A NOVEL APPROACH FOR MEASURING THERMAL CONDUCTIVITY OF ELECTRO-SPUN NANOFIBERS

The ability to replace metals, ceramics, and composites with polymer nanofibers can lead to more desirable applications in heat exchangers, energy storage, and biomedical fields. Research into thermal transport within polymer nanofibers has increased for such applications, with the need to develop new techniques to measure and understand such properties. Polymers with high thermal conductivity have been a growing asset in desired heat transfer devices. Much effort has gone into developing advanced thermal conductivity measurement techniques; however, there is still a lack of fundamental understanding between the relationship of structures and thermal transport properties in polymer nanofibers. The purpose of this thesis is to contribute to the further understanding of the thermal effects and conductivity capabilities of polymer nanofibers, and to supply a novel method of measurement to the research community.

nanofibers have shown thermal conductivity to be as high as 104 W/mK, which is larger than about half of the pure metals [1]. Due to the highly oriented polymer chains and enhanced crystalline structure, a fiber with high thermal conductivity can be produced using high voltage [2]. The best known way is to use the electrospinning method to create these nanofibers. Creating a device to measure these nanofibers is a challenge.

Thermal Transport Effects in Nanostructures
With an increased demand for small technological devices and processed materials, including structures on the nanometer scale, a greater understanding of thermal transport of nanoscale devices and individual nanostructures arise [3,4]. Novel thermal properties arise in low-dimensional nanostructures versus in bulk materials. Properties such as abnormal heat conduction, size dependence of thermal conductivity, phonon boundary internal and edge scatterings effect these onedimensional structures [5,6].

Thermal Properties of Polymer Nanofibers
Thermal transport within nonmetallic systems, such as polymers, transport heat via phonons. Phonons are a quasiparticle associated with vibration of a crystal lattice. While these phonons span a broad range of frequencies, those with mean free paths < 100nm at room temperature are typically the main contributors to thermal conduction [3]. However, structures within this scale no longer adequately follow the thermal transport for bulk models [7,8], unless taking into account the energies across boundaries [4].
The low thermal conductivity of bulk polymers comes from their structure due to random orientations and weak couplings of the polymer chains. Imperfections within bulk materials such as voids and entanglements, generate high thermal resistance between the crystalline domains; high frequency phonon modes within must transfer from high to lower frequencies modes across domains [9]. It has been found through many studies that polymer nanofibers with highly aligned polymer chains can have much higher thermal conductivities and Young's modulus than typical bulk values [1,2,3,9,10,11]. As the degree of crystallinity, increases so does the thermal conductivity [12,13,14].  [15,16]. This was suggested due to the relaxation times and residual solvents remaining within after the fiber has been spun; this can accelerate chain relaxation and lead to shorter relaxation times [17,18]. This can also be seen for polymers with high glass Transition temperature (Tg) values [19]. However, with polymers and polyesters with lower Tg values, such as PCL (Tg ∼ −60 • C), takes longer time to crystallize [20]. A gel-spinning process can fix this dilemma by applying a post-drawing once the fiber is in solid state below the melting temperature to prevent chain relaxation after orientation [21,15]; the design is seen in 1. voltage signal is used to excite the heater at a frequency w, periodic heating generates oscillations in the electrical resistance of the metal line at a frequency of 2w. In turn, this leads to a third harmonic (3w) in the voltage signal, which is used to infer the magnitude of the temperature oscillations" [22]. However, this technique is difficult to implement on a single nano-structure.
Another technique used a micro device that suspended carbon nanotubes between two silicon nitrite plates. Current was passed through one generating a heat input, where the induced temperature change was measured through the other [23]. These plates were fabricated using an electron beam, photolithography, metal coating, and etching. Each island consisted of a platinum thin film resistor. This served as a heater to increase the temperature. Since the tempera-ture changed with a change in their resistance, these could also be used to measure the temperature of each island [24]. Carbon nanotubes were then bridged between the two suspended islands using a similar method used to fabricate Atomic Force Microscope (AFM) scanning probe tips. This technique is used throughout industries to measure the thermal conductance of carbon nanotubes. However, it is not used for single polymer nanofibers. In more recent years two measurement methods were able to measure the thermal conductivity of a single polymer nanofiber: Shen [1] and Canetta [25]. Shen utilized a bi-material cantilever to directly draw a polymer wire from a droplet from a soldering iron tip. Shen constructed an optical system within a vacuum to measure the deflections from the bi-material cantilever once the tip of the iron was heated, seen in Figure 1.3. A fiber thermal conductivity could be known since the temperature of the heated tip and calculations of beam deflections due to temperature changes were known. Canetta used a slightly different approach. Instead of drawing out a fiber onto a cantilever he designed the dual-cantilever technique, seen in Figure   1.4. This technique utilized two fabricated bi-material cantilevers, one for laser heating and the other for deflection sensing. Similar to Shen, Canetta constructed an optical system within a vacuum. Polymer nanofibers were electrically spun and then bridged between both cantilevers. Once one cantilever was heated with a known power, the deflection of the sensing cantilever was known.

Proposed Method
In combination with both Shen and Canetta's designs, my novel design will use two cantilevers in forced actuation, as seen in Figure 1.5. This design will take a look at a new thermal conductivity measurement design that focuses on a mechanical system rather than an optical method. The reason to create such a design is to mitigate complexity in measuring polymer nanofibers by joining two independent subsystems. This allows the underlining bi-material cantilever and fiber sample to be easily loaded into a traditional AFM without modification or cost.
A sample plate, which is loaded into the AFM, includes a heated prong, bi-material cantilever, and electro-spun nanofiber, where as the AFM will only include a tipless silicon cantilever and operated normally. Both cantilevers will be placed tip to tip, once temperature is added to the system, the temperature sensitive bi-material cantilever will force the AFM's cantilever to deflect. The deflection will then be recorded, which will then be used to obtain the fibers thermal conductivity. This will be known as the Probe-to-Probe Technique. One of the most common and traditional methods to create nano sized polymer fibers is the use of an electro-spinning apparatus. Electro-spinning has regained more attention due to a surging interest in nanotechnology since various polymers with diameters down to submicrons or nanometers can be easily fabricated [26]. It is currently the only method that allows the fabrication of continuous fibers with diameters down to a few nanometers [17].
Electro-spinning is a process by which polymer nanofibers with diameters between 0.05 and 5 microns can be produced using an electro-statically driven jet of polymer solution using high voltage electric fields [27,28]. In general, a fluid solution is pumped through an electrode to a grounding plate or a collector plate.
When a high electric field is applied to the droplet of fluid, which may be a melt or solution coming out from the tip, this acts as one of the electrodes [29]. "When an external electric field is applied to a solution, the positive and negative ions in the polymer fluid tend to move in opposite directions. Negative ions are forced toward the positive electrode, and positive ions are forced toward the negative electrode [19]." This leads to the droplet deformation and finally to the ejection of a charged jet from the tip of the cone accelerating towards the counter electrode leading to the formation of continuous fibers [30,31].
At a traditional needle tip where the droplets or strands are produced, a Taylor Cone is formed. This is a consequence of electrical forces that form a conical protrusion [19,32]. This theory was first described by Taylor in 1964 [33].
As charged liquid is pumped out of the needle, surface tension keeps the droplet appended until the surface charges of the droplet are overcome by the grounding electrode. As the intensity of the electric field is increased, the hemispherical surface of the fluid at the tip of the needle elongates to form a conical shape known as a Taylor Cone. Taylor studied these electric fields E and their influence on surface tension σ, where ε o is the permittivity of free space (F m −1 ), r tanα is the curvature of a cone, and α is the half cone angle [34,35] [36]. The surface tension, viscoelasticity, and charge density within the ejected polymer are the key influences in proper fiber formation [37].
The discharged liquid solution jet undergoes an instability and elongation process, which allows the jet to become long and thin. Meanwhile, the solvent evaporates, leaving behind a charged polymer fiber. In the case of the melt, the discharged jet solidifies when it travels in the air [26]. The discharged polymer solution jet undergoes a whipping process wherein the solvent evaporates, leaving behind the charged polymer fiber, which tends to the grounding plate [28]. Angstroms utilizing an electron beam rather than a light source [40]. However, polymers exhibit weak electron scattering and poor contrast and act essentially like organic materials within this system. Secondly, polymers typically have low electrical conductivity leading to rapid accumulation of negative charges, which dramatically decreases resolution [41]. SEMs may also damage polymers due to their high power; AFMs only utilize the inter-atomic forces, a non-destructive technique for surface measurements [42]. The AFM will be used based on its ability to detect the small temperature-mechanical displacement variations implemented into the system and utility compared to previously mentioned instruments.

AFM Components
The main components of an AFM consist of a cantilever probe, piezoelectric scanner, force sense, laser, Proportional Integral and Derivative (PID) controller, and a photo diode signal detector, as seen in Figure 1.6 [43]. A probe is attached to a cantilever beam that is used to read the roughness of a surface as the probe scans in XY coordinates and interacts with the sample. As the cantilever is deflected due to forces exerted from the sample, a focused laser senses this change and sends a signal to the photo-diode detector. Once the detector notices this change, either the piezo stage moves to counter at the variation or stays static and records the bending moment, depending on the mode type. The AFM has a few different operating modes to use and consider for the novel measurement design. The operating modes are split into dynamic or static modes with attractive and net-repulsive interactions, contact or non-contact within modes.
Dynamic, or tapping modes, measure the changes in the vibrational properties of the cantilever measuring: resonance frequency, oscillation amplitude, and phase between the cantilever oscillations, simply intermittent contact. Static mode can operate either in applying constant force or constant deflection by just adjusting the z-piezo axis of the cantilever. However, in this case, since the sample will not be moving nor have the need for vibrational properties and the need for variable force change, the design will use a net-repulsive contact static mode, non-vibrating mode. The system will be held at a constant position and not allow the cantilever to re-position once deflection is noticed.

AFM System Parameters
Without constructing an external optical system as seen in prior designs, AFMs are programmed to be optimized, adjusted, and calibrated to certain preferences.
This allows the PID gains controller and High Voltage (HV) gains of the system piezos to adjusted to particular parameters, depending on the need.
A sample is typically fixed on the top of a 3-axes piezoelectric stage that moves the sample under the tip, where the movements X and Y are controlled by the computer that generates two synchronized voltage ramps (gains) [44]. Once in contact, a force transducer sensor senses the force between the tip and the surface, allowing the feedback controller to feed a signal from the transducer back to the piezoelectric stage. This allows the AFM to maintain a fixed force between tip and sample. Typically, the expansion coefficient for a single piezoelectric device is on the order of 0.1nm per applied volt. Therefore, if the voltage to excite the piezoelectric stage is 2 volts, then the material will expand about 0.2nm [39].
The z-piezoelectric stage and the force transducer are linearly mirrored.
Therefore to get the most data out of the AFM in use (can only record two devices at once), the XY piezo gains and force transducer gain will be zeroed. Zeroing the transducer gains allows the piezo stage not to respond to photo-diode signals.
Secondly, turning off the X and Y piezo stage gains hinders the scanning in X and Y directions, but still allows sensing detection from the photo-diode. Additionally, the PID gain controls can also be adjusted to scale the system properly, such that the feedback loop will respond quickly to topography changes. More on these settings will be discussed in Chapter 4.

Use of Bi-Material and Silicon Cantilevers
The reason for using a bi-material cantilever is because it has high sensitivity due to small dimensions and thermal mass [45]. A bi-material cantilever beam is comprised of two material layers. Typically bi-material cantilevers are made of silicon and coated with gold or aluminum to improve reflectively for AFM sensing.
Once a temperature change is introduced, the cantilever will deflect due to the difference in Coefficients of Thermal Expansion (CTE) of both layers. These types of bi-material cantilevers have shown the ability to detect deflections at 3 pm resolution with the measurement of temperature, optical power, and energy with 2 µK, 76 pW , and 15 f J resolution, respectively [46].
Typical silicon heated cantilevers have spring constants ranging from 0.01 to 10 N/m, and resonant frequencies ranging from 50 to 300 kHz and are calibrated using the thermal noise method [47]. Raman microspectroscopy characterizes them with spatial and temperature resolutions of 1 µm and 3 • C [47,48].
Additional research of laser heated cantilevers, shows that contact thermal conductivity is typically in the range 0.1-100 nW/mK; and for probe tip radii of 30 nm, a typical contact diameter is about 10 nm providing a contact force of 10 nN.
The spreading contact conductivity due to 10 nm contact on a polymer sample becomes 2 nW/mK [48]. This may be useful when determining conductivity of the cantilevers utilized within this thesis.
Furthermore, the heat transfer coefficient air gap h airgap may be noticed when determining heat transfer from fiber to cantilever. As seen in equation 1.2, the h airgap would be on the order of 10 kW/m 2 K for a tip height of 1-10 µm.
Similarly, the probe being utilized within this thesis has a 15µm diameter. In continuing works, this may be useful. Moreover, 10 kW/m 2 K will also be utilized for the heat transfer coefficient for the fiber.
More on the properties of the cantilevers can be seen in the Appendix.

Thesis Outline
The purpose of this thesis is to further understand the thermal effects and conductivity capabilities of polymer nanofibers and to supply a novel method of measurement to the research community. With the use of temperature sensitive bi-material cantilevers and AFMs, one can obtain the thermal conductivity of electro-spun nanofibers.
Chapter 2 reviews introductory thermodynamics, along with the theoret-ical design of the Probe-to-Probe measurement system. This design incorporates a two cantilever force interactive relationship. The reason to create such a design is to mitigate complexity in the measurement of polymer nanofibers by joining two independent subsystems. This allows the underlining bi-material cantilever and fiber sample to be easily loaded into a traditional AFM without modification or cost. Detailed modeling of this conjoined system design can be seen here.
Chapter 3 focuses on the experimental apparatus, fiber creation, and equipment used. It will review the electro-spinning apparatus used for fiber fabrication, the experimental setup and creation of sample plates, fiber appending, measurement and alignment, and AFM integration for the measurement process.
Chapter 4 describes the procedural measurement of the system and outlines how the system was calibrated, along with how each of the system settings was determined. This chapter will review introductory heat transfer and the thermal modeling of the system in use. As seen in figure 1.5 the system utilizes a heating prong, a silicon tipless AFM cantilever, and a bi-material AFM cantilever. As heat is sent through the base of the heating prong it is transferred through the fiber to the tip if the bi-material cantilever. Due to the thermal expansion of gold verse silicon nitride, the gold deflects and adversely bends the tipless AFM cantilever. The AFM cantilever deflects, and the laser from the AFM records this beam deflection.
As seen from natural beam deflection theory and Hooke's Law this deflection can be used to obtain the temperature output T ct from the fiber at the tip of the bimaterial cantilever. In congruence, the flux q f and temperature T f into the fiber from the heating prong is obtained through the use of thermocouples. The thermal conductivity of the nanofiber k f can be found after finding the temperatures at both ends of the fiber.
The system will be modeled as a one-dimensional time-independent steady state problem. This allows the modeling and measurements to be simplified. Subscripts p, f , and c, and directions x 1 , x 2 , and x 3 represent the prong, fiber, and bi-material cantilever respectively throughout this paper.

Heat Transfer and Conduction
Before reviewing the thermal models through the proposed system, the first step it to briefly explain heat transfer fundamentals for one-dimensional systems. Heat transfer by conduction is the flow of thermal energy within a solid and non-flowing fluids driven by a non-uniform temperature field. Heat transfer Q and work W are the two types of energy interactions that make up the internal energy E. These interactions are known as the first Law of Thermodynamics for a closed system or the conservation of energy: Internal energy is associated with the disorderly motion of molecules transferring kinetic and potential energies, where heat is transferred by conduction. Work is the transfer of energy resulting from a force acting through a distance and heat is the energy transferred as the result of a temperature difference [49]. In terms of unidirectional and per-time equivalent through a insulated beam with heat generation from equation 2.1, internal energy can be simplified as as a heat transfer rate q with some output change ∆x in terms of heat . This is also known as Fourier's Law of heat conduction which reads where k is the thermal conductivity and ∇T is the temperature gradient [50].
This is the assumption that the material is isotropic and homogeneous where the thermal conductivity is constant.

Thermal Modeling of a Fin
For both 1D beams and rods, or cantilevers and fibers respectively, the heat transfer through these systems can be modeled as fins including Newtons Law of Cooling. Steady state is defined as the process of unchanging time which will be considered zero since the collection of experimental data will be assumed to be steady state. The heat generated term will be considered the heat sink for cooling fins. Reviewing equation 2.5 and noting that in steady state all the heat that is being generated inside the fins must be transferred to the fluid around the fin, in this case the fluid is air. The temperature distribution T (x) reaches a steady state because the surface of the fins are bathed in an ambient temperature T ∞ where the heat transfer coefficient h is uniformly surrounding the perimeters P by a change of length ∆x. As seen in equation 2.7 the lateral heat convection can be written: The fins can be modeled as 1D because the conduction through the axis the y and radial directions respectively are much lower than in the lengths of each model. Using the dimensionless quantity of the Biot number, the thermal contact of the surface of the fins to the fluid can be shown to be 'poor', or that the fins are good thermal conductors. If the Biot number in equation 2.8 is << 1 over the length of the fin, than the boundary temperature is nearly the same as the temperature in the center of the fin. A typical AFM cantilever has a Boit number less than 10 −4 , confirming it can be modeled as 1D [48,51]. This also works for each subsystem (prong, fiber, and both cantilevers).
Again with assuming steady state and that each part of the system is modeled as individual fins, the heat balance for each fin problem will be similar. Combining equations 2.6 and 2.7 the heat balance equation becomes: Taking the lim ∆x→0 , equation 2.9,: Then substituting the modified Fourier Law as heat flux and taking the derivative with respect to x gives: Combining equations 2.10 and 2.11 the governing equation of a fin becomes: The constants m, P , and A will represent the perimeters and cross sectional area respectively in each of the three fin systems. For the heating prong and bi-material cantilever the perimeter P = w + s and the area A = ws, where w and s are the width and thickness respectively. For a fiber it is modeled as a cylinder so P f = πd where d is the measured diameter of the fiber. Therefore the general solution temperature profile of a fin is equation 2.14 where C a and C b will represent variable constants, a and b represent multiple constants.
This will be utilized throughout modeling each fin. After the boundary conditions are defined, the constants can be found.

Thermal Modeling of the Heating Prong
When it comes to appending a fiber to the system it is difficult to get it directly in the same position along the heating prong every time a new fiber is added. The best method is to theoretically model the flux at any point to the fibers position on the prong. Since the fiber can sit anywhere along the prong, the heat flux, temperature, and length to the fiber from the base, all vary. As seen in Figure. 2.2 the equation can be used to measure the heat flux at any position where q f (x 2 ) is the flux into the fiber, q p1 (x 1 ) is the input flux at the fiber location on the prong and q p2 (x 1 ) is the output flux to the rest of the prong. In this case A f = A p since fluxes in the contact cross sectional areas are the same. Since flux into the fiber is represented from equation 2.15, a change in temperature along the fiber will also need to be found. Equation 2.16 is the general form in which to solve for k f . The first part for modeling the system is to obtain the temperature profile of the heating prong at T p (l 1 ) = T f Cal without an appended fiber where l 1 is the length from the base of the prong to the fiber. From equation 2.14, T p (x 1 ) is represented as, Utilizing boundary conditions: Solving for constants C 1 and C 2 : Combining equation 2.17 and constants C 1 and C 2 at T p (l 1 ) = T f Cal and re-writing the profile in terms of T f Cal becomes: Again this temperature value is represented as the heat flux q p1 at any temperature T f Cal along the heating prong.
2.3.1 Heating Prong Fluxes q p1 (x 1 ) and q p2 (x 1 ) Obtaining fluxes q p1 and q p2 are the next steps to solving the thermal conductivity of a single nanofiber using equation 2.16. To find both fluxes q p1 and q p2 , the temperature profiles will be split into two separate systems as seen in Figure 2.2.
Again, the T f obtained here will be compared with the T f Cal to show similarity.
Heating Prong Flux, q p1 (x 1 ): The governing equation can be written as: where the temperature profile from equation 2.14 becomes Utilizing boundary conditions: After applying boundary conditions, the constants are found to be: Combining equations 2.16 and 2.3.1, and rearranging the constants solving for q p1 (l 1 ) gives: Heating Prong Flux, q p2 (x 1 ): The flux from the second part of the prong is similarly solved as q p1 however, with reverse direction. Referencing back to Figure 2.2 and equation T p2 (x 1 ) governing equation can be written as: where the temperature profile becomes Applying boundary conditions: Again this tip temperature T pt is when a fiber is appended. The temperature exiting and entering into the second portion of the prong and fiber should all be the same temperature T f Cal . After applying boundary conditions constants are found to be: Combining equations 2.16 and 2.31, and rearranging the constants solving for q p2 (l 2 ) gives: Modeling of a Single Nanofiber however, there is an additional ambient temperature change over the length of the fiber. This can be measured, seen in calibration from the prong to the bimaterial cantilever. Therefore the change of ambient temperature was treated as a linear decline since T ∞p at the prong has a higher temperature than T ∞c at the is written as is similar to a forcing function for this second order non-homogeneous differential equation, the temperature profile T f (x 2 ) will have to be split into a homogeneous and particular solutions [52]. Using where y = T f (x 2 ), y h is the homogeneous solution, and y p is the particular solution.
The solution to the homogeneous solution of the temperature profile is when y p = 0, which is similar to that of previous fin solutions. Allowing to be a linear function represented as where constants C 9 and C 10 can be simply solved as The particular equation solution reduces to: Adding the homogeneous equation 2.39 and particular 2.45 solutions to obtain equation 2.46, keeping C 9 and C 10 for simplification.
After applying boundary conditions: C 7 and C 8 can be found: Now that all the fluxes of equation (2.15) have been obtained, all but the thermal conductivity of the fiber k f and the tip temperature of the bi-material cantilever T ct are known.

Applied Temperature with AFM Integration
Since flux into the fiber is represented from equation 2.15, all the fluxes 2.29, 2.36, and 2.51 have been solved. Next is to obtain the output temperature T ct to be substituted into equation 2.51 where the thermal conductivity k f of the fiber can be found. This temperature is found through the bi-material cantilever deflection with an applied force at the tip. The applied force is due to the resistance from the above AFM cantilever; the natural beam deflection, where only thermal expansion is acting on it. Since the natural bi-material deflection is a function of the temperature profile T c (x 3 ) the next set of modeling will be split into three sections: temperature profile, natural beam deflection, and acting force displacement from Hooke's Law.

Bi-Material Cantilever Temperature Profile
Starting with the temperature profile of the bi-material cantilever, as seen in Figure 2.4, the x 3 direction is now from the base of the cantilever T co to the tip of the cantilever T ct of some L c . The cantilevers in use are a triangle form. However, this can be simply modeled as a rectangle [45].
Conducting the same steps as in section 2.2 for the thermal modeling of a fin the temperature profile is where m 2 c was solved when the width was much greater than the thickness t: Utilizing the boundary conditions the constants are solved as: where the complete temperature profile of the bi-material cantilever is:

Bi-Material Natural Beam Deflection Theory
This section will cover a bi-material natural beam deflection when there is no force added to the tip of the cantilever. In the below equations, three assumptions were made. One is that a linear strain distribution existed through the thickness of the beam, second that both materials are perfectly bonded at the interface, and third, that the temperature distribution within the cantilever is uniform. Upon temperature change, the bi-material cantilever will deflect due to the bending moment generated by thermal expansion (a) of the two materials. As seen similarly where E is Young's Modulus and T o is the temperature at which the cantilever has zero deflection throughout, ∂ 2 znat Integrating again using equation 2.63, d 2 is Instead of writing out the the deflection at some z(x 3 ) since only the deflection at the free end of the cantilever is needed, the full equation at z nat (L c ) becomes: This is the general form however, this model can be reduced when T co = T ∞c to equation 2.68. This assumption can be made since in this case the cantilever has non-uniform temperature distribution due to the heat at the tip of the cantilever.
This non-uniformity can be kept small from the temperature output of the fiber [53].
This deflection is only taking into account a natural beam deflection with no applied force at the free end. Solving equation 2.68 for T ct simplifies to:

AFM Displacements Individual Zero Deflection
Before discussing forced beam deflection, the individual AFM cantilevers in use will be reviewed to obtain a T o value. Again this value is temperature at which there is zero deflection, which is more complex to find depending on the cantilever in question considering each cantilever has an individual intrinsic bending moment.
If referring to a basic Pyrex-Nitride Probe (PNP) cantilever, which are probes that have silicon nitride cantilevers with very low force constants, then these are optimized for a bending less than 2 • at room temperature for a long cantilever, a 100um will be less theoretically, as referenced by NanoAndMore USA Corp. But due to the fact that nitride and gold have different thermal expansion coefficients each cantilever has an individual zero deflection on a different temperature and drift as a function of temperature.
The cantilever bending angle is the angle between the tangent to the cantilever at its free end (tip) and the support chip surface in degrees. A positive algebraic sign a is used for bending towards the detector side and a negative sign for bending towards the tip side. A rough estimation of the bending angle may be calculated from the cantilever length l and the total deflection h; this is only valid for small angles. This may be useful when dealing with the calibration seen in Chapter 4, however if a 2 • or larger angle was to be used in a theoretical case, the deflection determined can be subtracted from final displacements. However, we can assume that since AFM cantilevers are optimized to have near zero deflection in room temperature, this small angle will be considered zero deflection at room temperature. So 23 • C, which is a standard scientific value of room temperature, will be used for T o throughout calibrations and measurements. Moreover, T o may be optimized in calibration, dropping to zero, since all temperatures will be zeroed for simplification.

Forced Beam Deflection
As seen in Chapter 4 when the system is calibrated, the AFM cantilever will interact with the bi-material cantilever. Both cantilevers will interact at the horizontal plan with the AFM cantilever fixture tilt at 10 degrees. In calibration, the cantilevers will undergo two different forces. The first force interacts in less than a few angstroms, this is known as the repulsive force. This results from a charge overlap between both cantilever tips. This force is very localized, and involves only a few near field interacting atoms. The other force is up to hundreds of angstroms known as the attractive force, called the van der Waals (vdW) force. This results from a change in dipole moment induced interaction [58,59]. In calibration it can be seen that these cantilevers can be adjusted to account for these forces. The assumption can be made when knowing the 'jump-to-contact' distance an additional jog on the AFM can be applied to infer that the net force on the system can be zeroed and both cantilevers attend to a horizontal plane again minimizing large angle deformations.  Now that Hooke's Law has been applied it must be referenced back to natural beam deflection to obtain T ct needed considering a change of temperature.
When calibrated, natural beam deflection is already occurring. As seen in Figure   2.5, we can see that where the difference from natural deflection to the displaced bi-material beam is equivalent to the change in displacement from the AFM cantilever [61]. When a force is applied from varied temperature of the bi-material cantilever to the AFM cantilever a change in δ af m occurs, dδ af m , where additionally z nat changes, dz nat .
This dδ af m change is the recorded displacement form the AFM photo-diode laser.
Both these values will be compared later after obtaining calibrated and appended fiber measurements for zeroing and acquiring the T ct value. This chapter will review the electrospinning apparatus used for fiber fabrication, the experimental setup of sample plates, and AFM integration for the measurement process.

Plate Design
A transitioning device and system play a key role to measure the thermal conductivity of a single nanofiber. Thermally sensitive commercially obtained bimaterial cantilevers have been shown to successfully measure and conduct polymeric nanofibers [62]. In addition to their abilities, they are traditionally used in nano-instrumentation as actuators and sensors [63]. AFM's are typically useful in measuring small optical cantilever beam deflections and are able to detect these small thermal changes. However, with the use of an AFM, a few complications can arise. AFMs utilize a sensing laser to measure these small deflections. Depending on the AFM, the lasers power may differ however, this power may effect ambient temperature measurements due to radiation and adversely modeled cantilever beam deflections. Additionally, for the practicality and advantages that an AFM provides, it can be difficult to calibrate if one was to construct an apparatus in tandem. By designing and creating a simple sample plate device one can limit optical interference and to utilize an AFM as easily as possible.
The sample plates were created to account for a few parameters. The first is that the plates have to conform to the traditional use of the AFM where the convective heat from the soldering iron and heating prong will not effect the AFM laser or lack of hindrance from the thermocouples. This also includes the AFM's cantilever for the use of detection. Again since this thesis focuses on not using power lasers for heating, the sample plate utilizes separate cantilevers for detection.
The bi-material cantilever is attached to the sample plate with the sphere facing up and the other is a tipless silicon nitride cantilever already calibrated in the AFM before use as seen in Chapter 4.
The second parameter is the working distance from the bi-material cantilever to the heating prong. The prong is heated using a ZENY 862D+ soldering iron which has a working lowest temperature of 100 • C with an upper limit of 480 • C. Since the system will undergo calibration and appending fiber steps where the bi-material cantilever will deflect due to ambient variations, an optimal heating prong distance from the cantilever must be obtained. Moreover, a proper thermal resistance from the soldering iron and measurable ambient differences must be achieved. Since the lowest controlled temperature of the soldering iron is 100 • C, this will be the starting point value when measuring the temperature from the heating prong. Keeping in mind that the value of the prong base should not exceed the maximum melting temperature of the nanofiber once appended, where the melting temperature of the polymer. It was found that through measuring the experimental system at steady state and varying the soldering temperature that about 7mm was an optimal distance. This allowed the soldering iron and thermocouples not to interact with the workings of the AFM. Meaning that the ambient temperatures around the bi-material cantilever are very small compared to the heating temperature of the soldering iron. This calibration will be noted in   The bi-material cantilever is then appended with UV glue upside down to the AFM sample mount 7mm from the heating prong while also allowing the probe to hang well over the curvature of the mount. This is to avoid complications when loading a fiber to the probe and AFM for measurements. To not potentially break break a probe, the cantilever should be appended just before fiber appending.

Fiber Creation
As explained previously in section (1.1) morphology and fiber properties are key in obtaining uniform thermally conductive nanofibers. The molecular chain length and the molecular weight of a polymer is vital when choosing a "good" solvent to use [64,65]. The first step for creating nanofibers is to properly formulate a solution for different polymers needed for measurement. Depending on the polymer being used for spinning, there are certain solvent carriers that work best with the polymer. This is due to the ability of the solvent to fully dissolve the polymer.
The polymer to be used for spinning and alternatively measuring the thermal conductivity needs to be determined. There are many different polymers to choose from depending on the use. Electrospun nanofibers can be useful in such fields as electrical or even biomedical. In the biomedical field, these fiber matrices have shown morphological similarities to natural extra-cellular matrices, characterized by ultrafine continuous fibers, high surface-to-volume ratio, high porosity and variable pore-size distributions [66]. Solution properties such as solution viscosity, conductivity, dielectric constant, and surface tension may go into the decision on a polymer needed.
In the proposed experiment, the chosen polymer is polycaprolactone (PCL) with a density (p x ) of 1.145g/mL at room temperature and a melting point of 60 • C. The reason for using polycaprolactone is due to the biomedical usage. PCL is known as a synthetic biodegradable aliphatic polyester for uses in tissue engineering, scaffolds, nerve guides, and drug delivery systems [67]. Since acetone is the working solvent, PCL works best [68]. Additionally acetone is the low end of the spectrum of toxicity, where solvents such as dimethylformamide (DMF) or tetrahydrofuran (THF) have high ratings of such toxicity. Keep in mind the melting point of PCL, this factors into the heating limit of the heating prong and the fiber itself. There are additional solvent and polymer combinations that produce more consistent and uniform fibers, that are more thermally conductive and have a higher melting point. However, note that this is also not to say that the method of creating a solution, the spinning process, or measuring the thermal conductivity is any different. Further details of the produced fiber properties can be seen in the Appendix.
With the chosen polymer we can move on to calculating a proper equation to use for determining a useful weight of solute needed for the concentration. Using a concentration of the polymer solution as w/v%, a sought after typical percentage would be between 7.5% and 10% for this solution [68].
w v % = mass of the solute w x volume of the solution x 100 (3.1) The optimization of electrospun fibers versus electrospraying is fundamental for electrospinnability. The determination of how the fibers are produced depends a lot on the concentration of the solution itself. There is a certain minimal concentration value that must be reached, which, if below this value, electrospray-ing occurs or the formation of droplets only [69,70]. This is because under high voltage or electrical force the charge density of a droplets surface at the evaporation point increases, the coulomb repulsion overcomes the surface tension and several smaller droplets are formed. This is theorized by Rayleigh instability phenomena [71,72]. The higher the concentration the more stable the string of droplets become [64]. On the other hand, if the concentration is too high and thus becomes more viscous, uniform fibers will no longer be produced [73]. As the concentra- where n sp is the specific viscosity, c is the concentration of polymer solution, [n] is the intrinsic viscosity of the polymer, and K m is a constant and it is a measure of polymer-polymer and polymer-solvent interactions [74,77].
Knowing what concentrations to use, next is to determine how much solute is needed. As seen in equation 3.3 the weight (w x ) of polymer in grams to use can be found utilizing 3.1 and rewritten in a general form, where v y is the volume of solvent being used and P is the fraction of the concentration wanted.
In this experiment, 20mL of acetone and a 10% PCL concentration was

Electrospinning Apparatus
For a traditional electrospinning apparatus only a few general items are needed, a high voltage power supply up to 30kV, a syringe pump, electrode connected to a needle, and a grounding collector. As seen in Figure 3.3 is the constructed custom electrospinning apparatus.

Figure 3.3: Electrospinning Apparatus
Electrospinnging systems can be setup for sheet collection as well as for a single nanofiber. Many traditional methods utilize spinning for thin film matrices, using flat, cylindrical rotating drum, or even 3D columnar collector plates [78].
However, due to the instability and chaotic motion of the electrically charged fibers it can be a challenge to collect a single nanofiber. Additional methods such as magnetic fields can produce alignment of fibers [79] or the utilization of a drawing method, other than electrospinning, to combat this problem as seen from However, the applied voltage also needs to work in tandem with this flow rate.
The voltage applied depends on the distance of the needle to the collector. With this, a critical voltage V c can be expressed to determine the electric field needed to develop fluid instability. Taylor (1964) also showed that this can be expressed as: where H is the distance between the needle tip and collector, L is the length of the needle, R is the radius of the needle, and σ is the surface tension of the liquid [30,26,80]. Equation 3.4 may be useful in determining a proper applied voltage. A voltage around 10kV DC was found to be the workable amount to spin PCL polymer nanofibers. Once the fibers are spun, they can be collected, measured for dimensions, and appended to the sample plates.

Appending a Fiber
Once a web of fibers has been amended between both steel rods the next step is to append a single fiber to the sample plate. A calibrated 10x OMax microscope will be used to measure and aid in appending a single strand as seen in Figure 3.4.
The microscopes XY-axis slide is removed and a held positioning screw is added.
The alligator clip, of the sample plate, is clamped to the screw which aligns well in view under the microscopes camera. A pointed tweezer prong is attached to XYZ-axis stages; these can also be pivoted around the microscope for easier fiber loading. Loading tweezers are used to then grab suspended fibers from the steel rods. Depending on the thickness of the fibers produced the fibers my not naturally append to the tweezers by Van der Waal forces. In this case, UV glue and light source can be used on the tips to stick the desired fiber. The fiber is then brought to the microscope and attached as seen in Figure 3.5.
First, the fiber is attached to the heating prong, the glues' surface tension is able to hold this in place; than the fiber is attached to the microscope pointed tweezer. The UV light source is used on the both the heating prong and microscope tweezer to hold the fiber in place. The microscope tweezer is then swung slowly towards the cantilever, using the XYZ stages. Once close enough, the fiber will jump to the tip of the cantilever by Van der Waals forces as seen in Figure 3.6 (this cantilever is not the one used within the design, it is an alternate testing cantilever with probe). Figure 3.7 provides the final appended PCL fiber used in testing.
Alternative appended fibers can be seen in    b) Shows Van der Waal attractive force. c) and d) are alternate append fibers bridging both cantilevers. These were appended to a test bi-material cantilever sample plate and were not tested.
A soldering iron is used to cut the ends of the fiber leaving only the fiber suspended between the cantilever and the heating prong; keeping in mind the temperature of the iron since the polymers melting temperature is 60 • C. While carefully unloading the sample plate and placed into a carrying container, it is taken to the AFM for measurement.

AFM Setup
Once the sample plate with appended fiber is ready, it is moved to the AFM for

Methodology and Calibration
This chapter reviews the system settings and parameters used, along with how to conduct and measure the thermal conductivity of electrospun nanofibers. Before any measurements are to be recorded, the AFM cantilevers responses and settings will first need to be understood. The resonance frequency will initially be measured, to validate that its force constant is indeed lower than that of the bi-material cantilever. Once this is proven, the cantilever will be tested in contact with bi-material cantilevers silicon base to show baseline changes in nm/C values, seen in Chapter 5. This will first, show how the AFM cantilever will react under varied temperatures without the bi-material cantilever and secondly, indicate proper PID settings needed.
After the AFM cantilever parameters and settings have been determined, the Probe-to-Probe system will then be set up to determine the calibrated temperature vs deflection recordings; results of these recordings can be seen in Chapter 5.

Measuring Force Constant and Resonance Frequency
The force constant can be simply known by measuring the resonance frequency of the cantilever itself. The purchased cantilevers all come with supplied force constants and resonance frequencies. However, some cantilevers such as the tipless cantilevers used as the AFM cantilever provide a range between 0.02-0.77 N/m force constant will need to be lower than that of the bi-material cantilever to be able to see a response. The AFM itself has two different modes to work in, vibrating and contact modes. The vibrating mode will only be used to measure the resonant frequency of the tipless cantilever. Additionally, since the bi-material cantilever is coated in gold, the laser could not be aligned at all due to reflectively issues and therefore cannot be measured for confirmation.
The ratio from the resonant frequency to the force constant can be compared to indicate the proper force constant; if lower than the bi-material, the cantilever can be used. Equation 4.1 can also be used to obtain the resonance frequency [81].
Also in good practice, before recording all cantilever deflections, one should obtain the voltage-deflection V /nm response from the photo-diode, which additionally aids in the measured of force constant. Traditionally the force distance curve portion of the AFM would be used to determine this V /nm response however, this is not currently functioning properly; utilizing applied force and known force constant. Due to this limitation, alternate methods were determined.
These results can be seen in Chapter 5.

Setting System Parameters
The next step is to confirm how altering the PID and voltage gains effect the stability of the system with varied heat and its feedback loop. Using the same AFM cantilever with known force constant; several scans were performed using alternate PID and gain settings. The XY HV Gains are always set to zero to hinder any XY movement from the piezo stage. Alterations included system changes to Z Drive (1-15 HV Gain), XY percentage gains (1-100%) which represent the Z Sense response, varied PID settings, and varied feed back loop. As a side note, the Z Drive should mirror the Z sense values (force transducer verse the piezo stage sensor); either the Z Drive or Z Sense can be used for simplicity. A few objectives were to be determined after testing various settings: 1. The first was to determine if a feed back loop was needed; meaning that the system can either be held in constant deflection or the system can just be set to a user defined value and a change in voltage can be recorded. If one was to have no feedback loop, the Z Drive may be turned to zero for simplicity, and the Z Sense used.
2. The second was to confirm that the PID settings will accurately respond to temperature increase and that the stage or force transducer are not auto adjust/auto correct linearly showing a phantom force response. As an example, in the case of not using a feedback loop, the piezo stage may increase linearly without push back or balancing from the Z Drive. The piezo stage is programmed with a constant linear gain value as low as .0001V, this can not be changed to zero, however the other PID settings can be. As recommend from AFM Workshop, the PID settings should be set to zero leaving the gain value at its original .5V setting. Again, the fore-mentioned settings will indicate if these changes are accurate.
3. The third was to confirm that the stage piezo is not responding to conduction from the heated stage. To test this, the system is heated instantly to 400C and run for 40mins; no change in the Z Sense should be noticed. AFM piezos are also created to withstand conduction temperatures below 150C, so there should be no difference however, if a change is noticed between the Z Drive and Z Sense due to temperature increase, the Z Sense recordings will not be used.
By the end of testing key settings were defined. Starting with the first objective, the system should include a feedback loop. It was noticed that once in Probe-to-Probe contact, without a feedback loop, that the T-B and L-R photodiode sensors would shift without external heating; meaning the cantilevers were slipping. This may be due to an imperfect gluing of a bi-material cantilever on the sample plate or alignment of the AFM cantilever; working at nanoscale may provide these challenges.
With a change in the first objective, requiring a feedback loop, the second determined to simply use the original PID, Z Drive and Z Sense HV gain settings for this feedback. It is known that the higher the Z Drive HV gain the faster the response, however increasing voltage may have lead to more noise. Throughout testing various gain responses and feedback loops, the original PID and Z Drive values can be used. Lower values under 10 HV however, may lead to imprecise and weak contact. A linear response still existed creating noise within the system however, were considered very low and were to be subtracted out when comparing the calibrated recordings to the appended fiber recordings; results on this are seen in Chapter 5.
A major problem that surfaced from testing was that there existed a power draw from the soldering iron to the rest of the system. This was discovered when the soldering iron was turned on, a voltage gain was noticed within the Z Drive and Z Sense. Increasing the output temperature quickly on the soldering iron also showed this spike. This was a considerable problem. Since the soldering iron uses a power regulator to control the temperature supplied, it would also draw power from the AFM itself both from the Z piezo stage and from the force transducer once the cantilevers interacted. This type of error produced many problems with inconsistent drowned out data. This problem was easily fixed by disconnecting the soldering iron from the sample plate; only allowing ambient heating. The down side of doing this may lead to a lower power through the fiber in addition to having to increase the heating steps of the soldering iron to achieve higher ambient heating temperatures throughout. However, all temperatures within the system are still recorded the same way; so the current modeling is still accurate.
Referencing back to the third objective, it was determined that the stage piezo was not interacting with the change to external ambient temperature however, if the soldering iron was connected to the sample plate, differential power steps can be seen. Again, showcasing that the power regulator of the soldering iron plays a large role in noise drowning out any results.
The final system settings can be seen below. The AFM can only record two sensors at once. The T-B voltage and Z Drive will be recorded since the Z Sense was shown to mirror the Z Drive.

Procedural Setup and Alignment
This section reviews how the probe-to-probe system is set up and run however, it does not include how to initially set up the AFM for traditional use or laser alignment. On instructions on how to set up the AFM for traditional use, the AFM's operators manual can be referenced. Any alterations to the manual are mentioned. This also includes the procedural steps for the prior mentioned calibration parameters.
1. Setup a tipless AFM cantilever traditionally and align the laser normally.
2. Once the AFM cantilever has been placed into the AFM and laser aligned, the sample plate is placed onto the magnetic stage making sure of clearance to the AFM cantilever seen in Figure 3.10. Set the magnification focused to the small 100um length cantilever or base depending on either conducting a probe-to-probe or silicon base contact scan.
3. Prior to Probe-to-Probe contact, the AFM cantilever is jogged down to the silicon "under belly" or now the top surface of the bi-material cantilever as seen below. Both the XY axis sample plate and AFM cantilever Z axis will have to be turned and adjusted. This helps to figure out how close the cantilevers are to each other without accidentally breaking both cantilevers.
4. The XY HV gain is set to zero.
5. Set the AFM mode to vibrating and obtain the resonance frequency/force constant.
6. The automatic tip approach is used, and the AFM cantilever is set to jog slowly down to the surface at a rate of .3um/s to make contact. Again, depending on the surface, either Probe-to-Probe or base contact can be seen in Figure 4  The T-B sensor should read a steady 0.1V lower then the original value once in feedback; this can be set to other forced values however, it is not required.
If the L-R signal reads greater or less then its original value after contact, the cantilever is jogged up and the sample plates XY axis's adjusted. This is done until the L-R signal does not change after proper contact. In addition, it is possible during scanning that the stages' Z range will end. If the range of the stage ends, an increase in a T-B value will be noticed during a scan.
If this happens, stop the current scan, jog up, and slide the sample plate further centered on the magnetic stage. The entire scan will have to be started over.
7. Once contact has been made, the Topo Scan tab will be used. 10. After both scans, the soldering iron is turned off and cooled for 20mins, ready for another set of scans.

Measurement Results and Discussion
This chapter compares all theoretical and measured results to determine the thermal conductivity of the nanofibers. This also reviews recorded values for voltage and natural deflection calibration results.

Introduction
There exist two different models to consider when obtaining measurements, theoretical and measured/recorded. Calibrated values and recordings are first conducted in order to make sure recorded measurements match theoretical results.
The calibrated recordings of the Probe-to-Probe system without a fiber are compared to the measured/recorded deflections with an appended fiber.
The theoretical values are obtained using the modeled functions from Chapter 2, whereas the recorded values are sent to an Excel spreadsheet. Both the theoretical and measured/recorded values are generated/called in to a MATLAB program where depictions are shown. The theoretically calculated bi-material cantilever temperatures T ct , where T ct is output temperature from the fiber to the bi-material cantilever, will need to be theoretically generated since they can not be measured. A difference in T ct should be noticed once a fiber is appended.
The temperature into the fiber from the heating prong, T f , are also numerically generated; the approximation for this is seen in Chapter 2 but the values are shown further in this chapter. Once both T ct and T f temperatures are known, the heat conducted through the appended fiber k f will be known.

Deflection Results
The preceding recordings will showcase all measurements and theoretical results of cantilever deflection with and without a fiber. Figure 5.1 shows the average deflection with ambient temperature values reaching steady state after 8 minutes; averaging the last 2 minutes of the 10 minute individual stepped scans. Six full scan sets exist for the AFM-tipless cantilever natural deflection and calibrated Probe-to-Probe contact, and only two for the appended fiber probe-to-probe contact. A difference in the calibrated versus appended values can be seen; the following sections will delve in to the individual recordings and discussions. the slopes of these methods can still be compared to show that the AFM-tipless cantilever reacts properly within the system. Moreover, considering that the bimaterial cantilevers natural deflection could not be measured due to the gold reflectively and poor laser itself, the AFM-tipless cantilever was used.
The first method, part 1, was to only record the photo-diode's voltage output whereas part 2 only records the natural deflection, via a held constant forced deflection, zeroed voltage. Showcasing both these graphs as V /C and nm/C respectively, provides V /nm. To measure V /C, the AFM cantilever was held just above the sample plates' bi-material cantilever and heated. To measure nm/C, the   From these results, in Figure 5.2, one can see that the AFM cantilever actually does indeed respond to small temperature variations during a system scan.
Once the scan was complete and temperature turned off, one can notice that over time the cantilever will revert back to its original state via the photo-diode, which does not require a set scan to display. This is considerable considering that even a long silicon cantilever could be further utilized in measurements and future work. Figure 5.2 above not only shows how the photo-diode responds to deflections but also shows that there is no linear gain applied to the system from the stage since the nm/C slope is similar to V /C slope. More on this gain value is seen in Chapter 4, System Settings. Slope V /C in Figure 5.2 however, also represents very limited data points; more data should still be supplied.
One can also utilize the following second method to measure V /nm. The AFM functions traditionally by sensing a deflection response from the photo-diode, sending a signal to the force transducer and the piezo in the stage telling both systems to balance and hold a constant set deflection provided by the user. In this case, the AFM will be told not to hold this deflection and instead force the piezo stage to move the cantilever. This is done by jogging the cantilever down to make contact rather than utilizing the automatic tip approach which initiates the feedback loop response. The Z-Drive is turned on as well, but the piezo within the drive is told not to respond since the feedback loop is not used. Once the scan is started, the stage will supply a linear decrease pulling the contacted cantilever along with it. The scan will record both Z-Sense position and T-B signal in time. Dividing the slope of the photo-diode by the stage's movement provides the V /nm slope. However this method was not used or required within this thesis, the procedure can still applied to future works.  There are no statements from the purchased cantilever manufacturer that the silicon cantilever should respond to higher than normal variations in temperature, only that they should not deflect in regular ambient room temperature.
Traditional uses of AFM do not include increases to ambient normally, so scans would typically ignore this since ambient usually does not change. The system was also tested by removing the AFM cantilever, then heating the system; no change in the photo-diode was noticed. However, the actual theoretical result does match expected results seen from Chapter 1, showing silicon cantilever temperature res-olutions of about 1 µm and 3 • C. This is promising considering it showcases that the thermal model from Chapter 2 is near accurate. Figure 5.5 provides the average temperatures for all recorded scans at steady state during calibrated measurements.
Step 0 were temperatures recorded at room temperature and were just used as a test run; these values were also inconsistent, but were still removed.

Appended Fiber Deflection Recordings
After recording deflection and temperature values once a fiber was appended, the recorded slope must be subtracted from the recorded calibrated slope; this zeros out data to provide a true deflection verses change in ambient temperature solution seen in Figure 5    These temperatures will be used to determine the heat flux into the fiber to obtain the thermal conductivity of the fiber seen in the following section.
Recorded Temperatures with Appended Fiber • C Solder Temp Tambp  Tpo  Tpt  Tf  Tambc  Tct  Tct-Tambc   100  0  0  0  0  0  0  0   This may show that there is very limited heat conducted to the tip of the bimaterial cantilever, if none at all, or the use of ambient heating is overcoming any heat transfer. It is possible that Figure 5.1 may be providing an illusion of change even though there is no additive temperature to the cantilever. The thermocouple position was consistent and accurate between scans when adjusted to the system however, it is possible position was not ideal, and ambient recordings varied.

Obtaining Fiber Thermal Conductivity
Taking the temperature values from 5.1,  ). This shows that the thermal conductivity of the fiber may fall between 25 and 105 (W/mK). This is assuming that the recorded difference between T ambc and T ct , seen in Table 5.1, is indeed significant enough to showcase this range, but in a large ambient bath, with such similar values. Using a linear solution from the fluxes provided, the thermal conductivity was determined. Furthermore, the change in flux between each scan is larger as the system is heated; due to increases in temperatures, the gap between ambient temperatures increases, as seen from Figure 5.9. Given the testing parameters and constraints of the current system, one can state that highest recorded thermal conductivity is 103.67 (W/mK).

Summary and Future Work
The chapter details end thesis conclusions and takeaways along with design ideas that can be incorporated in future works.

Chapter Summaries
This section reviews each chapter, the resulting conclusions and details noticed.
Chapter 1 -Provided background information on nanoscale thermal transport, past measurement techniques, and defined the proposed Probe-to-Probe design.
Chapter 2 -Defined introductory heat transfer along with the detailed modeled design of the Probe-to-Probe system. It further provided listed equations to obtain unknowns such as system and fiber fluxes, system cantilever and prong temperatures, and forced and natural deflections, all of which required to solve for the thermal conductivity of an appended nanofiber.
Chapter 3 -Focused on the experimental apparatus, fiber and sample plate creation, and equipment used. System properties such as bi-material cantilever, AFM-tipless cantilever, and electrospun PCL nanofibers were obtained, which are further seen in the Appendix.
Chapter 4 -Reviewed the procedural measurement of the system and outlined how the system was calibrated, along with how each system setting were determined. It was found that the bi-material cantilevers' natural deflection to ambient changes could not be measured due to inert reflectively possibly due to the strength and focus of the laser. A voltage draw was also noticed when in a conductive heat transfer system. This was fixed by disconnecting the soldering iron from the sample plate and recording all temperatures as normal.   The Wheatstone bridge provides feedback, adjusts, and balances the bridge voltage in order to measure the probe's temperature (in TCM) or maintain a constant probe temperature (in CCM) [82]. Similarly to a traditional AFM, a SThM cantilever tip serves as a resistance thermometer, or as a heater in CCM mode. The resistivity of the SThM tip is then changed according to the temperature of the surface in contact; the tip to sample heat transfer changes the tip temperature, which used to calculate the temperature or thermal properties of the sample at contact [83]. This would be very useful when determining the thermal conductivity of a nanofiber rather than a planar surface. This may be a promising technique when used in tandem with a vacuum. Kim et al. have utilized an Ultra-High Vacuum (UHV)-based SThM technique that was capable of quantitatively mapping temperature fields with about 15 mK temperature resolution and 10 nm spatial resolution [84]. However, the thermal transport around the nanoscale contact area between a SThM probe tip and sample is still an issue in SThM [85].
The second option would be to mitigate all ambient temperature utilizing a vacuum with with the current system or an SThM. Again, this would limit thermal noise and excess heat into the system to provide only conductive heat transfer. Considering the issue seen in Chapter 4 with voltage draw, this calibration would have been more easily noticed if the system was not in an ambient bath skewing data.
The third would be to utilize alternate silicon or bi-material cantilevers.