Chapter IV: Ultralow thermal conductivity and mechanical resilience
4.2 Alumina nanolattice mechanical properties
conductivity to follow classical effective medium theories. To test this hypothesis, we performed finite element method (FEM) simulations on the representative unit cell shown in the inset of Figure 4.1b using the thermal conductivity of amorphous alumina reported in literature [71]. The measured 3ω data are consistent with the predictions of FEM and a thermal conductivity model developed for cellular solids [1]. In the low-density regime, a simple thermal resistor model gives a reasonable approximation [56].
The temperature-dependent thermal conductivities are also in agreement with FEM simulations, as depicted in Figure 4.2. The thermal conductivities of these nanolattices have similar temperature dependence, which suggests that phonon trans- port occurs by diffusion and classical size effects are not significant. The observed diffusive transport behavior is consistent with the expectation that vibrational mean free paths are on the order of a few nanometers in amorphous alumina [77, 78].
1.41
1.91
Figure 4.3: Experimental and computational stiffness values. The experiments wheret/r < (t/r)crit are marked by a thicker black outline. Power-law fits of the formE∗ = axbfor both experiments and simulations are shown as dotted lines. Due to imperfections in manufacturing, the experiments show a slightly higher scaling exponentb, and a vertical offset, in comparison to the simulations.
thickness, 0.5% relative density), we performed additionalin situexperiments using an InSEM (Nanomechanics Inc.) nanoindenter, where cyclic loading to ∼ 20%
strain allowed recoverability and deformation modes to be observed.
The uniaxial compression experiments provided stress-strain curves from which the Young’s modulus of each nanolattice was calculated. Slight misalignments between the indenter tip and the top of the lattice commonly caused a non-linear regime upon first contact, followed by a linear loading regime, which transitioned into sequential or catastrophic failure, depending on the t/r ratio. In order to minimize the effects of initial misalignments between the lattices and the indenter, the maximum slope of the linear loading regime was taken as the measured Young’s modulus.
Several identical samples were compressed for each set of parameters that were used for the thermal samples. Specifically, all samples had anr/lratio of 0.108,t/r ratios from 0.013 to 0.10, and spanned a relative density range from 0.78 to 4.1 %.
Figure 4.4: Periodic boundary condition simulations with a constant wall thickness but varyingr/l, showing optimal values ofr/lto maximize stiffness.
The relative density values were calculated using a CAD model (SolidWorks), using dimensions obtained from SEM micrographs. Figure 4.3 shows the stiffness values of all fabricated samples compared to finite element simulations with identical parameters (Abaqus FEA). The simulations consisted of a discretized octet-truss unit cell composed of S3R shell elements and a linear elastic material model, where the Young’s modulus and Poisson’s ratio of ALD alumina were used [81]. Periodic boundary conditions (PBC) were applied on all sides of the unit cell, and a uniaxial strain was enforced as a linear perturbation. The simulation size ranged from 110 000 to 310 000 elements depending on ther/landt/r parameters used.
The power-law fits of the formE∗ = axbin Figure 4.3 yielded scaling constants b= 1.41 and 1.91 for simulations and experiments, respectively. The vertical offset and the higher scaling of the experimental samples in comparison to the defect- free simulations can be attributed to imperfections such as wall waviness (on the order of a few nanometers), which has been shown to significantly decrease the stiffness of thin-walled hollow beams [79]. The non-linear scaling is consistent with the analysis done in that work, where the complex parameter space of hollow nanolattices is explained.
0 0.05 0.10 0.15 0.20 0.25 Strain
0 0.05 0.10 0.15 0.20 0.25
Stress (MPa)
6 5 4
3 2
1
b c d
a
Figure 4.5: (A) Stress and strain recorded during a 6-cycle compression test of a 24 nm wall thickness nanolattice. Curves are labeled with the cycle number. SEM images (B) before, (C) during, and (D) after the test show 98% recovery. The circular marker indicates the stress and strain of the partially compressed nanolattice shown in (C). Zoomed images (insets) illustrate the contribution of beam buckling, shell buckling, and fracture. Scale bars are 50 nm for (B, C, D) and 10 nm for insets.
Prior to manufacturing the samples, PBC simulations with constant wall thick- ness and unit cell size but varying beam radiusr were used to find an optimalr/l value at which the stiffness of circular-beam hollow octet-truss nanolattices was maximized. Figure 4.4 shows the resulting stiffness asr/l was varied from 0.014 to 0.14 and t/l remained at 0.007. The observed trend guided the design of the samples withr/l ≈0.1.
Figure 4.5a depicts the stress-strain data for cyclic compression of a 24 nm wall
thickness nanolattice. Images of the structure before, during, and after the test are shown in Figure 4.5b, c, and d. After yielding, the nanolattice experiences bursts of strain that correspond to layer-by-layer collapse. This serrated deformation signature is characteristic of competing failure modes—brittle fracture of tube walls, hollow beam buckling, and local shell buckling [69]. The strain bursts are caused by non-catastrophic brittle fracture of the ceramic walls at nodes, while the mechanism of recoverable deformation is elastic shell buckling as shown in Figure 4.5C. This explanation for recoverability is further supported by adapting the analysis by Meza et al. [69] to circular hollow beams. The localization of cracks at the lattice nodes causes the stiffness of the nanolattices to significantly decrease throughout cyclic loading, but the structure still recovers to 98% of its original dimensions after each cycle. We expect that recoverable behavior can occur at strains as large as 50% [69], which was not probed in this work.
Following the analysis done by Meza et al. [69], the dominating deformation modes for circular cross-section hollow nanolattices can be estimated. The three modes to be considered are material fracture, beam buckling, and shell buckling.
The critical stress values for each mode are
σfrac =σf, (4.1)
σbuckle = π2E I
Le2Atube, (4.2)
σshell = E p
3(1−ν2) t
rc
, (4.3)
respectively. Here, σf is the fracture strength, E is the Young’s modulus, and ν is the Poisson’s ratio of the constituent material. The cross-sectional area of the tube is denoted as Atube, its second area moment is I, and its effective length isLe
(depending on the boundary conditions). In the case of the octet-truss architecture, Le = L/2, where L is the actual length of the tube. The quantities t andrc refer to the wall thickness and the wall’s radius of curvature, respectively. For a circular hollow tube,rc =r, wherer is the tube radius.
Approximating the second area moment toI =πr3and the cross-sectional area to Atube = πrt, settingσbuckle = σfrac andσshell = σfrac, and solving forr/l andt/r
10-3 10-2 10-1 100 102
104 106 108
Specific modulus (Pa kg-1 m3 )
Thermal conductivity (W m-1K-1) Alumina nanolattices
Aerogels Elastomers
Polymers Ceramics Natural materials
Space shuttle tiles
Glasses
Figure 4.6: Material property plot of specific modulus versus thermal conductivity.
Dashed outlines indicate foams. For the same specific stiffness, our results demon- strate that nanolattices can achieve an order of magnitude lower thermal conductivity than polymer foams and porous ceramics used for space shuttle thermal protection systems. For the same thermal conductivity, nanolattices have almost two orders of magnitude higher specific stiffness than evacuated aerogels.
yields
r l
crit
= 1 2π
rσf
E , (4.4)
t r
crit
= σf
E p
3(1−ν2), (4.5)
where(r/l)critand(t/r)critare the critical ratios below which elastic Euler buckling and elastic shell buckling are expected to dominate over material fracture, respec- tively.