CHARACTERIZATION OF THE IN-PLANE AND CROSS-PLANE THERMAL CONDUCTIVITY OF POLYMER NANOFIBER THIN FILM UTILIZING THE 3-OMEGA METHOD

The thermal conductivity of Polycaprolactone (PCL) nanofiber thin films produced via an electrospinning method has been studied using the 3ω method. The production process utilized to produce the nanofibers used in this work were selected from a range of tested production processes with preference given to process parameters that produced consistently sized polymer nanofibers, nanofibers devoid of defects, and parameters that increased the general alignment of the resulting PCL nanofiber thin film. The effect of the void portion of a PCL nanofiber thin film is observed with increasing void percentange inversely affecting the effective thermal conductivity of the nanofiber thin film in both the in-planer and crossplaner directions, but increasing the portion of heat energy conducted along the in-planer direction. The data in this work concludes the thermal conductivity of the nanofibers tested is 20% higher in the in-planer direction than in the cross-planer direction, with the highest recorded thermal conductivity being 4.82 Wm−1K−1.

There exist methods today that do just this with polymers such as the process of electrospinning polymers into nanofibers, which will be discussed further in this work.
Polymers with high coefficients of thermal conductivity are becoming increasing important in today's world. In many fields and practices polymers are becoming increasing favored over other materials such as wood, metal, and ceramics due to its affordability and its ease to work with and shape into many configurations, and the field of thermal management is not ignoring polymers either. As technology pushes the size of components down to the nano-scale, a better understanding of thermal transport of nano-scale devices and individual nanostructures arise [1,2].
With increasing interest in the applications of polymer nanofibers, the scientific literature generated is accelerating in its ability to find novel uses for this material and understand its intricacies. Current applications for electrospun fibers as 1 well as their resulting fabrics include filtration [3,4], drug-delivery systems [5,6,7], and biomedical applications, namely in tissue engineering [8,9].

Thermal Transport in Nanoscale Structures
Thermal transport occurs in all electrical insulators and semi conductors through the transfer of energy in the form of phonons. These phonons transfer energy through lattice vibrations however themselves have no linear momentum.
In bulk materials, the thermal properties of a material are measured statistically, with bulk materials generally being samples with dimensions on scales from millimeters to kilometers [10]. For many materials their thermal properties at the bulk scale are higher than they are at the nanoscale, for some materials such as Silicon, significantly so [11]. There are several exceptions to this generality however, carbon is among the most notable one

Proposed Method
The process of measuring the thermal conductivity of materials at the micro and nanoscale is a difficult task. Just producing samples at this size is very time, labor, and material intensive. Along with this all quantification tools at this scale are specialized pieces of equipment meant to work at this specific scale. In this section, the different methods of sample production and measurement will be discussed as well as giving justification to the methods selected to complete this work.

Electrospinning
Electrospinning is mainly used to fabricate fibers with diameters ranging from tens of nanometers to a few microns in diameter [12]. Generally electrospinning is done through two different methods. These consist of solution electrospinning as well as melt-electrospinning. In melt-electrospinning, a polymer is heated to its 2 melting point and charged with a very high voltage to create a nanofiber that collects on a grounded collector [13,14]. In this version of electrospinning no solvents are needed meaning the solvent does not contaminate the final nanofiber product.
The other form of electrospinning is solution based electrospinning. This process is affected by more parameters than its melt-electrospinning variation, of which can be seen in Table 1, but allows for more control over the resulting fibers and for a wider range of workable polymers and enhancing materials. In this work however solution electrospinning is the method that was used to produce any nanofiber mentioned from here onward. In the solution electrospinning method, a polymer solution is pumped at a constant rate through a positively charged electrode, which then travels through a high voltage electric field to a grounded collector plate. This induced electric field causes the electrode to positively charge the polymer solution at its tip. The droplet of solution that collects at the tip of the electrode is thus deformed as it tries to repel the homogeneous positive ions it has collected. When the applied voltage reaches a critical value, the deformed droplet finally ejects a charged jet from the tip of the cone that accelerats towards the grounded collector. As this jet The ejection of a charged jet from the deformed droplet has been well studied and has been determined to be the result of the formation of a phenomena known as a Taylor Cone. This theory is first stated by Taylor in 1964 [16]. As the solution continues to collect and simultaneously repel itself at the tip of the electrode, the surface tension of the solution holds it together until the surface forces are overcome by the attractive force of the oppositely charged grounding electrode.
As this voltage differential increases, the droplet at the electrode will continue to elongate to a conical shape, a Taylor Cone. Taylor characterized the relationship between the electric field E, the solutions surface tension σ and the curvature of the Taylor Cone, r. In this relation 0 is the permittivity of free space, and α being the half angle of the cone.
Due to imperfections that cause concentrations of positively charged ions along the axis of the elongated polymer fiber that is ejected from the Taylor Cone, the fiber will experience a whipping process. This process further elongates the fiber and increases the rate of evaporation of the solvent. The formation of a Taylor Cone as the voltage differential reaches its critical value is depicted in Fig 1.

SEM Use
A scanning electron microscope (SEM) is a type of microscope that uses a focused beam of electrons to produce an image at a far greater magnification and resolution than traditional optical microscopes. Samples imaged through SEM are typically electrically conductive, as without this property negative charges quickly accumulate within a sample and create a sort of over-exposed image when observing the sample. Polymers are typically among such materials that have low electrical conductivity and produce poor resolution images in SEM machines. The high power electron beam can quickly damage the sample leaving visible scars on the sample and also rapidly accumulate negative charges from the electron beams [17]. There are however methods to get around this on SEM machines to produce high quality images of the orientation and morphology of electrospun polymer nanofibers. Through the use of a low-voltage scanning mode these nanofibers can be imaged [18]. The SEM used in this work is a Field Emission SEM purchased from Zeiss microscopy, Fig. 2.
In order to obtain these sharp images at the necessary size scale and resolution an unusually low electron beam power was used for the sample. Typically when using SEM for nano and micron scale microscopy a 3 keV electron beam is used. As per the concerns pointed out in this section however, this would cause far too much of a negative charge to build in the sample and would quickly damage the sample used in this work. An electron beam of power 1.1keV was used instead. with proper adjustments to the aperture and signal of the SEM the polymer nanofibers were able to be successfully imaged. There is still some minor scarring on the samples as a result of the electron beam, however SEM shows to be an effective method to evaluating the morphology of individual nanofibers and determining the alignment of nanofiber supported thin films. See Fig. 3 [19,20]. Despite the AFM's high accuracy in measuring surface topologies, the SEM will be used for the majority of the nanofiber thin film imaging in this work due to the images produced by SEM being more workable and being able to measure multiple characteristics from a single image.
6 Figure 3. SEM Imaging of polymer nanofibers at 6870x magnification

3ω Method
Measuring the thermal properties of micro and nanoscale materials is no simple process. Along with the challanges associated simply with the small geometry of the samples, when trying to measure the thermal conductivity of small solids, the heat transfer can very easily be dominated by heat transfer from convective sources as well as heat transfer through radiation.
A solution to this issue has been shown in the form of the 3ω method. This method is first described by David G. Cahill in 1990 [21]. This work uses a single wire like heating element powered by an alternating current. This heating element works as both the sample heater as well as a sample thermometer by measuring the frequency dependence of the sample's temperature oscillations. In this method the temperature of the sample is measured by the third harmonic of the voltage across the metal line. This occurs due to joule heating occuring at 2ω with an input current at angular frequency ω. As the temperature of the sample increases the resistance of the metal line will also increase making the resistance of the metal line also a function of ω, oscillating at 2ω. When multiplying the original input current oscillation by the resistance oscillation, as per Ohm's law (Eq.2) the resulting voltage oscilation will be at frequency 3ω.
The amplitude of the measured voltages at frequency 3ω can then be related back to the heat capacity as well as the thermal conductivity of the sample it is measuring. Along with the 3ω's novel method of measuring the thermal conductivity of thin films it also has several advantages over conventional thermal property measurement techniques. The 3ω method is capable of measuring the thermal conductivity of a sample at a very large range of temperatures, as extreme as 30 K to 750 K has already been proven viable [21]. In addition to operating at these temperature extremes the method also determines all of these thermal properties in as short a time as periods of the frequency of the input current into the 3ω metal line, which is no more than a few seconds. This totally bypasses the usual need of conventional thermal measurement techniques of keeping a sample at a controlled temperature for a long enough time to reach steady state, which can take hours to perform full tests. The 3ω heater design geometry used in this work is shown in The exact 3ω method used has several key modifications made to it when compared to Cahill's apparatus as described in his original work on the method.
The details of these modifications will be explained later, but what they accomplish is quite significant. The first outcome of these modifications is the introduction of a Wheatstone bridge in the 3ω test platform. Through the use of this simple circuit accurately [22]. The following modification is to deposit the 3ω test platform onto one substrate, and the test sample onto another. This method is necessary for the type of testing conducted in this work but has the added benefit of not requiring the testing to be conducted in a vacuum chamber due to reduced heat transfer through convection [23]. While these modifications have individual benefits, combined they also result in significant cost savings. Tests conducted without the ability to make these modifications frequently need to purchase custom equipment and have their 3ω test platform fabricated on printed circuit-boards. A full custom 3ω platform can cost $20,000 to produce [24], a cost that can be virtually completely ignored in this work.

Other Considered Thermal Conductivity Measurement Methods
Alongside the 3ω method of measuring the thermal conductivity of electrospun nanofibers. Among the other methods considered is the T-type method of measuring thermal conductivity. This method has successfully measured the thermal conductivity of fibers of many materials that are too thin to measure using conventional methods [25,26,27]. The T-type method is similar to the 3ω method in that both use a metallic calibration material with a known thermal conductivity to act as both a heater and a thermometer of the sample it is measuring. Unlike the 3ω method however the thermal conductivity of the material is measured using a direct current instead of an alternating current. The T-type thermal conductivity measurement method relies on measuring the one-dimensional steady-state heat conduction through a material. By measuring the thermal expansion of the sample material as well as the temperature at all three points on the T-type measurement platform the thermal conductivity of the sample can be determined. While this method is promising in measuring the thermal properties of materials which are too small to be measured by conventional methods, it specializes in measuring individual fibers, something that is not looking to be measured in this work. Along with this this method must take into account heat loss from radiation, for these reasons the 3ω method for measuring the thermal conductivity of a sample was chosen.
Another considered method for thermal measurements in this work were the Time Domain Thermoreflectance Method (TDTR) as well as the Frequency Domain Thermoreflectance Method (FDTR). Both of these methods are non contact techniques that use optical lasers to characterize the thermal properties of a sample. This laser focuses on a sample and provides energy to the sample which heats it and then measures the change in reflectively of the sample as it heats. In able to properly utilize this method however, a laser that operates at an incredibly short time constant must be used, this ensures that only the surface of the sample is heated to prevent any thermal contribution from the substrate the thin film is prepared on (picosecond resolution) [1].
Similar to most Scanning Electron Microscope (SEM) imaging, the samples measured in this method require a very thin layer (about 80 nm) coating of a high conductivity material. This method however requires this coating to use it as a reflective surface to measure the thermal properties, and as the polymer nanofibers used in this work are quite dull this coating is required. This method is quite intensive to set up but accurately characterizes the thermal conductivity of thin films [28,29]. This method has successfully characterized the thermal properties of bulk materials [30], liquid compounds [31], and conductive thin films without an additional metallic thin layer [32]. While this is a very powerful tool in measuring the cross-plane thermal conductivity of materials, no literature on it successfully measuring in-plane thermal conductivity's were found. This chapter will review the electrospinning apparatus used for fiber fabrication, as well as the 3ω method that is used to characterize the thermal conductivity of the polymer nanofiber supported thin films.

Nanofiber creation
The nanofibers used in this work are created through an electrospinning process. In this process a polymer is dissolved into a natural solvent at a certain weight percentage, defined by the following equation. When spinning the nanofibers, the solution is loaded into a 30 mL syringe which is further placed into a 2 channel syringe pump. Nanofibers spun in this work were spun using the syringe pump depicted in Fig. 5 with varying syringe flow rates ranging from 5 to 12 mL/h. The varying flow rates in this work are used to generate nanofibers with varying cross sectional areas, with slower flow rates producing nanofibers with smaller cross sections than the nanofibers produced at higher flow rates. The flow rates must remain high enough however to maintain a 14 Taylor Cone at the tip of the spinnerets. While not used particularly in this work this syringe pump has the capability of pumping two solutions simulataneously.
This is useful when producing nanofibers with a core-shell geometry which will be expanded upon in the future work section.
Using the prepared solution in a syringe placed into a syringe pump, an appropriate flow rate for the solution is determined. This is done by applying a flow rate and observing the spinneret electrode for the solution. The solution should form a drop at the tip of the spinneret electrode, when this drop falls, an appropriate flow rate will result in a drop reforming at the tip of the spinneret immediately.
In this work, a flow rate of 7ml/hr was determined as the minimum appropriate flow rate. After determining the appropriate flow rate of the solution, the critical voltage differential is determined for the proper formation of a Taylor Cone on the spinneret. Taylor expressed a relation between solution properties and spinneret properties for determining the critical voltage [1]. This is expressed as: Here H is used to describe the Tip to Target distance, L is the length of the needle, R is the radius of the needle, and σ is the surface tension of the solution.
This relation can be used to determine the critical voltage empirically. However the electrospinning apparatus used in this work allows for the critical voltage to be determined experimentally by simply increasing the voltage differential until a Taylor Cone forms at the tip of the spinneret and maintains a continuous jet between the spinneret electrode and the collector. The determined critical voltage in this work was found to be +20kV using a tip to target distance of 15 cm.    Table 2 in order to produce highly aligned polymer nanofibers.
The electrospinning spinneretts used in this work are 20 AWG 1/2" industrial dispensing tips purchased from CML supply. These needles are arranged in a 1 x 5 array orientation with all of the needles receiving the polymer solution in parallel.
These needles are all brought to a +18k voltage differential simultaneously and sweep across the collector drum to apply an even coat of electrospun nanofibers to the collector drum. The arrangement of the electrospinning spinnerets can be seen in Fig. 7.
To collect the polymer nanofibers, the TL-Pro-BM comes with a modular base collector that allows for the collector to be swapped out to a number of different

Temperature and Humidity Contribution
Reproducing nanofibers of comparable quality is also a difficult task to complete. Beyond the simple controllable variables of solution parameters as well as of study in the field of electrospinning as well as a large obstacle to the success of electrospinning in industry [2]. While the ambient conditions that the nanofibers were produced in in this work were not controlled, they were considered. Studies have been conducted to determine the effect of ambient conditions on the morphology of electrospun nanofibers taking into account ambient temperature and relative humidity.
The common finding when observing the effects of ambient temperature and relative humidity on the morphology of a nanofiber, is the formation of crater-like pores on the surface of the polymer nanofibers with increasing relative humidity.
Increasing the ambient temperature that the polymer nanofibers are produced in has a dampening effect on the formation of these pores, requiring an increase in relative humidity to produce the same porous morphology on a sample produced at a higher ambient temperature as it does one produced at a lower ambient temperature. There are two common explanations that describe the formation of these pores [2].
1. The pores are the result of water droplets that have formed on the surface of the polymer solution jet, with increasing frequency as the humidity increases as a result of a more probable chance of reaching the dew point of water.
2. The presence of water in the ambient air resulting in a phase separation in the polymer solution jet into polymer-rich and polymer-poor sections. This phase-change likely caused by the water-rich ambient air causing a temperature drop at specific points along the polymer solution jet.

Nanofiber Dimensions
During the creation of nanofibers, much of the determining factors on the diameter of resulting nanofiber occur in the gap between the spinneret and the collector [3]. As the nanofiber solution exits the spinneret, the jet will have a small segment were it travels in a straight line. This straight segment is immediately followed by a boil of increasing diameter caused by bending instabilities. Bending instabilities in the electrospinning process are described in Taylor's original work on the topic and are characterized for small perturbations [4], and have also been expressed as mutual Coulomb interactions caused by the uniformly distributed electrical charge in the polymer jet [5]. After several turns of this coil, the process repeats as a new bending coil forms in the jet causing a smaller coil to form about the path of the original coil. This process repeats again and will further continue until the electrically caused elongation of the polymer solution stops, which normally coincides with the solidification of the polymer nanofiber [6]. In The coils that form from the bending instabilities in the fiber are only one Figure 9. Bending instabilities in the polymer solution jet during a typical electrospinning process [6] example of the instabilities that occur in the formation of the polymer nanofibers.
In addition to these instabilities, branching and the formation of beads in the nanofibers also occurs in specific conditions. The first of these instabilities, branching, is the result of excess of charge by increasing the potential difference in the nanofiber solution [6]. These branches form outward from the center of the nanofiber solution and are dominated by the electric field that results from the positively charged nanofiber solution jet, but also effected by the voltage differential between the spinneret electrode and the collector. The resultant of these forces develops a branch that grows radially outward from the center of the nanofiber solution jet with a small downward component due to the potential difference between the spinneret electrode and the collector. The formation of these branches can be relatively controlled through a single process variable in the formation of these nanofibers [7]. The distance between the formed branches has been shown to decrease as the potential difference between the spinneret and the collector is increased. This leads to the conclusion that decreasing the potential difference between the spinneret and the collector to a point that still maintains a Taylor Cone results in far fewer, or possible elimination of branches from the polymer solution jet.
The second form of instability in nanofiber production is the formation of beads along the fiber. The formation of thin nanofibers with periodic relatively large beads is the result of a lower surface energy compared to maintaining a fiber of a consistent unbeaded cylindrical form [6,8]. The best practice to avoid this outcome is to maintain a high enough voltage differential between the spinneret electrode and the collector. Previous work into the phenomena [8], showed that by surrounding the air around the polymer solution jet with a neutralizing ion stream through a corona discharge leads to a significant increase in beaded fibers.
The conclusions drawn from this work show that by maintaining a sufficiently high potential difference between the spinneret and the collector the polymer solution is actively being stretched and forced into a consistent diameter until the point that the solution is fully evaporated and the nanofiber solidifies. Both the formation of beads and branches are in this work non-favorable instabilities in the formation of nanofibers and require opposite calibration of the potential difference between the spinneret and the collector to be avoided. Due to this the formation of one branching is favored over the formation of beading. A higher voltage potential is used as the formation of branching tends to be limited to the early regime of the polymer solution jet, the straight segment and the first bending instability.

Nanofiber Alignment
The orientation of a nanofiber sheet is largely determined by the method in which it is collected. The most common method of collecting nanofibers in such a way that they are oriented largely parallel to each other, is to collect them on a rotating collector. By grounding a rotating drum, the generally randomly oriented nanofibers will begin to orient themselves tangentially to the axis of rotation on the drum collector. The rotational velocity needed to produce aligned nanofibers is generally referred to as the critical uptake speed, and it is generally normalized into a tangential velocity in m/s rather than rotations per minute (RPM) to account for electrospinning apparatus of varying drum diameters. Various works have the critical uptake speed for electrospun nanofibers to be in the range of several meters per second [6]. However electrospun PCL nanofibers specifically have been found to have a critical uptake speed of about 1.7 m/s [2]. Other works have been published showing SEM images of aligned fibers collected at incredibly high uptake velocities, almost 10 times the previous value at 15.7 m/s [9]. When the uptake velocity reaches much higher than 2.5 m/s the air around the collector becomes more turbulent, negatively impacting the alignment of the nanofibers as they reach the collector. This is reversed however at even higher uptake velocities as the collector pulls the fiber onto itself at a much higher rate, stretching it in the process.

3ω test platform fabrication
The 3ω test platform for this work is produced through a photo lithography and sputtering dual process. In total eight 3ω metal lines of various geometries and sizes were produced, each meant to test the cross plane thermal conductivity, the trans-axial cross plane thermal conductivity or the in-plane thermal conductivity of the nanofiber thin film.
The production of the metal lines started with a simple calculation to determine the necessary line thicknesses from estimates of the expected thermal resistance of the thin film and the thickness of the thin film produced. To determine the necessary sizes of the 3ω metal line, the following equation was used, where b 23 is one half of the metal line width and d f is the thickness of the film it is measuring. The resultant of this equation is a unitless numbers that describes if the joule heating from this process will be in-planer or cross planer. heating [10,11]. This term can be thought of as the heating thickness regime, and it defines if the joule heater is considered a thick or thin heater.
When this term is large, above 0.5, with a b d f of 2.5 or greater [12], the heating will be entirely dominated by cross planer heating. The opposite extreme is also true however when ω t is small, under 0.1, where the joule heating will result entirely in in-planer heating. [10,11]. Due to all of the nanofibers being oriented parallel to each other, the sample may also be turned 90 • to determine the trans-axial thermal conductivity. Figure 10. ω t value at various thin film thicknesses and void percentages Due to many of these numbers being estimates based on Eq. A.10 and averages determined from expected thin film thickness the metal line values were chosen to be well below the 0.1 narrow heater regime distinction. Metal line widths (ω t ) of 5 µm, 10 µm, 25 µm, and 50 µm were ultimately chosen going forward in the 3ω heater design. Figure 10 demonstrates the ω t values for each metal line thickness, demonstrating the range of nanofiber supported thin film physical properties that this measurement method will be compatible with. The physical characteristics of the nanofiber supported thin film will mostly be determined by the morphology of the nanofibers produced in the electrospinning apparatus. With the measurements available at the time of producing the metal lines, it could be said with confidence that the 3ω heaters would be able to accurately characterize the in-plane thermal conductivity of any nanofiber thin film that had a thickness greater then 2 µm as well as the cross-planer thermal conductivity of any thin film with a thickness less than 20 µm.
The eight 3ω heater geometries that were chosen were digitally produced in Solidworks, and the drawing file that contained the geometries was sent to JD Photodata for them to produce a photomask at such small geometries. The 3ω heaters were printed onto 3" silicon <100> n-type test grade wafers purchased from University Wafer.

Photolithography Process
Due to the incredibly small geometries being produced for the 3ω test platform all photolithography conducted in this work was completed in the University of Rhode Island's clean room. Each 3ω heater geometry was patterned onto a single silicon wafer. The wafer first goes through a prep process to ensure proper exposure to uv light during the photolithography process as well as to ensure proper adhesion of the sputtered gold material that would inevitably be deposited onto it. The first preparatory process is to clean all of the wafers in an acetone rinse and plasma Following the cleaning and sterilization of the silicon wafer, each wafer is spin coated in layers of OmniCoat TM purchased from MicroChem. This product acts as a sacrificial layer after the wafers are coated and exposed in SU-8 to aid in the lift off process. To ensure that the lithography process across each 3ω heater on each silicon wafer was consistent, the exact same procedure was used to best control the end product as possible. The procedure was as follows. A pre-cleaned silicon wafer is taken and centered with a spin coater. To ensure each wafer is centered as best and consistantly as possible, a small 3" diameter centering tool that contoured itself to the 3" silicon wafer as well as the vacuum chuck below it was used ( Fig.  Figure 12. A) The computer controlled spin coater from Brewer Science used for spin coating the silicon wafers in this work. In the highlighted area is the 3D printed wafer centering tool used to center wafers to maximize the spin coater's effectiveness. B) The wafer oven from Brewer Science used in this work to soft bake and post exposure bake the silicon wafers.

12)A.
A pipette is used to dispense 3 mL of OmniCoat TM onto the center of the silicon wafer. The spin coater is loaded with a formula appropriate for the silicon wafer's size, in this case the recipe was an initial solution spreading spin off 500 RPM for 5 sec with an acceleration of 100 RPM/s. This is immediately followed by a secondary spin of 3000 RPM for 30 sec with acceleration of 300 RPM/s. The wafer is then promptly removed from the spin coater and inserted into a preheated oven held at 200 • C for 1 min, Fig. 12B. The wafer is removed from the oven and allowed to cool to room temperature. According to MicroChem's data sheet on OmniCoat TM , the coating should be 13 nm thick after the baking process.
This process is repeated twice more to ensure a thick enough sacrificial layer of OmniCoat TM . This is completed for all eight silicon wafers used to produce the 3ω heaters.
The next process is completed to apply a layer of SU-8 negative photore-27 sist purchased form MicroChem and expose it to the prepared photomask. The OmniCoat TM prepared silicon wafer is reinserted into the spin coater and centered.
A pipette is used to dispense 3 mL of SU-8 2015 type photoresist onto the center of the wafer. The spin coater is loaded with a two step recipe starting with a solution spreading spin of 500 RPM for 10 seconds with an acceleration of 100 RPM/s. This is immediately followed by a spin of 2000 RPM for 30 seconds with an acceleration of 300 RPM/s, this rate is determined by the SU-8 data sheet depicted in Fig 13. This will result in an SU-8 layer 15 µm thick. lowing the exposure the silicon wafer is given a post exposure bake for 4 minutes at 95 • C. Subsequent to the exposure process, all of the SU-8 photoresist that has been exposed to the UV light is now highly cross-linked and resistant to conventional removers. This allows for the SU-8 developer to remove all of the photoresist that was shielded from the UV light by the photomask. An OmniCoat TM developer is now used to remove the now exposed OmniCoat TM that was protected by the photomask, leaving the bare surface of the silicon wafer where the photomask cast its shadow during the exposure process. The process of preparing, applying, exposing, and developing the photoresist as well as the successive process of sputtering and liftoff can be observed in Fig. 15. Figure 15. The development process of creating the 3ω heaters used in this work. The process as follows is, coating the substrate in OmniCoat TM , coating, exposing, and developing the SU-8 with the negative photomask, developing the OmniCoat TM , sputtering the 3ω heater, lift-off of the SU-8 to leave only the 3ω heater on the substrate.

3ω Heater Sputtering
The now exposed and developed wafers are ready to be sputtered in order to gain their metallic thin line 3ω heater. Each wafer is inserted into a sputtering system with a 4 in gold target and treated for 15 minutes. This produces a uniform coating of 200nm of gold directly onto the silicon wafer substrate, as well as the remaining cross-linked UV exposed SU-8 Photoresist. Only the gold that has adhered to the silicon wafer is needed so the rest is removed in a lift-off process. The entire silicon wafer is submerged in PG Remover purchased from MicroChem. This solution attacks and degrades the OmniCoat TM sacrificial layer which sequentially removes the developed SU-8 layer that is adhered to it. The only remaining coating on the silicon wafer is the 3ω metal heater that was patterned onto the wafer through the photolithography process.

Applying the Thin Film
After creating a nanofiber thin film, for its thermal conductivity to be measured, it must be appended to a 3ω test platform. In the original 3ω procedures presented by Cahill in his original paper [12] the 3ω heater is printed directly on the Figure 16. Attachment pressure in use to append the nanofiber thin film to the 3ω heater test subject through photolithography. For measuring the thermal conductivity of a nanofiber supported thin film however this is not possible. Firstly the surface of a thin film is far too porous to go through any of the photolithography process, and secondly the solvents used in the photolithography process would attack and destroy the polymers used to produce the thin film. There does exist methods however of appending a sample made of porous materials to a 3ω heater produced on an appropriate medium to measure its thermal conductivity none the less [13].
In order to apply the nanofiber thin film to the 3ω heater, the thin film will be pressed onto the heater at a constant pressure. This attachment pressure will have some relation to the thermal conductivity as increasing the attachment pressure should have an inverse effect on the thermal contact resistance between the 3ω heater and the nanofiber thin film. Figure 16 demonstrates this process.
Samples are measured at various attachment pressures to allow for consistent tests at lower attachment temperatures, as when the attachment pressure reaches a critical value, the thermal conductivity test becomes a destructive test for the nanofiber samples. The nanofibers begin to lose their shape and are flattened by the surface they are pushed against as demonstrated in Fig. 17. This plastic deformation of the nanofibers creates inconsistencies between measurements as it decreases the nanofiber film thickness at which the measurements are calibrated for and also decreases the void ratio of the nanofiber thin film. This is reflected in Eq.
A.13. The attachment pressure of each nanofiber thin film is applied by the flat surface of a Standard SEM pin stub purchased from Ted Pella Microscopy. These SEM pin stubs are of standard shape and have a 12.7mm diameter as well as a 3.2mm pin. These SEM pin stubs (PELCO Tabs) are the same sample mounts used to perform the SEM microscopy of the nanofiber thin film samples as they serve both purposes equally well. The PELCO Tabs provide a flat surface that when pressed into the 3ω heater will allow to easily calculate the attachment pressure that is being applied to the nanofiber thin film sample. Under the weight of only the PELCO Tabs the nanofiber thin films will experience an attachment pressure of 6.2 Pa, the minimum attachment pressure that will be analyzed in this work.

3ω Circuit Design
For the 3ω heater to be able to conduct the experiment, it needs to have a compatible circuit designed around it. While the math and theory that determine the thermal conductivity of the nanofiber thin film sample is relatively straightforward, the output voltages from the 3ω test platform will be very low and must be compared to a reference voltage to filter out the small voltage oscillations from the general noise produced by the alternating current [12].
The circuit is arranged in a Wheatstone bridge in order to determine the non-equilibrium voltage v 3ω . This non-equilibrium voltage is able to be compared to the input voltage of the circuit. When one side of a Wheatstone bridge is powered by a voltage supply, and the node opposite it is on a common ground to the power supply, the two remaining nodes may have their voltages compared. This can be easily accomplished through careful selection of the resistors in the 3ω circuit, and further so by using a potentiometer for R 3 to be able to finely tune the Wheatstone Bridge. Referring to Fig. 18 the resistance of R 0 can be evaluated for all time values and represented as a function of ω through the following relation.
On the opposite side of this relation however, by knowing the resistance of all of the resistors in the Wheatsone bridge as well as using a known voltage input, the voltage out to the Lock-in Amplifier can be determined.
The circuit begins with an alternating current source. For this work a fre- Figure 18. Schematic of the 3ω experimental circuit [13] quency synthesizer is used to produce a sine wave. This sine wave goes in parallel to a Lock-in Amplifier to be used as a reference signal and also across the metal bridge. After going across the metal bridge the non-equilibrium voltage, v 3ω , is also measured by the Lock-in Amplifier. This non-equilibrium voltage can be related to the 3ω voltage, V 3ω using the following relation made possible by the Wheatstone bridge [14].
Knowing now the 3ω voltage, the temperature relation of the thin film sample is able to be determined through the relation described by Cahill in his original work on the 3ω method [12].
In Cahill's original work the 3ω method is used to find the thermal conductivity of the substrate the 3ω heater is deposited on when the whole system is isolated in a vacuum chamber. Due to the modifications of this method the 3ω heater temperature is divided by 2 because only half of the power supplied by the 3ω heater is being applied to the material of interest. The result of this assumption is: In this relation, V is the amplitude of the voltage sine wave applied to the 3ω heater, R 0 is the measured voltage across the 3ω heater at room temperature with no voltage oscillations, and dT dR is the temperature dependence of R 0 . This calibration of dT dR is completed by measuring the temperature of the metal line at a low oscillation, and describing it as a function of the metal line's resistance.
Knowing the values of ∆T which represents the temperature oscillations of the nanofiber sample, the thermal conductivity of the sample may be determined through either a one-dimensional (1D) or two-dimensional (2D) thermal model depending on if the thickness of the metal line is much thicker than the thickness of the nanofiber thin film or is comparable to the thickness of the nanofiber thin film. Both cases are evaluated using different 3ω heaters to determine both the in-plane and cross-plane thermal conductivity of these nanofiber thin films.

CHAPTER 3 Methodology and Calibration
This chapter is to review the parameters of the nanofibers produced as well as the parameters of the 3ω circuit used to measure the thermal conductivity of the nanofiber thin films. In order to conduct the experiments, first the produced nanofibers are examined under the scanning electron microscope to evaluate the size and quality of the nanofibers produced. With this information the the modeling presented in Chapter 2 can now be more accurately performed by knowing the degree of alignment as well as the void ratio of the thin film.

Nanofiber Diameter Measurements
The size of the nanofiber thin films are evaluated in the Scanning Electron Microscope in order to evaluate the size and quality of the nanofiber thin films. The scans are performed at various magnifications and electron excitations to find the appropriate measurement settings, and thin films from different electrospinning apparatus settings. In Fig. 19 the polymer nanofibers being examined were produced using the BL-TM-Pro robotic electrospinning apparatus with a solution flow rate of 9ml/hr and a voltage differential of +22kV. Two different electrospinning run times at these settings are also examined, the first for 15 minutes and the second at 45 minutes. In order to determine an appropriate nanofiber diameter distribution, each nanofiber thin film is imaged in two different locations measuring the diameter of all nanofibers that cross the horizontal mid-line of the SEM image. The first observation of these images is that the nanofibers produced span at a relatively large range of diameter sizes with this production run having the largest variation as well as having some of the largest individual fibers, with the largest being up to 1 order of magnitude larger than smallest nanofiber in the same image. With Figure 19. Distribution of nanofiber diameters when prepared at a voltage differential of +22kV and a syringe flow rate of 9ml/hr this information, the nanofibers tend to have the highest number of occurrences in the 500 to 1000 nanometer range. The goal of this section is to determine the parameters of this individual electrospinning apparatus that will produce the most consistent nanofibers in the given machine settings. There exists many studies on relating the surface tension of the polymer solution, the solution flow rate as well as the voltage differential to the radius of the jet diameter during electrospinning [1,2], but obtaining experimental data specific to the parameters and equipment used in this work is important in validating the models and assumptions made in this work. Figure 20 shows polymer nanofibers produced in the same time intervals however the potential difference between the spinneret electrode and the collector was altered to +20 kV, and the flow rate remains at 9 ml/hr. These samples share a similar range of nanofiber diameters from fibers <500 nm up to 3000 nm in diameter. The difference from the original calibration production sample is that these fibers on average are smaller in diameter and have fewer occurrences of larger diameter nanofibers.
The final nanofiber diameter distribution calibration production sample was run with both a lower potential difference between the spinneret electrode and the collector as well as a lower pump rate, 20 kV and 7 ml/hr respectively. Figure   21 shows the nanofiber diameter distribution of this production run. This production run produced nanofibers with the most consistent nanofiber diameters, with a large concentration of nanobers in the 500 to 1000 nm range. All three of these production samples show randomly oriented nanofibers rather than aligned nanofibers despite being collected on a rotating drum collector. The likely cause for this is that the collector drum was rotating at a speed that caused turbulent air flow around the drum effecting the orientation of the nanofibers as they were This can be seen much more clearly in Fig. 23. These samples have a comparable nanofiber diameter distribution to that of the previous test so it is the sample production parameter that is selected to be used for the final thermal conductivity testing and analysis completed in this work. The nanofiber thin films are marked in such a way that they can be visibly oriented in such a way that the main direction of the fibers is in the desired direction and can be easily rotated in either direction To determine the degree of alignment as well as the void ratio of the nanofiber thin film test samples used in this work, an imaging software tool called ImageJ is used. This software makes taking measurements such as length of fibers and their orientation easy to manage. It is also very versatile and allows for measuring the area of an image and giving a description of the area of an image whose pixels are within a certain gray-scale value. This is used to calculate the void ratio of a nanofiber thin film.

3ω Heater Resistances
The 3ω metal lines that are used in this work to determine the thermal conductivity of the polymer nanofibers are quite fragile and require care when working The second method practiced was to use thermal tape to connect the wires and the pad, however this also failed and was a destructive method as well resulting in the lift-off of the gold 3ω metal line in some cases. The final approach to attaching wire leads to the 3ω metal line pads was to use micro-manipulators. This was by far the most successful attempt at attaching the wire leads, however in some cases the copper wire was much harder than the gold that is being attached to, resulting in scratching the gold metal line. This method was used for the entirety of the work, however further techniques are explored and recommended to ensure non 45 destructive attachment to the metal line in a future section.
The 3ω heaters produced in this work are all produced to act as both heaters and thermometers for the samples they are measuring. In order to accomplish this they are each produced with 4 electrical pad connection terminals. One pair of the terminals is used to be integrated into the 3ω circuit while the other goes directly to the lock-in amplifier to measure the oscillating resistance and voltage drop across the 3ω heaters. When designing each 3ω heater the expected resistance of the 3ω heater was calculated using Pouillet's law.
In this equation, R is the resistance of the 3ω heater, ρ is the material resistivity of gold, l is the length of the 3ω heater (7 mm) and A is the cross sectional area of the 3ω heater. A was calculated using the width of the selected metal line in the 3ω heater (5 µm, 10 µm, 25 µm, 50 µm) and using 200 nm as the height. The electrical resistivity of sputtered gold films has been well studied [6] and gold thin films with thicknesses above 300Å will have the same electrical resistivity as gold in its bulk form. The silicon wafer does not contribute to the electrical resistance of the 3ω heater, as the electrical resistance per unit length of the silicon wafer is at least ×10 6 that of the gold film due to the native oxide growth on the silicon wafer. The resistance of each 3ω heater is on the scale of 10's of Ohms.
When working with the 3ω method it is important to perform several measurements on the metal line to ensure that the line and equations are properly calibrated to the 3ω test platform. The first and arguably most important value to be calibrated is the temperature coefficient of resistance (α). This relation is determined using the following relation: In this equation, R 0 is the electrical resistance in Ohms across the 3ω metal line and dR/dT is the dependence of temperature on the electrical resistance across the metal line. This is accomplished by first measuring the electrical resistance of the 3ω metal line at what will be its normal operating temperature, and to then heat the 3ω metal line and measure the change in resistivity as the temperature rises.
In order to characterize dR/dT calibration curves of each 3ω metal line used in this work must be generated. These calibration curves are generated by first placing a silicon wafer onto a hot plate and measuring the resistance between its pads.
The temperature of the metal plate is raised and allowed 10 minutes to reach its new steady state. At this point the resistance of the metal line is measured again, this is completed enough times to generate a plot of the metal lines resistance over a large temperature range (x100 larger than it will experience during its normal operation) and the slope of the resulting trendline is taken. This is the dT /dR value for this metal line. Au), α is negative and this is due to strain gauge effects which outpaces the thermal effect of the resistivity of a metal outlined in Eq. 11 [7]. The remaining calibration lines for the 3ω heaters used in this work are represented in Fig. 25 and in Fig.   26. Figure 25 is the calibration curve for the second 10 µm metal line used in this work and it shares a very similar calibration curve to the one determined from Fig. 24, the similarities here will all likely come from both of these metal lines being prepared on the same silicon wafer. The success rate for producing the 3ω metal lines used in this work is rather low, for that reason multiple metal lines were produced on each silicon wafer to ensure the success of at least one it is still usable in this work with the generated calibration curve.

3ω Testing Procedure
With the fabricated and calibrated 3ω metal lines and 3ω circuit designed, the testing of the nanofiber thin films can be completed. To conduct the tests, an This process is completed again turning the nanofiber thin film 90 • to now measure the thermal conductivity of the nanofiber in the opposite orientation.
This entire process can be completed at as many different frequencies as needed to generate a curve that verifies the ω voltage component which is later approximated as a temperature differential and used to calculate the thermal properties of the polymer nanofiber thin film.
The final parameter to gather data for in this work is to measure the impact of the thermal contact resistance (R c ) between the polymer nanofiber thin film and the 3ω metal line. In most macroscopic systems, R c is ignored due to its generally insignificant contribution to a system, however when working with dimensions at the scale of 100's of nanometers this loss can play a larger roll in the overall heat transfer and losses in a system. This test is conducted by running these procedures again with one distinction. The attachment pressure of the nanofiber thin film is now recorded to measure the impact it has on the calculated thermal conductivity of the polymer nanofiber thin film. The attachment pressure is measured by using 52 an increasing mass to push the nanofiber thin film into the 3ω heater with more force, and keeping the pressure at which it is applied constant through the use of the SEM pin mount that the nanofiber thin film is attached to.  Through this procedure the lock-in amplifier is able to simply read the RMS voltages at the two nodes it is connected to. By subtracting these two voltages, one is left with the ω voltage component of the 3ω metal line. This procedure is conducted twice, one with only the substrate and a second time with the thin film included. This will result in a ∆T s for the substrate only measurement, and a ∆T s+f for the substrate and nanofiber thin film measurement. The following relation is used to determine the thermal conductivity of the thin film by modifying

List of References
Fourier's Law. parallel to one another. It is known that this was not the case for the sample tested in this work and that the nanofiber thin film was produced with nanofibers that were simply more aligned than that of a randomly oriented nanofiber thin film. The model can simply be modified to have a component that simulates nonperfectly aligned nanofibers and then predict the differences for the in-planer and cross-planer thermal conductivities.

CHAPTER 5
Summary and Future Work

Chapter Summaries
This section reviews each chapter, summarizing the methods used, results found and conclusions made as well as any relevant details of interest.
1. This chapter provides background on thermal transport at the macro and micro scale. In addition to this specific measurement techniques of the thermal conductivity measurement at the micro/nanoscale are discussed, as well as defining the proposed modifications to the 3ω method which makes this work possible.
2. This chapter looks at the results of thermally modeling a nanofiber thin film as a set of thermal resistors arranged in both series and parallel. Through this approximation it can be concluded that the axial thermal conductivity of the nanofiber thin film will be orders of magnitude greater than the trans-axial and even cross planer thermal conductivity.
3. This chapter outlines the processes and procedures used in the electrospinning process to generate the necessary fibers for this experiment, as well as outlining the process used in fabricating the 3ω heater system. Beyond just the fabrication of the 3ω heaters mentioned in this section, a process for calibrating the temperature dependance of these metal heaters is also outlined in this section.

Future Work
While the results found in this work show that the thermal conductivity of a nanofiber supported thin film is far greater than the thermal conductivity of the same polymer in bulk form, there still exists many avenues that this work can be expanded upon. The first planned extension of the work is to repeat the process again with the inclusion of metallic nanoparticles into the fiber by using a core shell spinneret, Fig. 32. Theoretically this should greatly increase the thermal conductivity of the thin film in the axial direction due to the contribution of high thermal conductivity nanoparticles. But this can be difficult to achieve experimentally. When spun in a core shell fashion nanofibers occasionally have an issue where they will form beads, creating areas in the nanofiber void of any metallic nanoparticles. The chance of this increases as the dielectric constant of the composite material also increases. This constant is likely to increase with the introduction of any metallic nanoparticles. [1,2]. It is also likely that by using a more conductive core solution than the shell solution, it will be pulled faster and result in breakages.
The benefit to the 3ω measurement platform however is that it is applicable through a very wide range of thin film structures and varying levels of thermal conductivity measurements. What worked for the non nanoparticle infused nanofiber thin films should also work for the nanoparticle infused thin films. The goal of this work is to determine the thermal conductivity of the stack when the nanofibers are all oriented in the same direction, alternating perpendicularly to each other, and randomly oriented one atop another. Once again the 3ω method can be used to measure the thermal conductivity of the nanofiber stack, but in this work other methods such as parallel plate thermal conductivity testing methods will be considered.
The 3ω test platform used to measure the thermal conductivity of the polymer nanofiber thin films in this work were designed using dimensions that were shown to be successful in other works [3,4,5]. However they quickly failed by the mechanical lift off process that occurred during the testing portion in this work. In order to work around this, the best method that was explored was to implement the 3ω heater into a printed circuit board (pcb). By adding the entire Wheatstone bridge in Fig. 18 including a 10 turn potentiometer, the system will be made much more robust and also be able to complete 3ω measurements without the fear of the To start analyzing the thermal conductivity of the nanofibers, the energy balance equation must first be used and solved with the models boundary problems to identify important terms.
Because the heat transfer is unidirectional and there is no internal heat generation, the equation simplifies to the following: The variables can further be non-dimensionalized Defining the boundaries of the system allows for the differential equation to be solved for all values within the boundaries, thus: Differentiating with respect to these variables as well as applying the boundary conditions results in the following temperature relationship This temperature relationship can be used in the Fourier Equation to define the heat flux through the material at its steady state. Looking at the terms that have an effect on the total heat flux through the material, keeping the temperature sources and the distance constant, only the effective thermal conductivity of the nanofiber thin film can be changed to effect the heat flux.

A.2 Thermal Resistivity Model
To compare the cross-plane thermal conductivity (λ x ) and the in-plane thermal conductivity (λ y ) a thermal resistor model can be analyzed. The thermal resistance of a material or composite is described as being directly proportional to the distance between the high temperature (T H ) and low temperature (T L ) sources, and inversely proportional to the cross sectional area (A) and effective thermal conductivity per unit length (k ef f ) of the material. Of these terms the effective thermal conductivity is the most important term as it is dependent on the temperature differential being oriented in the trans-axial direction of the nanofiber thin film, or oriented in the axial direction of the nanofiber thin film.
To define k ef f the thermal conductivity of both the nanofiber and the void between them as well as a ratio of void to nanofiber volume is needed. These terms are defined as k f , k a , and φ for the thermal conductivity of the fiber, thermal conductivity of air, and void percentage respectively. Inserting these into equation A.10 for the axial case seen in  Model of materials acting as thermal resistors in parallel for when the temperature differential is applied axially to the thin film. B) Model of materials acting as thermal resistors in serial for when the temperature differential is applied trans-axially.
If ∆y and ∆x are assumed to be equal, Eq. A.11 can be divided by Eq. A.12 to create a relation between the thermal resistance of a well aligned composite material thin film in its axial and trans-axial directions.
This relationship shows that the resistance in the axial direction will reduce rapidly in comparison to the trans-axial thermal resistance with increasing k f . Most notably in this relation however is that the cross-sectional area term, A, has no effect when comparing the two thermal resistances. This shows that it is not dependent on how many or how thick the fibers are, but only the fraction of the thin film that is fiber or air.

A.3 3ω Thermal Calculations
The 3ω heaters are responsible for determining the thermal characteristics of the measured polymer thin films. This is completed by integrating the 3ω test platform into a specialized circuit. This circuit contains a reference current, whose current is the same as the 3ω input current, and a Wheatstone bridge to balance the circuit and determine the difference between the 3ω signal and the reference signal.
The reference current is provided by a signal generator. This signal generator is able to have the frequency of the given signal modulated as needed to provide signals at various voltages and frequencies. Thermal conductivity measurements are typically conducted by comparing the differences in a sample's reaction to a signal given at two different frequencies, f 1 and f 2 . All voltages applied in this work are RMS voltages.
The thermal conductivity of a nanofiber thin film is first measured as a change in temperature of the sample, ∆T caused by the oscillating joule heating exerted on the sample by the 3ω heater. This is described by the following relation.
In this relation, dT dR is the differentiated calibration of the metal line, R is the average resistance of the 3ω heater, V is the V RM S of the input signal into the 3ω heater, and V 3 is the voltage differential across the 3ω heater. V 3 is evaluated at two different frequencies in order to determine the thermal conductivity of the measured sample. The thermal conductivity of the nanofiber thin film can be found through the following relation [1].

A.4 Thermal Circuit For In-Plane and Cross-Plane Heating
The heat conduction experienced when heating a nanofiber thin film can be modeled as a thermal circuit. The power input is represented by P with half of the energy going into the silicon wafer that the 3ω heater is deposited onto and the other half going into the polymer nanofiber on top of it. The steady-state heat conduction for this situation in the frequency domain is expressed as: in this expression a = jωρc/k. j is the imaginary component of an alternating current, ρ is the material density, c is the specific heat, k is the thermal conductivity, and l is the thickness of the medium. P represents the power applied, and ω is the angular frequency. The temperature variation with respect to the input frequency is described as: ∆T (ω) = P tanh(al) ak (A.20) Thermal impedence may then be determined by dividing Eq. A.20 by P resulting in [2]: From the definition of electrical impedance, thermal impedance can also be inferred through its relation to the thermal resistance, thermal capacitance, and the input heater is the contact resistance between the resistor and the polymer nanofibers.
In most situations the contact resistance between two materials is ignored because it is of a much smaller magnitude then all other equivalent resistances. This is the case between the 3ω heater and the silicon wafer as the heater is well applied to the silicon wafer. The contact resistance in this circuit is not known, but the variation in contact resistance will be explained through the use of varying attachment pressures used to force the polymer nanofibers into the 3ω heater. Previous work has shown an inverse relationship between the attachment pressure of a specimen and the thermal contact resistance when using the 3ω method [2]. In the case of cross-planer heating the thermal resistance expression for the nanofibers and the aluminum backing are different because the polymer nanofibers are a normal material in heat transfer and the aluminum backing is considered a semi-infinite medium. This leaves the thermal resistance for the polymer nanofibers as: leaving the thermal capacitance to be described as: When taking into account the assumption that there is no longer a ∆T across the entirety of the medium (The temperature distribution of the medium has some points with T =T ∞ ) the medium can be considered semi-infinite. As this happens Using the ∆T obtained from previous equations on the 3ω method, the nonnegligible contact resistance between the polymer nanofibers and the 3ω heater can now be evaluated and further be related to its attachment pressure to determine the relation there.

A.4.2 Thermal Circuit for In-Planer Heating
When using a 3ω heater that produces purely in-planer heating the thermal circuit simplifies to totally exclude the aluminum backing. In this scenario the polymer nanofibers are now assumed to be a semi-infinite medium. With this assumption not enough energy penetrates into the polymer nanofibers to reach the aluminum backing as the temperature distribution equals the ambient temperature at the edges of the polymer nanofiber thin film. As is seen in Fig. A.3 the thermal circuit simplifies to include only the thermal resistance of the silicon wafer, the polymer nanofiber thin film, and the contact resistance between the medium and the 3ω heater, as well as the thermal capacitance of the silicon wafer and the polymer nanofiber thin film. The relation in this case for the thermal contact resistance between the polymer nanofiber thin film and the 3ω heater changes little. It is now simply determined through the thermal impedance of the two mediums in parallel rather than 3 mediums in a complex circuit. The thermal resistance for both the polymer nanofiber thin film as well as the silicon wafer in this configuration can be determined using Eq. A.25, and the thermal capacitance is determined using Eq.
A.26. The reason why the polymer nanofibers can be considered a semi-infinite medium in this situation goes back to the relation tanh(al). With a being directly related to the input frequency supplied by the frequency synthesizer and l being the thickness of the medium (polymer nanofiber thin film in this case), using a higher input frequency as well as a thicker polymer nanofiber thin film, tanh(al) → 1.
In order to calculate the thermal contact resistance when using an in-plane heating configuration the ∆T must first be determined using methods discussed earlier in this work, and relating it to Eq. A.28. However for this scenario Z t (ω) is found using a different relation than used in the cross-planer heating scenario.
To find Z t (ω) in this scenario the following relation is used.
where Z pcl can be found calculated as the inverse of the thermal resistance of the polymer nanofiber thin film plus the product of the complex conjugate times the thermal capacitance of the polymer nanofiber thin film.
Using the same methods to calculate the thermal contact resistance found in the previous section, the thermal contact resistance between the 3ω heater and the polymer nanofiber thin film can also be analytically determined.
As seen in both the in-planer and cross-planer thermal circuit, the thermal inertia of the mediums through which heat is transferred plays a significant roll.

71
The thermal inertia of a material is defined as follows.