Tunable Thermoelastic Anisotropy in Hybrid Bragg Stacks with Extreme Polymer Confinement

Abstract Controlling thermomechanical anisotropy is important for emerging heat management applications such as thermal interface and electronic packaging materials. Whereas many studies report on thermal transport in anisotropic nanocomposite materials, a fundamental understanding of the interplay between mechanical and thermal properties is missing, due to the lack of measurements of direction‐dependent mechanical properties. In this work, exceptionally coherent and transparent hybrid Bragg stacks made of strictly alternating mica‐type nanosheets (synthetic hectorite) and polymer layers (polyvinylpyrrolidone) were fabricated at large scale. Distinct from ordinary nanocomposites, these stacks display long‐range periodicity, which is tunable down to angstrom precision. A large thermal transport anisotropy (up to 38) is consequently observed, with the high in‐plane thermal conductivity (up to 5.7 W m−1 K−1) exhibiting an effective medium behavior. The unique hybrid material combined with advanced characterization techniques allows correlating the full elastic tensors to the direction‐dependent thermal conductivities. We, therefore, provide a first analysis on how the direction‐dependent Young's and shear moduli influence the flow of heat.


Section S0. Summary of Experimental Section
Sample preparation. The synthetic clay sodium fluorohectorite (Hec, [Na0.5] inter [Mg2.5Li0.5] oct [Si4] tet O10F2) was delaminated by immersing it into Millipore water (0.5 wt%). The aqueous PVP solution (1 wt%) was added in the desired weight ratio. The suspension was mixed for at least one day in an overhead shaker. The homogeneity of the suspension was crosschecked by SAXS measurements. Self-supporting films were prepared using a fully automatic spray coating system. Every spraying cycle is followed by a drying cycle of 90 s at a temperature of 55 °C. We prepared pure PVP, pure Hec, and four hybrid Hec/PVP films. The self-supporting films were characterized by thermogravimetric analysis, XRD, and TEM. Additional information about the sample preparation can be found in Section S1.
In-plane thermal conductivity measurements. Lock-in thermography measures the temperature spreading across the free-standing samples upon thermal excitation by a focused laser beam with a modulated intensity. To prevent convective heat losses, the experiments are conducted in a vacuum chamber. The amplitude and phase data are extracted from the radial temperature distribution. The thermal diffusivity is then fitted by the slope method for thermally thin films. With the density, determined by helium pycnometry, and the specific heat, determined by differential scanning calorimetry (DSC), the thermal conductivity can be calculated. More details are provided in Section S2.
Cross-plane thermal conductivity measurements. The photoacoustic method uses a modulated laser beam to periodically heat a transducer layer in intimate contact with the sample. The heat is converted into an acoustic wave propagating into a gas tight cell above the sample, which is filled with helium at 20 psi. A sensitive microphone detects the phase shift between the acoustic signal and the modulated laser by a lock-in amplifier. The frequencydependent phase shift is then compared to a multilayer model, assuming one-dimensional heat transfer. Therefrom, the total layer resistance is obtained. With the film thickness determined by AFM, the effective thermal conductivity is calculated. More details are provided in Section S2.
Brillouin light spectroscopy (BLS). BLS measures the phonon dispersion, ω(q), by detecting the Doppler frequency shift, ω, of the inelastically scattered light by sound waves ("phonons") with a wave vector, q. We recorded BLS spectra utilizing three scattering geometries (transmission, reflection, and backscattering) and two polarization configurations of the incident (λ = 532 nm) and scattered light (polarized VV, depolarized VH), which allowed us to establish the nature of the observed phonons. We varied the incident angle to obtain the directiondependent sound velocities necessary for the determination of the anisotropic elasticity. The elastic stiffness tensors were obtained from the representation of the direction-dependent sound velocities by the Christoffel equation assuming transverse isotropy. The characteristic Young's moduli, shear moduli, and Poisson's ratios of the Bragg stacks were subsequently calculated. More details can be found in Section S3. Section S1. Sample preparation

Film preparation
For the delamination, the synthetic Hec was immersed into Millipore water (0.5 wt%). The complete delamination was studied by small angle X-ray scattering (SAXS). [2] The aqueous PVP solution (1 wt%) was added in the desired weight ratio. The suspension was mixed for at least 1 day in the overhead shaker. Homogeneity of the suspensions was cross-checked by SAXS measurements. All SAXS data were measured using the small-angle X-ray system "Double Ganesha AIR" (SAXSLAB, Denmark). The X-ray source of this laboratory-based system is a rotating anode (copper, MicroMax 007HF, Rigaku Corporation, Japan) providing a micro-focused beam. The data were recorded by a position sensitive detector (PILATUS 300 K, Dectris). To cover the range of scattering vectors between 0.004-1.0 Å -1 , different detector positions were used. The measurements of the suspensions were done in 1 mm glass capillaries (Hilgenberg, code 4007610, Germany) at room temperature. To improve the detection limit of the inhouse machine, the suspensions were first concentrated to » 10 wt% by centrifugation at 10000 rpm for 1 hour. A rational basal series was found for all suspensions, indicating that all clay nanosheets were separated to exactly the same distance by water and PVP ( Figure S2). Within experimental errors reaggregation to stacks of Hec can be excluded by the absence of a reflection typical for crystalline hydrated Hec phases at q = 0.65-0.41 Å -1 . The self-supporting films were prepared by spray coating. The fully automatic spray coating system was equipped with a SATA 4000 LAB HVLP 1.0 mm spray gun (SATA GmbH & Co. KG, Germany). Suspensions were sprayed on a corona-treated polyethylene terephthalate (PET) foil (optimont 501, bleher Folientechnik, Germany). The spraying and nozzle pressure were set constant at values of 2 and 4 bar, respectively. The round per flat fan control was set to 6 with a flow speed of 3 mL s -1 . The distance between the spraying gun and the substrate was 17 cm. The thickness of the suspension layer applied in one spraying step is » 2 µm which corresponds to » 20 nm dry nanocomposite film thickness. For drying the suspension layer, the sample is stopped under infrared lamps until evaporation of the solvent is complete. After every spraying cycle, a drying cycle of 90 s with a temperature of 55 °C took place. The spraying/drying cycle is repeated until the desired barrier film thickness of 50 µm is obtained. Afterwards, the film was dried at 100 °C for 3 days and peeled off from the PET foil to achieve self-supporting films. For characterization by photoacoustic analysis thinner films on the order of a few µm were spray coated onto clean 1 mm thick microscopy glass slides. In total, we prepared six samples, which are denoted as Hec0/PVP100, Hec23/PVP77, Hec31/PVP69, Hec40/PVP60, Hec51/PVP49, and Hec100/PVP0, respectively (Table S1). Throughout the manuscript, all samples are referred to by the nominal volume fractions (vol%) of Hec and PVP, respectively. To rule out compositional changes during spray coating, these ratios were cross-checked (Table S1) by thermogravimetric analysis (TGA), using a Linseis STA PT 1600 (Linseis Messgeräte GmbH, Germany). Changes in mass observed upon heating in synthetic air up to 800 °C were attributed to the combustion of PVP. X-ray diffraction (XRD) patterns for the films were recorded in Bragg-Brentano-geometry on an Empyrean diffractometer (PANalytical B.V.; the Netherlands) using Cu Kα radiation (λ = 1.54187 Å). The self-supporting films were placed on glass slides (Menzel-Gläser; Thermo Scientific). Prior to the XRD measurements, the samples were dried at 100 °C in a vacuum chamber for one week.   As a measure of the quality of the one-dimensional crystallinity of the films, the coefficient of variation (CV) and the full width at half maximum (FWHM) were determined (Table S1). Large CV-values (» 3% [3] ) and large FWHM indicate non-rationality of the diffraction pattern as caused by a random interstratification of different interlayer heights. Assuming PVP and Hec densities of 1.2 g cm -3 and 2.7 g cm -3 , respectively, nominal d-spacings can be calculated for the various volume ratios (dnominal in Table S1). [4] They agree reasonably with those obtained from XRD measurements (dXRD in Table S1). In analyzing the cross-plane thermal conductivity using the series resistance model, we used the dXRD values.

Section S2. Thermal measurements
For the determination of the in-plane and cross-plane thermal conductivity, the density and the specific heat are needed. Therefore, Helium pycnometry and differential scanning calorimetry (DSC) were used. We determined the in-plane thermal diffusivity by lock-in thermography, and the cross-plane thermal conductivity by the photoacoustic method. Prior to the measurements, the samples were dried at 100 °C in a vacuum chamber for one week.

Helium pycnometry
The density of the samples was measured by helium pycnometry. Therefore, an Ultrapyc 1200e (Quantachrome Instruments) was used. Prior to each measurement the volume of the empty measurement cell was measured. Afterwards, small pieces of the free-standing films were weighed into the sample cell with a nominal volume of 1.8 cm³. One hundred runs were performed to determine the volume of the films at room temperature. By knowing the mass (measured on a fine balance) and the volume, the density of the samples was calculated.

Differential scanning calorimetry
The specific heat was determined by DSC measurements according to the ASTM E1269 standard. The samples were freeze ground for better processability and contact to the DSC pans. The measurements were performed on a TA instruments Discovery DSC 2500. An isothermal step (1h, 100 °C) was conducted prior to the measurement to ensure dry conditions. Then two heating cycles were used and only the second cycle evaluated. The temperature profile ranged from -40 to 250 °C using a heating rate of 20 K min -1 with a nitrogen flow of 50 mL min -1 .

Lock-in thermography
The in-plane thermal diffusivity of free-standing Hec/PVP stack films was obtained by lock-in thermography (LIT). The self-built LIT set-up is shown in Figure S9. The sample is heated by a laser beam (Genesis MX 532-1000 SLM OPS, Coherent, λ = 532 nm) focused onto the sample surface by a lens of 150 mm focal length. The intensity of the laser is modulated using a shutter (SH05/M, Thorlabs) controlled by a shutter controller (SC10, Thorlabs). The emitted infrared (IR) radiation of the sample surface is detected by an Infratec VarioCAM HD research IR camera (spectral window: 7.5-14 µm). The camera is equipped with a close-up lense. In this configuration, the minimum spatial resolution is 29 µm (working distance: 33 mm). Since heat losses to the environment lead to an overestimation of the thermal diffusivity [5] all samples were measured under vacuum conditions (~3x10 -3 mbar). Furthermore, all samples were coated with a 20 nm carbon layer for enhanced laser absorption. The coating of the sample was facing to the IR camera. LIT measurements were performed using Infratec's IRBISactiveonline software. Measurements were conducted at several lock-in frequencies between 0.219 and 1.765 Hz (depending on the Hec/PVP composition). Furthermore, each measurement was averaged over several (600-2000) lock-in periods to enhance the signal to noise ratio, with the first 60-100 periods being discarded. The software calculates automatically the amplitude and phase of the temperature oscillations ( Figure S10). The red point marks the midpoint of the laser excitation and thus the midpoint of the radial distribution. A self-written Python script is used to extract radial profiles for the phase and amplitdue images ( Figure S11). The thermal diffusivity is calculated from the phase and amplitude slopes according to the slope method of a thermally thin film [5b] : . (S1) Here, is the slope of the linear relation of the phase and the radial distance r, is the slope of the linear relation of the natural logarithm of the amplitude T multiplied by the square root of the radial distance r, flockin is the lock-in frequency, and α|| is the in-plane thermal diffusivity.
Three films have been measured for each Hec/PVP composition. An average in-plane thermal diffusivity and a standard deviation were calculated for each composition, as summarized in Table S2.

Sample
In-plane thermal diffusivity (mm 2 s -1 ) The in-plane thermal conductivity was calculated from the in-plane thermal diffusivity (α||), density (ρ), and specific heat (CP) as

Photoacoustic method
The cross-plane thermal conductivity was determined by the photoacoustic method. It uses the photoacoustic effect to determine the thermal properties of a sample. Therefore, a modulated laser beam (λ = 488 nm) periodically heats the sample. For good absorption of the laser energy a thin Au transducer layer (~ 150 nm) was coated on the sample surface. For photoacoustic characterization, the samples were spray-coated on a glass substrate. The layout of the measurement cell above the sample is shown in Figure S12A. The gas tight cell is filled with a helium pressure of 20 psi. The microphone (Bruel&Kjaer, 4398-A-011) connected to the cell measures an acoustic wave, which is induced by the periodic heat conduction from the transducer layer surface to the gas phase. As shown in Figure S12B, the microphone is linked to a lock-in amplifier with integrated signal generator (Zurich instruments, HF2LI). The signal generator controls the electro-optic modulator (EOM, Conoptics, M25A) and therefore the frequency of the modulated laser beam. The acoustic signal detected by the microphone is then transferred into a phase and amplitude shift in relation to the modulation of the incident laser beam. The phase shift is detected as a function of the frequency in a range from 110 Hz to 4000 Hz. The signal is then normalized with the phase shift signal of a thermally thick glass sample (1 mm) with known thermal properties. A representative frequency sweep is shown in Figure S13. The red line indicates the best fit according to the fitting procedure presented by Singh et al.. [6] They used the generalized multilayer model of Hu et al. [7] assuming onedimensional heat transfer. The unknown fitting parameters are the contact resistance between the gold layer and the sample, the thermal diffusivity of the sample, and the contact resistance between sample and substrate. For thin films the fit is not very sensitive to the individual parameters, but to the total layer resistance. Therefore, only the total layer resistance is reported. From the total layer resistance it is possible to calculate the effective thermal conductivity by dividing with the thickness. The thickness of the samples was determined by AFM measurements. The values of the total layer resistance and sample thickness are summarized in Table S3. For each Hec/PVP ratio samples with three different thicknesses were measured to exclude influences from the sample thickness.

Section S3. Brillouin light spectroscopy
Brillouin Light Spectroscopy (BLS) measures the phonon dispersion, i.e., the wave vector, q, dependent sound velocity, , by detecting the Doppler frequency shift, f, in laser light inelastically scattered by sound waves ("phonons"). Since the phonon modes in question are thermally populated, the Bragg condition for light scattering is satisfied independent of the scattering angle, in contrast to the kindred time-domain ("picosecond ultrasonic") techniques that require external phonon injection via special sample preparation. Since the latter typically involves deposition of non-transparent metallic transducer films, they also do not readily allow for transmission and backscattering measurements, further limiting the potential of these techniques for probing anisotropic materials.
Assuming a planar sample, BLS can be conducted in three scattering geometries: transmission, reflection, and backscattering. For angle-dependent measurements in the transmission and reflection geometries, the laser source (λ = 532 nm) was mounted on a goniometer and rotated around the sample, similar to a wide-angle X-ray (WAXS) setup. For the transmission geometry (top-right inset to Figure 3B), the propagation vector, q, of the sound wave lies in the plane of the sample and its modulus, , is independent of the refractive index according to , with β being the light incident angle. For the reflection geometry, the Bragg condition is fulfilled for sound waves with q along the normal to the sample plane, so that , with n being the refractive index of the sample. [8] In order to find n, several points at different β were acquired, and then fit linearly; n was determined under the constraint that the fit has to pass through the origin ( Figure S14A), and the obtained values are reported in Table S4. In contrast to the transmission geometry, the range of the dispersion relationship that can be scanned by varying is more restricted due to refraction of the laser beam at the air-sample boundary. Finally, backscattering measurements can be conducted, where the incident and scattered light traverse the same path, and the sample is mounted on a rotating stage with angular gradation marks. Only a single dispersion point, corresponding to q directed opposite to ki in the film and of a magnitude, , is accessible in the backscattering geometry, but all possible directions can be probed. Because backscattering measurements use the same lens for focusing the incident and collimating scattered light, the alignment of the setup is generally much easier, so that all the backscattering spectra were acquired with microscope objectives (typically 10x) to reduce the spectrum accumulation time.
From the given formulas, it is easy to see that the Brillouin frequency shift does not exceed , where v is sound velocity. These frequencies render viscoelasticity effects negligible and are detected using a high-resolution six-pass Tandem Fabry Perot (TFP) interferometer (JRS Instruments, Switzerland) optimized for the 1-50 GHz range. Values still further out on the dispersion relationship can be accessed using, for example, picosecond ultrasonic interferometry (PUI), but at the expense of great effort and without any gain in information for the problem at hand. [9] Finally, BLS also offers direct access to the shear moduli, simply by analyzing scattered light in different polarizations: VV (i.e., vertical incident and vertical scattered light) corresponds to quasi-longitudinal and quasitransverse phonon modes, and VH (i.e., vertical incident and horizontal scattered light) to the pure-transverse phonon mode (for transversely isotropic samples, HV does not show a transverse peak). Before the BLS measurements, the samples were dried at 100 o C in a vacuum chamber for one week to remove any residual water content.   Based on the BLS-measured direction-dependent sound velocities, we obtained the elastic tensor within the framework of the Christoffel equation. [10] , where ρ is the density of the sample, v is the sound velocity, is the displacement vector, and (i, k = 1, 2, 3) is the Christoffel stress, which is defined as Here, Cijkl denotes an element of the 4 th rank elastic tensor, and represents the phonon propagation direction.  [ , , ] n n n = n For a transversely isotropic material, the elastic tensor, in the Voigt notation, has the following form. [11] , and contains five independent stiffness constants. After some algebra, the sound velocities of the Q-L, Q-T, and P-T modes along a direction defined by α can be represented as follows, where, (S10) . (S11) Through nonlinear χ 2 fitting of the BLS-measured sound velocities (i.e., v vs. α) with Equations (S6)-(S8), we obtained the elastic stiffness constants as well as their uncertainties. [12] The χ 2 is defined as , (S12) where vi, fit and vi, exp are the fitted and experimental sound velocities, respectively, Δvi, exp is the uncertainty of the measured sound velocity, and the summation is over all experimental sound velocities. By considering the uncertainties of the measured angles, refractive indices, and phonon frequencies, we estimated Δvi, exp to be 0.02vi, exp. We imposed the following constraints for the elastic stiffness constants [13] : (1) , (2) , and (3) , which ensure positive Young's and shear moduli. The availability of experimental data for all the Q-L, Q-T, and P-T modes allows unique determination of C11, C12, C13, C33, and C44.
After that, we calculated the engineering mechanical properties, [11] which include the in-plane Young's modulus with the direction-dependent thermal conductivity results, we used subscripts, "||" and "⊥", to represent directions parallel and perpendicular to the sample film, respectively, rather than directions parallel and perpendicular to the "3"-axis (i.e., the symmetry axis), as seen in typical analysis of transversely isotropic materials. The relevant mechanical properties are summarized in Tables S4 and S5.

Section S5. Uncertainty analysis
The uncertainties (standard deviations) of the data reported in this work were analyzed by taking into account the instrument accuracy, repetitive measurements, and propagation of uncertainties. The sound velocities were estimated to have an error bar of 2%. The uncertainties of the elastic stiffness constants were determined according to Zgonik et al.. [12] The uncertainties of the engineering mechanical properties were calculated according to principles of uncertainty propagation. For instance,