Negative Correlation Between Thermal and Electrical Conductivity in Epsilon‐Negative Nanocomposites

Epsilon‐negative materials (ENMs) hold promise for the advancement of the next generation of electronic devices. Most epsilon‐negative materials strongly correlate with metal properties, which limits their applications in electronic packaging. Instead, achieving a negative permittivity in the insulating state is expected to show the decoupling of electrical and thermal conductivities, and experimental demonstration of this behavior is lacking. In this study, multi‐walled carbon nanotubes (MWCNTs)@polydopamine (PDA)‐silver/polyimide (PI) nanocomposites are engineered to achieve weakly negative permittivity, which is attributed to the localized plasma oscillations. The PDA layer and nano‐Ag are exploited to confine electrons with MWCNTs for improving energy transport while perturbing directional current, thereby realizing high thermal conductivity and low electrical conductivity. This work provides insights into the fundamental nature of heat and charge transport in epsilon‐negative systems.


Introduction
[3] Nevertheless, this will inevitably result in the conversion of electromagnetic energy into heat as electronic devices evolve toward miniaturization, high integration, and high power density, leading to a reduction in the device's lifespan and reliability when using these materials.Therefore, high thermal conductivity is critical for such devices.Due to the Wiedemann-Franz (WF) law, thermal conductivity is proportional to electrical conductivity at a certain temperature. [4,5]For material with high thermal conductivity, this will lead to a high electrical conductivity which will cause electromagnetic interference and is detrimental to electronic devices.As a result, new requirements are called on packaging materials (including substrate material, thermal interface material, and cover plate) in terms of lighter, thinner, higher thermal conductivity, and electrical insulation. [6,7]onsequently, how to engineer the material to have ENMs with charge insulating and thermal metallic state becomes an urgent topic.
Plasma oscillations in transport are an unambiguous signature of negative permittivity.Most ENMs in normal conditions hold a positive relation between thermal conductivity and electrical conductivity.0] When electrons are confined to a narrower region (one dimension), the plasma will collectively oscillate locally when subjected to an external sinusoidal force.Although electron transport carries charge and energy (heat), localized plasma oscillations dramatically affect the charge and thermal conductance individually.The localized collective excitations significantly increase the local state density of the electrons, which can enhance the strength of coupling between electrons and phonons, resulting in effective energy transfer. [11,12]On the other hand, in local plasmas, increased Coulomb repulsion between the electrons due to their reduced spatial freedom introduces a frictional dissipation that causes strong perturbation to directional currents, resulting in finite conductivity even with the increased electron density. [13]As a consequence, the positive relation between thermal conductivity and electrical conductivity is generally violated.This phenomenon has been observed in a variety of materials with plasmonic properties, including graphene, and quasi-onedimensional conductors. [14,15]ecent studies have shown that multi-walled carbon nanotubes (MWCNTs) exhibit negative permittivity and plasmon behavior. [16]The plasmonic properties of MWCNTs strongly depend on their size, shape, and surface chemistry. [17,18]By controlling these factors, it is possible to tune the plasmonic properties of MWCNTs.[21] First, surface modification has been explored using the mussel-inspired polydopamine (PDA) method to tailor the molecular structure of the filler. [22]Second, Ag is utilized to attach to the surface of the PDA layer because it has a Coulomb blocking to restrict electron movement and has high thermal conductivity. [23,24]The presence of abundant-OH and -NH 2 groups on the PDA layer provides a molecular platform for the electrostatic attraction of precursors (Ag + ), which are subsequently anchored onto MWCNTS@PDA surfaces through reduction to form MWCNTs@PDA-Ag. [25,26] Third, polyimide (PI) opted to be exploited as the matrix material due to its outstanding thermal stability, mechanical prop-erties, electrical insulation, and corrosion resistance. [27,28]e successfully engineered the MWCNTs@PDA-Ag/PI nanocomposites and realized high thermal and low electrical conductivity.

Localized Plasma Oscillations and Enms with High Thermal and Low Electrical Conductivity
Plasma oscillations are the collective behavior of electrons when a perturbation is introduced, for example, by applying an external electric field, electrons would rapidly oscillate at certain equilibrium positions (Figure S1, Supporting Information). [29]When plasma oscillations induced within the conducting network constructed by naked MWCNTs are cut off through the PDA layer, which confines electrons within MWCNTs, resulting in the generation of localized plasma oscillations (Figure 1a).To better understand localized plasma oscillations and how they affect thermal and electrical conductivity in ENMs, we analyze the electrons' state in localized mode.Here, we assumed the momentum of electrons to be distributed in two orthogonal directions along the a and b axis, as shown in Figure S2 (Supporting Information).Acting together with the Coulomb force and the applied electric field force, the free electrons would rapidly oscillate along the a-axis at certain equilibrium positions.As this is a shortrange process, it results in limited electrical conductivity.Heatcarrying electrons frequently interact with other particles, such as carbon atoms and defects, leading to strong plasmon-phonon interactions that facilitate efficient energy transfer. [30,31]With consideration of electron speed along the b-axis, the oscillations propagate to other regions, creating the localized plasma oscillations within the MWCNTs.Electrons confined in MWCNTs are subject to significant self-repulsion, which causes strong perturbation to the orientation of the electron flow.This confinement leads to enhanced thermal conductivity but reduced electrical conductivity (Figure 1b).The programming of electron confinement within respective MWCNTs is depicted in Figure 1c.The schematic for the preparation of MWCNTs@PDA-Ag/PI nanocomposites is shown in Figure S3 (Supporting Information).In our approach, the MWCNTs are first acid-treated to remove the possible functional groups.Subsequent coating, a layer of PDA was coated onto the MWCNTs through self-polymerization of dopamine, creating an electrical barrier that prevents electrons from migrating among the tubes. [32]The PDA layer is rich in catechol and secondary amino groups, endowing it with a strong reducing capability that facilitates the transfer of Ag + to form nano-Ag. [33] Consequently, nano-Ag is decorated onto the PDA layer to produce MWCNTs@PDA-Ag without requiring additional reducing agents. [34]The electrical conductivity at 100 kHz and thermal conductivity of the resultant nanocomposites at room temperature are presented in Figure 1d.The MWCNTs-filled nanocomposites show an electrical conductivity of 1.29 × 0 −1 S cm −1 , while MWCNTs@PDA-filled nanocomposites exhibit a lower electrical conductivity of 3.9 × 10 −3 S cm −1 , which is two orders of magnitude smaller than the former.The drastic reduction in electrical conductivity is attributed to the disruption of electron interactions between nanotubes by the PDA layer.However, the electrical conductivity of MWCNTs@PDA-Ag-filled nanocomposites shows no noticeable change.Despite the expected linear decrease in thermal conductivity as per the WF law, an opposite trend was observed.The thermal conductivity of MWC-NTs@PDA/PI nanocomposites was significantly enhanced compared to MWCNTs/PI nanocomposites.This enhancement is at-tributed to the formation of localized plasma modes due to the confinement of electrons, resulting in a significant increase in the local density of states.These modes can interact strongly with phonos, leading to an increased energy transfer between them and enhanced thermal conductivity.The introduction of Ag onto the PDA layers further improves the thermal conductivity due to the high thermal conductivity of Ag itself.The sizedependent thermal conductivity of Ag nanoparticles is also noted, with smaller particles having higher thermal conductivity due to the increased number of surface atoms, enhancing phonon scattering and heat transfer. [35,36]These results demonstrate that the MWCNTs@PDA-Ag/PI nanocomposites exhibit both high thermal conductivity and low electrical conductivity.

Surface Chemical Analysis
We begin with conducting the surface chemical analysis of the resulting samples.Figure 2a displays the XRD patterns of MWC-NTs, MWCNTs@PDA, and MWCNTs@PDA-Ag.The diffraction peaks observed at 25.5 o and 43.9°correspond to the (002) and (100) planes of MWCNTs, which are typical graphitized MWC-NTs features. [37]The absence of crystalline peaks upon PDA introduction confirms that no pure PDA phase was formed.The vertical red lines at the bottom of the XRD represent the standard PDF card number for Ag.The XRD pattern of MWCNTs@PDA-Ag displays new peaks at 38.1 o , 44.5 o , 64.7 o , and 77.3 o , which correspond to the (111), (200), (220), and (311) planes of facecentered cubic silver, respectively. [38]This is consistent with prior studies on Ag+ reduction, indicating successful decoration of nano-Ag on the PDA surface.Additionally, XPS was utilized to further analyze the surface chemistry of the modified MWC-NTs (Figure 2b,c).Compared with the MWCNTs, the presence of an N 1s peak at ≈400 eV and an enhanced intensity of O 1s peak at ≈532 eV in MWCNTs@PDA and MWCNTs@PDA-Ag confirm the successful PDA coating. [39]Furthermore, the XPS spectrum of MWCNTs@PDA-Ag presents two peaks at 367 and 373 eV, which correspond to the Ag 3d 5/2 and Ag 3d 3/2 peaks of Ag nanoparticles, respectively. [40]Figure 2d-f directly illustrates the morphologies of surface-modified MWCNTs by TEM.PDA-modified MWCNTs display a rough, light-dark surface, indicating that a PDA layer ≈10 nm was successfully coated on the MWCNT surface.After decorating with Ag nanoparticles, many nano-Ag nanoparticles are homogeneously embedded in the PDA layers.These results provide evidence of the successful preparation of MWCNTs@PDA-Ag.SEM images of fractured cross-sections of the PI-based nanocomposites demonstrate the emergence of MWCNTs in the polymer matrix, and filler-matrix debonding and nanoparticle agglomeration are observed, as shown in Figure 3.In contrast, SEM images of MWC-NTs@PDA/PI and MWCNTs@PDA-Ag/PI were found to show fillers embedding into the polymer matrix, agglomeration and voids, and filler-matrix debonding reduced.These results indicate that the compatibility between the surface-modified MWC-NTs and the PI matrix was greatly improved.

Negative Permittivity in the Case of Free and Localized Plasma Oscillations and Electrical Conductivity Properties
To understand how the structure of negative dielectric materials affects heat transfer and electrical conductivity, we analyze the changes in electron state in both cases through the analysis of the key parameters, such as plasma frequency, damping factor, and relaxation time.Figure 4 presents the dielectric and electrical properties of PI-based nanocomposites filled with MWCNTs, MWCNTs@PDA, and MWCNTs@PDA-Ag at room temperature.The real permittivity of the nanocomposites is shown in Figure 4a,b, and the dispersion of the permittivity can be effectively controlled by varying the type of fillers.The epsilon-negative behavior observed in these nanocomposites arises from the plasma oscillation of electrons in terms of electrodynamics. [41]Mesoscopically, the negative permittivity of MWCNTs/PI nanocomposite is attributed to the conductive network built by naked MWCNTs, with the plasma flowing through the conducting path called free plasma oscillations.This behavior can be accurately described by the Drude model: [42] The plasma frequency  p is primarily determined by the electron average concentration n and the effective mass m, while the damping factor   represents a measure of the rate at which the oscillations decay, and is related to the scattering of electrons.Note that the  p is derived based on the assumption that the mass of atoms is significantly greater than that of the electrons.The permittivity becomes negative when the frequency  is below  p , and the negative effect increases as the frequency decreases.The experimental data are in excellent agreement with a theoretical model, revealing the presence of a low-frequency plasmonic state. [43]The strength and width of the negative permittivity region depend on the magnitude of  p and   .A larger  p or a smaller   will result in a stronger and broader negative permittivity region.Conversely, a smaller  p or a larger   will lead to a weaker and narrower negative permittivity region.In comparison to MWCNTs/PI nanocomposites, the negative permittivity of MWCNTs@PDA/PI and MWCNTs@PDA-Ag/PI nanocomposites show significant suppression.The absolute values are reduced by three orders of magnitude and become weakly frequency-dependent.The presence of the PDA layer effectively severs conductive pathways and confines the plasma within MWCNTs, thereby impeding the long-range movement of plasma and inducing a localized plasma oscillations mode. [44,45]e conducted an analysis of the variation of  p and   in the localized mode.When electrons are confined in thin wires by the PDA layer, both the number density and effective mass of electrons are simultaneously enhanced, so it is ambiguous to determine the change of  p based on equation (2). [46]However, an increase   is determined from the small slope of ′() curve in Figure 4b, which is consistent with theoretical predictions that localized mode has strong electron-electron interactions.Additionally, the introduction of nano-Ag could increase  p and   , but enhanced   makes more contributions to a reduction in negative permittivity compared to  p . [47]The dielectric losses of PI-based nanocomposites are shown in Figure S4 (Supporting Information).Compared with MWCNTs/PI nanocomposites, the dielectric losses of MWCNTs@PDA-Ag/PI nanocomposites are significantly increased, enhancing the attenuation of electromagnetic waves.Turning now to the electrical conductivity that shows the itinerant nature of the charge, as is displayed in Figure 4c.The frequency-dependent electrical conductivity at room temperature exhibits significantly distinct for three different samples.For MWCNTs/PI nanocomposites, the electrical conductivity drastically decreased with frequency due to the skin effect, indicating a signature of metal-like behavior.The behavior can be welldescribed by the following equation, [48]  ac where  dc = ne 2 /mdenotes DC electrical conductivity and  is the relaxation time that is inversely proportional to   .The fitting result in Figure 4c is consistent with the experimental data, providing strong evidence that  ac is primarily contributed by a large number of free electrons from the conducting network.In this case,  is relatively large, which results in a large  dc , while  is responsible for reducing  ac at high frequencies.Compared to the MWCNTs-filled nanocomposite, the electrical conductivity of filled-MWCNTs@PDA nanocomposite is significantly decreased to 3.9 × 10 −3 S cm −1 at 100 kHz because the PDA layer disrupts the electron interactions between nanotubes. [49]The  ac becomes nearly frequency-independent throughout the test band, while  dc dominates across the conductivity.When electrons are confined within MWCNTs,  become smaller, thereby producing smaller  dc and .In the low-frequency limit (<<1),  ac ≃  dc . [50]Importantly, the resulting MWCNTs@PDA can be regarded as one-dimension or quasi-one-dimension exhibiting strong electron-electron interactions, leading to strong perturbation to directionally electri-cal flow. [51,52]The electrical conductivity of MWCNTs@PDA-Agfilled nanocomposite slightly increased, likely due to the accumulation and interconnection of Ag nanoparticles.To highlight the nano-Ag effect against electron confinement, nanocomposites' negative permittivity and electrical conductivity were analyzed via the application of DC bias, as depicted in Figure 4d-f.The nano-Ag has the nature of Coulomb blocking, which limits the ability of electrons to move freely regardless of the magnitude of the DC bias. [53]It is found that the real permittivity and electrical conductivity of MWCNTs@PDA-Ag-filled nanocomposite showed a weak perturbation under an elevated external DC bias.As a comparison, the enhanced negative permittivity and electrical conductivity of MWCNTs@PDA/PI nanocomposite are presented in Figure S5 (Supporting Information).These results demonstrate that the MWCNTs@PDA-Ag/PI nanocomposites maintained weakly negative permittivity and low electrical conductivity even when subjected to a 2 V DC bias, revealing the stability of electron confidence by the PDA layer and nao-Ag.

Conclusions
In conclusion, the MWCNTs@PDA-Ag were synthesized through self-polymerization of DA and electroless Ag plating, and the resulting MWCNTs@PDA-Ag/PI nanocomposites were expected to exhibit weak epsilon-negative behavior, low electrical conductivity, high thermal conductivity, and high dielectric loss.The weakly epsilon-negative behavior was attributed to the localized plasma oscillation modes.The PDA layer and nano-Ag successfully confine electrons within MWCNTs.Compared to MWCNTs/PI nanocomposites, the negative permittivity and electrical conductivity of MWCNTs@PDA-Ag/PI nanocomposites were markedly suppressed, while their thermal conductivity was significantly enhanced.This work establishes the MWCNTs@PDA-Ag/PI system as having potential applications in electromagnetic attenuation and provides insights into the fundamental nature of heat and charge transport in an epsilon-negative system.

Figure 1 .
Figure 1.a) Schematic of transition from plasma oscillations induced by conducting network to localized mode induced by single MWCNTs.b) The schematic presents the variation of heat and charge current in ENMs.c) Preparation process of MWCNTs@PDA-Ag.PDA layer is through the selfpolymerization of DA and nano-Ag is by reducing Ag+.d) Electrical conductivity at 100 kHz and thermal conductivity of nanocomposites filled with different fillers.

Figure 2 .
Figure 2. a) XRD patterns of three samples.b) Full XPS spectra and c) the spectra from 350 to 420 eV of three samples.d-f) TEM images of the three samples, where the PDA layer is ≈10 nm and black dots represent nano-Ag.

Figure 3 .
Figure 3. SEM images for the fractured cross-sections of PI-based nanocomposites containing different types of fillers, namely a) MWCNTs, b) MWC-NTs@PDA, and c) MWCNTs@PDA-Ag.

Figure 4 .
Figure 4. a,b) Real permittivity spectrums and c) electrical conductivity spectrums of PI-based nanocomposites with as-received MWCNTs, MWC-NTs@PDA, and MWCNTs@PDA-Ag without DC bias, and the red lines indicates the negative permittivity and electrical conductivity fitted by the Drude model, respectively.d) Real permittivity spectrums and e) electrical conductivity spectrums of MWCNTs@PDA-Ag/PI nanocomposites with DC bias.f) Electrical conductivity value at 100 kHz of MWCNTs@PDA-Ag/PI nanocomposites.