Tailoring thermal conductivity via three-dimensional porous alumina

Three-dimensional anodic alumina templates (3D-AAO) are an astonishing framework with open highly ordered three-dimensional skeleton structures. Since these templates are architecturally different from conventional solids or porous templates, they teem with opportunities for engineering thermal properties. By establishing the mechanisms of heat transfer in these frameworks, we aim to create materials with tailored thermal properties. The effective thermal conductivity of an empty 3D-AAO membrane was measured. As the effective medium theory was not valid to extract the skeletal thermal conductivity of 3D-AAO, a simple 3D thermal conduction model was developed, based on a mixed series and parallel thermal resistor circuit, giving a skeletal thermal conductivity value of approximately 1.25 W·m−1·K−1, which matches the value of the ordinary AAO membranes prepared from the same acid solution. The effect of different filler materials as well as the variation of the number of transversal nanochannels and the length of the 3D-AAO membrane in the effective thermal conductivity of the composite was studied. Finally, the thermal conductivity of two 3D-AAO membranes filled with cobalt and bismuth telluride was also measured, which was in good agreement with the thermal model predictions. Therefore, this work proved this structure as a powerful approach to tailor thermal properties.

The first anodization process is performed in sulfuric acid electrolyte whose concentration is 0.3 M at a constant voltage of 25 V with a temperature in the vicinity of 1-2 ºC during 24 hours.
As a result of the first anodization, a not ordered porous alumina layer is formed. However, the pores at the alumina-aluminum interface are hexagonally distributed. The pores, form hemispherical concaves on the aluminum, which tend to arrange together with the pores. The first alumina layer was removed by chemical etching in an aqueous solution of CrO 3 + H 3 PO 4 and a pulsed anodization process was carried out as published by Martín et al 2 During a typical pulse anodization process, mild Anodization (MA) and hard anodization (HA) constant voltage steps are periodically alternated [2][3][4][5] . However, Martín and co-workers while constant voltage is applied in MA steps, the current is limited to a defined value during HA steps and thereof the voltage is not kept constant. The material is grown thus at a defined current during HA periods. Then, the obtained AAO is selectively etched. As a consequence, the aluminum oxide grown under HA step is removed selectively, so that transversal nanochannels are formed, establishing thus a 3D porous network. In this work, the length of the pulse (t HA ) is established in 2 s and the length of the MA (t MA ) steps is fixed to 180s.

Electrodeposition of cobalt and bismuth telluride
The electrodeposition was made with a three electrode cell with a platinum mesh as counter electrode, a Ag/AgCl 3M KOH reference electrode and a working electrode. In the case of the cobalt film, the working electrode was a glass substrate evaporated with 5 nm of chromium and 150 nm of gold. In the case of the 3D structure, a 3D AAO template coated with 5 nm chromium and 150 nm of gold was used. This template was then glued with silver paint to a copper holder, and covered with varnish to define the area of the template exposed to the electrochemical bath and avoid deposition in the holder. For cobalt electrodeposition, the electrochemical bath was composed by a 0.4 M boric acid and 0.1 M cobalt sulphide heptahydrate (Sigma Aldrich). The electrochemical deposition was carried out at a controlled temperature of 25º C. The film was deposited at -0.78 V applied voltage versus Ag/AgCl for 15 hours, giving rise to a 40 μm thickness film. In the case of the 3D AAO structure, the potential was applied in pulses, which consisted in -0.8 V applied voltage versus Ag/AgCl for 1 second and 0 A for 0.1 seconds. A total of 18000 pulses were applied, that is, 5 hours of effective deposit. For bismuth telluride electrodeposition 0.9·10 -2 M Bi 3+ , 1·10 -2 M HTeO 2+ and 1 M HNO 3 were used as the electrolyte solution. The reactants were purchased from Sigma Aldrich Bismuth pieces (99.999%), Sigma Aldrich (99.999%) Tellurium powder, and Panreac 65% nitric acid. The pulsed electrodeposition was performed between 0.015 V for 1 second as the on time pulse and an off time pulse of zero current density for 0.1 seconds. For a total of 52363 pulses, 14.5 hours of effective deposit. The temperature of the electrodeposition was fixed at 4º C.

Morphological characterization
The pore diameter and thickness of the films has been studied by a Philips® XL305-FEG field emission scanning electron microscope (FE-SEM). The transversal channels were examined by a FE-SEM Hitachi one model SU8000 with TE detector operated at 0.5-3 kV.

Density measurements
The measurements were carried out by Archimedes' principle with the assistance of an auxiliary liquid (ethanol) with a well-known density (ρ et ). A balance XS105DU from Mettler Toledo was employed to weigh the 3D sample in air (W air ) and in ethanol (W et ) so that the skeletal density is ultimately assessed by using the equation: Several measurements were performed so that the related uncertainty to the mean value in the density is diminished, standing lower than 5%.

Specific heat measurements
The measurements were performed with a Discovery DSC from TA instruments from -10ºC to 60ºC so that the 3D alumina did not experience any physical alteration. Some tests were carried out in order to guarantee the accuracy of the results, which finally presented an uncertainty lower than 5%.

Porosity
The porosity analysis and the specific surface area measurement were carried out using N 2 adsorption isotherms (Micromeritic, ASAP 2020 MICROPORE DRY Analyzer). The porosity is calculated from the pore volume obtained from the BET experiment by means of the following equation: where is the pore volume, which is obtained from the BET experiment and the skeletal alumina volume, , is calculated from the mass, m, and the skeletal density, , of the alumina ( = ⁄ ) 6 .

Thermal conductivity measurements
The photoacoustic technique was employed to extract the thermal conductivity of the AAO empty and cobalt-filled samples along the direction parallel to the longitudinal channels of the 3D membrane. An incident modulated radiation impinges on the sample so that it is periodically heated and cooled. Therefore, the air in contact with the sample expands and contracts as a thermal piston thus, creating acoustic waves which are sensed by a microphone. The effective thermal conductivity of the sample is calculated by comparing the laser signal with the acoustic signal. The data reduction is performed by applying a multilayer model developed by Hu et al. 7 The experimental setup is described elsewhere 6,8 and used to measure from thick and thin films to porous nanostructures 6,8,9 . In order to ensure the absorption of the laser beam, an 80 nm titanium layer was deposited via electron-beam evaporation onto the 3D sample.

Porosity, density and specific heat
In order to calculate the porosity of the 3D structure it is necessary to take into account both the longitudinal pores and the transversal nanochannels. The top-porosity which arises from the longitudinal pores can be calculated by the top SEM image analysis.
The overall porosity of the 3D AAO membrane is experimentally studied by BET via nitrogen sorption. The isotherm obtained indicates extensive mesoporosity for the 3D membrane with an average pore size of 35 nm, specific surface area of 1.6 m 2 /g and a pore volume of 0.014 cm 3 g -1 . The average pore size is reasonable since the pore diameter of the longitudinal pores is around 40 nm and the transversal nanopores pore diameter is in the vicinity of 30 nm. These BET experiments provided a value of porosity of 40 ± 6 % for this sample. This value is coherent since it is higher than the porosity corresponding to the longitudinal channels analysed from the SEM images which results to be 31.8%.
The skeletal density of the 3D AAO membrane, , resulted to be 2.770 ± 0.054 g·cm -3 which is in good agreement with an exhaustive recently study which established a value of 2.788 g·cm -3 for AAO membranes prepared under the sulfuric acid solution 6 . The composite density, , of the 3D AAO membrane with air inside the pores can be calculated by the rule of mixtures which is given by the following equation: where P is the overall porosity of the sample and is the density of the air whose value is taken as 1.275 · 10 -3 g·cm -3 at room temperature (300 K). This rule yields to a composite density value of 1.662 ± 0.169 g·cm -3 . Figure 1 shows the specific heat of the sample as a function of the room temperature showing an increase when increasing the temperature. The room temperature value of 0.885 J·g -1 ·K -1 was taken for the data reduction for the 3D membrane. Figure S1. Specific heat as a function of the temperature. The red point depicts the specific value used for the thermal conductivity data reduction.
In the case of the 3D alumina membrane filled with cobalt and bismuth telluride Eq. S3 is used to calculate de the density of the composite by taking account the filler material density, and its porosity. The composite specific heat was determined by the following expression Where ( ) is the product of the filler material density and specific heat and ( ) is the product of the density and specific heat of the alumina membrane. The cobalt and bismuth telluride density were and 8.86 g·cm -3 and 7.74 g·cm -3 , respectively, whereas the used specific heats were 0.421 J·g -1 ·K -1 and 0.194 J·g -1 ·K -1 for cobalt and bismuth telluride, respectively 10,11 .

Uncertainty analysis
The density, specific heat and top-porosity errors were determined as the random uncertainty within and 95% confidence interval as this error was higher than the instrumental error or other error sources: where x ̅ is the mean value of the measured property, t n-1 is the Student-t distribution, n is the number of trials and σ is the standard deviation.
The experimental error associated with the thermal conductivity measurement not only includes the random error, , but also the systematic error, E sist , given by the lock-in amplifier and those related with the density, specific heat and thickness of the sample. The general formula used to quantify it is given by: Δκ com = √ (E ran ) 2 + (E sist ) 2 + (κ com Once the Δκ com is obtained, Δκ AAO is calculated by taking into account Δκ com and the porosity error. The error associated with the overall porosity is given by the BET technique which for this type of sample is taken as 15% for the pore volume combined with the error corresponding with the later calculation from Eq. (S2) where the sample mass and the density are involved.