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PAPER Qian Wang et al. TiC2: a new two-dimensional sheet beyond MXenes

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DOI: 10.1039/C7NR00327G

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Received 00th January 20xx, Accepted 00th January 20xx DOI: 10.1039/x0xx00000x www.rsc.org/

Guilong Wang,*a,b Chongda Wang,b Jinchuan Zhao,b,c Guizhen Wang,d Chul B. Park*b and Guoqun Zhao*a The superinsulating material plays a pivotal role in achieving the sustainable development of our modern world by improving the energy efficiency, and reducing the energy consumption and CO2 emission. The nanocellular polymer foam has been expected as a promising superinsulating material, but as yet not achieved. The understanding of thermal transport through the nanocellular foam is crucial for developing this superinsulating material. Hereby, we report an accurate mathematical model for the first time to quantitatively estimate the thermal transport through the nanocellular polymer foam. This is realized by taking into account the phonon scattering effect, the Knudsen effect and the thin-film interference effect in modeling the thermal transport through the solid conduction, gas conduction and thermal radiation, respectively. We demonstrate the quantitative relation between the cellular structure and the equivalent thermal conductivity, and present the optimum cellular structure scope for achieving the superinsulating performance. In particular, the significance of thermal radiation in the nanocellula polymer foam is emphasized. This mathematical model offers a very useful tool for deeply understanding thermal transport through the nanocellular polymer foams, and guiding the development of the new generation of the superinsulating material.

1. Introduction A large amount of fossil energy consumption each year brings serious global problems, such as environmental pollution, global warming, energy shortage, climate change and loss of biodiversity. Therefore, pursuing sustainable development has become one of the most important and urgent tasks of the human society nowadays. Basically, renewable energy development and energy conservation are the two main ways to achieve sustainable development. Currently, renewable energy resources account for only less than 15% of total world energy demand although their future are promising.1 Therefore, energy conservation innovation for improving energy utilization efficiency has become more vital than ever in recent years. Thermal insulation plays a crucial role in energy conservation, and is the key to reduce energy consumption, particularly in the residential and commercial building sector, where energy consumption accounts for over 40% of the total energy consumption and more than 20% CO2 emissions.2 Due to the superior insulating performance, high mechanical properties, long durability, and low cost, polymer foams have

a. Key

Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, School of Materials Science and Engineering, Shandong University, Jinan, Shandong 250061, China. b. Microcellular Plastics Manufacturing Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5T3G8, Canada. c. Centre for Precision Engineering, School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China. d. Key Laboratory of Chinese Education Ministry for Tropical Biological Resources, Hainan University, Haikou, Hainan 570228, China.

become one of the most important thermal insulation materials, and widely used in various industrial sectors, such as building and construction, transportation, petrochemical and aerospace. Polymer foams are currently the second major thermal insulation building materials after mineral wool. They have a typical thermal conductivity of 30–40 mW/m·K, which is lower than that of the mineral wool (40–50 mW/m·K), and show a much more promising and remarkable future. However, due to the limitation of the relatively high air thermal conductivity (26 mW/m·K) and thermal radiation, the regular polymer foam’s thermal conductivity has already been closed to the theoretical minimum value. To further reduce the foam’s thermal conductivity below the value of air for the increasing high requirement of thermal insulation in today’s society with the themes of green and sustainable development, it is necessary to significantly reduce the gas’s thermal conductivity within the foams. Actually, this can be easily achieved by replace the air with some large-molecular gases, such as chlorofluorocarbons, hydrochlorofluoro-carbons and hydrofluorocarbon, which have much low thermal conductivity ( 0.9) (Fig. 8a). For the polymer properties used in this section, it is worth noting that merely increasing the expansion ratio of the nanocellular foam is impossible to achieve the super thermal insulation with an effective thermal conductivity less than 25 mW/m·K. This is due to the strong thermal radiation in nanocellular foams with large expansion ratios (Fig. 8c). This finding contradicts the conventional view that super thermal insulation is available with low-density nanocellular foams.4,5 The conventional view is not very reasonable because it unilaterally emphasizes the reduced gaseous thermal conduction due to Knudsen effect while it neglects the significant increase of thermal radiation in nanocellular foams due to the thin-film interference effect. The dependence of the foam’s effective thermal conductivity on the cell size under various void fraction levels is shown in Fig. 9a. For each fixed void fraction, the total effective thermal conductivity first decreases slowly, then increases rapidly, and finally decreases sharply with the reducing of the cell size. At the first stage, when the cell size is at a relatively large level, the thermal conductivity contributed by the thermal conduction remains nearly constant with the reducing of the cell size (Fig.

9b). Meanwhile, the thermal conductivity contributed by View Article Online DOI:the 10.1039/C7NR00327G thermal radiation decreases gradually with reducing of the cell size (Fig. 9c), thus leading to the decreased total effective thermal conductivity. The decreased thermal radiation at this stage is owing to the increase in the number of the cell walls (or polymer slabs) while the almost constant or even increased wave-averaged reflectance of the single cell wall (Fig. 9d). At the second stage, with the further reducing of the cell size, the wave-averaged reflectance of the single cell wall begins to decrease sharply (Fig. 9d). Although the number of the cell walls grows gradually with the reducing of the cell size, the growth rate is much slower compared with the decline rate of the waveaveraged reflectance (Fig. 9d). Thus, the radiative thermal conductivity begins to increase rapidly with the reducing of the cell size at this stage (Fig. 9c). Meanwhile, although the thermal conduction contributed by the polymer and the air begins to decrease gradually mainly due to the Knudsen effect (Fig. 9b), the decline rate is much slower than the increase rate of the thermal radiation. Therefore, the total effective thermal conductivity begins to increase significantly with the reducing of the cell size at the second stage. At the last stage, as the total effective thermal conductivity increases to a certain upper limit, it begins to reduce sharply with the further reducing of the cell size (Fig. 9a). This is mainly because of the rapid decline of the radiative thermal conductivity (Fig. 9c), which is resulted from the rapid increase in the cell wall number while just slow decrease in the wave-averaged reflectance of the single cell wall (Fig. 9d). In another aspect, the decrease of the thermal conduction with the further reducing of the cell size also contributes to the rapid decline of the total effective thermal conductivity (Fig. 9b). With the decrease of the cell size, as the reduced thermal conductivity contributed by thermal conduction (Fig. 9b) just compensates the increased radiative thermal conductivity (Fig. 9c), the minimum total effective thermal conductivity is obtained for the polymer foam with a fixed void fraction (Fig. 9a). To achieve the best thermal insulation performance for each fixed void fraction, the larger the void fraction is, the larger the optimum cell size is. It is worth noting that the microcellular foams exhibits better thermal insulation performance than the nanocellular foams (Fig. 9a). This is mainly because the strong thin-film interference effect in nanocellular foams significantly reduces the reflectance of the cell walls to the infrared electromagnetic waves and thus results in a high radiative thermal conductivity (Fig. 9c & 9d). It clearly demonstrates that merely reducing the cell size to the nanometre scale is not guaranteed to obtain the super thermal insulation. Thermal radiation will be a critical issue for the low-density nanocellular foams to achieve super thermal insulation performance. Taking into account that the polymer used in this section does not absorb any electromagnetic waves (κs = 0), it is concluded that mere reflections at the cell wall interfaces cannot effectively block the thermal radiation in nanocellular foams. Besides the reflection, absorption is another way to block thermal radiation. Its effect on the thermal transport in polymer foams will be discussed in the next section.

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As discussed in the last section, thermal radiations will become very significant in nanocellular foams. Thermal radiations in the foams can be blocked either by reflections at the cell wall surfaces or by absorptions as they transfer through the cell walls. The polymer’s optical property at infrared wavelength range must play a significant role in affecting thermal radiation transfer in the foams. The optical property herein refers to the complex refractive index which is composed of the real part (refractive index) and the imaginary part (absorption index). The refractive index reflects the reflection performance of the polymer to the infrared waves while the absorption index reflects the absorption performance of the polymer to the infrared waves. To clarify the relationship between the polymer’s optical property and the thermal transport in the polymer foams, two cases were carried out via numerical simulations. In the first case, various refractive indexes (ns = 1.2, 1.4, 1.6, 1.8, 2.0) with a constant absorption index (κs = 0) were used. In the other case, various absorption indexes (κs = 0.00, 0.0025, 0.005, 0.01, 0.03) with a constant refractive index (ns = 1.6) were used. The other parameters used in simulation were the same as those used in the last section. Fig. 10a plots the dependence of the total effective thermal conductivity of the nanocellular foam (δc = 100 nm) on the void fraction under various refractive indexes. A larger refractive index can reduce the total effective thermal conductivity, especially for the nanocellular foams with large expansion ratios. This is because the increased refractive index can reduce the radiative thermal conductivity (Fig. 10b) by enhancing the reflectance of the single cell wall (Fig. 10c). For the nanocellular polymer foams with a large void fraction, the radiative heat transfer is very significant because the strong thin-film interference effect dramatically reduces the reflectance of the thin cell walls. Thus, it is very effective to reduce the total effective thermal conductivity by blocking more thermal radiations with the increased refractive index. For each

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refractive index, there is a minimized effective thermal View Article Online 10.1039/C7NR00327G conductivity that can be achieved byDOI: optimizing the void fraction. The larger the refractive index is, the larger the corresponding optimum void fraction is and the lower the minimum effective thermal conductivity is (Fig. 10a). Although the minimum effective thermal conductivity decreases gradually with the increasing of the refractive index, the decrease tendency becomes more and more slow (Fig. 10d). Notably, even if the refractive index increases to 2.0, the total effective thermal conductivity is still as high as 28.7 mW/m·K. Taking into account that the regular polymer’s refractive index is in the range between 1.3 and 1.7, it is infeasible to achieve a super thermal insulation of the nanocellular polymer foams by merely enhancing the polymer’s reflection to thermal radiations.

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3.4 Super thermal-insulation nanocellular polymer foams To clarify more clearly the relationship between the thermal conductivity and the cellular structure, we further calculated the colour contour isotherms of the thermal conductivity as a function of the cell size and the void fraction under various absorption coefficients. For the non-absorption polymer (Fig. 12a), the minimum thermal conductivity is 31.6 mW/m·K, which is obtained with the cell size of around 125 μm and with the void fraction of about 0.969. It is clearly demonstrated that it is infeasible for the non-absorption polymer foam to achieve the superinsulating property by reducing the cell size from a microscale to a nanoscale. This is because the reduced gaseous thermal conductivity due to the Knudsen effect is much smaller than the increased radiative thermal conductivity due to the thin-film interference effect which significantly reduces the reflectance of the cell walls to thermal radiations. The thermal radiation is so strong for the non-absorption low-density nanocellular foams that an extremely high thermal conductivity region occurs at the top-left corner of the thermal conductivity contour diagram (Fig. 12a). With the increase of the infrared absorption coefficient, there is a region with much lower thermal conductivity gradually occurring in the thermal conductivity contour diagram (Fig. 12b, 12c & 12d). For the polymer with an absorption coefficient of 0.005 (Fig. 12b), the minimum thermal conductivity that can be achieved is as low as 20.1 mW/m·K with

the cell size of 50 nm and with the void fraction of around 0.905. View Article Online DOI: 10.1039/C7NR00327G Clearly, it is feasible to obtain the superinsulating property (< 25 mW/m·K) with a cell size less than 100 nm and with a void fraction in the range between 0.85 and 0.95 (Fig. 12b). The minimum thermal conductivity that can be obtained reduces continuously by further increasing the infrared absorption coefficient (Fig. 12c & 12d). The minimum thermal conductivities are 15.1 mW/m·K and 9.8 mW/m·K corresponding the infrared absorption coefficients of 0.01 and 0.03, respectively. To obtain the minimum thermal conductivity, the corresponding void fraction increases gradually with the infrared void fraction increasing. Moreover, the cellular structure scope to obtain the superinsulating property (< 25 mW/m·K) is widened with the increase of the infrared absorption coefficient. a

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Fig. 11a plots the dependence of the total effective thermal conductivity of the nanocellular foams (δc = 100 nm) on the void fraction under various absorption indexes. When the void fraction is less than 0.6, the absorption index has little influence on the total effective thermal conductivity because the thermal radiation in this situation is very weak (Fig. 11b). This minor radiative heat transfer in the nanocellular polymer foams with a very small void fraction has been discussed in the literature.9,44 As the void fraction becomes larger, the absorption index plays a more and more significant role in affecting the total effective thermal conductivity (Fig. 11a). The larger the absorption index is, the smaller the total effective thermal conductivity is. This is because the thermal radiation for large void fractions becomes very significant (Fig. 11b) and the increased absorption index can effectively block the thermal radiations by enhancing the net reflectance of the cell walls (Fig. 11c). It is worth noting that there is a minimum effective thermal conductivity than can be achieved for each absorption index by optimizing the void fraction of the nanocellular foam (Fig. 11a). Absolutely, the larger the absorption index is, the smaller the minimum effective thermal conductivity is (Fig. 11d). With an absorption index of 0.03, the minimum effective thermal conductivity is as low as 13.6 mW/m·K. It demonstrates that a superinsulating property can be obtained with the nanocellular foams by using a polymer matrix with a strong infrared-wave absorption capability. However, to achieve such low thermal conductivity, the void fraction of the nanocellular foam (δc = 100 nm) should be as large as 0.96. Currently, it is still impossible even in the laboratory to fabricate the nanocellular foam with such a large void fraction.10,11,45

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It is concluded that the nanoscale cell size simultaneously with an appropriate void fraction is necessary but not sufficient to obtain the superinsulating property. The infrared absorption coefficient of the polymer matrix also plays a crucial role in the thermal conductivity of the nanocellular polymer foams. The larger the infrared absorption coefficient is, the wider the cellular structure scope is to achieve the superinsulating property. Although the optimum combination of the cell size and void fraction for minimizing the thermal conductivity depend upon the infrared absorption coefficient, the target cell size should be at least less than 200 nm and the ideal void fraction should be in the range between 0.9 and 0.95. Taking into account these conditions, the polymer matrix such as PMMA which has a strong infrared absorption capability and simultaneously has a potentiality to fabricate low-density nanocellular foams will be the ideal candidate for preparing the superinsulating polymer foams. In recent years, a lot of studies have been conducted to prepare nanocellular PMMA foams toward generating a new superinsulating material.5,7 However, the nanocellular foam’s void fraction is currently still too small to achieve the superinsulating property. Recently, Costeux45 reported the fabrication of nanocellular PMMA foams with the average cell size as small as 100 nm and with the void fraction as large as 0.85. This is currently the largest void fraction of the


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Nanoscale International Postdoctoral Exchange FellowshipView Program of Article Online DOI: 10.1039/C7NR00327G China, China Postdoctoral Science Foundation (CPSF), the Fundamental Research Funds of Shandong University, and the Consortium of Cellular and Micro-Cellular Plastics (CCMCP) for their financial support of this project

Notes and references 1 2 3 4


4. Conclusions We have developed a theoretical model for quantitatively estimating the thermal transport through the polymer foam without using any fitting parameters, which, to the best of our knowledge, is the first mathematical model so far fitting for the nanocellular polymer foams. The mathematical model can accurately predict the thermal conductivity of the polymer foams in a wide range of cellular structures under various conditions. We clearly demonstrate the quantitate relationships between the cellular structure, the foam component’s property, and the thermal-insulation performance of the polymer foam. In particular, the optimum cellular structure scope is presented for achieving the superinsulating performance with the nanocellular foam. The larger the void fraction is, the larger the critical cell size is for minimizing the thermal conductivity. The thermal conductivity can be effectively decreased by increasing either the refractive index or the absorption coefficient of the polymer matrix. Thermal radiation becomes very significant in the nanocellular polymer foam because the strong thin-film interference effect could dramatically reduce the infrared reflectance of the cell wall. In this situation, the infrared absorption of the polymer matrix plays a crucial role in blocking thermal radiation through the nanocellular polymer foam. By enhancing the infrared absorption capacity, the nanocellular polymer foam simultaneously with a reasonable void fraction will perform a superinsulating property. The potential minimum thermal conductivity of the nanocellular polymer foam reduces with the increase of the polymer matrix’s absorption coefficient. The cellular structure scope of the superinsulating nanocellular polymer foams can be widened by increasing the absorption coefficient. To achieve a superinsulating property with the nanocellular polymer foam, the target cell size should be less than 200 nm and the ideal void fraction should be in the range between 0.9 and 0.95. The mathematical model provides a useful tool for deeply understanding thermal transport in the nanocellular polymer foams, and guiding the development of the new generation of the superinsulating material.

Acknowledgements The authors are grateful to the National Natural Science Foundation of China (NSFC, 51405267, 11564011), Shandong Provincial Natural Science Foundation (ZR2014EEQ017), the

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nanocellular polymer foams (δc ≤ 100nm) that can be achieved by foaming process. Given that the PMMA/silica composite used for fabricating the nanocellular foam has the same properties with the common pure PMMA, the nanocellular foam’s thermal conductivity predicted by our model is still a little larger than 26 mW/m·K. Therefore, the nanocellular foam’s void fraction should be increased further to obtain the superinsulating property. The mathematical model developed in this study provides a very useful tool for guiding the design of the superinsulating nanocellular polymer foams in the future.



Nanoscale

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DOI: 10.1039/C7NR00327G


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