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Significantly High Thermal Rectification in an Asymmetric Polymer Molecule Driven by Diffusive versus Ballistic Transport Hao Ma, and Zhiting Tian Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b02867 • Publication Date (Web): 07 Dec 2017 Downloaded from http://pubs.acs.org on December 10, 2017
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Significantly High Thermal Rectification in an Asymmetric Polymer Molecule Driven by Diffusive versus Ballistic Transport Hao Ma1 and Zhiting Tian1, 2 ∗ 1
Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA 2 Macromolecules Innovation Institute, Virginia Tech, Blacksburg, Virginia 24061, USA
Abstract Tapered bottlebrush polymers have novel nanoscale polymer architecture. Using non-equilibrium molecular dynamics simulations, we show that these polymers have the unique ability to generate thermal rectification in a single polymer molecule and offer an exceptional platform for unveiling different heat conduction regimes. In sharp contrast to all other reported asymmetric nanostructures, we observed that the heat current from the wide end to the narrow end in tapered bottlebrush polymers is smaller than that in the opposite direction. We found that a more disordered to less disordered structural transition within tapered bottlebrush polymers is essential for generating non-linearity in heat conduction for thermal rectification. Moreover, the thermal rectification factor increased with device length, reaching as high as ~70% with a device length of 28.5nm. This large thermal rectification with strong length dependence uncovered an unprecedented phenomenon – diffusive thermal transport in the forward direction and ballistic thermal transport in the backward direction. This is the first observation of radically different transport mechanisms when heat flow direction changes in the same system. The fundamentally new knowledge gained from this study can guide exciting research into nanoscale organic thermal diodes.
Keywords Tapered bottlebrush polymer Thermal rectification Diffusive transport Ballistic transport
Introduction ∗
Corresponding author. Electronic mail:
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Like electronic diode behavior, thermal rectification is a phenomenon in which heat transports preferentially in one direction. The ability to control heat flow is central to phononics, where thermal devices such as the thermal diode1, 2, thermal transistor3, thermal memory4, and thermal logic gate5 have been proposed. Thermal rectification in nanostructured materials has been attracting significant attention. Beside applications in phononics, nanoscale thermal rectification can also play important roles in nanoscale calorimeters and thermal management of microelectronics.6 In bulk materials, thermal rectification often happens at bi-material junctions7-11. Nanostructures enable thermal rectification in single materials without heterojunctions. Using atomistic simulations, thermal rectification has been observed in asymmetric structures such as carbon nanotubes12, carbon nanohorns13, carbon nanocones14, graphene nanoribbons15, 16, threedimensional diamond pyramids17, and Si nanowires18. The experimental evidence of nanoscale thermal rectification was reported in asymmetrically mass-loaded nanotubes6 and vanadium dioxide beams19. A fundamental understanding of asymmetric thermal transport is still in its infancy. The underlying mechanism for nanoscale thermal rectification is under debate and it should, perhaps, not be considered to be one-fold. Sufficient conditions for thermal rectification in nanostructures were found to vary over a wide range including vibrational spectrum mismatch13-15, asymmetric coupling with thermal contacts17, 20, phonon localization21-23, phonon lateral confinement23, standing wave formation24, and presence of solitons25. Few studies26-28 have been conducted on thermal rectification in organic systems. Most soft organic systems contain amorphous components or are purely amorphous. Thermal transport in amorphous materials can be much more complicated than occurs in crystalline solids. Allen and Feldman classified vibrational modes in amorphous solids into three categories: propagons, diffusons, and locons.29 Propagons are propagating, delocalized modes (i.e., phononlike), diffusons are non-propagating, delocalized modes, and locons are non-propagating, localized modes.30, 31 However, their roles in thermal rectification remain unexplored. In this work, we studied thermal rectification in a novel nanoscale organic material – tapered bottlebrush polymers. Bottlebrush polymers are comprised of a linear polymer backbone with densely grafted side-chain polymers of a single molecular weight. Tapered bottlebrush polymers are composed of side-chain polymers of systematically varied molecular weights that can be tailored to produce a cone-shaped asymmetric macromolecule at the nanoscale32-36. We used large-scale molecular dynamics simulations to investigate the heat current in tapered bottlebrush polymers along opposite directions. We followed convention and defined the forward direction as being from the wide end to the narrow end and the backward direction as being from the narrow end to the wide end. We discovered an unusual thermal rectification effect that had a larger heat flux in the backward direction, which we attribute to the structural disordered gradient within the same molecule and the subsequent non-linear heat transfer regime it leads to. We observed the strong length dependence of thermal rectification ratio, which uncovers a new mechanism – diffusive thermal transport in the forward direction and ballistic thermal transport in the backward direction. The freedom to fully tune the polymer chains offers a unique opportunity to explore the different modes of fundamental thermal transport in one system and actively control the heat flow. This is the first time that thermal transport has been shown to switch from a diffusive to a ballistic regime as the heat flow direction flips within the system. Knowledge gained from this work provides useful insight into and guidance on optimization of 2 ACS Paragon Plus Environment
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thermal rectification in tapered bottlebrush polymers and, more importantly, lays the groundwork for using asymmetric polymers as thermal diodes. Method We built all the structures in the BIOVIA Materials Studio37. The chemical structure of tapered bottlebrush polymers is shown in Figure 1. We used polynorbornene (PNb) as the backbone and polystyrene (PS) as the sidechains.38-40 We first performed geometry optimization using the Forcite module with Polymer Consistent Force-Field (PCFF) in the Materials Studio (Figure 2(a)). PCFF, a member of the ab initio consistent force-field family, is intended for applications in polymers and organic materials. It is a class II potential which includes anharmonic bonding terms 41, 42.
Figure 1. Chemical structure of a PNb-graft-PS bottlebrush polymer where “n” denotes the number of backbone units and “m” denotes the number of side-chain units. We performed non-equilibrium molecular dynamics (NEMD) simulations with PCFF potential using the LAMMPS package43. We set the timestep to be 0.5fs. We first relaxed the system in an NVT ensemble for 5 10 timesteps with the Nose-Hoover thermostat44, 45. Afterwards, NEMD was performed for another 8 10 steps with the Berendsen thermostat 46 as the heat baths (Figure 2(a)). We used the backbone atoms as the heat baths after careful investigation of different choices of heat baths as shown in the Supporting Information (SI). Note that the conformation during the MD simulations is different from the initial conformation before the MD simulations (Figure 2(b)). Because there are only weak van der Waals interactions among the sidechains, the sidechain atoms vibrate at larger amplitudes than backbone atoms. Also, the different sidechain lengths provide an even larger degree of freedom for sidechains to vibrate in tapered bottlebrush polymers compared to their bottlebrush polymer counterparts where steric effects restrict the space of sidechain atoms. We chose ∆ 100 to reduce thermal noise. We set 200 to make sure the whole system (including the hot side) is well below the glass transition temperature of PS (around 360K). The temperature range studied in this work is on the same order of the Debye temperature of PS (190~230 )47. Since we are concerned with temperature differences and relative variations of heat current, quantum correction was not included. We set the hot and cold side temperatures to be ∆ and ∆ , respectively. The heat current (the energy transported across the system per unit time) due to the temperature bias is calculated as , where is the total energy that has been 3 ACS Paragon Plus Environment
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subtracted from the cold heat bath. We used heat current in the cold bath because the heat current in the hot bath fluctuates at least twice as much (Figure S3 in SI). We then defined thermal rectification ratio ( ) as " $" ! #" % & 100% (1) %
where the subscript ‘+’ indicates heat transport in the forward direction and ‘-’ indicates heat transport in the backward direction.
Figure 2. (a) Optimized structure of a tapered bottlebrush polymer device (n=30, m=30:1:1) with heat baths and fixed ends before MD simulations. Green denotes the backbone chain and pink denotes the sidechains. Red and blue denote heat baths and each heat bath contains 40 atoms; black denotes fixed boundaries. Yellow arrows denote heat flux directions. (b) A snapshot of a tapered bottlebrush polymer device (n=30, m=30:1:1) from Visual Molecular Dynamics (VMD)48 during MD simulations.
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X-ray diffraction (XRD) pattern is a primary technique for determining the degree of crystallinity in polymers. XRD intensity is calculated based on a mesh of reciprocal lattice nodes defined by the entire simulation domain using a simulated radiation of wavelength (. The XRD intensity, I, at each reciprocal lattice point is computed from the structure factor, F, using the equations: -∗-
) *+ !,&
/
0!1& ∑/ 4?@ !AB& CDE !B&?FG@ !B&
|L| A
(2) (3) (4) (5)
where 1 is the location of the reciprocal lattice node, ;4 is the position of each atom, 34 is the atomic scattering factors, *+ is the Lorentz-polarization factor, and , is the scattering angle of diffraction.49 The radius of gyration (Rg) is a measure of the size of a polymer coil. If we compare two polymers with the same chain length, a larger radius of gyration indicates a more stretched structure (Figure S4 in SI). The Rg of a particular group at a given time is defined as = MNA ∑F PF !;F ;Q &A (6) O where M is the total mass of the group, rcm is the center of mass position of the group, and the summation is over all atoms in the group. We calculated Rg in LAMMPS. Results and Discussion Direction of Thermal Rectification The major and counterintuitive finding is that heat current in the backward direction is bigger, i.e., > < $ , unlike > > $ that has been found in all asymmetric nanostructures previously reported,12-15, 18, 23 although their temperature profiles (Figure S6 in SI) are all similar. We investigated structural disorder first. XRD intensities were calculated on a three-dimensional mesh of reciprocal lattice nodes. Meshes defined by the simulation domain must at least contain periodic boundary conditions in one direction. Because we only have a single molecule with fixed boundary conditions, we cannot directly calculate the XRD of tapered bottlebrush polymers. To characterize the crystallinity of both the wide and narrow ends of our tapered bottlebrush polymers, we built two separate bottlebrush polymers (BB-2 and BB-25) in the Materials Studio with the same chemical structure as shown in Figure 1. BB-2 and BB-25 both have a backbone length of n=10 but have different sidechain lengths: BB-2 (m=2) and BB-25 (m=25). We put 4 parallel chains in a periodic box and calculated their XRD patterns in LAMMPS after fully relaxing the systems. XRD patterns of BB-2 and BB-25 are shown in Figure 3a. Clearly, BB-2, with short sidechains, has more sharp peaks, indicating higher crystallinity. BB-25, with long sidechains, gives broader peaks that are characteristic of a more disordered amorphous phase. We also calculate XRD pattern of BB-2 and BB-25 with similar number of atoms in the simulation box, which shows similar trend. (Figure S5 in SI) Comparison 5 ACS Paragon Plus Environment
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between BB-2 and BB-25 indicates that decreasing sidechain length in a tapered bottlebrush polymer can form a structural transition from a more disordered phase to a less disordered one. To further validate the existence of structural transition in tapered bottlebrush polymers, we directly calculated Rg at both ends of the tapered bottlebrush polymer (n=30, m=30:1:1). More specifically, we calculated Rg of m=30th-27th in the wide end and m=4th-1st in the narrow end, respectively, and obtained an Rg (wide end)=1.70nm and an Rg (narrow end)=0.75nm. If the sidechains are stretched straight, their length ratio is ~10, but the Rg ratio is only ~2. This means that the sidechains at the wide end are more entangled and more disordered than those at the narrow end, which is consistent with our XRD results. Both XRD and Rg calculations confirm that there is a structural transition from the more disordered wide end to the less disordered narrow end, as sketched in Figure 3b.
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Figure 3. (a) The XRD pattern of BB-2 and BB-25 obtained from LAMMPS. Blue denotes BB-2 and Red denotes BB-25. (b) Schematic of the more disordered to less disordered structural transition in a tapered bottlebrush polymer.
How does the structural disorder gradient give rise to thermal rectification? Thermal conductivity and its temperature dependence vary from the more disordered wide end to the less disordered narrow end, which leads to a non-linear heat transfer regime with an inseparable thermal conductivity dependence on both temperature and space. This non-linear regime is a necessary condition for thermal rectification10, 50, 51. In the temperature range that we considered in this work (100-300K), the more disordered long sidechain polymers showed nearly temperature independence, while the less disordered short sidechain polymers had positive temperature dependence (Figure 4). Given 200 and ∆ 100 , thermal conductivity of the wide end in the forward and backward directions is very close, while thermal conductivity for the narrow end in the forward direction is smaller than that in the backward direction. Overall, the backward thermal conductivity is higher and results in a larger heat current in the backward direction.
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Figure 4. Temperature dependence of BB-2 single chain and BB-10 single chain at 100 K~300 K. Blue denotes BB-2 and red denotes BB-10. BB-10 has the same backbone length as BB-2 but has a different sidechain length: m=10. At the vibrational mode level, one possible interpretation is that the disorder asymmetry leads to different localization levels. We speculate that the wide end has more locons than the narrow end. Despite lack of good methods to directly identify locons within tapered bottlebrush polymers, we previously demonstrated that even BB-2 has significantly more locons than PNb (backbone) in frequency ranges of 50-90 THz due to the existence of sidechains, as shown in participation ratio plots of Figure 3d in Reference 40. Locons do not significantly contribute to thermal conductivity52, could trap energy53 and become the bottleneck for heat conduction. Localization asymmetry can thus contribute to the spatial dependence of thermal conductivity23. Length Dependence of Thermal Rectification Using gradual change in sidechain length, we investigated the effect of the device length by setting n=10, 18, 30, 36, and 48. Interestingly, η continues to increase as the device length increases (Figure 5a). Remarkably, the thermal rectification ratio can reach as high as !69.8 ± 2.6&% at ∆ 100 for n=48 and m=48:1:1 at a device length of 28.5nm. These findings raise several fundamental questions: How can we explain the strong length dependence? How could the relatively small thermal conductivity and weak temperature dependence generate such a large thermal rectification? Since the cross sectional area varies gradually from one to the other end, we defined the effective cross-sectional area, A, of tapered bottlebrush by averaging the cross-sectional area over length. We used the averaged cross sectional area of tapered bottlebrush polymer and calculated thermal conductivity. We plotted k as a function of device length, L, in Figure 5b. The structural asymmetry ratio is defined by the ratio of longest sidechain length to the shortest sidechain length. When L
