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Dimensional crossover of heat conduction in amorphous polyimide nanofibers

Dimensional crossover of heat conduction in amorphous polyimide nanofibers Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 National Science Review 5: 500–506, 2018 RESEARCH ARTICLE doi: 10.1093/nsr/nwy004 Advance access publication 9 January 2018 PHYSICS Center for Phononics and Thermal Energy Science, School of Dimensional crossover of heat conduction in amorphous Physics Science and Engineering, Tongji polyimide nanofibers University, Shanghai 200092, China; 1,2,3 1,2,3 4,5 1,2,3 China-EU Joint Lan Dong ,QingXi , Dongsheng Chen ,Jie Guo , Center for 1,2,3,6 1,2,3 4 1,2,3,∗ Nanophononics, Tsuneyoshi Nakayama , Yunyun Li , Ziqi Liang , Jun Zhou , School of Physics 1,2,3,∗ 7,∗ Xiangfan Xu and Baowen Li Science and Engineering, Tongji University, Shanghai 200092, China; ABSTRACT Shanghai Key The mechanism of thermal conductivity in amorphous polymers, especially polymer fibers, is unclear in Laboratory of Special Artificial comparison with that in inorganic materials. Here, we report the observation of a crossover of heat Microstructure conduction behavior from three dimensions to quasi-one dimension in polyimide nanofibers at a given Materials and temperature. A theoretical model based on the random walk theory has been proposed to quantitatively Technology, School of describe the interplay between the inter-chain hopping and the intra-chain hopping in nanofibers. This Physics Science and model explains well the diameter dependence of thermal conductivity and also speculates on the upper limit Engineering, Tongji University, Shanghai of thermal conductivity of amorphous polymers in the quasi-1D limit. 200092, China; Department of Keywords: dimensional crossover, thermal conductivity, nanofiber Materials Science, Fudan University, polyethylene nanowires fabricated by the improved Shanghai 200433, INTRODUCTION China; College of nanoporous template wetting technique, due to Polymers are widely used materials due to their fas- Mathematics and the high chain orientation arising from crystallinity Physics, Shanghai cinating properties such as low mass density, chem- [13,14]. More recently, Singh et al. demonstrated University of Electric ical stability and high malleability, etc. [1]. Unfor- that better molecular chain orientation could also Power, Shanghai tunately, the relatively low thermal conductivity of 200090, China; improve the thermal conductivity when polymer −1 −1 polymer, which is in the range of ∼0.1 Wm K Hokkaido University, fibers remain amorphous [ 4], which indicates −1 −1 to ∼0.3 Wm K [2–4], limits its application in Sapporo 060-0826, that it is also of significance to study the intrinsic thermal management. The low thermal conductiv- Japan and mechanism of thermal conductivity in amorphous Department of ity of polymer is considered to be one of the major polymer. All these pioneering works indicate that Mechanical reasons for the thermal failure in electronic devices Engineering, the thermal properties in polymers are highly related [5,6]. Therefore, thermally conductive polymers are University of to their microscopic configurations, and thermal highly demanded for heat dissipation in microelec- Colorado, Boulder, CO conductivity is limited by the molecular orientation tronic and civil applications. 80309-0427, USA and the inter-chain scatterings [ 15,16]. In contrast to common wisdom, polymer It has also been found that through molecular dy- nanofibers hold surprisingly high thermal con- Corresponding namics (MD) simulation that the chain conforma- authors. E-mails: ductivity; some of them are even comparable to tion would strongly influence thermal conductivity xuxiangfan@tongji. that in some metals or even silicon. Choy and his edu.cn; zhoujunzhou@ [17,18]. However, very few theories have been pro- co-workers carried out the pioneering theory and tongji.edu.cn; posed to quantitatively study the structure depen- experiments to demonstrate that the alignment of [email protected] dence of thermal conductivity in amorphous poly- molecular chains could enhance the thermal con- mers because of their complex intrinsic structure. ductivity along the alignment direction [7–9]. The Received 21 August Alternatively, theories for amorphous inorganic ma- increase of the thermal conductivity is attributed 2017; Revised 12 terials such as heat transfer by diffusons [ 19,20], December 2017; to the increase of the degree of crystallinity in the minimum thermal conductivity model [21–23] Accepted 6 January subsequent experimental works [1,10–12]. Cai et al. and the phonon-assisted hopping model [24–26] also observed thermal conductivity enhancement in The Author(s) 2018. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. All rights reserved. For permissions, plea se e-mail: [email protected] Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 501 (a) (c) (b) (d) (e) (f) Figure 1. Schematic picture of the electrospinning setup and details of amorphous PI nanofibers. (a) Schematic of anisotropic quasi-1D thermal diffusion in nanofibers with small diameters. All the molecular chains are aligned along the fiber axis. The blue arrow denotes the hopping between neighboring localization centers within the chain and only intra-chain hopping could happen in this case. (b) Schematic of the quasi-isotropic thermal diffusionin nanofibers with large diameters. The molecular chains are randomly oriented and entangled with each other. Heat carriers hop equally to every directi on and there is also the possibility of inter-chain hopping, denoted by the red arrow. (c) The electrospinning setup. Nanofiber was collected on the two suspended membranes (inset of Fig. 1c), which act as heater and temperature sensor during the thermal conductivity measurement. (d) 3D structural map of PI. (e) SEM image of PI nanofiber. The scale bar is 10 μm. The red circle marks the position of a single PI nanofiber. (f) Enlarged SEM image of the PI nanofiber shown in (e). The scale bar is 30 nm. have been borrowed to qualitatively explain the ther- conductivity, it is straightforward to look into the mal conductivity of amorphous polymers [4,27,28]. diameter dependence of thermal conductivity in Compared to the unique type of hopping in inor- nanofibers through spinning or ultra-drawing [ 7,8], ganic amorphous materials, there are two types of during which processes the entanglement of chains hopping processes in bulk polymers, i.e., intra-chain could be much reduced by adjusting controllable pa- and inter-chain hopping processes, which together rameters such as the static-electrical field and draw with their interplay play an important role in the ratio [29,30]. In this paper, we systematically in- heat conduction. Therefore, the mechanism of the vestigate the microstructure dependence of thermal enhancement of thermal conductivity in polymer conductivity in polyimide (PI) nanofibers for dif- nanofibers and the upper limit of such enhancement ferent diameters. The diameters of the obtained PI when the polymer is stretched are not yet totally nanofibers range from 31 nm to 167 nm (see Table clear. S1) and the lengths of the obtained PI nanofibers are illustrated in Table S2. Molecular chains in thin nanofibers tend to align along the fiber axis with less RESULTS AND DISCUSSION entanglement, as Fig. 1a demonstrates, while chains in thicker nanofibers are randomly oriented and en- Thanks to the development of experimental tech- tangled with each other, as is illustrated in Fig. 1b. niques, it is possible to characterize the thermal Figure 1c presents a schematic diagram of the conductivity of ultra-thin polymer nanofibers. To electrospinning setup. Due to the static-electrical test the microstructure dependence of thermal Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 502 Natl Sci Rev, 2018, Vol. 5, No. 4 RESEARCH ARTICLE nanofibers with different diameters is on the order −10 of 1 × 10 W/K (Fig. 2a). To eliminate the effect of thermal radiation, a blank suspended device was used to probe standard thermal radiation in a wide temperature range. The measured thermal radiation between the two suspended membranes in the blank device is around ∼100 pW/K at room temperature. This result is a few times lower than that observed by Pettes et al.[36], probably due to the better vacuum level, which would reduce the air conduction and convection. In order to illustrate the effects of ther- mal contact resistance, two approaches were used to simulate the temperature distribution of the sus- pended membranes and calculate the thermal con- tact resistance at the platinum/PI nanofiber inter- face. These two approaches verified that the thermal contact resistance held a negligible contribution of Figure 2. Thermal transport of PI nanofibers with different diameters as a function of the total measured thermal resistance (see Figs S3 temperature. (a) Thermal conductance of PI nanofibers. The blue rhombus points exhibit and S4 and Table S4 in the sections entitled ‘Finite the thermal radiation measured by the differential circuit configuration with high vac- element simulations (COMSOL Multiphysics 5.2)’ −8 uum (on the order of 1 × 10 mbar). (b) Thermal conductivity of PI nanofibers excluding and ‘Thermal contact resistance’ in the online sup- thermal radiation for two samples: No. II d = 37 nm, L = 14.8 um; No. IX d = 167 nm, plementary data). L = 15.2 um, respectively (the morphology details of other samples are illustrated in The measured thermal conductivity increases Tables S1 and S2 in the online supplementary data). Solid lines are tted fi by κ∼T with λ = 0.84 ± 0.14 and 0.31 ± 0.02 for samples with diameters d = 37 nm and 167 nm, monotonously with temperature T, which is a typi- respectively. Error bars are estimated based on uncertainties associated with the fiber cal feature of amorphous material, as Fig. 2b shows. diameter and temperature uncertainty (see Tables S1 and S3 in the section entitled The amorphous character of PI may arise from the ‘Thermal conductivity uncertainty’ in the online supplementary data). defects and random bond angles within the chain, as well as the complex entanglement between chains. Furthermore, we find that the thermal conductiv- force introduced by high electrical voltage, sus- ity varies with temperature as κ∼T , where λ varies pended PI nanofibers were formed across two SiN membranes. These two SiN membranes were cov- from 0.31 ± 0.02 to 0.84 ± 0.14 as the diameter ered by platinum (i.e., the electrical ground). This changes. For a thick nanofiber with diameter d = 0.31 was the key step where molecular chains tend to 167 nm, the power law dependence T agrees with align along the axis of the nanofiber. There might be the experimental measurement from Singh’s group several PI nanofibers passing through the gap in the [4]. As the diameter decreases, the power index ap- middle of the device after electrospinning. In our ex- proaches 1, which coincides with the prediction of periments, only one nanofiber is left; the others will the hopping mechanism [24–26]. be cut by a nanomanipulator with a tungsten needle To look inside into the intrinsic dominant (see Fig. S1). Figure 1d depicts a 3D structural map mechanism of thermal transport, the diameter dependence of thermal conductivity at room of PI. It shows large conjugated aromatic bonds in a temperature is systematically investigated and the PI structure, which could help to enhance the ther- results are shown in Fig. 3. The thermal conductivity mal conductivity of PI nanofibers [ 31]. of PI nanofibers is close to that of bulk PI when Thermal conductivity along the fiber axis was the diameters are larger than 150 nm. It increases measured by the traditional thermal bridge method dramatically as the diameter decreases, and reaches [32–34] (Fig. 1e and f). The whole device was placed an order of magnitude larger than that in bulk PI in a cryostat with high vacuum on the order of 1 × −8 when the diameters are smaller than 40 nm. A simi- 10 mbar to reduce the thermal convection. To in- lar result was also observed in electrospun nylon-11 crease the measurement sensitivity, the differential nanofibers [ 12], which suggests that the stretching circuit configuration (see Fig. S2a in the section en- process could induce more ordered molecular titled ‘The differential circuit configuration’ in the chains in polymers, confirmed by high-resolution online supplementary data) was used and the mea- wide-angle X-ray scattering. surement sensitivity of the thermal conductance in- creased from ∼1 nW/K to 10 pW/K (see Fig. S2b To describe the diameter dependence of the ther- in the section entitled ‘The differential circuit config- mal conductivity quantitatively, we propose a the- uration’ in the online supplementary data) [33,35]. oretical model based on random walk theory to In our experiment, the thermal conductance of PI incorporate the diffusion of phonons through the Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 503 Table 1. Fitting parameters obtained by fitting the experi- mental data in Fig. 3a with Equation (1). −1 −1 κ (Wm K ) d (nm)  (nm) quasi−1D 0 63.0 ± 1.639 ± 13 62 ± 31 83.6 ± 2.234 ± 12 66 ± 36 The diameter dependence of Z is described by an empirical function Z = [2f(d) + 1] [f(d) + 2], where f(d) = 1 − 2/{1 + exp[(d − d )/]} for d > d . In the above expression, d is the criti- 0 0 cal diameter under which the diffusion converges to quasi-1D,  is the changing rate of the tran- sition from quasi-1D to 3D and  denotes the average number of inter-chain hopping sites. The average nearest inter-chain neighbor  should be determined from the real configuration of polymer Figure 3. Dimensional crossover of thermal conductivity of PI nanofibers at room tem- chains. From complex network theory, the num- perature. (a) The diameter and length details of PI nanofibers are illustrated in Tables ber of nearest inter-chain neighbors should be 6– S1 and S2 in the online supplementary data. The gray shadowed bar represents the −1 −1 thermal conductivity of bulk PI within the range of 0.1–0.3 Wm K . The rhombus 10 [38]. For further validation, numerical simula- (left axis and bottom axis) represents the experimental data. The olive solid line (left tions in generating polymer chains are required. The axis and bottom axis) and pink dashed line (left axis and bottom axis) are tted fi by Equa- current form of f(d) could successfully describe tion (1) with different values of the parameter . Red triangles (right axis and upper the transition from 3D to quasi-1D. When d = d , axis) denote thermal diffusivity obtained from the random walk simulation (details of f(d) = 0, meaning that the system is quasi-1D and the random walk simulation are included in Fig. S5 and Table S5 in the section entitled there is no inter-chain hopping. When d approaches ‘Schematic of the diameter confinement effect on the coordination number’ in the on- infinity, f( d) saturates to 1, corresponding to the 3D line supplementary data). (b) Dual-logarithm thermal conductivity versus diameter. The system. We should stress that our empirical function olive line is tted fi by Equation (1). is definitely not unique but it is one of the best ones (as is always the case in inverse problems) that fits hopping mechanism. We are aware that the lattice the experimental data optimally. The thermal con- vibrations in disordered systems without periodic- ductivity is expressed by κ = αρC , where α is ity do not have dispersion but the terminology of thermal diffusivity, ρ is mass density and C is spe- a ‘phonon’ is still usable to describe energy quanta. cific heat capacity [ 39], thus thermal conductivity Considering a complex network with a large num- is inversely proportional to Z (details are given in ber of entangled macromolecular chains, phonons the section entitled ‘Diameter dependence of aver- transport across this complex network through the age coordination number’ in the online supplemen- hopping process. Note that there are two different tary data): types of hopping: intra-chain hopping, in which a phonon hops between localization centers within a 2κ quasi−1D single chain, as shown in Fig. 1a, and inter-chain κ (d) = . (1) hopping, in which a phonon hops from one chain to another chain, as shown in Fig. 1b. According to the random walk theory, the thermal diffusiv- Note that f(d) = 0 when d ≤ d , and the ther- ity along the fiber axis is defined as [ 37] α = mal conductivity converges to κ . This means quasi−1D 1 2 ¯ ¯ R , where Z is the average effective coordi- that the thermal conductivity could not increase in- tot nation number along the fiber axis, R is the aver- finitely with decreasing fiber diameter. There ex- age hopping distance and  is the temperature- ists an upper limit for the thermal conductivity of tot dependent total hopping rate. In our simplified electrospun PI, corresponding to the 1D intra-chain model, we do not consider the difference between diffusion, where the average effective coordination the hopping rates of the inter- and intra-chain hop- number along the fiber axis is 2. In this limit, all poly- ping processes. Note that the inter-chain hopping mer chains are well aligned and the inter-chain in- process usually happens at cross links of chains, in teractions are negligible. To verify the validity of our which case the hopping distance is negligible com- model, we fit the experimental results with Equation pared to that of the intra-chain hopping process; (1). The fitting parameters are listed in Table 1.Our it is reasonable to assume that R is mainly deter- model fits well with the experimental data, as the mined by the intra-chain hopping distance R . lines in Fig. 3a show. intra Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 504 Natl Sci Rev, 2018, Vol. 5, No. 4 RESEARCH ARTICLE We also do a random walk simulation and obtain densed matter physics and thermal measurements the dimensionless thermal diffusivity. The details of on much thinner polymer fibers. the simulation are included in the section entitled ‘Random walk simulation’ in the online supplemen- tary data. The exact value of the thermal diffusivity of EXPERIMENTAL SECTION 2 −1 bulk PI is α = 0.0775 mm s , estimated from bulk Thermal conductivity measurement the observed thermal conductivity of bulk PI κ = PI −1 −1 The PI nanofibers fabricated by the electrospin- 0.12 Wm K , density of bulk PI ρ = 1.42 × PI 3 −3 ning method served as bridge to connect two 10 kg m , and specific heat of bulk PI C = pPI 3 −1 −1 platinum/SiN membranes (Fig. 1e). These two 1.09 × 10 Jkg K [40]. Nanofibers with di- x membranes were regarded as thermometers. A DC ameters smaller than d exhibit quasi-1D thermal current of a slow change step combined with an AC transport behavior, while nanofibers with diame- current (1000 nA) was added to one of the mem- ters much larger than d +  tend to behave like branes serving as a heater resistor (R , the left plat- bulk polymers. A crossover of heat conduction from h inum coil shown in Fig. 1c). The DC current was quasi-1D to 3D is only apparent in the range d ≤ applied to provide Joule heat and also to increase d d + . The magnitude of critical diameter d ∼ 0 0 its temperature (T ). The AC current was used to and parameter  is related to the radius of gyration h measure the resistance of R .Meanwhile,anACcur- R of macromolecular chains, which is typically on rent of the same value was applied to another mem- the order of tenths of nanometers [41]. R is deter- brane serving as sensor resistor (R , the right plat- mined by the structure of monomers, the bond an- inum coil shown in Fig. 1c), to probe the resistance gle between monomers, the length of a single chain, of R . The Joule heating in R gradually dissipated and the process condition such as applied voltage in s h through the six platinum/SiN beams and the PI electrospinning. When d > d +  ≥ 2R ,bulk- 0 g nanofiber, which raises the temperature in R (T ). like polymer nanofibers can be realized since macro- s s At steady state, the thermal conductance of the PI molecular chains can easily gyrate. When d < d , nanofibers ( σ ) and the thermal conductance of the macromolecular chains can hardly gyrate and the PI suspended beam (σ ) could be obtained by chains prefer to lie along the fiber axis. l σ = T + h s CONCLUSIONS A crossover of heat conduction from 3D to quasi- and 1D has been observed experimentally in amorphous polymer nanofibers obtained from electrospinning. l s σ = , PI This behavior has been quantitatively explained by T − h s a model based on random walk theory in which both inter-chain and intra-chain hopping processes where T and T indicate the temperature rise h s are considered. Two important fitting parameters, in R and R , and Q is the Joule heat applied to the h s i.e., d and , are obtained as the characterization 0 heater resistor and one of the platinum/SiN beams. length of the dimensional transition. Our theory suc- cessfully testifies that the hopping mechanism based Electrospinning on the random walk picture is valid and it is use- To fabricate nanoscale PI fibers with controllable ful to explain the diameter dependence of thermal diameters and chain orientations, we utilized elec- conductivity in nanofibers. Nevertheless, there are trospinning using a commercialized electrospinner. still many open questions deserving further investi- The solvent, a mixture of PI and dimethylformamide gation. First, the temperature dependence of ther- (DMF) solution, was prepared with concentrations mal conductivity has not been well explained and it from 45% to 80%, followed by all-night stirring to requires deeper and quantitative understanding of guarantee complete mixing of the PI molecules and the inter-chain thermal transport mechanism. Sec- DMF solvent. The diameter of the PI nanofiber ond, the four parameters in the empirical function should increase with increasing PI molecule weight require validation from further simulations and ex- ratio. periments. For example, κ is closely related quasi−1D to the thermal conductivity of a single chain, and it can be obtained from molecular dynamics; , d SUPPLEMENTARY DATA and  are determined by the configuration of poly- Supplementary data are available at NSR online. mer chains, which requires research on polymer con- Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 505 cated by a nano-molding technique. Heat Trans Eng 2013; 34: ACKNOWLEDGEMENTS 131–9. Special thanks go to the technical engineer Ning Liu from ZEISS, 15. Liu J and Yang R. Tuning the thermal conductivity of polymers who helped to obtain the high-resolution SEM image. In addition, with mechanical strains. Phys Rev B 2010; 81: 174122. 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Dimensional crossover of heat conduction in amorphous polyimide nanofibers

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Copyright © 2022 China Science Publishing & Media Ltd. (Science Press)
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2095-5138
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10.1093/nsr/nwy004
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Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 National Science Review 5: 500–506, 2018 RESEARCH ARTICLE doi: 10.1093/nsr/nwy004 Advance access publication 9 January 2018 PHYSICS Center for Phononics and Thermal Energy Science, School of Dimensional crossover of heat conduction in amorphous Physics Science and Engineering, Tongji polyimide nanofibers University, Shanghai 200092, China; 1,2,3 1,2,3 4,5 1,2,3 China-EU Joint Lan Dong ,QingXi , Dongsheng Chen ,Jie Guo , Center for 1,2,3,6 1,2,3 4 1,2,3,∗ Nanophononics, Tsuneyoshi Nakayama , Yunyun Li , Ziqi Liang , Jun Zhou , School of Physics 1,2,3,∗ 7,∗ Xiangfan Xu and Baowen Li Science and Engineering, Tongji University, Shanghai 200092, China; ABSTRACT Shanghai Key The mechanism of thermal conductivity in amorphous polymers, especially polymer fibers, is unclear in Laboratory of Special Artificial comparison with that in inorganic materials. Here, we report the observation of a crossover of heat Microstructure conduction behavior from three dimensions to quasi-one dimension in polyimide nanofibers at a given Materials and temperature. A theoretical model based on the random walk theory has been proposed to quantitatively Technology, School of describe the interplay between the inter-chain hopping and the intra-chain hopping in nanofibers. This Physics Science and model explains well the diameter dependence of thermal conductivity and also speculates on the upper limit Engineering, Tongji University, Shanghai of thermal conductivity of amorphous polymers in the quasi-1D limit. 200092, China; Department of Keywords: dimensional crossover, thermal conductivity, nanofiber Materials Science, Fudan University, polyethylene nanowires fabricated by the improved Shanghai 200433, INTRODUCTION China; College of nanoporous template wetting technique, due to Polymers are widely used materials due to their fas- Mathematics and the high chain orientation arising from crystallinity Physics, Shanghai cinating properties such as low mass density, chem- [13,14]. More recently, Singh et al. demonstrated University of Electric ical stability and high malleability, etc. [1]. Unfor- that better molecular chain orientation could also Power, Shanghai tunately, the relatively low thermal conductivity of 200090, China; improve the thermal conductivity when polymer −1 −1 polymer, which is in the range of ∼0.1 Wm K Hokkaido University, fibers remain amorphous [ 4], which indicates −1 −1 to ∼0.3 Wm K [2–4], limits its application in Sapporo 060-0826, that it is also of significance to study the intrinsic thermal management. The low thermal conductiv- Japan and mechanism of thermal conductivity in amorphous Department of ity of polymer is considered to be one of the major polymer. All these pioneering works indicate that Mechanical reasons for the thermal failure in electronic devices Engineering, the thermal properties in polymers are highly related [5,6]. Therefore, thermally conductive polymers are University of to their microscopic configurations, and thermal highly demanded for heat dissipation in microelec- Colorado, Boulder, CO conductivity is limited by the molecular orientation tronic and civil applications. 80309-0427, USA and the inter-chain scatterings [ 15,16]. In contrast to common wisdom, polymer It has also been found that through molecular dy- nanofibers hold surprisingly high thermal con- Corresponding namics (MD) simulation that the chain conforma- authors. E-mails: ductivity; some of them are even comparable to tion would strongly influence thermal conductivity xuxiangfan@tongji. that in some metals or even silicon. Choy and his edu.cn; zhoujunzhou@ [17,18]. However, very few theories have been pro- co-workers carried out the pioneering theory and tongji.edu.cn; posed to quantitatively study the structure depen- experiments to demonstrate that the alignment of [email protected] dence of thermal conductivity in amorphous poly- molecular chains could enhance the thermal con- mers because of their complex intrinsic structure. ductivity along the alignment direction [7–9]. The Received 21 August Alternatively, theories for amorphous inorganic ma- increase of the thermal conductivity is attributed 2017; Revised 12 terials such as heat transfer by diffusons [ 19,20], December 2017; to the increase of the degree of crystallinity in the minimum thermal conductivity model [21–23] Accepted 6 January subsequent experimental works [1,10–12]. Cai et al. and the phonon-assisted hopping model [24–26] also observed thermal conductivity enhancement in The Author(s) 2018. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. All rights reserved. For permissions, plea se e-mail: [email protected] Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 501 (a) (c) (b) (d) (e) (f) Figure 1. Schematic picture of the electrospinning setup and details of amorphous PI nanofibers. (a) Schematic of anisotropic quasi-1D thermal diffusion in nanofibers with small diameters. All the molecular chains are aligned along the fiber axis. The blue arrow denotes the hopping between neighboring localization centers within the chain and only intra-chain hopping could happen in this case. (b) Schematic of the quasi-isotropic thermal diffusionin nanofibers with large diameters. The molecular chains are randomly oriented and entangled with each other. Heat carriers hop equally to every directi on and there is also the possibility of inter-chain hopping, denoted by the red arrow. (c) The electrospinning setup. Nanofiber was collected on the two suspended membranes (inset of Fig. 1c), which act as heater and temperature sensor during the thermal conductivity measurement. (d) 3D structural map of PI. (e) SEM image of PI nanofiber. The scale bar is 10 μm. The red circle marks the position of a single PI nanofiber. (f) Enlarged SEM image of the PI nanofiber shown in (e). The scale bar is 30 nm. have been borrowed to qualitatively explain the ther- conductivity, it is straightforward to look into the mal conductivity of amorphous polymers [4,27,28]. diameter dependence of thermal conductivity in Compared to the unique type of hopping in inor- nanofibers through spinning or ultra-drawing [ 7,8], ganic amorphous materials, there are two types of during which processes the entanglement of chains hopping processes in bulk polymers, i.e., intra-chain could be much reduced by adjusting controllable pa- and inter-chain hopping processes, which together rameters such as the static-electrical field and draw with their interplay play an important role in the ratio [29,30]. In this paper, we systematically in- heat conduction. Therefore, the mechanism of the vestigate the microstructure dependence of thermal enhancement of thermal conductivity in polymer conductivity in polyimide (PI) nanofibers for dif- nanofibers and the upper limit of such enhancement ferent diameters. The diameters of the obtained PI when the polymer is stretched are not yet totally nanofibers range from 31 nm to 167 nm (see Table clear. S1) and the lengths of the obtained PI nanofibers are illustrated in Table S2. Molecular chains in thin nanofibers tend to align along the fiber axis with less RESULTS AND DISCUSSION entanglement, as Fig. 1a demonstrates, while chains in thicker nanofibers are randomly oriented and en- Thanks to the development of experimental tech- tangled with each other, as is illustrated in Fig. 1b. niques, it is possible to characterize the thermal Figure 1c presents a schematic diagram of the conductivity of ultra-thin polymer nanofibers. To electrospinning setup. Due to the static-electrical test the microstructure dependence of thermal Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 502 Natl Sci Rev, 2018, Vol. 5, No. 4 RESEARCH ARTICLE nanofibers with different diameters is on the order −10 of 1 × 10 W/K (Fig. 2a). To eliminate the effect of thermal radiation, a blank suspended device was used to probe standard thermal radiation in a wide temperature range. The measured thermal radiation between the two suspended membranes in the blank device is around ∼100 pW/K at room temperature. This result is a few times lower than that observed by Pettes et al.[36], probably due to the better vacuum level, which would reduce the air conduction and convection. In order to illustrate the effects of ther- mal contact resistance, two approaches were used to simulate the temperature distribution of the sus- pended membranes and calculate the thermal con- tact resistance at the platinum/PI nanofiber inter- face. These two approaches verified that the thermal contact resistance held a negligible contribution of Figure 2. Thermal transport of PI nanofibers with different diameters as a function of the total measured thermal resistance (see Figs S3 temperature. (a) Thermal conductance of PI nanofibers. The blue rhombus points exhibit and S4 and Table S4 in the sections entitled ‘Finite the thermal radiation measured by the differential circuit configuration with high vac- element simulations (COMSOL Multiphysics 5.2)’ −8 uum (on the order of 1 × 10 mbar). (b) Thermal conductivity of PI nanofibers excluding and ‘Thermal contact resistance’ in the online sup- thermal radiation for two samples: No. II d = 37 nm, L = 14.8 um; No. IX d = 167 nm, plementary data). L = 15.2 um, respectively (the morphology details of other samples are illustrated in The measured thermal conductivity increases Tables S1 and S2 in the online supplementary data). Solid lines are tted fi by κ∼T with λ = 0.84 ± 0.14 and 0.31 ± 0.02 for samples with diameters d = 37 nm and 167 nm, monotonously with temperature T, which is a typi- respectively. Error bars are estimated based on uncertainties associated with the fiber cal feature of amorphous material, as Fig. 2b shows. diameter and temperature uncertainty (see Tables S1 and S3 in the section entitled The amorphous character of PI may arise from the ‘Thermal conductivity uncertainty’ in the online supplementary data). defects and random bond angles within the chain, as well as the complex entanglement between chains. Furthermore, we find that the thermal conductiv- force introduced by high electrical voltage, sus- ity varies with temperature as κ∼T , where λ varies pended PI nanofibers were formed across two SiN membranes. These two SiN membranes were cov- from 0.31 ± 0.02 to 0.84 ± 0.14 as the diameter ered by platinum (i.e., the electrical ground). This changes. For a thick nanofiber with diameter d = 0.31 was the key step where molecular chains tend to 167 nm, the power law dependence T agrees with align along the axis of the nanofiber. There might be the experimental measurement from Singh’s group several PI nanofibers passing through the gap in the [4]. As the diameter decreases, the power index ap- middle of the device after electrospinning. In our ex- proaches 1, which coincides with the prediction of periments, only one nanofiber is left; the others will the hopping mechanism [24–26]. be cut by a nanomanipulator with a tungsten needle To look inside into the intrinsic dominant (see Fig. S1). Figure 1d depicts a 3D structural map mechanism of thermal transport, the diameter dependence of thermal conductivity at room of PI. It shows large conjugated aromatic bonds in a temperature is systematically investigated and the PI structure, which could help to enhance the ther- results are shown in Fig. 3. The thermal conductivity mal conductivity of PI nanofibers [ 31]. of PI nanofibers is close to that of bulk PI when Thermal conductivity along the fiber axis was the diameters are larger than 150 nm. It increases measured by the traditional thermal bridge method dramatically as the diameter decreases, and reaches [32–34] (Fig. 1e and f). The whole device was placed an order of magnitude larger than that in bulk PI in a cryostat with high vacuum on the order of 1 × −8 when the diameters are smaller than 40 nm. A simi- 10 mbar to reduce the thermal convection. To in- lar result was also observed in electrospun nylon-11 crease the measurement sensitivity, the differential nanofibers [ 12], which suggests that the stretching circuit configuration (see Fig. S2a in the section en- process could induce more ordered molecular titled ‘The differential circuit configuration’ in the chains in polymers, confirmed by high-resolution online supplementary data) was used and the mea- wide-angle X-ray scattering. surement sensitivity of the thermal conductance in- creased from ∼1 nW/K to 10 pW/K (see Fig. S2b To describe the diameter dependence of the ther- in the section entitled ‘The differential circuit config- mal conductivity quantitatively, we propose a the- uration’ in the online supplementary data) [33,35]. oretical model based on random walk theory to In our experiment, the thermal conductance of PI incorporate the diffusion of phonons through the Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 503 Table 1. Fitting parameters obtained by fitting the experi- mental data in Fig. 3a with Equation (1). −1 −1 κ (Wm K ) d (nm)  (nm) quasi−1D 0 63.0 ± 1.639 ± 13 62 ± 31 83.6 ± 2.234 ± 12 66 ± 36 The diameter dependence of Z is described by an empirical function Z = [2f(d) + 1] [f(d) + 2], where f(d) = 1 − 2/{1 + exp[(d − d )/]} for d > d . In the above expression, d is the criti- 0 0 cal diameter under which the diffusion converges to quasi-1D,  is the changing rate of the tran- sition from quasi-1D to 3D and  denotes the average number of inter-chain hopping sites. The average nearest inter-chain neighbor  should be determined from the real configuration of polymer Figure 3. Dimensional crossover of thermal conductivity of PI nanofibers at room tem- chains. From complex network theory, the num- perature. (a) The diameter and length details of PI nanofibers are illustrated in Tables ber of nearest inter-chain neighbors should be 6– S1 and S2 in the online supplementary data. The gray shadowed bar represents the −1 −1 thermal conductivity of bulk PI within the range of 0.1–0.3 Wm K . The rhombus 10 [38]. For further validation, numerical simula- (left axis and bottom axis) represents the experimental data. The olive solid line (left tions in generating polymer chains are required. The axis and bottom axis) and pink dashed line (left axis and bottom axis) are tted fi by Equa- current form of f(d) could successfully describe tion (1) with different values of the parameter . Red triangles (right axis and upper the transition from 3D to quasi-1D. When d = d , axis) denote thermal diffusivity obtained from the random walk simulation (details of f(d) = 0, meaning that the system is quasi-1D and the random walk simulation are included in Fig. S5 and Table S5 in the section entitled there is no inter-chain hopping. When d approaches ‘Schematic of the diameter confinement effect on the coordination number’ in the on- infinity, f( d) saturates to 1, corresponding to the 3D line supplementary data). (b) Dual-logarithm thermal conductivity versus diameter. The system. We should stress that our empirical function olive line is tted fi by Equation (1). is definitely not unique but it is one of the best ones (as is always the case in inverse problems) that fits hopping mechanism. We are aware that the lattice the experimental data optimally. The thermal con- vibrations in disordered systems without periodic- ductivity is expressed by κ = αρC , where α is ity do not have dispersion but the terminology of thermal diffusivity, ρ is mass density and C is spe- a ‘phonon’ is still usable to describe energy quanta. cific heat capacity [ 39], thus thermal conductivity Considering a complex network with a large num- is inversely proportional to Z (details are given in ber of entangled macromolecular chains, phonons the section entitled ‘Diameter dependence of aver- transport across this complex network through the age coordination number’ in the online supplemen- hopping process. Note that there are two different tary data): types of hopping: intra-chain hopping, in which a phonon hops between localization centers within a 2κ quasi−1D single chain, as shown in Fig. 1a, and inter-chain κ (d) = . (1) hopping, in which a phonon hops from one chain to another chain, as shown in Fig. 1b. According to the random walk theory, the thermal diffusiv- Note that f(d) = 0 when d ≤ d , and the ther- ity along the fiber axis is defined as [ 37] α = mal conductivity converges to κ . This means quasi−1D 1 2 ¯ ¯ R , where Z is the average effective coordi- that the thermal conductivity could not increase in- tot nation number along the fiber axis, R is the aver- finitely with decreasing fiber diameter. There ex- age hopping distance and  is the temperature- ists an upper limit for the thermal conductivity of tot dependent total hopping rate. In our simplified electrospun PI, corresponding to the 1D intra-chain model, we do not consider the difference between diffusion, where the average effective coordination the hopping rates of the inter- and intra-chain hop- number along the fiber axis is 2. In this limit, all poly- ping processes. Note that the inter-chain hopping mer chains are well aligned and the inter-chain in- process usually happens at cross links of chains, in teractions are negligible. To verify the validity of our which case the hopping distance is negligible com- model, we fit the experimental results with Equation pared to that of the intra-chain hopping process; (1). The fitting parameters are listed in Table 1.Our it is reasonable to assume that R is mainly deter- model fits well with the experimental data, as the mined by the intra-chain hopping distance R . lines in Fig. 3a show. intra Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 504 Natl Sci Rev, 2018, Vol. 5, No. 4 RESEARCH ARTICLE We also do a random walk simulation and obtain densed matter physics and thermal measurements the dimensionless thermal diffusivity. The details of on much thinner polymer fibers. the simulation are included in the section entitled ‘Random walk simulation’ in the online supplemen- tary data. The exact value of the thermal diffusivity of EXPERIMENTAL SECTION 2 −1 bulk PI is α = 0.0775 mm s , estimated from bulk Thermal conductivity measurement the observed thermal conductivity of bulk PI κ = PI −1 −1 The PI nanofibers fabricated by the electrospin- 0.12 Wm K , density of bulk PI ρ = 1.42 × PI 3 −3 ning method served as bridge to connect two 10 kg m , and specific heat of bulk PI C = pPI 3 −1 −1 platinum/SiN membranes (Fig. 1e). These two 1.09 × 10 Jkg K [40]. Nanofibers with di- x membranes were regarded as thermometers. A DC ameters smaller than d exhibit quasi-1D thermal current of a slow change step combined with an AC transport behavior, while nanofibers with diame- current (1000 nA) was added to one of the mem- ters much larger than d +  tend to behave like branes serving as a heater resistor (R , the left plat- bulk polymers. A crossover of heat conduction from h inum coil shown in Fig. 1c). The DC current was quasi-1D to 3D is only apparent in the range d ≤ applied to provide Joule heat and also to increase d d + . The magnitude of critical diameter d ∼ 0 0 its temperature (T ). The AC current was used to and parameter  is related to the radius of gyration h measure the resistance of R .Meanwhile,anACcur- R of macromolecular chains, which is typically on rent of the same value was applied to another mem- the order of tenths of nanometers [41]. R is deter- brane serving as sensor resistor (R , the right plat- mined by the structure of monomers, the bond an- inum coil shown in Fig. 1c), to probe the resistance gle between monomers, the length of a single chain, of R . The Joule heating in R gradually dissipated and the process condition such as applied voltage in s h through the six platinum/SiN beams and the PI electrospinning. When d > d +  ≥ 2R ,bulk- 0 g nanofiber, which raises the temperature in R (T ). like polymer nanofibers can be realized since macro- s s At steady state, the thermal conductance of the PI molecular chains can easily gyrate. When d < d , nanofibers ( σ ) and the thermal conductance of the macromolecular chains can hardly gyrate and the PI suspended beam (σ ) could be obtained by chains prefer to lie along the fiber axis. l σ = T + h s CONCLUSIONS A crossover of heat conduction from 3D to quasi- and 1D has been observed experimentally in amorphous polymer nanofibers obtained from electrospinning. l s σ = , PI This behavior has been quantitatively explained by T − h s a model based on random walk theory in which both inter-chain and intra-chain hopping processes where T and T indicate the temperature rise h s are considered. Two important fitting parameters, in R and R , and Q is the Joule heat applied to the h s i.e., d and , are obtained as the characterization 0 heater resistor and one of the platinum/SiN beams. length of the dimensional transition. Our theory suc- cessfully testifies that the hopping mechanism based Electrospinning on the random walk picture is valid and it is use- To fabricate nanoscale PI fibers with controllable ful to explain the diameter dependence of thermal diameters and chain orientations, we utilized elec- conductivity in nanofibers. Nevertheless, there are trospinning using a commercialized electrospinner. still many open questions deserving further investi- The solvent, a mixture of PI and dimethylformamide gation. First, the temperature dependence of ther- (DMF) solution, was prepared with concentrations mal conductivity has not been well explained and it from 45% to 80%, followed by all-night stirring to requires deeper and quantitative understanding of guarantee complete mixing of the PI molecules and the inter-chain thermal transport mechanism. Sec- DMF solvent. The diameter of the PI nanofiber ond, the four parameters in the empirical function should increase with increasing PI molecule weight require validation from further simulations and ex- ratio. periments. For example, κ is closely related quasi−1D to the thermal conductivity of a single chain, and it can be obtained from molecular dynamics; , d SUPPLEMENTARY DATA and  are determined by the configuration of poly- Supplementary data are available at NSR online. mer chains, which requires research on polymer con- Downloaded from https://academic.oup.com/nsr/article/5/4/500/4794956 by DeepDyve user on 12 July 2022 RESEARCH ARTICLE Dong et al. 505 cated by a nano-molding technique. Heat Trans Eng 2013; 34: ACKNOWLEDGEMENTS 131–9. Special thanks go to the technical engineer Ning Liu from ZEISS, 15. Liu J and Yang R. Tuning the thermal conductivity of polymers who helped to obtain the high-resolution SEM image. In addition, with mechanical strains. Phys Rev B 2010; 81: 174122. Lan Dong wants to thank the support and company from Tongren 16. Zhang T, Wu XF and Luo TF. Polymer nanofibers with outstand- Liu in the last few years when pursuing her Ph.D degree in Tongji University. ing thermal conductivity and thermal stability: fundamental link- age between molecular characteristics and macroscopic ther- mal properties. J Phys Chem C 2014; 118: 21148–59. FUNDING 17. Zhang T and Luo TF. Role of chain morphology and stiffness in This work was supported by the National Natural Science Foun- thermal conductivity of amorphous polymers. J Phys Chem B dation of China (11674245, 11334007 and 11505128), the 2016; 120: 803–12. program for Professor of Special Appointment (Eastern Scholar) 18. Wei XF, Zhang T and Luo TF. Chain conformation-dependent at Shanghai Institutions of Higher Learning (TP2014012), thermal conductivity of amorphous polymer blends: the impact the Shanghai Committee of Science and Technology in China of inter- and intra-chain interactions. 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