Crystalline polymer nanofibers with ultra-high strength and thermal conductivity

Polymers are widely used in daily life, but exhibit low strength and low thermal conductivity as compared to most structural materials. In this work, we develop crystalline polymer nanofibers that exhibit a superb combination of ultra-high strength (11 GPa) and thermal conductivity, exceeding any existing soft materials. Specifically, we demonstrate unique low-dimensionality phonon physics for thermal transport in the nanofibers by measuring their thermal conductivity in a broad temperature range from 20 to 320 K, where the thermal conductivity increases with increasing temperature following an unusual ~T1 trend below 100 K and eventually peaks around 130–150 K reaching a metal-like value of 90 W m−1 K−1, and then decays as 1/T. The polymer nanofibers are purely electrically insulating and bio-compatible. Combined with their remarkable lightweight-thermal-mechanical concurrent functionality, unique applications in electronics and biology emerge.


Fabrication of polyethylene microfiber from the gel
In this fabrication method, we first produce the PE gel. To make the PE gel, we used 0.8 wt % ultra-high molecular weight polyethylene (UHMWPE) powder (average molecular weight 3 -6 × 10 6 g mol -1 purchased from Sigma Aldrich) and mixed it with a decalin solvent. The mixture is heated on a hot plate to 145 ˚C. It becomes transparent and viscous as the PE powder dissolves in the solvent. A glass rod is used to constantly stir the solution. To avoid oxidation and subsequent molecular degradation, this process is carried out inside an argon filled glove box. The solution is then quenched in a room temperature water bath, and the gel forms.
After gel preparation, a two-stage tip drawing method is used to form a microfiber, PEMF. A 5 mm × 5 mm silicon chip with a thin film heater attached on the backside is used to heat the gel to 120 -130 C. A hot plate placed 1 cm below the silicon chip heats the overhead air to 90 C to prepare for hot-stretching. The fabrication setup is shown in Supplementary Fig. 2. The translucent gel turns to a transparent solution as it is heated to 130 C. As the solution turns clear, a sharp glass tip (10 µm) is used to draw a short length (several hundreds of microns) of PEMF.
The fiber undergoes stress-induced crystallization. During crystallization, the decalin evaporates, aided by the convective current from the hot plate. Decalin syneresis further facilitates crystallization 8 . The PEMF is then further drawn to a length of 1 cm after which it is quenched to room temperature to minimize relaxation of the extended chain. After the drawing is complete, PEMF is placed on a sample collector as shown in Supplementary   Fig. 3a, where it remains under tensile stress. The collected PEMF is locally heated using a micro heater to make nanofibers as shown in Supplementary Fig. 3b and described in main text. Here, we estimate the maximum temperature attained by the fiber during local heating. First, we simulated the temperature profile of the micro heater in ANSYS. Electrothermal modeling with a potential bias 0.7 V was applied to the heater, same as in our experiment. Both convection and radiation heat losses were considered in the simulation. In microsystems, the boundary layer gets smaller, thus, heat transfer coefficient (h) greatly increases. Here, we used ℎ = 1000 W m -2 K -1 , emissivity = 0.1 in our simulation 9 . Supplementary Fig. 4 shows the simulated temperature profile of the heater.
Supplementary Figure 4: Temperature profile along the heater from ANSYS simulation.
Next, we added a PEMF with diameter 3 µm and set a 20 µm gap between the heater and the fiber. The thermal conductivity of the PEMF was modeled using gel spun microfibers with = 20 W m -1 K -1 4 . We used emissivity value of the PEMF as 0.2 10

Estimation of strain rate during local drawing
The molecular alignment within the fiber is correlated with increasing strain rate. We estimate that the strain rate in local drawing reaches up to 1400 s -1 compared to ~1000 s -1 in electrospinning 2 and ~1 s -1 in gel spinning 11 . Strain rate is defined as ∆ ( 1 ∆ ) ⁄ where ∆ is the extension of the fiber segment of length 1 within ∆ seconds. after undergoing high strain rate elongation (diameter measured post measurement using Scanning electron microscopy (SEM)).
To estimate the strain rate during the local drawing, we used two consecutive image frames, as shown in Supplementary Fig. 6 ⁄ .
The fiber starts thinning at lower strain rate from 3 µm to around 1.5 µm before undergoing peak strain rate drawing. The video was obtained at an average frame rate of 6.3 frames per second.

Positioning PENF on the MEMS devices
As the fiber diameter decreases, it becomes more challenging to precisely locate and position it onto the MEMS test platforms. Electron microscopes cannot be used because of the sensitivity of polymer samples to electron beam radiation 12,13 , which impairs the mechanical and thermal property enhancement 3,14 . Therefore, the manipulation of the PENF is limited to optical methods.
The sample must also be properly aligned. For example, during tensile testing an unwanted bending moment can cause the loadcell to rotate and the fiber force will be underestimated. The miniscule sample size, the limited resolution of the optical microscope and unpredictable forces such as van der Waals, triboelectric and capillary forces due to moisture are factors that make it challenging to successfully place nanofibers. Instead of manipulating the PENF directly, the sample collector with a mounted PENF is maneuvered as shown Supplementary Fig. 7. The microfiber in the undrawn section is used as a reference to align the nanofiber. After aligning the PENF, the sample was cut by local melting a suitable distance away from the region tested using a micro heater. This is performed under an optical microscope using a 20X objective.

SAED analysis of PENF
During the high strain rate fabrication step, PE crystals can undergo a phase transformation from the orthorhombic to a monoclinic phase 15 . Most SAED images indicated only the orthorhombic phase. One SAED pattern, Supplementary Fig. 8, reveals the monoclinic phase, in the orthorhombic matrix. We did not evaluate exact weight fraction of monoclinic phase, however, we expect it to be small. It has been measured for a highly drawn PE microfiber using Wide angle X-Ray scattering (WAXS) to be 4.3 % 16 .

PE powder characterization using DSC
The PE powder (Sigma Aldrich, Mw = 3 -6 × 10 6 g mol -1 ) used as the control was characterized using differential scanning calorimetry (DSC) to understand its initial crystallinity, as seen in Supplementary Fig. 9.
Here c is the % crystallinity of the sample, ∆ s is the enthalpy of fusion of the sample and ∆ f is the enthalpy of fusion of pure crystal 289.3 J g -1 . The enthalpy of fusion of PE powder was obtained to be 157.1 J g -1 . Therefore, the crystallinity is 54.3 %.

Structural damage of PENF during Raman characterization
Supplementary Fig. 10 shows Raman spectra of a PENF obtained at the same spot. They essentially overlap (within 5 %), suggesting that the fiber undergoes minimal structural damage during the characterization.

PE powder crystallinity characterization from micro-Raman
Following the method of Strobl and Hagedorn 18 , the crystallinity of unoriented PE powder can be determined using micro Raman as follows: where 1415 is the integral area under 1415 cm -1 Raman band and (1295+1305) is the integral area of 1295 cm -1 and 1305 cm -1 Raman bands, which act as an internal standard. A spectrum is shown in Supplementary Fig. 11.  An average crystallinity of 57.1 % was obtained from the analysis of Raman bands of PE powder, in good agreement with the crystallinity obtained from the DSC analysis. The small difference of 3% could be because the DSC measurement is an average of bulk while micro Raman measurement is for a local area of a powder particle.

Noise equivalent thermal conductance of platinum resistance thermometer (PRT) micro devices
The thermal conductance assessment is limited by the ability to measure the temperature rise in the sensing island (denoted by subscript s) accurately. Following the sensitivity analysis of Shi et al. 19 , the noise-equivalent thermal conductance ( s ) in the sensing island can be shown as where b is the total thermal conductance (W K -1 ) of the supporting SiNx beams of an island, ∆ h is the temperature rise in the heating island, ∆ s is the temperature rise in the sensing island and s is the noise-equivalent temperature (K) of the sensing island. Also, where is the noise-equivalent resistance (Ω) of the sensing island. Through measurement of a precision 1 MΩ resistor, s s ⁄ was found to be ~7.5 × 10 -5 . The temperature coefficient of resistance (TCR) of the PRT was found to be ~2.6 × 10 −3 K -1 at 150 K and ~2.1 × 10 −3 K -1 at 300 K. Therefore, s is ~29 mK at 150 K and ~36 mK at 300 K. The temperature fluctuation in the cryostat is ~10 mK after waiting for the global temperature to stabilize for an hour at temperatures above 150 K.
The thermal conductance of the suspended PRT beams, b , is 70 -90 nW K -1 at 150 K and 90 -110 nW K -1 at 300 K. The temperature difference between heating and sensing islands (∆ h − ∆ s ) is kept within 5 K. The s of the measurement system can be now found from Supplementary Equation 4. At 300 K, the s is ~0.85 nW K -1 and at 150 K it is ~0.52 nW K -measurements.

Background thermal conductance
Because the thermal conduction experiment was carried out at a high vacuum level of 2 × 10 -7 Torr and the temperature difference between the islands was limited to 5 K, we expect negligible heat transfer from residual gas and radiation. A background thermal conductance, bkgd , measurement of an empty micro thermal device in the same experimental conditions verifies that bkgd is within the s as shown in Supplementary Fig. 12.
Supplementary Figure 12: Background thermal conductance of an empty device with gap 5 µm.

Uncertainty analysis in thermal conductivity
The thermal conductivity of an individual PENF is given by where is its thermal conductance and and are its length and cross-sectional area. Using the uncertainty propagation rule, the error in can be expressed as where is the uncertainty. The values for ( ) 2 and (2 ) 2 are obtained from the SEM images as described later. From the analysis of Shi et al. 19,20 , where is the heat transferred to the heating island. The error in can be written as When a dc current is applied to the heating island, it can be shown that the heat transferred to the heating island is equivalent to where h is the resistance of the heating island and l is the resistance of a current carrying SiNx beam. The error in DC power in the heating island is given by, DC current up to 20 µA is applied with a Keithley 2400 source measurement unit with a high accuracy of ±5 nA. Thus, the uncertainty (<0.07 %) due to heat input into the heating island ( ) is low.
The error in temperature rise in the heating or sensing island can be shown to be For a typical sample with diameter of 100 nm and length 8 µm, the standard deviation in the diameter is found to be around 4 % and negligible in length. Then, the total error in the thermal conductivity at 300 K is ( ) = √(2.99 × 10 −2 ) 2 + (2 × 4 × 10 −2 ) 2 = 8.54 %.

Thermal contact resistance
The thermal conductance measurement system uses two-probes and includes thermal contact resistance. Thermal contact resistance between the sample and the islands can result in an underestimation of the sample's . The total thermal resistance of the system at steady state is the sum of thermal contact resistance between the heating island and the nanofiber (Rc,hi), the intrinsic thermal resistance of nanofiber ( s ) and the thermal contact resistance between the sensing island and the nanofiber (Rc,si). Therefore, the total thermal resistance ( tot = 1 s ⁄ ) can be written as tot = s + c hi + c si (13) The heating (h) and sensing (s) sides have the same geometrical properties and same contact mechanism, so we assume that c hi ≈ c si ≈ c . Then To measure the sample's accurately, 2 should be negligible compared to sample's intrinsic thermal resistance ( s = 4 s s 2 s ⁄ ). To achieve 2 c ≪ s , either s can be increased or 2 c can be lowered. For high samples, such as PENF, intrinsic thermal resistance ( s ) can be increased by making the nanofiber long and the cross-section area small. However, this will also reduce the heat flux significantly. Consequently, the temperature rise of sensing side will be lessened, making accurate measurement difficult. Alternatively, thermal contact resistance can be decreased to make 2 c ≪ s . A platinum or graphite coating, using electron beam or focused ion beam (FIB), has been extensively used in literature to reduce 2 c 22,23 . However, high-energy electron/ion beam amorphized our sample and reduced the thermal conductivity enhancement to bulk as shown in Supplementary Fig. 13.

Supplementary
where ⊥ (0.33 W m -1 K -1 26 ) is the radial thermal conductivity of the nanofiber sample, d is the diameter of the sample, w is the contact width of nanofiber on the substrate and kpt (70 W m -1 K -1 ) is the thermal conductivity of platinum.

Contact width estimation
We need to know the contact width ( ) between a nanofiber sample and an island to calculate 2 c using Supplementary Equations 15 and 16. Elastic plane strain analysis can be applied to obtain assuming the van der Waals force represents the applied force 24,25 .
For an elastic cylinder on a flat substrate, Bahadur et al. 25 derived due to elastic deformation from the van der Waals force given as: where Fvdw is the van der Waals force per unit contact length between the nanofiber and the island, and Em is the effective modulus defined as

Load cell stiffness estimation of the Nanotractor Platform
The stiffness of the loadcell, as shown in Fig. 5a of the main text, was estimated from the dimensions of the loadcell beams measured from the SEM images. Four identical fixed-guided beams determine its stiffness. The load cell was designed such that it was stiff enough to exert sufficient force to test the sample until failure and compliant enough to have good force resolution. The load cell beam stiffness is given by L = ℎ 3 3 ⁄ where ℎ are Young's modulus, height, width and length of the beam respectively.

Error analysis in stress using the Nanotractor Platform
Tensile strength ( ts ) is the maximum stress that the sample can withstand before failing. It can be expressed as ts = max (21) where max is the maximum force. This can be rewritten as The total uncertainty in ts is also estimated using the uncertainty propagation rule:  Increasing the surface energy of PE is the most common way at the macroscale to enhance PE adhesion. The surface energy can be altered by flame discharge, corona discharge, acid treatment and plasma treatment 38 . These surface treatments are destructive which will alter the surface chemistry and property of the sample via oxidation. The length scale of the surface where these surface treatments affect the material is similar to the diameter of our sample 38 . Thus, these methods are not viable. Surface roughening is another method that has been frequently used at the macroscale. The glues within the rough surface provide additional force due to mechanical locking. However, with our sample size mechanical roughening of the fiber is not possible.
Adhesion of an epoxy matrix with PE can also be enhanced by mixing reactive graphitic carbon nanofibers (r-GNF) 39 . However, dispersion is always an issue when mixing r-GNF with glues.
There have been previous measurements where an adhesive was used to fix the polyethylene nanofiber during an AFM based three-point bending test. This was successful because the applied force was three orders of magnitude smaller and tested for very small displacement 40 .
Here, we introduce a mechanical locking method to increase the gripping force.

Evaluation of glue adhesion and performance of dog bone
From composites theory 41 , the maximum force (Fmax) up to which a sample with a diameter ds was tested before it slipped can be used to obtain the shear strength ( shear ) of glue/PE, given as where lglue is the length of the nanofiber on which glue is applied. In Supplementary Fig. 16a  where the same three regions have been measured. A region with distinct patterns of debris on silicon wafer was chosen so it would be easy to find in both AFM and SEM. The SEM measured diameters are 10.7 %, 11.9 % and 11.9 % larger than those from AFM. Supplementary Fig. 17d also shows the morphological damage in PE nanofiber. The chains in the as-drawn fiber are scissioned by the irradiation. The observed ripples show the sheesh-kebab structure that evolves thereafter.  Unlike thermal measurements, the PENF breaks during strength measurements. When imaged in SEM, it coils rapidly due to irradiation near the free end. In the literature, the diameter of polymer fibers has been measured from an untested section 14 . However, this is not feasible here because the fiber exhibits tapering. Hence, the diameter is measured within the test section by SEM. As discussed in the manuscript, the fiber fails by ductile failure (low L ) or by extreme necking (high L ). In the first case, the fiber diameter was measured (10 µm) away from the fractured end as shown in Supplementary Fig 19a. In a crystalline PEMF, the failure strain was 6.5 % at a strain rate comparable to this work (10 -3 s -1 ) 42 . So, assuming constant volume deformation, the diameter of the fiber decreases by 3.2%. However, the controlled experiment ( Supplementary Fig 17) showed that the measured diameter of the fiber was overestimated by 12 % due to irradiation. The value used to calculate strength is from SEM without correction, and therefore the reported strength is a lower bound. In case of extreme necking, the local region can undergo large non-uniform deformation.
Therefore, the diameter in this case was measured close to the clamped region as shown in Supplementary Fig 19c. Supplementary Figure 19: Diameter measurement for evaluation. a) Fractured sample. b) Zoom in of inset of a. c) Extremely necked sample. The diameter is measured far away from necked region.
The coiling is prominent in freely suspended sample compared to the sample on the substrate. Scale bars, 5 µm (a,c), 1 µm (b), 500 nm (d).
a b c Extreme necking d

Extreme necking
As stated in the manuscript, the samples tested with the high stiffness loadcell experienced extreme necking. Under electron beam irradiation, the free end of the PENF coils rapidly.
Therefore, a measurement was performed using AFM on an extremely necked region of a mechanically probed sample. The AFM height values of 7, 1.4 and 1.2 nm in Supplementary   Fig. 20b are obtained from regions 1, 2 and 3 in Supplementary Fig. 20a. This shows that the fibers deform by extreme necking rather than a typical local fracture with a short neck region.

Cross section of s fabricated by local heating
We have assumed in this work that the fiber cross-section is circular. So, the PENF diameter in the main text was reported based on the width measured using SEM. However, upon measuring the height using atomic force microscopy (AFM), we found that the sample cross section is indeed non-circular at SEM measured widths below 150 nm, where the measured height using  Fig. 21. However, when calculating the 's of the PENFs from the measured thermal conductances, we assumed the fibers to have a cylindrical shape with uniform diameters that correspond to the widths measured by SEM. The same situation is true for ts measurements as well. Hence, the and ts reported in the manuscript should be taken as a lower bound for the PENF.

Shape of a PENF along the length evaluated by SEM
In our and measurements, we assume that the PENFs have a cylindrical shape that can be measured by SEM. Supplementary Fig. 22 shows a typical diameter measurement along the fiber length using SEM of a PENF fabricated using local drawing. It can be seen that the diameter is fairly uniform. The sample excludes the dog bone and the tapered region shown in Supplementary Fig. 15. For thermal measurements, since the test length is within 10 µm, we often obtain relatively uniform diameter. Samples with standard deviation of nanofiber diameter larger than 15% are excluded from thermal measurements. However, for mechanical measurements the nominal gage length is 30 µm, so the sample always has some nonuniformity.
Supplementary Figure 22: Diameter profile of a typical PENF fabricated using local drawing.