ZT*.

In order to unveil the size effect on the intrinsic physical properties of TE materials, the measurement of nanowire without interference from either the matrix or external contacts is imperative. Furthermore, the TE properties measurement as well as structural characterization on single nanowire is also crucial in terms of the accuracy and reliability of the resulting *ZT* value. The low accuracy may occur because every single nanowire, although in the same batch synthesis process, may have a different structure or TE properties. However, measuring all properties on single nanowire is still challenging due to the unavailability of NW microchip which is compatible for all type measurements.

In this chapter, the synthesis and structural characterization of PbTe NWs as well as the preparation of newly design NW's microchip to resolve the above mentioned problem will be described in the first section. For synthesizing the NW, an alternative free-catalyst technique, the stress-induced growth method will be introduced. In the second section, we will discuss about their TE properties (*S*, *σ*, and *κ*).

Type of TE NW used in this chapter was single-crystalline PbTe NW. PbTe is a semiconductor with an energy band gap of 0.31 eV at 300 K [11–14]. In recent years, it has been found that PbTe is one of the superior TE materials in the temperature range of 400–900 K. This material has a large Seebeck coefficient, a very low *κl* (2.2 W m−1 K−1 at 300 K) [15] and a good electrical conductivity when appropriately chemical doped [16]. The synthesis of low-dimensional PbTe NWs has been intensively explored in the past decades [17–27]. Besides that, there have also been some studies on the TE properties of PbTe-based nanostructures such as PbTe/PbSeTe quantum dot superlattice [28] and PbTe NWs [20] with room temperature *ZT* value 0.75 and 0.0054, respectively.

### **2. Synthesis and structural characterization of PbTe NWs**

### **2.1. Synthesis of PbTe NWs**

can harvest renewable energy for power generation and thermal sensing application [1–3]. The efficiency of TE materials is evaluated based on the dimensionless figure of merit *ZT*,

coefficient, the electrical conductivity, the electronic thermal conductivity, and the lattice

Theoretically, a reduction in dimensionality from three dimensions to one dimension yields a dramatically increased electronic density of states (DOS) at the energy band edges and a decreased thermal conductivity. As a result, the thermoelectric PF and assuredly *ZT* value enhances [4, 5]. Numerous studies have reported that the enhancement of *ZT* value in nanomaterials is the result of quantum confinement effects and increased surface phonon scatter-

Due to the nanometer scale effect, it is believed that in NWs comprising TE materials, such as PbTe, the value of *S* is higher than its bulk counterpart [8, 9]. According to the certain carrierscattering assumptions, the enhancement of *S* value occurs because of the sharp increase in the local DOS around the Fermi level, which also can be interpreted as an increased local DOS effective mass (md\*). However, with the overall benefit of such an improvement in *S*, nanosize will cause a declining in carrier mobility (*μ*) and thus affect to the decreasing the ZT value. This is occurs because the increased local DOS usually leads to a heavier transport-effective mass of carriers. In the most well-known high-temperature TE, the carriers are predominantly scattered by phonons. [10] Increasing the *S* is an obvious goal for obtaining a high efficiency TE materials. Nevertheless, other changes in transport properties often sacrifice the *σ* correlated with an increase in the *S* and thus do not ultimately lead to an

In order to unveil the size effect on the intrinsic physical properties of TE materials, the measurement of nanowire without interference from either the matrix or external contacts is imperative. Furthermore, the TE properties measurement as well as structural characterization on single nanowire is also crucial in terms of the accuracy and reliability of the resulting *ZT* value. The low accuracy may occur because every single nanowire, although in the same batch synthesis process, may have a different structure or TE properties. However, measuring all properties on single nanowire is still challenging due to the unavailability of NW micro-

In this chapter, the synthesis and structural characterization of PbTe NWs as well as the preparation of newly design NW's microchip to resolve the above mentioned problem will be described in the first section. For synthesizing the NW, an alternative free-catalyst technique, the stress-induced growth method will be introduced. In the second section, we will discuss

Type of TE NW used in this chapter was single-crystalline PbTe NW. PbTe is a semiconductor with an energy band gap of 0.31 eV at 300 K [11–14]. In recent years, it has been found that PbTe is one of the superior TE materials in the temperature range of 400–900 K. This material has a large

when appropriately chemical doped [16]. The synthesis of low-dimensional PbTe NWs has been

(2.2 W m−1 K−1 at 300 K) [15] and a good electrical conductivity

, and *κl*

denotes the thermal power or Seebeck

*σ* is defined as the power factor (PF).

), where *S*, *σ*, *κe*

which is written as *S2*

ing [3, 6, 7].

improvement in *ZT*.

*σT*/(*κe* + *κl*

150 Impact of Thermal Conductivity on Energy Technologies

thermal conductivity, respectively. The quantity of *S2*

chip which is compatible for all type measurements.

about their TE properties (*S*, *σ*, and *κ*).

Seebeck coefficient, a very low *κl*

In this work, the synthesis of single-crystalline PbTe NWs via a stress-induced method is described elsewhere [27], in a way similar to the on-film formation (ON–OFF) growth of other semiconductor NWs. [29] Briefly, the PbTe was made by mixing elemental Pb (Alfa Aesar, −200 mesh, 99.9%) and Te (Alfa Aesar, −325 mesh, 99.999%) inside a carbon-coated silica tube. After that, the resulting mixtures inside the tube was vacuumed up to 10−6 Torr, sealed, and slowly heated to 1000°C over a period of 12 hours. On the top of the heating process, the temperature was held for 4 hours, and then cooled down to room temperature. The resulted ingot was then cut by a diamond saw into a disc shape with a diameter of 10 mm. Before fabricating the PbTe films, the disc target and substrates were both ultrasonically cleaned in acetone, isopropanol, and then rinsed with deionized water. The PbTe films were prepared by depositing the synthesized PbTe ingot on single-crystal SiO2 /Si (100) substrates in a pulsed laser deposition (PLD) system (LPX Pro 210). The base pressure of the vacuum system was 5.0×10−7 Torr. The excimer laser was applied for 15 min at room temperature with energy and frequency of 140 mJ and 10 Hz, respectively. The substrate rotation speed was approximately 10 rpm. With the all mentioned set up, the total thickness of the PbTe films was about 20 nm. To synthesis PbTe NWs, the PbTe films were sealed in a vacuumed quartz tube below 5×10−6 Torr, annealed at 450°C for 5 days, and then cooled slowly to room temperature. During the annealing process, the NWs grew from the film to release the compressive stress caused by the difference in thermal expansion coefficients between the PbTe film (19.8×10−6/°C) and the SiO2 /Si substrate (0.5×10−6/°C)/ (2.4×10−6/°C).

### **2.2. Microchip preparation**

As mentioned above, the challenge in working with TE NW was the measurement of thermoelectric properties as well as structural analysis on specific single-crystalline nanowire, to get a high accuracy of *ZT* value. Here, in order to solve that problem, a novel design of microchip was discussed. The preparation processes of the measurement platform are shown in **Figure 1**. First, the silicon (Si) wafer with Si3 N4 layer (**Figure 1a**) on both sides was covered with the photoresist by using spin coating method (**Figure 1b**), followed by standard photolithography processes. After that the exposure soluble photoresist can be developed by the developer. The Si wafer was then put into the reactive ion etching system (RIE, Model: ANELA DEM-451T) for dry etching. The unprotected Si3 N4 will be etched by the reactive ion. Then, the wafer is immersed into potassium hydroxide (KOH) solution for wet etching. The silicon which is exposed to KOH will be etched and leaving a thin Si3 N4 membrane for further processes.

**Figure 1.** Schematic Si3 N4 membrane template preparation: (a) the silicon wafer with Si3 N4 on the both sides, (b) substrate is spin coated with photoresist, (c) Photoresist is exposed to a rectangular pattern with ultraviolet light, (d) soluble photoresist can be developed by the developer, (e) removing Si3 N4 layer by reactive ion etching system (RIE), (f) dip the wafer into a bath of KOH for wet etching to create a cavity and leave a suspended Si3 N4 membrane, and (g) strip the photoresist.

The standard photolithography processes were used to define the contact pads of the measurement platform. As shown in **Figure 2**, first, Si wafer with Si3 N4 membrane is covered with photoresist material by spin coating, followed by exposure, evaporation, and lift-off process. The developed primary measurement platform is then ready to be used.

Rnw represents resistance between point 2 and 3, resistance of Cr/Au electrodes, resistance of contacted NW to electrode, and NW resistance, respectively. The resistance of R2point and Rnw were measured by two and four-point probes. Whilst, in order to obtain the contact resistance as Rcont = (R2point - Rnw)/2, Rlead (~10 Ω) was neglected. The measurement result of R2point was about 6‒7% from Rnw value. Since the power dissipation at the contacts is much smaller than the minimum power for 3ω signal, the resultant contact resistance supposedly does not affect the third harmonic signal. Furthermore, the contact metal pads act as large thermal reservoirs where the temperature is kept constant at the initial temperature during

spin coated with photoresist, (c) photoresist can be exposed to a pattern by ultraviolet light, (d) soluble photoresist will be developed by the developer, (e) the Cr/Au electrodes are deposited by thermal evaporator, (f) lift-off the photoresist

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline…

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153

membrane, (b) substrate is

**Figure 2.** Schematic of depositing outer electrodes of the template: (a) silicon wafer with Si3

by acetone, and (g) measurement platform is ready to be used.

In addition, the prepared nanowire exhibited a linear current–voltage (I-V) curve wherein indicated the Ohmic contact response; the current range within 0 to 100 µA indicating the resistivity of NW follows the Ohm's law. An Ohmic contact is an electrical junction between two conductors which has a linear current–voltage (I-V) curve following the Ohm's law. Low resistance Ohmic contacts are applied to facilitate the flow of charge in both directions between the two conductors, without blocking from the excess power dissipation due to voltage thresholds. The contact quality has a contribution to both an electrical and a thermal effect, such as, if the electrical contact resistance is too high, the third harmonics measurement will be influenced by the heat dissipation occurred at the contacts due to Joule heating. Furthermore, incorrect selection of a working frequency will develop an error experiment results. Therefore, the AC impedance measurements with no electrical artifacts involved were

the experiment.

utilized to choose a correct working frequency.

The flow chart of the suspending process of a nanowire on the measurement platform (microchip) was shown in **Figure 3**. Resistance temperature detectors (RTDs) and current leads were fabricated on the primary measurement platform by electron beam lithography [E-Beam Writer System, Model: Elionix ELS-7000 (100 keV)]. The Si3 N4 membrane was removed by inductively coupled plasma etching system (ICP, Model: Elionix EIS-700) to open the window. After that, the single nanowire from the PbTe thin film was picked up by a tungsten needle (dw = 100 nm) under a binocular optical microscope and placed across on two RTDs of a Si3 N4 microchip, where both ends of the nanowire attached to the current leads. In order to improve thermal and electrical contacts between the nanowire and the contact pads, the contacts of the nanowire on thermometers and current leads were covered with a thin layer of Platinum (Pt) using a focused ion beam (FIB) [DBFIB-SEM, FEI NOVA-600].

To examine how good the prepared contact, the contact resistance of all four points (point 1, 2, 3, and 4) as depicted in **Figure 4** were measured. First, the known direct current (DC) was applied between contacts 1 and 4 (4 probes) and then the voltage drop across contact 2 and 3 (2 probes, see **Figure 4**) was measured. The total resistance of two-point probe configuration is expressed as R2point = Rlead + 2Rcontact + Rnw. Wherein, R2point, Rlead, Rcontact, and Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline… http://dx.doi.org/10.5772/intechopen.76635 153

**Figure 2.** Schematic of depositing outer electrodes of the template: (a) silicon wafer with Si3 N4 membrane, (b) substrate is spin coated with photoresist, (c) photoresist can be exposed to a pattern by ultraviolet light, (d) soluble photoresist will be developed by the developer, (e) the Cr/Au electrodes are deposited by thermal evaporator, (f) lift-off the photoresist by acetone, and (g) measurement platform is ready to be used.

The standard photolithography processes were used to define the contact pads of the mea-

is spin coated with photoresist, (c) Photoresist is exposed to a rectangular pattern with ultraviolet light, (d) soluble

N4

membrane template preparation: (a) the silicon wafer with Si3

photoresist material by spin coating, followed by exposure, evaporation, and lift-off process.

The flow chart of the suspending process of a nanowire on the measurement platform (microchip) was shown in **Figure 3**. Resistance temperature detectors (RTDs) and current leads were fabricated on the primary measurement platform by electron beam lithography [E-Beam

inductively coupled plasma etching system (ICP, Model: Elionix EIS-700) to open the window. After that, the single nanowire from the PbTe thin film was picked up by a tungsten needle (dw = 100 nm) under a binocular optical microscope and placed across on two RTDs

to improve thermal and electrical contacts between the nanowire and the contact pads, the contacts of the nanowire on thermometers and current leads were covered with a thin layer of

To examine how good the prepared contact, the contact resistance of all four points (point 1, 2, 3, and 4) as depicted in **Figure 4** were measured. First, the known direct current (DC) was applied between contacts 1 and 4 (4 probes) and then the voltage drop across contact 2 and 3 (2 probes, see **Figure 4**) was measured. The total resistance of two-point probe configuration is expressed as R2point = Rlead + 2Rcontact + Rnw. Wherein, R2point, Rlead, Rcontact, and

microchip, where both ends of the nanowire attached to the current leads. In order

N4

N4

layer by reactive ion etching system (RIE), (f) dip the

N4

membrane is covered with

on the both sides, (b) substrate

N4 membrane, and (g) strip the

membrane was removed by

surement platform. As shown in **Figure 2**, first, Si wafer with Si3

wafer into a bath of KOH for wet etching to create a cavity and leave a suspended Si3

Writer System, Model: Elionix ELS-7000 (100 keV)]. The Si3

of a Si3 N4

**Figure 1.** Schematic Si3

photoresist.

N4

152 Impact of Thermal Conductivity on Energy Technologies

photoresist can be developed by the developer, (e) removing Si3

The developed primary measurement platform is then ready to be used.

Platinum (Pt) using a focused ion beam (FIB) [DBFIB-SEM, FEI NOVA-600].

Rnw represents resistance between point 2 and 3, resistance of Cr/Au electrodes, resistance of contacted NW to electrode, and NW resistance, respectively. The resistance of R2point and Rnw were measured by two and four-point probes. Whilst, in order to obtain the contact resistance as Rcont = (R2point - Rnw)/2, Rlead (~10 Ω) was neglected. The measurement result of R2point was about 6‒7% from Rnw value. Since the power dissipation at the contacts is much smaller than the minimum power for 3ω signal, the resultant contact resistance supposedly does not affect the third harmonic signal. Furthermore, the contact metal pads act as large thermal reservoirs where the temperature is kept constant at the initial temperature during the experiment.

In addition, the prepared nanowire exhibited a linear current–voltage (I-V) curve wherein indicated the Ohmic contact response; the current range within 0 to 100 µA indicating the resistivity of NW follows the Ohm's law. An Ohmic contact is an electrical junction between two conductors which has a linear current–voltage (I-V) curve following the Ohm's law. Low resistance Ohmic contacts are applied to facilitate the flow of charge in both directions between the two conductors, without blocking from the excess power dissipation due to voltage thresholds. The contact quality has a contribution to both an electrical and a thermal effect, such as, if the electrical contact resistance is too high, the third harmonics measurement will be influenced by the heat dissipation occurred at the contacts due to Joule heating. Furthermore, incorrect selection of a working frequency will develop an error experiment results. Therefore, the AC impedance measurements with no electrical artifacts involved were utilized to choose a correct working frequency.

**Figure 3.** Flow charts of the suspending process of a nanowire on the measurement platform.

### **2.3. Structural characterization of PbTe NWs**

The size and crystalline structure of PbTe NWs were characterized by scanning electron microscope (SEM) and transmission electron microscope (TEM). The SEM image of grew PbTe NWs on the substrate (**Figure 5a**) reveals that the length and the diameter of NWs were ranging from 5 to 70 µm and 50 to 300 nm, respectively. The NWs with length about 70 µm was picked up, placed, and contacted on the microchip as shown in **Figure 5b**. The prepared contacts have a resistance about 425–430 Ω, wherein showed nearly Ohmic contacts. The microchip consisting of PbTe NW displayed in **Figure 5b** was used for complete structural analysis and TE properties measurements. The representative TEM image showed that the employed PbTe NWs has a diameter of 75 nm (**Figure 5c**) and this size was further convinced by the scanning transmission electron microscope (STEM) result (**Figure 6a**). In addition, the TEM image and a corresponding selected-area electron diffraction (SAED) pattern (inset of the top left of **Figure 5c**) revealed

**Figure 5.** (a) SEM image of PbTe NWs grew on the surface of the PbTe thin film, (b) SEM images of a PbTe NW suspended

microchip, (c) low-magnification TEM images of a PbTe NW, inset of the top left of figure: the SAED pattern (at the [001]

zone axis), inset of the bottom right of figure: a high-resolution TEM image of a PbTe NW.

template, inset: the Pt/C thermal contact between the PbTe NW and 10-nm Cr/50-nm Au electrodes on a Si3

N4

**Figure 4.** A schematic setup of four-point probe for measurement of electrical resistance (*R*), Seebeck coefficient (*S*) and

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the thermal conductivity (*κ*) by 3ω method.

on a Si3 N4 Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline… http://dx.doi.org/10.5772/intechopen.76635 155

**Figure 4.** A schematic setup of four-point probe for measurement of electrical resistance (*R*), Seebeck coefficient (*S*) and the thermal conductivity (*κ*) by 3ω method.

**Figure 5.** (a) SEM image of PbTe NWs grew on the surface of the PbTe thin film, (b) SEM images of a PbTe NW suspended on a Si3 N4 template, inset: the Pt/C thermal contact between the PbTe NW and 10-nm Cr/50-nm Au electrodes on a Si3 N4 microchip, (c) low-magnification TEM images of a PbTe NW, inset of the top left of figure: the SAED pattern (at the [001] zone axis), inset of the bottom right of figure: a high-resolution TEM image of a PbTe NW.

**2.3. Structural characterization of PbTe NWs**

154 Impact of Thermal Conductivity on Energy Technologies

The size and crystalline structure of PbTe NWs were characterized by scanning electron microscope (SEM) and transmission electron microscope (TEM). The SEM image of grew PbTe NWs on the substrate (**Figure 5a**) reveals that the length and the diameter of NWs were ranging from 5 to 70 µm and 50 to 300 nm, respectively. The NWs with length about 70 µm was picked up, placed, and contacted on the microchip as shown in **Figure 5b**. The prepared contacts have a

**Figure 3.** Flow charts of the suspending process of a nanowire on the measurement platform.

resistance about 425–430 Ω, wherein showed nearly Ohmic contacts. The microchip consisting of PbTe NW displayed in **Figure 5b** was used for complete structural analysis and TE properties measurements. The representative TEM image showed that the employed PbTe NWs has a diameter of 75 nm (**Figure 5c**) and this size was further convinced by the scanning transmission electron microscope (STEM) result (**Figure 6a**). In addition, the TEM image and a corresponding selected-area electron diffraction (SAED) pattern (inset of the top left of **Figure 5c**) revealed

**Figure 6.** (a) STEM images of a PbTe NW. The line profiles show that the Pb (blue line) and Te (red line) are homogeneously distributed throughout the NW, (b) and (c) elemental mapping showing the uniform distribution of Pb and Te along the NW, respectively.

from NW to the microchip substrate that affects the S and ρ measurement was assured zero. Furthermore, in order to eliminate convectional heat loss, all measurements were carried out

**Figure 7.** Temperature dependence of (a) electrical resistivity, (b) Seebeck coefficient, and (c) power factor of a 75 nm

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Four-point probe method was applied for ρ measurement. In this work, an AC current flowed to the NW via electrode 1. The voltage (V) and current (I) difference between electrode 1 and 2 was measured through the voltage leads. A pair of current leads into the NW to determine the root mean square of the voltage difference of a pair of voltage leads. By substituting the obtained V and I value to the V = I•R formula, the resistance (R) value is obtained. We get the R of NW, the ρ value could be attained by applying the formula of ρ = R•A / L where A and L are cross-section area of the wire and length between a pair of voltage leads, respectively. **Figure 7a** showed the measurement results of NW resistivity ρ at temperatures range 300–350 K. This figure demonstrated that the resistivity ρ of a 75-nm PbTe NW was temperature dependence which indicated a semiconducting behavior. The resistivity of a PbTe NW at near room temperature was 6.55 × 10−4 Ω m, which 43 times greater than the bulk counterpart (1.52 × 10−5 Ω m) [14]. This is probably due to the surface

For the Seebeck effect measurement, the characterization was based on the voltage and temperature difference generated between electrodes 3 and 4 (**Figure 5b**). To generate the temperature gradient between those two electrodes, the heater with frequency 1ω and magnitude equals to *I•Sin(ωt)* was applied at one end of the NW. In this experiment, the sample was employed as the sensor of the thermometer as well, thus, temperature coefficient of electrical resistance of them are needed to be calibrated at first. By applying a DC current to the sensor and measuring the change of the voltage difference at frequency 2ω between the two ends of the sensor, the resistance change of the sensor would be known. After obtaining the temperature coefficient of electrical resistance, the temperature gradient created between two ends of the NW would be gained, because the observed heat is proportional to the square of the current multiplied by the

that the frequency of 2ω was applied to heat the heater. As the heater is heated at frequency 2ω, the sample temperature and sensor resistance would fluctuated at frequency 2ω as well. By

(ωt)•R, where Q and R denote the observed heat and

α = [(1-cos2α)/2], it means

•sin2

the electrical resistance of the NW, respectively. Mathematically, sin2

in a high vacuum of at least lower than 2×10−6 Torr.

scattering of charge carriers [30].

PbTe NW.

electrical resistance of the NW, Q ∝ I2

that the PbTe NWs were high-quality single crystals with a growth along the [100] direction. While, the lattice fringes of the smooth PbTe were separated by 0.33 nm (inset of the bottom right of **Figure 5c**). This is consistent with a periodicity along the [200] direction with lattice constants of approximately 6.549 Å, which are approximately 1.2% higher than the bulk counterpart (a = 6.47 Å).

The chemical composition of the PbTe NWs was studied by using energy dispersive X-ray spectroscopy (EDS). The EDS line scan profile, shown in **Figure 6a**, revealed the uniform spatial distribution of the Pb and Te elements throughout the NW. This was further confirmed by using a STEM to map elements across the NW [**Figure 6b** and **c**]. The EDS point scanning experiments of the NWs quantitatively confirmed that Pb and Te are present in an average atomic ratio of 54.19 and 45.81% (Table in **Figure 6**), respectively. In addition, the EDS data also revealed that the atomic ratio of Pb/Te ≈ 1.18, with no impurities. As a result, the stoichiometric composition of the individual NW was Pb1.08Te0.92.

### **3. Thermoelectric properties**

As mentioned before that the microchip with a rectangular window which shown in **Figure 5b** was employed to measure the TE properties of NW, i.e. electrical resistivity (ρ) and Seebeck coefficient (S). In this experiment, the PbTe NW was placed across two resistance thermometers, hot side thermometer Th (a gold wire parallel line to the heater between contact electrodes 7 and 8 which is marked as red arrows in **Figure 5b**) and cold side thermometer Tc (a gold wire between the contact electrode 9 and 10, **Figure 5b**) with both ends of the NW attach to the current leads (electrode 1 and 2, **Figure 5b**). To measure the V1ω and V3ω signal, electrodes 3 and 4 (voltage leads) were connected to lock-in amplifier. The thermal conduction

**Figure 7.** Temperature dependence of (a) electrical resistivity, (b) Seebeck coefficient, and (c) power factor of a 75 nm PbTe NW.

from NW to the microchip substrate that affects the S and ρ measurement was assured zero. Furthermore, in order to eliminate convectional heat loss, all measurements were carried out in a high vacuum of at least lower than 2×10−6 Torr.

that the PbTe NWs were high-quality single crystals with a growth along the [100] direction. While, the lattice fringes of the smooth PbTe were separated by 0.33 nm (inset of the bottom right of **Figure 5c**). This is consistent with a periodicity along the [200] direction with lattice constants of approximately 6.549 Å, which are approximately 1.2% higher than the bulk coun-

**Figure 6.** (a) STEM images of a PbTe NW. The line profiles show that the Pb (blue line) and Te (red line) are homogeneously distributed throughout the NW, (b) and (c) elemental mapping showing the uniform distribution of Pb and Te along the

The chemical composition of the PbTe NWs was studied by using energy dispersive X-ray spectroscopy (EDS). The EDS line scan profile, shown in **Figure 6a**, revealed the uniform spatial distribution of the Pb and Te elements throughout the NW. This was further confirmed by using a STEM to map elements across the NW [**Figure 6b** and **c**]. The EDS point scanning experiments of the NWs quantitatively confirmed that Pb and Te are present in an average atomic ratio of 54.19 and 45.81% (Table in **Figure 6**), respectively. In addition, the EDS data also revealed that the atomic ratio of Pb/Te ≈ 1.18, with no impurities. As a result, the stoichiometric composition of the individual NW was

As mentioned before that the microchip with a rectangular window which shown in **Figure 5b** was employed to measure the TE properties of NW, i.e. electrical resistivity (ρ) and Seebeck coefficient (S). In this experiment, the PbTe NW was placed across two resistance thermometers, hot side thermometer Th (a gold wire parallel line to the heater between contact electrodes 7 and 8 which is marked as red arrows in **Figure 5b**) and cold side thermometer Tc (a gold wire between the contact electrode 9 and 10, **Figure 5b**) with both ends of the NW attach to the current leads (electrode 1 and 2, **Figure 5b**). To measure the V1ω and V3ω signal, electrodes 3 and 4 (voltage leads) were connected to lock-in amplifier. The thermal conduction

terpart (a = 6.47 Å).

NW, respectively.

156 Impact of Thermal Conductivity on Energy Technologies

Pb1.08Te0.92.

**3. Thermoelectric properties**

Four-point probe method was applied for ρ measurement. In this work, an AC current flowed to the NW via electrode 1. The voltage (V) and current (I) difference between electrode 1 and 2 was measured through the voltage leads. A pair of current leads into the NW to determine the root mean square of the voltage difference of a pair of voltage leads. By substituting the obtained V and I value to the V = I•R formula, the resistance (R) value is obtained. We get the R of NW, the ρ value could be attained by applying the formula of ρ = R•A / L where A and L are cross-section area of the wire and length between a pair of voltage leads, respectively. **Figure 7a** showed the measurement results of NW resistivity ρ at temperatures range 300–350 K. This figure demonstrated that the resistivity ρ of a 75-nm PbTe NW was temperature dependence which indicated a semiconducting behavior. The resistivity of a PbTe NW at near room temperature was 6.55 × 10−4 Ω m, which 43 times greater than the bulk counterpart (1.52 × 10−5 Ω m) [14]. This is probably due to the surface scattering of charge carriers [30].

For the Seebeck effect measurement, the characterization was based on the voltage and temperature difference generated between electrodes 3 and 4 (**Figure 5b**). To generate the temperature gradient between those two electrodes, the heater with frequency 1ω and magnitude equals to *I•Sin(ωt)* was applied at one end of the NW. In this experiment, the sample was employed as the sensor of the thermometer as well, thus, temperature coefficient of electrical resistance of them are needed to be calibrated at first. By applying a DC current to the sensor and measuring the change of the voltage difference at frequency 2ω between the two ends of the sensor, the resistance change of the sensor would be known. After obtaining the temperature coefficient of electrical resistance, the temperature gradient created between two ends of the NW would be gained, because the observed heat is proportional to the square of the current multiplied by the electrical resistance of the NW, Q ∝ I2 •sin2 (ωt)•R, where Q and R denote the observed heat and the electrical resistance of the NW, respectively. Mathematically, sin2 α = [(1-cos2α)/2], it means that the frequency of 2ω was applied to heat the heater. As the heater is heated at frequency 2ω, the sample temperature and sensor resistance would fluctuated at frequency 2ω as well. By knowing the temperature gradient and also measuring the thermoelectric voltage of two ends of the sample, S can be calculated by implementing the formula: S = (△V) / (△T).

The experiment results of Seebeck effect of a 75-nm PbTe NW shown in **Figure 7b** revealed that the S value was temperature dependence. The S value increased with increasing the temperature. In addition, as depicted in **Figure 7b** that the S of PbTe NWs at various temperature measurements has a negative sign which indicated for n-type semiconductor material. Those negative sign appears because their electrons have a much higher µ than holes and dominate the electronic transport properties [31, 32]. At temperature of 300 K, the S value for the 75-nm NW was −54.76 µV K−1, which is about 69% lower compare to the bulk counterpart [14]. This result may due to the consequences of structural imperfections, such as antisite defects inside PbTe NW (i.e., the creation of one vacancy at the tellurium site) [14, 33].

Theoretically, for almost all materials, the trend of ρ was closely correlated with the S. It was consistent with our experimental resultts (**Figure 7a** and **b**) which showed that the smaller ρ had a higher S value. Conversely, due to s = 1/ρ, thus s value increases when S is increased. The PF (S2 σ) calculation results of PbTe NWs as a function of temperature are plotted in **Figure 7c**. The PF increased gradually when the temperature increased, and this result was mainly due to the influence of ρ trends. At the temperature about 300 K the S2 σ values were 4.58 µW m−1 K−2.

The thermal conductivity of the 75 nm PbTe nanowire was measured by the self-heating 3ω method [35] in the temperature range of 300–350 K. In the experiment, the V3ω as a function of frequency will be attained. As shown in **Figure 9a**, the V3ω was dependent on the frequency. The V3ω reduce significantly by increasing the frequency, however, the reduction become slightly at a frequency above 10 Hz. The relation between thermal conductivity, V3ω and fre-

**Figure 9.** (a) Frequency dependence of V3ω at 300–350 K for 75 nm PbTe NW and (b) the solid line is predicted relation

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0 3 *R R*′ *L* \_\_\_\_\_\_\_\_\_\_\_

\_\_\_\_\_\_\_\_

1 + (2)2 (1)

1 + *Gx*2 (2)

κS) (ωγ →0). To further validate the extraction

ω versus I0

follows the I0

3 , as

*π*4 *S* √

√ \_\_\_\_\_\_

where *I* and *ω* denote the amplitude and frequency of the alternating current applied on nanowire, R and R' are the resistance and derivative of resistance at corresponding temperature, κ is the thermal conductivity, *S* is the cross section, and γ is the characteristic thermal

**Figure 9b** shows the fitting result of V3ω to frequency of a 75 nm PbTe NW at 300 K. The thermal conductivity κ of the nanowire can be derived from the intercept of the fitting value

of thermal conductivity by 3ω method, the variation of 3ω signals toward the input current

By substituting all acquired data from self-heating 3ω experiment to Eq. (1), the thermal conductivity, κ of a 75 nm PbTe NW at range temperature of 300–350 K were 0.96–0.72 Wm−1 K−1 (**Figure 11a** and **b**), which is approximately 2.40–3.19 times lower than the bulk counterpart (κ = 2.3 W m−1 K−1). Likewise, the κ value at room temperature was 0.96 W m−1 K−1, which is

3

LRR')/(π4

and frequency was studied. The result shows that V3

exhibited in the equation of **Figure 10**, which mean it agree with the Eq. (1).

quency is described in the Eq. (1).

at a certain temperature V3ω = (4*I*0

which simplified as:

V3ω α 1/√(1 + (2ωγ)2

time constant.

amplitude I0

*V*3*ω* = 4 *I*

) at 300 K for a 75 nm PbTe NW.

*y* = \_\_\_\_\_ *K*

Comparing the S value as a function of carrier density of various PbTe bulk obtained by Harman et al., **Figure 8** show that the S was closely correlated with the carrier concentration (n), Our results was consistent with the other previous experiments [14, 34] in which a smaller n had a higher S value.

**Figure 8.** Absolute value of the thermoelectric power or Seebeck coefficient (S) of various PbTe samples [14] as a function of carrier density; electrons (n) or holes (h) at room temperature. The solid red stars denote the Pb1.08Te0.92 samples with 75, 142, and 217 nm diameter wires [27, 34].

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline… http://dx.doi.org/10.5772/intechopen.76635 159

**Figure 9.** (a) Frequency dependence of V3ω at 300–350 K for 75 nm PbTe NW and (b) the solid line is predicted relation V3ω α 1/√(1 + (2ωγ)2 ) at 300 K for a 75 nm PbTe NW.

The thermal conductivity of the 75 nm PbTe nanowire was measured by the self-heating 3ω method [35] in the temperature range of 300–350 K. In the experiment, the V3ω as a function of frequency will be attained. As shown in **Figure 9a**, the V3ω was dependent on the frequency. The V3ω reduce significantly by increasing the frequency, however, the reduction become slightly at a frequency above 10 Hz. The relation between thermal conductivity, V3ω and frequency is described in the Eq. (1).

$$V\_{3\omega} = \frac{4\,I\_0^3 R \,R' L}{\pi^4 \,\text{s} \,\text{S} \sqrt{1 + (2\alpha \eta)^2}}\tag{1}$$

which simplified as:

knowing the temperature gradient and also measuring the thermoelectric voltage of two ends

The experiment results of Seebeck effect of a 75-nm PbTe NW shown in **Figure 7b** revealed that the S value was temperature dependence. The S value increased with increasing the temperature. In addition, as depicted in **Figure 7b** that the S of PbTe NWs at various temperature measurements has a negative sign which indicated for n-type semiconductor material. Those negative sign appears because their electrons have a much higher µ than holes and dominate the electronic transport properties [31, 32]. At temperature of 300 K, the S value for the 75-nm NW was −54.76 µV K−1, which is about 69% lower compare to the bulk counterpart [14]. This result may due to the consequences of structural imperfections, such as antisite defects inside

Theoretically, for almost all materials, the trend of ρ was closely correlated with the S. It was consistent with our experimental resultts (**Figure 7a** and **b**) which showed that the smaller ρ had a higher S value. Conversely, due to s = 1/ρ, thus s value increases when S is increased.

**Figure 7c**. The PF increased gradually when the temperature increased, and this result was

Comparing the S value as a function of carrier density of various PbTe bulk obtained by Harman et al., **Figure 8** show that the S was closely correlated with the carrier concentration (n), Our results was consistent with the other previous experiments [14, 34] in which a smaller

**Figure 8.** Absolute value of the thermoelectric power or Seebeck coefficient (S) of various PbTe samples [14] as a function of carrier density; electrons (n) or holes (h) at room temperature. The solid red stars denote the Pb1.08Te0.92 samples with

σ) calculation results of PbTe NWs as a function of temperature are plotted in

σ values were

of the sample, S can be calculated by implementing the formula: S = (△V) / (△T).

158 Impact of Thermal Conductivity on Energy Technologies

PbTe NW (i.e., the creation of one vacancy at the tellurium site) [14, 33].

mainly due to the influence of ρ trends. At the temperature about 300 K the S2

The PF (S2

4.58 µW m−1 K−2.

n had a higher S value.

75, 142, and 217 nm diameter wires [27, 34].

$$y = \frac{K}{\sqrt{1 + Gx^2}} \tag{2}$$

where *I* and *ω* denote the amplitude and frequency of the alternating current applied on nanowire, R and R' are the resistance and derivative of resistance at corresponding temperature, κ is the thermal conductivity, *S* is the cross section, and γ is the characteristic thermal time constant.

**Figure 9b** shows the fitting result of V3ω to frequency of a 75 nm PbTe NW at 300 K. The thermal conductivity κ of the nanowire can be derived from the intercept of the fitting value at a certain temperature V3ω = (4*I*0 3 LRR')/(π4 κS) (ωγ →0). To further validate the extraction of thermal conductivity by 3ω method, the variation of 3ω signals toward the input current amplitude I0 and frequency was studied. The result shows that V3 ω versus I0 follows the I0 3 , as exhibited in the equation of **Figure 10**, which mean it agree with the Eq. (1).

By substituting all acquired data from self-heating 3ω experiment to Eq. (1), the thermal conductivity, κ of a 75 nm PbTe NW at range temperature of 300–350 K were 0.96–0.72 Wm−1 K−1 (**Figure 11a** and **b**), which is approximately 2.40–3.19 times lower than the bulk counterpart (κ = 2.3 W m−1 K−1). Likewise, the κ value at room temperature was 0.96 W m−1 K−1, which is

**Figure 10.** The 3rd harmonic voltage signal V3ω as function of the extraction current amplitude I0 for a 75 nm PbTe NW at 300 K. Red solid line represents the cubic relationship of V3ω and I0 .

approximately 58% lower than the typical reported value of κ = 2.3 W m−1 K−1 for bulk PbTe. This decline mainly due to the carrier concentration difference or size effect. [36]

For the purpose of calculating the κl value, the electron thermal conductivity κe need to be determined. The κe value is calculated by using Eq. (3) (Wiederman-Franz law, where the Lorenz number L = 2.44 × 10−8 W.Ω.K−2), while the κl is gained from subtracting the κ with κe value. The values of κl of a 75 nm PbTe NW at 300 K was 0.95 W m−1 K−1, which is 57% lower than the PbTe bulk (κl = 2.2 W m−1 K−1). [16] As reference, the lattice contribution (κl ) of superlattice thin films PbSe0.98Te0.02/PbTe at room temperature was 0.35 W m−1 K−1 [28].

$$\kappa\_o = \text{L.or.T} \tag{3}$$

436 nm) [19] including our recent work (d = 75 nm) which were plotted in **Figure 11b**, the κ value decreases gradually as its diameter shrinks. The enhanced phonon boundary scattering has a considerable effect in reducing the κ value of NW. Hence, it is suggested to have an effect on suppressing the phonon transport through the NWs as well [40, 41]. Theoretically, those phenomena would cause the *ZT* value of NW higher than the bulk counterpart, provided that the electronic properties were not degraded by the nanostructure. However, based upon all the above measurement results, i.e. S (−54.76 – ‒68.80 µV K−1) [27], σ (1526.19–1878.68 S m−1) [27] and κ (0.96–0.72 W m−1 K−1), the obtained *ZT* value of a 75 nm PbTe NW at 300–350 K are in the range of ∼1.4‒4.3 x 10−3 (**Figure 12**) and it is still much lower than the ZT of PbTe bulk which is approximately ∼ 0.25 at 300 K

( ) of a 75 nm PbTe NW and (b) size-dependent κ properties of individual 75 nm PbTe NW (compared

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline…

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161

( ) and lattice thermal

**Figure 11.** (a) Measured thermal conductivity κ( ), electron thermal conductivity κe

and maximal ~ 0.8 at 700 K [42].

to the reported 182 nm, 277 nm and 436 nm [19].

conductivity κl

The lattice thermal conductivity has a lower limit wherein the phonon mean free path becomes comparable to the lattice spacing of the atoms [37]. Alternatively, the lattice thermal conductivity of a material can be determined by using Eq. (4), where C, ν, andι represent the heat capacity per unit volume, the speed of sound in the material, and the mean free path of the phonons, respectively. By applying the atom spacing of PbTe as the minimum ι and substituting the values of the ν and C to Eq. (4), the κl of PbTe is around 0.2 W m−1 K−1. Repeating this calculation for a variety of compounds, the lowest possible value of κl are in the range of 0.1–0.2 W m−1 K−1 [38].

$$\kappa\_{\text{l}} = 1/3.\text{C.v.u} \tag{4}$$

As the size of the nanowire approaches the ι (the median phonon free path in PbTe is about 42 nm) [39], the κ value will drop due to the increased phonon scattering. According to the reported κ of individual PbTe NW with various diameters (d = 182, 277, and

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline… http://dx.doi.org/10.5772/intechopen.76635 161

approximately 58% lower than the typical reported value of κ = 2.3 W m−1 K−1 for bulk PbTe.

.

κe = L.σ.T (3)

The lattice thermal conductivity has a lower limit wherein the phonon mean free path becomes comparable to the lattice spacing of the atoms [37]. Alternatively, the lattice thermal conductivity of a material can be determined by using Eq. (4), where C, ν, andι represent the heat capacity per unit volume, the speed of sound in the material, and the mean free path of the phonons, respectively. By applying the atom spacing of PbTe as the minimum ι and sub-

κl = 1/3.C.ν.ι (4)

As the size of the nanowire approaches the ι (the median phonon free path in PbTe is about 42 nm) [39], the κ value will drop due to the increased phonon scattering. According to the reported κ of individual PbTe NW with various diameters (d = 182, 277, and

value is calculated by using Eq. (3) (Wiederman-Franz law, where the

of a 75 nm PbTe NW at 300 K was 0.95 W m−1 K−1, which is 57% lower

need to be

for a 75 nm PbTe NW at

) of super-

are in the range of

is gained from subtracting the κ with κe

of PbTe is around 0.2 W m−1 K−1. Repeating

This decline mainly due to the carrier concentration difference or size effect. [36]

**Figure 10.** The 3rd harmonic voltage signal V3ω as function of the extraction current amplitude I0

For the purpose of calculating the κl value, the electron thermal conductivity κe

than the PbTe bulk (κl = 2.2 W m−1 K−1). [16] As reference, the lattice contribution (κl

lattice thin films PbSe0.98Te0.02/PbTe at room temperature was 0.35 W m−1 K−1 [28].

this calculation for a variety of compounds, the lowest possible value of κl

Lorenz number L = 2.44 × 10−8 W.Ω.K−2), while the κl

300 K. Red solid line represents the cubic relationship of V3ω and I0

160 Impact of Thermal Conductivity on Energy Technologies

stituting the values of the ν and C to Eq. (4), the κl

determined. The κe

value. The values of κl

0.1–0.2 W m−1 K−1 [38].

**Figure 11.** (a) Measured thermal conductivity κ( ), electron thermal conductivity κe ( ) and lattice thermal conductivity κl ( ) of a 75 nm PbTe NW and (b) size-dependent κ properties of individual 75 nm PbTe NW (compared to the reported 182 nm, 277 nm and 436 nm [19].

436 nm) [19] including our recent work (d = 75 nm) which were plotted in **Figure 11b**, the κ value decreases gradually as its diameter shrinks. The enhanced phonon boundary scattering has a considerable effect in reducing the κ value of NW. Hence, it is suggested to have an effect on suppressing the phonon transport through the NWs as well [40, 41]. Theoretically, those phenomena would cause the *ZT* value of NW higher than the bulk counterpart, provided that the electronic properties were not degraded by the nanostructure. However, based upon all the above measurement results, i.e. S (−54.76 – ‒68.80 µV K−1) [27], σ (1526.19–1878.68 S m−1) [27] and κ (0.96–0.72 W m−1 K−1), the obtained *ZT* value of a 75 nm PbTe NW at 300–350 K are in the range of ∼1.4‒4.3 x 10−3 (**Figure 12**) and it is still much lower than the ZT of PbTe bulk which is approximately ∼ 0.25 at 300 K and maximal ~ 0.8 at 700 K [42].

**Conflict of interest**

**Author details**

Bandung, Indonesia

**References**

Dedi1,3\*, Indah Primadona2

The authors declare no competing conflict of interests.

\*Address all correspondence to<EMAIL_ADDRESS>
3 Institute of Physics, Academia Sinica, Taipei, Taiwan

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10.1103/PhysRevB.62.4610

10.1039/C0JM02755C

, Ping-Chung Lee3

, Chi-Hua Chien3

Structural and Thermoelectric Properties Characterization of Individual Single-Crystalline…

1 Research Center for Electronics and Telecommunication, Indonesian Institute of Sciences,

2 Research Unit for Clean Technology, Indonesian Institute of Sciences, Bandung, Indonesia

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and Yang-Yuan Chen3

http://dx.doi.org/10.5772/intechopen.76635

163

**Figure 12.** Figure of merit *ZT* for a n-type 75 nm PbTe NW.

#### **4. Conclusions**

In summary, we have demonstrated a new technique for structural characterization and TE properties measurement of individual single-crystalline PbTe NW by using a novel design of microchip. In this work, the single PbTe NW grown by the stress-induced method was employed for four different type of characterization simultaneously: structural characterization, Seebeck coefficient, electrical conductivity, and thermal conductivity. The structural characterization revealed that the synthesized PbTe NWs were single crystals with a growth along the [100] direction. While the TE properties (S, σ, and κ) of a 75 nm single-crystalline PbTe NW at room temperature are −54.76 µV K−1, 1526.19 S m−1, and 0.96 W m−1 K−1, respectively. Based on those results, the experimental calculation of *ZT* value of its NW was ∼1.4‒4.3 x 10−3 at 300–350 K range. This technique provided high accuracy and reliable ZT value of individual single-crystalline TE NW. Thus, the sizedependent study of TE properties is very feasible. In this study, the κ value in growth [100] direction PbTe NW is size-dependent, wherein indicates that thermal transport through the individual PbTe nanowires is limited by boundary scattering of both electrons and phonons.

### **Acknowledgements**

Technical support was provided by the Core Facilities for Nanoscience and Nanotechnology at the Institute of Physics of the Academia Sinica in Taiwan. This study was funded by the National Science Council of Taiwan (grant NSC 100-2112-M-001-019-MY3).

### **Conflict of interest**

The authors declare no competing conflict of interests.

### **Author details**

Dedi1,3\*, Indah Primadona2 , Ping-Chung Lee3 , Chi-Hua Chien3 and Yang-Yuan Chen3

\*Address all correspondence to<EMAIL_ADDRESS>
1 Research Center for Electronics and Telecommunication, Indonesian Institute of Sciences, Bandung, Indonesia

2 Research Unit for Clean Technology, Indonesian Institute of Sciences, Bandung, Indonesia

3 Institute of Physics, Academia Sinica, Taipei, Taiwan

### **References**

**4. Conclusions**

**Figure 12.** Figure of merit *ZT* for a n-type 75 nm PbTe NW.

162 Impact of Thermal Conductivity on Energy Technologies

phonons.

**Acknowledgements**

In summary, we have demonstrated a new technique for structural characterization and TE properties measurement of individual single-crystalline PbTe NW by using a novel design of microchip. In this work, the single PbTe NW grown by the stress-induced method was employed for four different type of characterization simultaneously: structural characterization, Seebeck coefficient, electrical conductivity, and thermal conductivity. The structural characterization revealed that the synthesized PbTe NWs were single crystals with a growth along the [100] direction. While the TE properties (S, σ, and κ) of a 75 nm single-crystalline PbTe NW at room temperature are −54.76 µV K−1, 1526.19 S m−1, and 0.96 W m−1 K−1, respectively. Based on those results, the experimental calculation of *ZT* value of its NW was ∼1.4‒4.3 x 10−3 at 300–350 K range. This technique provided high accuracy and reliable ZT value of individual single-crystalline TE NW. Thus, the sizedependent study of TE properties is very feasible. In this study, the κ value in growth [100] direction PbTe NW is size-dependent, wherein indicates that thermal transport through the individual PbTe nanowires is limited by boundary scattering of both electrons and

Technical support was provided by the Core Facilities for Nanoscience and Nanotechnology at the Institute of Physics of the Academia Sinica in Taiwan. This study was funded by the

National Science Council of Taiwan (grant NSC 100-2112-M-001-019-MY3).

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DOI: 10.1038/am.2012.52

164 Impact of Thermal Conductivity on Energy Technologies

S0921-4526(02)01187-0

p. 441

[35] Lu L, Yi W, Zhang D.L. 3ω method for specific heat and thermal conductivity measurements. The Review of Scientific Instruments. 2001;**72**:2996-3003. DOI: 10.1063/1.1378340

**Chapter 10**

Provisional chapter

**Nonlinear Radiative Heat Transfer of Cu-Water**

Nonlinear Radiative Heat Transfer of Cu-Water

**the Influence of Particle Shape**

the Influence of Particle Shape

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74807

Abstract

unsteady rotating flow

1. Introduction

K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha

K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha

**Nanoparticles over an Unsteady Rotating Flow under**

DOI: 10.5772/intechopen.74807

A 3D study on Cu-water-rotating nanofluid over a permeable surface in the presence of nonlinear radiation is presented. Particle shape and thermophysical properties are considered in this study. The governing equations in partial forms are reduced to a system of nonlinear ordinary differential equations using suitable similarity transformations. An effective Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique is applied to attain the solution. The effects of flow parameters on the flow field and heat transfer characteristics were obtained and are tabulated. Useful discussions were carried out with the help of plotted graphs and tables. It is found that the rate of heat transfer is more enhanced in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles. Higher values of rotating parameter enhance the velocity profile and corresponding boundary layer thickness. It has quite the opposite behavior in angular velocity profile. Further, unsteady parameter increases

the velocity profile and corresponding boundary layer thickness.

Keywords: particle shape effect, nonlinear radiation, Cu-water nanoparticles,

The interaction of thermal radiation has increased greatly during the last decade due to its importance in many practical applications. We know that the radiation effect is important under many isothermal and nonisothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then radiation

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Nanoparticles over an Unsteady Rotating Flow under

#### **Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under the Influence of Particle Shape** Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under the Influence of Particle Shape

DOI: 10.5772/intechopen.74807

K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha K. Ganesh Kumar, B.J. Gireesha and S. Manjunatha

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74807

#### Abstract

[35] Lu L, Yi W, Zhang D.L. 3ω method for specific heat and thermal conductivity measurements. The Review of Scientific Instruments. 2001;**72**:2996-3003. DOI: 10.1063/1.1378340

[36] Ioffe AF. Semiconductor Thermoelements and Thermoelectric Cooling. London: Info-

[37] Slack GA. The thermal conductivity of nonmetallic crystals. In: Ehreneich H, Seitz F, Turnbull D, editors. Solid State Physics. Vol. 34. New York: Academic Press; 1979. pp. 1-71.

[38] Goldsmid HJ. A New Upper Limit to the Thermoelectric Figure-of-Merit. In: Rowe DM, editor. Thermoelectrics Handbook: Macro to Nano. CRC Taylor & Francis; 2006.

[39] Dames C, Chen G. Thermal Conductivity of Nanostructured Thermoelectric Material. In: Rowe DM, editor. Thermoelectrics Handbook: Macro to Nano. CRC Taylor & Francis;

[40] Li D, Wu Y, Kim P, Yang P, Majumdar A. Thermal conductivity of individual silicon nanowires. Applied Physics Letters. 2003;**83**:2934-2936. DOI: 10.1063/1.1616981

[41] Abouelaoualim D. Size effects on nanowire phonon thermal conductivity: A numerical investigation using the Boltzmann equation. Acta Physica Polonica A. 2007;**112**:49-54 [42] Tritt TM, Subramanian MA. Thermoelectric materials, phenomena, and applications: A

bird's eye view. MRS Bulletin. 2006;**31**:188-198. DOI: 10.1557/mrs2006.44

search; 1957

p. 10.1-10.8

2006. p. 42.1-42.11

DOI: 10.1016/S0081-1947(08)60359-8

166 Impact of Thermal Conductivity on Energy Technologies

A 3D study on Cu-water-rotating nanofluid over a permeable surface in the presence of nonlinear radiation is presented. Particle shape and thermophysical properties are considered in this study. The governing equations in partial forms are reduced to a system of nonlinear ordinary differential equations using suitable similarity transformations. An effective Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique is applied to attain the solution. The effects of flow parameters on the flow field and heat transfer characteristics were obtained and are tabulated. Useful discussions were carried out with the help of plotted graphs and tables. It is found that the rate of heat transfer is more enhanced in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles. Higher values of rotating parameter enhance the velocity profile and corresponding boundary layer thickness. It has quite the opposite behavior in angular velocity profile. Further, unsteady parameter increases the velocity profile and corresponding boundary layer thickness.

Keywords: particle shape effect, nonlinear radiation, Cu-water nanoparticles, unsteady rotating flow

### 1. Introduction

The interaction of thermal radiation has increased greatly during the last decade due to its importance in many practical applications. We know that the radiation effect is important under many isothermal and nonisothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then radiation

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

could become important. The knowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with a sought characteristic. Magnetohydrodynamic 3D flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation has been examined by Hayat et al. [1]. Shehzad et al. [2] proposed the nonlinear thermal radiation in 3D flow of Jeffrey nanofluid. Refs. [3–11] are some of the works associated with stretching sheet problem of thermal radiation.

the surface velocity in x and y directions, respectively, and wwð Þ x; t is the wall mass flux velocity in the z-direction as represented in Figure 1. Under these conditions, the governing

Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under…

∂z � 2Ωv ¼ � 1

þ 2Ωv ¼ � 1

∂2 T ∂z2 þ

u ¼ uwð Þ¼ x; t v ¼ 0, w ¼ 0, T ¼ Tw at z ¼ 0,

where velocity components in x, y and z directions are u, v and w, constant angular velocity of the Nano fluid is Ω, dynamic viscosity of the Nano fluid is μnf , density of the nanofluid is rnf , thermal diffusivity of the nanofluid is αnf , T is temperature of nanofluid and wall temperature

r ∂p ∂x þ μnf rnf ∂2 u

r ∂p ∂y þ μnf rnf ∂2 v

> 1 rcp nf

u ! 0, v ! 0, w ¼ 0, T ! T∞ as z ! ∞ (6)

∂qr

∂z ¼ 0, (1)

http://dx.doi.org/10.5772/intechopen.74807

169

∂z2 , (2)

∂z2 , (3)

∂z2 , (4)

∂z : (5)

∂u ∂x þ ∂v ∂y þ ∂w

equations can be written as:

∂u ∂t þ u ∂u ∂x þ v ∂u ∂y þ w ∂u

∂v ∂t þ u ∂v ∂x þ v ∂v ∂y þ w ∂v ∂z

> ∂w ∂t þ u

∂T ∂t þ u

Boundary conditions for the problem are,

Figure 1.

Nanotechnology has been widely utilized in the industries since materials with the size of nanometers possess distinctive physical and chemical properties. Nanofluids are literally a homogeneous mixture of base fluid and the nanoparticles. Common base fluids embody water, organic liquids, oil and lubricants, biofluids, polymeric solution and other common liquids. Nanoparticles are created from totally different materials, like oxides, nitrides, carbide, ceramics metals, carbons in varied (e.g., diamond, graphite, carbon nanotubes, fullerene) and functionalized nanoparticles. Nanofluids have novel properties that are potentially helpful in several applications in heat transfer, as well as microelectronics, pharmaceutical processes, heat exchanger, hybrid-powered engines, domestic refrigerator, fuel cells, cooling/vehicle thermal management, nuclear reactor agent, in grinding, in space technology, ships and in boiler flue gas temperature reduction. Choi [12] was the first who composed the analysis on nanoparticles in 1995. Later, Maiga et al. [13] initiated the heat transfer enhancement by using nanofluids in forced convective flows. The laminar fluid flow which results from the stretching of a flat surface in a nanofluid has been investigated by Khan and Pop [14]. Recently, a number of researchers are concentrating on nanofluid with different geometries; see [15–19].

Experimental studies have shown that the thermal conductivity of nanofluids is determined by the parameters related to: nanoparticles, concentration, size, spherical and nonspherical shapes, agglomeration (fractal-like shapes), surface charge and thermal conductivity, base fluids (e.g., thermal conductivity and viscosity), nanofluids (e.g., temperature), the interfacial chemical/physical effect or interaction between the particles and base fluid and others. For more details, readers are referred to the studies [20–25].

Impact of nonlinear thermal radiation on 3D flow and heat transfer of Cu-water nanoliquid over unsteady rotating flow have been considered. The heat transfer characteristics are studied in the presence of different particle shapes, thermophysical properties and nonlinear thermal radiation. The principal equations of continuity, momentum, energy and mass equations are transferred into a set of nonlinear similarity equalities by applying the appropriate transformations. The condensed equalities are solved numerically, and the impacts of relevant parameters are discussed through plotted graphs and tables.

### 2. Mathematical formulation

An unsteady laminar flow over a permeable surface in a rotating nanofluid is considered in this study. The copper-water motion is 3D due to Coriolis force in the present problem. The Cartesian coordinates are x, y and z where the rotation of the nanofluid is at an angular velocity Ωð Þt about the z-axis, and time is denoted as t. Let uwð Þ¼ x; t bx 1�δt and vwð Þ x; t represent

the surface velocity in x and y directions, respectively, and wwð Þ x; t is the wall mass flux velocity in the z-direction as represented in Figure 1. Under these conditions, the governing equations can be written as:

$$
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0,\tag{1}
$$

$$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - 2 \overline{\Omega} v = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{\mu\_{\eta\zeta}}{\rho\_{\eta\zeta}} \frac{\partial^2 u}{\partial z^2} \tag{2}$$

$$\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + 2\overline{\Omega}v = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{\mu\_{nf}}{\rho\_{nf}} \frac{\partial^2 v}{\partial z^2} \tag{3}$$

$$\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial z} + \frac{\mu\_{\eta f}}{\rho\_{\eta f}} \frac{\partial^2 v}{\partial z^2},\tag{4}$$

$$
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = a\_{\eta \prime} \frac{\partial^2 T}{\partial z^2} + \frac{1}{\left(\rho c\_p\right)\_{\eta \prime}} \frac{\partial q\_r}{\partial z}.\tag{5}
$$

Boundary conditions for the problem are,

could become important. The knowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with a sought characteristic. Magnetohydrodynamic 3D flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation has been examined by Hayat et al. [1]. Shehzad et al. [2] proposed the nonlinear thermal radiation in 3D flow of Jeffrey nanofluid. Refs. [3–11] are some of the works associated with stretching sheet prob-

Nanotechnology has been widely utilized in the industries since materials with the size of nanometers possess distinctive physical and chemical properties. Nanofluids are literally a homogeneous mixture of base fluid and the nanoparticles. Common base fluids embody water, organic liquids, oil and lubricants, biofluids, polymeric solution and other common liquids. Nanoparticles are created from totally different materials, like oxides, nitrides, carbide, ceramics metals, carbons in varied (e.g., diamond, graphite, carbon nanotubes, fullerene) and functionalized nanoparticles. Nanofluids have novel properties that are potentially helpful in several applications in heat transfer, as well as microelectronics, pharmaceutical processes, heat exchanger, hybrid-powered engines, domestic refrigerator, fuel cells, cooling/vehicle thermal management, nuclear reactor agent, in grinding, in space technology, ships and in boiler flue gas temperature reduction. Choi [12] was the first who composed the analysis on nanoparticles in 1995. Later, Maiga et al. [13] initiated the heat transfer enhancement by using nanofluids in forced convective flows. The laminar fluid flow which results from the stretching of a flat surface in a nanofluid has been investigated by Khan and Pop [14]. Recently, a number

of researchers are concentrating on nanofluid with different geometries; see [15–19].

more details, readers are referred to the studies [20–25].

eters are discussed through plotted graphs and tables.

2. Mathematical formulation

Experimental studies have shown that the thermal conductivity of nanofluids is determined by the parameters related to: nanoparticles, concentration, size, spherical and nonspherical shapes, agglomeration (fractal-like shapes), surface charge and thermal conductivity, base fluids (e.g., thermal conductivity and viscosity), nanofluids (e.g., temperature), the interfacial chemical/physical effect or interaction between the particles and base fluid and others. For

Impact of nonlinear thermal radiation on 3D flow and heat transfer of Cu-water nanoliquid over unsteady rotating flow have been considered. The heat transfer characteristics are studied in the presence of different particle shapes, thermophysical properties and nonlinear thermal radiation. The principal equations of continuity, momentum, energy and mass equations are transferred into a set of nonlinear similarity equalities by applying the appropriate transformations. The condensed equalities are solved numerically, and the impacts of relevant param-

An unsteady laminar flow over a permeable surface in a rotating nanofluid is considered in this study. The copper-water motion is 3D due to Coriolis force in the present problem. The Cartesian coordinates are x, y and z where the rotation of the nanofluid is at an angular

1�δt and vwð Þ x; t represent

velocity Ωð Þt about the z-axis, and time is denoted as t. Let uwð Þ¼ x; t bx

lem of thermal radiation.

168 Impact of Thermal Conductivity on Energy Technologies

$$\begin{aligned} \mu = \mu\_w(\mathbf{x}, t) = \upsilon = 0, w = 0, T = T\_w &\text{ at } z = 0, \\ \mu \to 0, \upsilon \to 0, w = 0, T \to T\_\infty &\text{ as } z \to \infty \end{aligned} \tag{6}$$

where velocity components in x, y and z directions are u, v and w, constant angular velocity of the Nano fluid is Ω, dynamic viscosity of the Nano fluid is μnf , density of the nanofluid is rnf , thermal diffusivity of the nanofluid is αnf , T is temperature of nanofluid and wall temperature is Tw, T∞ denotes temperature outside the surface (Table 1).

Figure 1. Influence of Ω on f 0 ð Þ η .

The radiative heat flux expression in Eq. (5) is given by:

$$q\_r = -\frac{16\sigma^\*}{3k^\*} T\_\omega^3 \frac{\partial T}{\partial z} \tag{7}$$

Using Eqs. (2)–(6) and (10), we can have

1 1 � ϕ � �2:5 1 � ϕ þ ϕ rs

1 1 � ϕ � �2:5 1 � ϕ þ ϕ rs

ð Þ rcp f

fð Þ¼ 0 0, f 0

The transformed boundary conditions are as follows:

f 0

b is rotation rate, λ ¼ δ

Cfx ¼ τwx ru2

> ∂u ∂z � �

τwx ¼ μnf

Using Eqs. (16) and (17), we obtain

1 � ϕ � �2:5 f

where Ω ¼ ω

to η.

obtain:

ffiffiffiffiffiffiffiffi Rex p Cfx ¼ 1

where Rex ¼ uwx

parameter, Pr ¼ αnf

knf

kf 1 � ϕ þ ϕ ð Þ rcp s

rf

rf

� � � � 1 þ Rð Þ 1 þ ð Þ θw � 1 θ 3

000

00 � f 0 g � fg0

� f 02 � ff 0

� 2Ωg þ λ η

� 2Ωf þ λ η

θ

ð Þ! η 0, gð Þ! η 0, θ ηð Þ! 0 as η ! ∞ (14)

b is unsteadiness parameter R ¼ 16σ∗T3

υnf is Prandtl number and primes denote the differentiation with respect

wð Þ yt , Nux ¼ xqw

and qw ¼ �knf

ð Þ0 and Nux ffiffiffiffiffiffiffiffi

Rex p ¼ knf kf

kfð Þ Tw � T∞

∂T ∂z � �

h i � Pr λ η

Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under…

ð Þ¼ 0 1, gð Þ¼ 0 0, θð Þ¼ 0 1 at η ¼ 0

Rosca et al. [11] mentioned that the pressure term ð Þp can be integrated from Eq. (4); thus, we

The physical quantities of interest in this problem are the skin friction coefficients in x and y

∂w ∂z � rw2

p ¼ υr

directions, Cfx and Cfy as well as the local Nusselt number Nux which are defined as:

ru2

∂v ∂z � �

1 � ϕ � �2:5 g

z¼0

0

wð Þ xt , Cfy ¼ τwy

, τwy ¼ μnf

Surface shear stress τwx, τwy and surface heat flux qw are defined as:

z¼0

00ð Þ0 , ffiffiffiffiffiffiffiffi Rex p Cfy ¼ 1

υ is local Reynolds number.

2 f 00 þ f 0 h i � � ¼ 0, (11)

2 g 0 þ g h i � � ¼ 0, (12)

> 2 θ 0 � fθ

0 h i ¼ 0, (13)

http://dx.doi.org/10.5772/intechopen.74807

171

∞

2 þ c (15)

3kf k∗ is radiation

, (16)

þ qrz¼0 (17)

w � �θ

� �,

0 ð Þ0

(18)

� 1 þ Rθ3

� � � � f

� � � � g

where σ∗ and k∗ are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively, and in view to Eq. (7), Eq. (4) reduces to:

$$
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = \frac{\partial}{\partial y} \left[ \left( \alpha\_{\eta f} + \frac{16 \sigma^\* T\_{\text{ess}}^3}{3 \left( \rho c\_p \right)\_{\eta f} k^\*} \frac{\partial T}{\partial Z} \right) \right] \frac{\partial T}{\partial y}. \tag{8}
$$

Parameters μnf , rnf and αnf are interrelated with nanoparticle volume fraction; ϕ and can be defined as:

$$\begin{split} \rho\_{\textit{nf}} &= \rho\_{\textit{f}} \left( 1 - \phi + \phi \left( \frac{\rho\_{\textit{s}}}{\rho\_{\textit{f}}} \right) \right), \mu\_{\textit{nf}} = \frac{\mu\_{\textit{f}}}{\left( 1 - \phi \right)^{2.5}}, \alpha\_{\textit{nf}} = \frac{k\_{\textit{nf}}}{\left( \rho \mathbf{c}\_{\textit{p}} \right)\_{\textit{nf}}} \\ \left( \rho \mathbf{c}\_{\textit{p}} \right)\_{\textit{nf}} &= \left( \rho \mathbf{c}\_{\textit{p}} \right)\_{\textit{f}} \left( 1 - \phi + \phi \left( \frac{\left( \rho \mathbf{c}\_{\textit{p}} \right)\_{\textit{s}}}{\left( \rho \mathbf{c}\_{\textit{p}} \right)\_{\textit{f}}} \right) \right), \frac{k\_{\textit{nf}}}{k\_{\textit{f}}} = \frac{\left[ k\_{\textit{s}} + (m-1)k\_{\textit{f}} \right] - (m-1)\phi \left( k\_{\textit{f}} - k\_{\textit{s}} \right)}{\left[ k\_{\textit{s}} + (m-1)k\_{\textit{f}} \right] + \phi \left( k\_{\textit{f}} - k\_{\textit{s}} \right)} \end{split} \tag{9}$$

where volumetric heat capacity of the solid nanoparticles is rcp � � s , volumetric heat capacity of the base fluid is rcp � � f , volumetric heat capacity of the nanofluid is rcp � � nf , thermal conductivity of the nanofluid is knf , thermal conductivity of the base fluid is kf , thermal conductivity of the solid nanoparticles is ks, nanoparticle volume fraction is ϕ, density viscosity of the base fluid is rf and dynamic viscosity of the base fluid is μf (Table 2).

Now, we introduce similarity transformations:

$$u = \frac{b\mathbf{x}}{1 - \delta t} f'(\eta), \text{ } v = \frac{b\mathbf{x}}{1 - \delta t} g(\eta), \text{ } w = -\sqrt{\frac{b\mathbf{x}}{1 - \delta t}} f(\eta), \text{ } \eta = \sqrt{\frac{b\mathbf{x}}{\nu(1 - \delta t)}} z, \text{ } \theta(\eta) = \frac{T - T\_m}{T\_w - T\_m} \tag{10}$$

with T ¼ T∞ð Þ 1 þ ð Þ θw � 1 θ and θw ¼ Tw T∞ , θw > 1 is the temperature ratio parameter.

Table 1. Thermophysical properties of water and nanoparticles.

Table 2. Values of the empirical shape factor for different particle shapes.

Using Eqs. (2)–(6) and (10), we can have

The radiative heat flux expression in Eq. (5) is given by:

respectively, and in view to Eq. (7), Eq.  Thermophysical properties of water and nanoparticles.

Table 2. Values of the empirical shape factor for different particle shapes.

ð Þ η , v ¼ bx

with T ¼ T∞ð Þ 1 þ ð Þ θw � 1 θ and θw ¼ Tw

nf ¼ rcp � �

defined as:

rcp � �

the base fluid is rcp

u ¼ bx 1 � δt f 0 qr ¼ � 16σ∗

3k∗ T3 ∞ ∂T

αnf þ

16σ∗T3 ∞

" # !

3 rcp � � nf k ∗ ∂T ∂Z

rcp � � nf

ks þ ð Þ m � 1 kf

� � s

¼ ks þ ð Þ m � 1 kf

where σ∗ and k∗ are the Stefan-Boltzmann constant and the mean absorption coefficient,

Parameters μnf , rnf and αnf are interrelated with nanoparticle volume fraction; ϕ and can be

, knf kf

, volumetric heat capacity of the nanofluid is rcp

ity of the nanofluid is knf , thermal conductivity of the base fluid is kf , thermal conductivity of the solid nanoparticles is ks, nanoparticle volume fraction is ϕ, density viscosity of the base

> ffiffiffiffiffiffiffiffiffiffiffiffi bx 1 � δt

fð Þ η , η ¼

r

Copper (Cu) 385 8933 400 Water 997.1 4179 0.613

Particle shapes Sphere Tetrahedron Column m 3 4.0613 6.3698

1 � ϕ � �2:5 , αnf ¼ knf

, μnf ¼ μf

� � s rcp � � f

∂z (7)

∂T

� � � ð Þ m � 1 ϕ kf � ks

� � þ ϕ kf � ks

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bx υð Þ 1 � δt

s

T∞ , θw > 1 is the temperature ratio parameter.

r cp k

∂y : (8)

� �

nf , thermal conductiv-

(9)

� �

, volumetric heat capacity of

z, θ ηð Þ¼ T � T∞

Tw � T∞

(10)

$$\frac{1}{\left(1-\phi\right)^{2.5}\left(1-\phi+\phi\left(\frac{\rho\_\ast}{\rho\_\dagger}\right)\right)}f''-\left[f'^2-f'-2\Omega g+\lambda\left(\frac{\eta}{2}f'+f'\right)\right]=0,\tag{11}$$

$$\frac{1}{\left(1-\phi\right)^{2.5}\left(1-\phi+\phi\left(\frac{\rho\_\*}{\rho\_{\ell}}\right)\right)}\mathbf{g}''-\left[f\,\mathbf{g}-f\mathbf{g}'-2\Omega f+\lambda\left(\frac{\eta}{2}\mathbf{g}'+\mathbf{g}\right)\right]=\mathbf{0},\tag{12}$$

$$\frac{k\_{n\!f}}{k\_{\!f}\left(1-\phi+\phi\left(\frac{\left(\rho c\_{\!\!r\!}\right)\_{\!\!s}}{\left(\rho c\_{\!\!r\!}\right)\_{\!\!f}}\right)}\left[1+R(1+(\theta\_{w}-1)\theta)^{3}\theta\right]-\Pr\left[\lambda\frac{\eta}{2}\theta^{\prime}-f\theta^{\prime}\right]=0,\tag{13}$$

The transformed boundary conditions are as follows:

$$\begin{aligned} f(0) &= 0, \dot{f}'(0) = 1, \ddot{g}(0) = 0, \theta(0) = 1 \quad \text{at} \ \eta &= 0\\ \dot{f}'(\eta) &\to 0, \ddot{g}(\eta) \to 0, \theta(\eta) \to 0 \quad \text{as} \ \eta \to \infty \end{aligned} \tag{14}$$

where Ω ¼ ω b is rotation rate, λ ¼ δ b is unsteadiness parameter R ¼ 16σ∗T3 ∞ 3kf k∗ is radiation parameter, Pr ¼ αnf υnf is Prandtl number and primes denote the differentiation with respect to η.

Rosca et al. [11] mentioned that the pressure term ð Þp can be integrated from Eq. (4); thus, we obtain:

$$p = \upsilon \rho \frac{\partial w}{\partial z} - \frac{\rho w^2}{2} + c \tag{15}$$

The physical quantities of interest in this problem are the skin friction coefficients in x and y directions, Cfx and Cfy as well as the local Nusselt number Nux which are defined as:

$$\mathbf{C}\_{\text{fr}} = \frac{\mathbf{r}\_{w\text{rx}}}{\rho u\_w^2(\text{xt})}, \quad \mathbf{C}\_{\text{fy}} = \frac{\mathbf{r}\_{w\text{y}}}{\rho u\_w^2(\text{yt})}, \text{ Nu}\_{\text{x}} = \frac{\mathbf{x}\eta\_w}{k\_f(T\_w - T\_\text{se})},\tag{16}$$

Surface shear stress τwx, τwy and surface heat flux qw are defined as:

$$
\pi\_{wx} = \mu\_{\eta'\_f} \left(\frac{\partial u}{\partial z}\right)\_{z=0}, \tau\_{wy} = \mu\_{\eta'\_f} \left(\frac{\partial v}{\partial z}\right)\_{z=0} \text{ and } q\_w = -k\_{\eta'\_f} \left(\frac{\partial T}{\partial z}\right) + qr\_{z=0} \tag{17}
$$

Using Eqs. (16) and (17), we obtain

$$\sqrt{\text{Re}\_{\text{x}}}\mathbb{C}\_{\text{ft}} = \frac{1}{\left(1-\phi\right)^{2.5}}\boldsymbol{f}^{\prime}(0), \ \sqrt{\text{Re}\_{\text{x}}}\mathbb{C}\_{\text{ft}} = \frac{1}{\left(1-\phi\right)^{2.5}}\boldsymbol{g}^{\prime}(0) \text{ and } \frac{\text{Nu}\_{\text{x}}}{\sqrt{\text{Re}\_{\text{x}}}} = \frac{k\_{\text{nf}}}{k\_{\text{f}}}\left(-\left[1+R\theta\_{\text{w}}^{3}\right]\boldsymbol{\theta}^{\prime}(0)\right). \tag{18}$$

where Rex ¼ uwx υ is local Reynolds number.

### 3. Numerical method

Numerical solutions of nonlinear coupled differential Eqs. (11)–(13) subject to the boundary conditions (14) constitute a two-point boundary value problem. Due to coupled and highly nonlinear nature, which are not amenable to closed-form solutions; therefore, we resorted to numerical solutions. In order to solve these equations numerically, we follow most efficient fourth-fifth order Runge-Kutta-Fehlberg integration scheme along with shooting technique. In this method, it is most important to choose the appropriate finite values of η∞. The asymptotic boundary conditions at η∞ were replaced by those at η8 in accordance with standard practice in the boundary layer analysis.

Figure 1 portrays the effect of Ω on velocity profile f

compared to tetrahedron- and sphere-shaped nanoparticles.

Figures 3 and 4 depict the effect of λ on the f

Figure 2. Influence of Ω on gð Þ η .

Figure 3. Influence of λ on f

0 ð Þ η . 0

corresponding thickness of the boundary layer enhance with larger values of Ω. This is because the larger value of Ω parameter leads to higher rotation rate as compared to stretching rate. Therefore, the larger rotation effect enhances velocity field. Figure 2 shows the impact of Ω on angular profile gð Þ η . From this figure, one can see that gð Þ η reduces for larger values of Ω. Further, it is noticed that rate of heat transfer is larger in column-shaped nanoparticles when

Nonlinear Radiative Heat Transfer of Cu-Water Nanoparticles over an Unsteady Rotating Flow under…

0

figures that an increase in λ decreases the momentum boundary layer thickness resulting in velocity decrease. It is also noted that gð Þ η decreases smoothly with the increase in the

ð Þ η . The velocity profile and

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http://dx.doi.org/10.5772/intechopen.74807

ð Þ η and gð Þ η profile. It is clear from both the

### 4. Result and discussion

To get a clear insight into the physical situation of the present problem, numerical values for velocity and temperature profile are computed for different values of dimensionless parameters using the method described in the previous section. The numerical results for the local Nusselt number are presented for different values of the governing parameters in Table 3.

Table 3. Numerical values of Nusselt number for different physical parameters.

Figure 1 portrays the effect of Ω on velocity profile f 0 ð Þ η . The velocity profile and corresponding thickness of the boundary layer enhance with larger values of Ω. This is because the larger value of Ω parameter leads to higher rotation rate as compared to stretching rate. Therefore, the larger rotation effect enhances velocity field. Figure 2 shows the impact of Ω on angular profile gð Þ η . From this figure, one can see that gð Þ η reduces for larger values of Ω. Further, it is noticed that rate of heat transfer is larger in column-shaped nanoparticles when compared to tetrahedron- and sphere-shaped nanoparticles.

Figures 3 and 4 depict the effect of λ on the f 0 ð Þ η and gð Þ η profile. It is clear from both the figures that an increase in λ decreases the momentum boundary layer thickness resulting in velocity decrease. It is also noted that gð Þ η decreases smoothly with the increase in the

Figure 2. Influence of Ω on gð Þ η .

3. Numerical method

172 Impact of Thermal Conductivity on Energy Technologies

the boundary layer analysis.

4. Result and discussion