# Origin of high thermal conductivity in disentangled ultra-high molecular
weight polyethylene films: ballistic phonons within enlarged crystals

Taeyong Kim *  — Division of Engineering and Applied Science, California
Institute of Technology, Pasadena, California 91125, USA  Stavros X.
Drakopoulos *  — Department of Materials, Loughborough University,
Loughborough LE11 3TU, United Kingdom  Sara Ronca *  — Department of
Materials, Loughborough University, Loughborough LE11 3TU, United Kingdom
Austin J. Minnich *  —<EMAIL_ADDRESS>Division of Engineering and
Applied Science, California Institute of Technology, Pasadena, California
91125, USA

(August 27, 2024)

###### Abstract

The thermal transport properties of oriented polymers are of fundamental and
practical interest. High thermal conductivities ($\gtrsim 50$ Wm-1K-1) have
recently been reported in disentangled ultra-high molecular weight
polyethylene (UHMWPE) films, considerably exceeding prior reported values for
oriented films. However, conflicting explanations have been proposed for the
microscopic origin of the high thermal conductivity. Here, we report a
characterization of the thermal conductivity and mean free path accumulation
function of disentangled UHMWPE films (draw ratio $\sim 200$) using cryogenic
steady-state thermal conductivity measurements and transient grating
spectroscopy. We observe a marked dependence of the thermal conductivity on
grating period over temperatures from 30 – 300 K. Considering this
observation, cryogenic bulk thermal conductivity measurements, and analysis
using an anisotropic Debye model, we conclude that longitudinal atomic
vibrations with mean free paths around 400 nanometers are the primary heat
carriers and that the high thermal conductivity for draw ratio $\gtrsim 150$
arises from the enlargement of extended crystals with drawing. The mean free
paths appear to remain limited by the extended crystal dimensions, suggesting
that the upper limit of thermal conductivity of disentangled UHMWPE films has
not yet been realized.

## I Introduction

Thermally conductive polymers are of interest for fundamental materials
science as well as applications such as thermal management Chen et al.
(2016a); Mark (2007); Chen et al. (2016b); Prasher (2006); Wei et al. (2021);
Huang et al. (2018). Although the thermal conductivity of unoriented polymers
is generally less than 1 Wm-1K-1 Mark (2007), early works reported orders of
magnitude increase in uniaxial thermal conductivity of oriented samples,
including polyethylene (PE) Hansen and Bernier (1972); Burgess and Greig
(1975), polyacetylene Piraux et al. (1989), and polypropylene Choy and Greig
(1977); Choy et al. (1980). In particular, the reported thermal conductivity
of oriented polyethylene ranged from $\sim$ 14 Wm-1K-1 Choy et al. (1978) for
draw ratio $DR=25$ up to $\sim$ 40 Wm-1K-1 for solution processed PE with a DR
of 350 Choy et al. (1999). The enhancement was attributed to various
mechanisms including increased chain alignment along the drawing direction
Hansen and Bernier (1972); Liu et al. (2015), phonon focusing in the
elastically anisotropic crystalline phase Pietralla et al. (1989);
Mergenthaler and Pietralla (1994), and increased crystallinity Gibson et al.
(1977); Choy et al. (1978). Recently, thermal conductivity values around
$20-30$ Wm-1K-1 and $\gtrsim 60$ Wm-1K-1 have been reported in PE microfibers
Liu et al. (2015); Wang et al. (2013) and nanofibers Shen et al. (2010);
Shrestha et al. (2018), respectively. In macroscopic samples, the introduction
of disentangled ultra-high molecular weight polyethylene (UHMWPE) films
Rastogi et al. (2011, 2005) with higher crystallinities and less entangled
amorphous regions compared to prior samples has led to reports of high thermal
conductivities exceeding 60 Wm-1K-1 Ronca et al. (2017); Xu et al. (2019).
Several recent studies have also reported high thermal conductivity up to 20 –
30 Wm-1K-1 in a diverse set of polymers besides PE, including
polybenzobisoxazole Wang et al. (2013), polyethylene oxide Lu et al. (2017),
and amorphous polythiophene Singh et al. (2014).

Knowledge of the structural changes that occur upon drawing from nascent PE
aid in identifying the origin of the high thermal conductivity values, and
extensive studies have characterized the atomic and nanoscale structure of PE
films at different DR. The nascent structure consists of spherulites Bunn and
Alcock (1945) which are in turn composed of unoriented stacked lamella in
which folded chains are bridged by intra- and inter-lamella tie molecules
Keller (1968). The initial crystalline fraction is on the order of $\sim
60-70$%, as measured using nuclear magnetic resonance (NMR) Wilson III and
Pake (1953) or heat capacity measurements Peterlin and Meinel (1965), and from
SAXS the crystalline domains have long periods $\sim 10-30$ nm Fischer and
Schmidt (1962); KOBAYASHI and KUROKAWA (1962); Yeh and Geil (1967) with the
corresponding size of the crystalline domain inside the unit being around 90%
of the long period Geil et al. (1964). On drawing, Peterlin proposed a
sequence of processes occur in which stacked lamella transition to micro-
fibrils and eventually to chain-extended crystals Peterlin (1971). More
precisely, on initial drawing, the lamellae begin to align and a crystalline
micro-fibril structure bridged by amorphous domains or tie molecules emerges
as the lamellae are fragmented. Subsequently, for $10\lesssim DR\lesssim 50$
the micro-fibrils aggregate with concurrent tautening of the tie-molecules and
marginal changes in the crystallinity. Finally, for $DR>50$, chain extension
leads to an extended crystal phase formed from the aggregated micro-fibrils
and tie-molecules. The density of states and dispersion of atomic vibrations
in the crystalline phase have been characterized by various inelastic
scattering techniques Danner et al. (1964); Safford and Naumann (1967);
Holliday and White (1971); Feldkamp et al. (1968); Twisleton et al. (1982);
Heyer et al. (1984); Mermet et al. (2003).

Experimental evidence in support of the above picture has been obtained using
various techniques such as transmission electron microscopy (TEM) Smith et al.
(1985); Brady and Thomas (1989), SAXS Van Aerle and Braam (1988), wide angle
x-ray scattering (WAXS) Van Aerle and Braam (1988), and NMR Hu and Schmidt-
Rohr (2000); Litvinov et al. (2011), among others Magonov et al. (1993); Heyer
et al. (1984). For instance, the formation of the micro-fibrils via lamellae
fragmentation is consistent with a lack of a clear trend of crystallite size
with DR for DR $\lesssim 10$ Zubov et al. (1992); Van Aerle and Braam (1988).
The subsequent unfolding and tautening of tie-molecules along with the
aggregation of microfibrils is consistent with an initial rapid increase in
crystallinity, elastic modulus and orientation factor with DR below DR 20
Anandakumaran et al. (1988) followed by a marginal increase of only a few
percent up to DR as large as 200 Anandakumaran et al. (1988); Van Aerle and
Braam (1988); Stein and Norris (1956). Evidence for the existence of extended
crystal was obtained using various complementary methods such as SAXS and WAXS
Litvinov et al. (2011); Zubov et al. (1992); Tang et al. (2007), TEM Brady and
Thomas (1989); Smith et al. (1985), and NMR Hu and Schmidt-Rohr (2000). These
various techniques indicated that the diameter of the extended crystal is
$\sim 10-20$ nm and of longitudinal dimension around 100 – 250 nm or greater
Litvinov et al. (2011).

Morphological changes under drawing identified from the above structural
studies have provided insight into origin of increase in thermal conductivity.
Below DR $\sim 50$, the uniaxial thermal conductivity is observed to
monotonically increase with DR Choy et al. (1980, 1993). Various effective-
medium type models have been proposed to interpret this increase in terms of
changes in the crystallinity and crystalline orientation Hennig (1967); Choy
and Young (1977); Takayanagi et al. (1964). Although these models are
generally successful in explaining measured thermal conductivity data, the
actual transport processes may differ from those assumed by effective medium
theory because of the presence of phonons that are ballistic over multiple
crystallites. Evidence of such processes has been reported even in partially
oriented PE samples with low DR $\sim 7.5$ using transient grating
spectroscopy (TG) Robbins et al. (2019).

For disentangled UHMWPE of DR $\gtrsim 150$, the increase in thermal
conductivity is difficult to interpret using the above models because the
thermal conductivity is observed to increase on average by factor of $\sim 20$
% despite a lack of detectable change in crystallinity or chain orientation
Choy et al. (1978, 1980, 1999); Ronca et al. (2017); Xu et al. (2019).
Conflicting explanations have been proposed to account for these observations.
For instance, Xu et al. used the isotropic helix-coil model to conclude that
the thermal conductivity of the amorphous phase ($\kappa_{a}$) must be as high
as 16 Wm-1K-1 to explain the high thermal conductivity for samples with DR
$\sim 60$ Wm-1K-1 Xu et al. (2019). On the other hand, Ronca et al. used the
same model to conclude that the high thermal conductivity for $DR\gtrsim 150$
originates from the enlargement of the extended crystal dimensions Ronca et
al. (2017). The discrepancy is difficult to resolve by bulk thermal
conductivity measurements because the properties of the crystalline and
amorphous phases cannot be independently measured. As a result, the physical
origin of the high thermal conductivity of disentangled UHMWPE remains
unclear.

Here, we report measurements of the thermal conductivity and mean free path
accumulation function of disentangled UHMWPE films (DR $\sim 200$) using
cryogenic thermal conductivity measurements and transient grating
spectroscopy. The thermal conductivity exhibits a marked grating dependence,
indicating the presence of ballistic heat-carrying atomic vibrations over the
length scale of a grating period. We interpret the TG and cryogenic thermal
conductivity measurements using an anisotropic Debye model that describes heat
transport by longitudinal atomic vibrations. The analysis indicates that the
heat is nearly entirely carried by this branch, with values of the
temperature-independent mean free paths being around 400 nm up to several THz.
Comparing these results to those of our prior study of disentangled UHMWPE
films of lower DR, we find that the high thermal conductivity for DR $\gtrsim
150$ can be attributed to the presence of enlarged extended crystals. As the
phonon MFPs appear to be limited by the dimensions of the extended crystals,
our study suggests that disentangled UHMWPE films with higher thermal
conductivity may be realized in samples with larger extended crystals.

## II Experiment

Figure 1:  (A) Schematic illustration of transient grating formation and
temperature profile in disentangled UHMWPE films. Pump laser pulses
impulsively generate a spatial grating on the sample from which probe beams
diffract. (B) Optical image of disentangled UHMWPE film (DR = 196). (C) SEM
image of the disentangled UHMWPE film. Extended fibers over tens of $\mu m$
are visible. (D) Representative TG signal versus time for grating period
$L=9.8$ $\mu m$ at 300 K. The signal is an average of 3$\times$104 repetitions
at a single location on the sample. The measurement was conducted at multiple
locations (see Supplementary Material Sec. 1 for additional data). The thermal
diffusivity is obtained as the time constant of the exponential decay. (E)
Thermal conductivity versus angle between draw direction and thermal gradient
defined by the grating $L=9.8$ $\mu$m. The 0° (90°) indicates heat flow
direction parallel (perpendicular) to the draw direction. The maximum thermal
conductivity is around 40 Wm-1K-1 along the chain, while the perpendicular
value is comparable to that of unoriented PE.

We measured the in-plane thermal conductivity of disentangled UHMWPE films
using TG, as schematically illustrated in Fig. 1. The TG setup is identical to
that described in Ref. Robbins et al. (2019). Briefly, a pair of pump pulses
(wavelength 515 nm, beam diameter 530 $\mu$m, pulse duration $\sim$ 1 ns,
pulse energy 13 $\mu$J, repetition rate 200 Hz) is focused onto the sample to
impulsively create a spatially sinusoidal temperature rise of period $L$ and
wave vector $q=2\pi/L$. The grating relaxes by heat conduction, and its decay
is monitored by the heterodyne measurement of a diffracted CW signal beam and
reference probe beam (wavelength 532 nm, beam diameter 470 $\mu$m, CW power
900 $\mu$W, chopped at 3.2 % duty cycle to reduce steady heating on the
sample). The samples are disentangled UHMWPE films synthesized using the same
procedure given in Ref. Rastogi et al. (2011), but with higher draw ratio
($DR=196$) achieved by rolling ($\times 7$) and stretching ($\times 28$).
Figure 1 shows an optical image of the film that is of centimeter scale
dimension laterally and of thickness around 30 micrometers, as measured using
calipers. A scanning electron microscope (SEM) image is given in Fig. 1. In
both images, highly oriented fibers extending over tens of microns are
visible. Since PE is transparent to visible light, Au nanoparticles (diameter:
$\sim 2-12$ nm Drakopoulos et al. (2020); concentration: 1 wt%) were added as
an optical absorber. The concentration of the Au was selected to minimize the
effect of the filler on the thermal conductivity while enabling the formation
of a thermal grating on the sample Robbins et al. (2019). Experimental
characterization of similar samples using polarized light microscopy, among
other methods, indicates that the nanoparticles are oriented in linear chains
in the amorphous regions. Drakopoulos et al. (2020)

A representative TG signal measured at grating period $L=9.8$ $\mu m$ is shown
in Fig. 1. As described in the Supporting Information of Ref. Robbins et al.
(2019), the signal consists of an initially fast decay (time constant
$\lesssim 1$ ns) followed by a slower decay (time constant $\gtrsim 10$ ns).
The initial fast decay is attributed to the thermal relaxation of the Au
nanoparticles, while the subsequent slower decay corresponds to thermal
conduction in the film. Following the procedure in Ref. Robbins et al. (2019),
we fit the signal with a multi-exponential function; the time constant of the
slower decay yields the thermal diffusivity of the sample. Because the initial
signal from the nanoparticles exhibits a short time constant compared to their
thermal signal, the influence of the nanoparticle signal on the fitted thermal
diffusivity is negligible. The signal-to-noise ratio (SNR) of the present
measurements is generally less than 20, which is about 30% of that reported in
Ref. Robbins et al. (2019) for $DR=7.5$ samples . This decrease is because the
highly oriented samples scatter visible light intensely owing to the increased
inhomogeneity over length scales comparable to the optical wavelength, as
evidenced by AFM images perpendicular to the fiber alignment direction (see
Supplementary Material Sec. 4). Nevertheless, TG is able to measure the
thermal signal with adequate SNR because only the diffracted signal due to the
spatial refractive index profile at the grating wave vector is measured, and
the scattered light intercepted by the detector that does not arise from
diffraction can be subtracted from the final signal using a heterodyning
procedure Johnson et al. (2013). The thermal conductivity was calculated from
the measured thermal diffusivity using the heat capacities of linear PE in
Ref. Chang (1974).

The in-plane thermal conductivity versus angle between the fiber alignment
direction and the thermal gradient is shown in Fig. 1. The thermal
conductivity is $\sim$ 40 Wm-1K-1 at 0° ($\kappa_{\parallel}$, parallel to
grating), and decreases to 0.4 Wm-1K-1 at 90° ($\kappa_{\bot}$, perpendicular
to grating). The value along the draw direction is in reasonable agreement
with that obtained on a sample without Au nanoparticles using the laser flash
method Ronca et al. (2017). The value of $\kappa_{\bot}\sim 0.4$ Wm-1K-1 is
close to that of unoriented PE, which is attributed to the heat conduction by
interchain van der Waals interactions Choy (1977). The angle-dependent thermal
conductivity was fitted by a geometric model Zolotoyabko (2011) with the
thermal conductivity along the two principal directions as input. Good
agreement between the model and the data is observed.

Figure 2:  (A) Thermal diffusivity along the chain axis versus temperature for
grating period $L=9.8$ $\mu m$. An increase in the thermal diffusivity is
observed as temperature decreases. (B) Thermal conductivity along the chain
axis versus temperature measured by TG for $L=9.8$ $\mu m$ (blue triangles:
cooling, red triangles: heating) ) and PPMS (yellow circles). The thermal
conductivity is approximately constant from $\sim 300-220$ K, below which the
thermal conductivity decreases. The trend and corresponding values from PPMS
and TG with $L=9.8$ $\mu m$ are in reasonable agreement, suggesting that
phonon mean free paths are less than $\sim L/2\pi\sim 1.5$ micrometers.
Calculated thermal conductivity versus temperature obtained using Eq. 1 (solid
purple line: bulk; dashed green line: $L=9.8$ $\mu m$).

The thermal transport properties can be further examined by measuring the
temperature dependence of the thermal diffusivity and conductivity. The bulk
thermal diffusivity along the chain direction versus temperature obtained from
TG with $L=9.8$ micrometers between 30 – 300 K is shown in Fig. 2. The
diffusivity exhibits a linearly increasing trend with decreasing temperature,
a qualitatively similar trend as that reported in microscale crystalline
fibers Liu et al. (2015). The corresponding $\kappa_{\parallel}$ versus
temperature is shown in Fig. 2. Within the uncertainty of the measurement, the
thermal conductivity remains constant from room temperature to $\sim 220$ K,
below which the thermal conductivity decreases.

The macroscopic dimensions of the present samples permits additional
characterization of the thermal conductivity down to $\sim 3$ K using a
Physical Property Measurement System (PPMS) (See Supplementary Material Sec. 3
for further details). The results are shown in Fig. 2. The measured values and
the trend of the bulk thermal conductivity are consistent with that obtained
from TG. The cryogenic thermal conductivity values on a logarithmic scale are
given in Fig. 3. The values exhibit two distinct temperature dependencies with
a transition at around 10 K.

Figure 3:  Cryogenic bulk thermal conductivity versus temperatures below 100 K
from TG (filled colored triangles), PPMS (yellow circles) and calculated
(purple solid line). Representative literature data of thermal conductivity
versus temperature for semi-crystalline PE with various DRs are also plotted
as open symbols (extruded thin film, DR 4.4, Ref. Burgess and Greig (1975);
solution-cast thin film, DR 40, Ref. Nysten et al. (1995); solution-cast
macroscopic fiber, DR 150, in Ref. Fujishiro et al. (1998). As temperature
decreases, the trend of measured thermal conductivity exhibits a transition
from $\sim T^{2}$ to $\sim T$ near 10 K.

The temperature dependence of the thermal diffusivity and conductivity
provides insight into the origin of the high thermal conductivity in the
present samples. First, the thermal diffusivity is observed to depend on
temperature, ruling out a constant relaxation time for all phonon
polarizations as suggested in Ref. Shrestha et al. (2018). Second, the
measured trend of thermal conductivity versus temperature is consistent with
structural scattering being the dominant scattering mechanism. Above 10 K, the
trend is generally consistent with those of previously reported data on PE
films of various DR as shown in Fig. 3, although the thermal conductivity of
the present sample is consistently higher. Below 10 K, a weaker trend with
temperature is observed compared those exhibited by other samples in the same
temperature range. This difference may be due to the disentangled nature of
the present sample compared to prior solution-cast films and will be the topic
of future study. Third, within the uncertainty of the measurements, the
measured bulk thermal conductivity is in reasonable agreement with the TG data
for $L=9.8$ $\mu m$, indicating phonon mean free paths are shorter than $\sim
9.8/2\pi\sim 1.5$ micrometers Minnich (2012).

Figure 4:  (A) Measured TG signal versus time (symbols) for grating period
$L=1.5\leavevmode\nobreak\ \mu m$, corresponding to $q^{2}\sim 17.5$ $\mu
m^{-2}$, along with the best fit (solid red line) and predicted decay
estimated using the thermal conductivity obtained for $L=9.8$ $\mu$m (dashed
black line). The signal is an average of $5\times 10^{4}$ shots measured at a
single location. The signal clearly decays slower than predicted based on the
thermal conductivity measured at larger grating period, indicating a departure
from diffusive thermal transport. (B) TG signal decay rate versus $q^{2}$. The
measured decay rates for $q^{2}\gtrsim 0.3$ $\mu m^{-2}$ deviate from that
predicted from the thermal conductivity measured at $L=9.8$ micrometers. (C)
Experimental thermal conductivity versus grating period at selected
temperatures from experiments (magenta symbols: 300 K; blue symbols: 100 K)
along with the calculation (orange solid line: 300 K; green dotted line: 100
K). The thermal conductivity exhibits a marked dependence on grating period
and is nearly independent of temperature. Error bars indicate 95% confidence
intervals obtained using the procedure given in Ref. Robbins et al. (2019).

Despite these constraints from the temperature-dependent bulk thermal
transport properties, the microscopic properties of the heat-carrying atomic
vibrations remain underdetermined. To gain further insight, we exploited the
ability of TG to systematically vary the induced thermal gradient over
micrometer length scales by tuning the grating period. If heat-carrying
phonons propagate ballistically over the grating period, the thermal decay is
slower than that predicted from the bulk thermal conductivity Johnson et al.
(2013); Minnich (2012). We have previously used this approach to identify
ballistic phonons over nanocrystalline domains in disentangled UHMWPE samples
of lower DR Robbins et al. (2019).

We apply this approach to the present samples, measuring the thermal
conductivity along the chain axis versus grating period at temperatures of 300
K, 220 K, 100 K, and 35 K. The measured TG signal for a grating period of
$L=1.5$ $\mu m$ is shown in the inset of Fig 4. The decay is clearly slower
than expected based on the bulk thermal conductivity value, indicating the
presence of ballistic phonons on the length scale of the grating period.
Measurements of the decay rate versus $q^{2}$ for all the grating periods at
300 K are given in Fig. 4. The measured decay rate is close to that predicted
by the bulk thermal conductivity at $q^{2}\leq 0.3$ $\mu m^{-2}$, above which
the decay rate is slower by up to a factor of 5 at the smallest grating
period.

The corresponding thermal conductivity versus grating period at 300 K and 100
K is shown in Fig. 4. The thermal conductivity exhibits a marked dependence on
grating period up to $\sim 10$ $\mu$m, a value comparable to that observed in
other covalent crystals with higher thermal conductivity such as silicon
Ravichandran et al. (2018); Johnson et al. (2013). Compared to PE samples of
lower DR $\sim 7.5$ Robbins et al. (2019), the observed trend is significantly
more pronounced, indicating the presence of heat-carrying phonons with longer
MFPs in the present samples. The observed grating dependence lacks a clear
temperature dependence for the temperatures considered here (see Supplementary
Material Sec. 3 for additional data). This finding indicates the dominance of
the structural scattering, consistent with the trend of the bulk thermal
conductivity versus temperature in Fig 2.

## III Low energy Debye model

We now construct a model to interpret the measurements in Figs 2, 3, and 4. We
begin by noting that Fig. 4 indicates that phonons with MFPs on the order of
hundreds of nanometers carry the majority of the heat. Phonons with such long
MFPs are likely from the LA branch owing to its high group velocity $v_{c}\sim
16-17$ km s-1 Mermet et al. (2003); Holliday and White (1971); Twisleton et
al. (1982); Feldkamp et al. (1968); Pietralla (1996). Therefore, the marked
dependence of grating period in Fig. 4 implies that nearly all heat is carried
by the LA branch. Further, the grating-period dependent thermal conductivity
in Fig. 4 does not exhibit a temperature dependence, indicating that the MFPs
are independent of temperature. We therefore construct an anisotropic Debye
model Bowman and Krumhansl (1958); Chen et al. (2013) for the heat conducted
by this branch and use the data to constrain the frequency-dependence of the
LA branch relaxation time. The marked elastic anisotropy of PE can be
accounted for to good approximation by assuming the group velocities all point
along the chain axis Pietralla et al. (1989). Therefore, the thermal
conductivity measured in TG ($\kappa_{i}$) can be expressed as

$\kappa_{i}=\int_{0}^{\omega_{ab}}S(x_{i})\left[C_{1}(\omega)v_{c}\Lambda(\omega)\right]d\omega+\int_{\omega_{ab}}^{\omega_{c}}S(x_{i})\left[C_{2}(\omega)v_{c}\Lambda(\omega)\right]d\omega$
(1)

where $q_{i}=2\pi L_{i}^{-1}$, $x_{i,s}=q_{i}\Lambda_{s}(\omega)$,
$S(x_{i,s})$ is the anisotropic phonon suppression function for an arbitrary
phonon dispersion Minnich (2015), $C_{1}(\omega)$ and $C_{2}(\omega)$ refer to
the heat capacity terms in Eq. 11b of Ref. Chen et al. (2013).

We now specify the numerical values for this model. The c-axis velocity of the
longitudinal polarization is $v_{c}\sim 17$ km s-1 Holliday and White (1971);
Feldkamp et al. (1968); Twisleton et al. (1982). The velocity along a
perpendicular crystal axis was reported as $v_{ab}\sim 1.35$ km s-1 Holliday
and White (1971) from inelastic neutron scattering and $\sim 4.5$ km s-1 from
an ab-initio calculation Shulumba et al. (2017) which may be reflective of the
monocrystals present in disentangled UHMWPE Rastogi et al. (2011). We roughly
estimate $v_{ab}\sim 3$ km s-1. We consider $\omega_{c}$ to be a
characteristic frequency at which the $c$-axis longitudinal velocity decreases
below the Debye velocity and ultimately tends towards zero. Roughly, we
estimate $\omega_{c}\sim 10$ THz; the analysis below is not sensitive to this
choice. This choice then determines $\omega_{ab}=1.8$ THz. The corresponding
maximum wave vector magnitude is $\sim 6$ Å-1.

We now seek to identify the function $\Lambda(\omega)$ that best explains the
temperature and grating dependence of the thermal conductivity. To constrain
the MFP function, we note that at $\sim 1$ THz, the magnitude of the MFP can
be estimated using the dominant phonon approximation Zeller and Pohl (1971)
and the cryogenic thermal conductivity measurements in Fig 3. We use the
cryogenic thermal conductivity at around 12 K, the minimum temperature that is
still on the $T^{2}$ trend of the thermal conductivity. We find
$\Lambda\sim\kappa/C_{l}v_{l}\sim 340$ nm at $\sim 1$ THz, where $\kappa\sim
2.45$ Wm-1K-1 and $C_{l}\sim 4.2\times 10^{-4}$ Jcm-3K-1 is the computed heat
capacity of the longitudinal branch at 12 K.

We now consider various MFP profiles versus frequency that yield the best
agreement with the experiments in Figs. 2, 3, and 4 by adjusting the MFP
profile subject to the above constraint. After extensive comparison, we found
that the best fit is obtained using a constant $\Lambda\sim 350-400$ nm up to
$\sim 4.5-6$ THz, beyond which the MFP decreases as $\sim\omega^{-4}$,
although similar power laws also yield reasonable agreement. (See
Supplementary Material Sec. 3 for further details on results using other
candidate profiles).

The resulting computed bulk thermal conductivity using this profile is
presented in Figs. 2 and 3. In Fig. 2, the bulk thermal conductivity exhibits
qualitative agreement with the measurements, producing the observed magnitude
and trend. The calculated thermal conductivity in TG with $L=9.8$ $\mu$m is
close to bulk value, consistent with the good agreement between the TG and
PPMS measurements. The grating period dependence of the thermal conductivity
is also qualitatively reproduced by the calculation in Fig. 4, and Fig. S2 in
Supplementary Material Sec. 2, although quantitative disrepancies exist.

The calculated cryogenic thermal conductivity is shown in Fig. 3. Although a
similar qualitative trend is observed above $\sim 10$ K, the agreement is
worse below this temperature. This discrepancy could be attributed to heat
conduction by other types of atomic vibrations which may make the primary
contribution to thermal conductivity below $\sim 10$ K.

## IV Discussion

We now discuss our findings in context with prior studies of thermal
conduction in oriented PE films. We first consider the MFP value of $\sim 400$
nm for frequencies in the THz range. For frequencies around 1 THz, prior
values inferred from literature data using the dominant phonon approximation
at cryogenic temperatures are $\sim 80$ nm using $\kappa\sim 0.34$ Wm-1K-1 for
DR 40 Nysten et al. (1995), as estimated using LA specific heat of $\sim
2.8\times 10^{-4}$ Jcm-3K-1 at 10 K using the model described above. This
value is comparable to an estimated value of $\sim 60$ nm for DR 6 at 10 K
Mergenthaler and Pietralla (1994). At higher frequencies $\sim 6$ THz, IXS has
been used to obtain a MFP of $\sim 50$ nm for DR 5.5 Mermet et al. (2003),
which is in reasonable agreement with the values inferred from transport
studies. The MFPs obtained from TG measurements on a DR 7.5 sample in our
previous study ranged from 10 - 200 nm Robbins et al. (2019). All of these
values are comparable to but distinctly smaller than the present value,
expected as the present samples exhibit higher thermal conductivity. The
decrease in MFP at frequencies exceeding $\sim 7$ THz is also consistent with
the increase in broadening reported in IXS Mermet et al. (2003), although this
increase occurred at higher frequencies in their data.

An average MFP for all LA phonons of $\sim 7$ nm was estimated from a study of
PE microfibers Wang et al. (2013). This value is considerably smaller than
those above; as noted in that work, this value is an underestimate owing to
the use of the total heat capacity which is greater than that of the
longitudinal polarization only. Finally, an ab-initio study of PE predicted
MFPs of up to 200 nm Shulumba et al. (2017). This prediction thus appears to
be an underestimate and is likely due to insufficient grid density to sample
the stiff LA branch. This observation implies that the predicted thermal
conductivity in that work is also an underestimate of the upper limit of
thermal conductivity in crystalline PE.

Our findings help to explain the origin of high thermal conductivity in
disentangled UHMWPE films. Recent studies have provided conflicting
explanations for the high thermal conductivity, with Xu et al. Xu et al.
(2019) attributing it to the high thermal conductivity of the amorphous phase
Robbins et al. (2019) but Ronca et al. Ronca et al. (2017) attributing it to
increased extended crystal dimensions. Our data and analysis are consistent
with the latter explanation. Compared to the MFPs of a DR 7.5 sample in our
prior study,Robbins et al. (2019) the present MFPs are clearly larger, as
would be expected if the extended crystal dimensions have increased. The MFPs
in both samples exhibit a clear maximum in the low THz frequencies and are
independent of temperature, suggesting phonon scattering is predominantly due
to reflections at crystalline domain boundaries. Further, evidence exists for
the presence of extended crystals with length on the order of hundreds of
nanometers from NMR and TEM Hu and Schmidt-Rohr (2000); Litvinov et al.
(2011); Brady and Thomas (1989), values that are compatible with the MFPs
obtained here. We infer that heat conduction in disentangled UHMWPE is
primarily due to longitudinal atomic vibrations that are ballistic within the
extended crystal phase, scattered primarily by reflections at the boundaries
between the crystals. Our results therefore support the hypothesis of Ref.
Ronca et al. (2017) in which the high thermal conductivity compared to the
prediction including only crystallinity and anisotropy factors for DR $\gtrsim
180$ (Fig. 7 in Ref. Ronca et al. (2017)) was attributed to the enlargement of
extended crystal dimensions.

Finally, we discuss the implications of our findings for realizing PE films of
higher thermal conductivity. Because the MFPs appear to be limited by the size
of the extended crystals, our study indicates that the thermal conductivity of
PE films has not yet reached its upper limit. The practical challenge is
synthesizing disentangled UHMWPE films with larger extended crystal
dimensions. Such films would be expected to have higher thermal conductivity
of an amount proportional to the increase in crystalline dimension.

## V Summary

In summary, we have characterized the thermal conductivity and mean free path
accumulation function of disentangled UHMWPE films with draw ratio of nearly
200 at various temperatures. We find that heat in PE films is nearly entirely
carried by longitudinal atomic vibrations with MFPs in the sub-micron range
and limited by the dimensions of the extended crystal phase. The high thermal
conductivity at $DR\gtrsim 150$ exceeding the expected value considering
orientation and crystallinity can be attributed to increased dimensions of the
extended crystals. Considering the MFPs appear to remain limited by extended
crystal dimensions, our work indicates that the thermal conductivity of
disentangled UHMWPE films has not yet reached its upper limit.

###### Acknowledgements.

The authors thank Bolin Liao and Wenkai Ouyang for assistance with PPMS
measurements. This work was supported by the Office of Naval Research under
Grant Number N00014-18-1-2101.

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