Polyethylene Nanofibers with Very High Thermal Conductivities · the nanofiber and QC the heat...

SUPPLEMENTARY INFORMATIONdoi: 10.1038/nnano.2010.27

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Polyethylene Nanofibers with Very High Thermal Conductivities

Sheng Shen1, Asegun Henry1, Jonathan Tong1, Ruiting Zheng1,2 and Gang Chen1*

1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2Key Laboratory of Radiation Beam Technology and Materials Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University; Beijing, 100875, China

* Email: [email protected]

1. Fabrication of ultradrawn polyethylene nanofibers

A decalin solution, containing 0.8 wt. % ultrahigh molecular weight polyethylene,

is prepared by heating the polymer-solvent mixture at 145 oC. To prevent the degradation

of polyethylene, the dissolution was carried out under nitrogen. The solution is then

quenched in water to form a gel. The nanofiber pulling system is illustrated in Fig. S1.

The fabrication of the ultradrawn polyethylene nanofibers in the current work includes

two steps. First, a small sample of wet gel is heated by heater 1. After reaching 120 oC,

the heater is turned off to reduce the evaporation of the solvent from the gel and a 100-

200 μm long suspended fiber is rapidly drawn using a sharp tungsten tip or an AFM

cantilever, which is fixed on a motorized stage. Second, heater 2, located underneath the

fiber and heater 1, is used to heat the fiber and surrounding air to ~ 90 oC. After several

seconds, when the two ends of the fiber dry out and solidify, further drawing is conducted

by moving the tungsten tip or the AFM cantilever at a speed of ~ 1 μm/s to achieve the

higher draw ratios.

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2. Stretching ratios of the ultradrawn nanofibers

To estimate the draw ratio in Step 1, we approximate the polyethylene gel, before

deformation, as a thick cylinder with diameter D1 and length L1 (Fig. S2), and assume

that the deformation caused by drawing in Step 1 only occurs in the thick cylinder. In

reality, the two ends are conically shaped, but for approximation purposes, we neglect the

change in volume within the conical ends. The end of the AFM cantilever used in the

drawing process is ~ 4 μm wide. We assume the diameter of the initial cylinder is D1 = ~

6 μm, accounting for the heated polymer gel flowing over the cantilever end due to the

capillary force when the AFM cantilever dips into it. After the drawing in Step 1, the

length of the polyethylene fiber was measured to be L2 = 100-200 μm and the diameter,

D2=1.5-2.5 μm (Fig. S2). Based on volume conservation within the cylindrical section,

we can calculate L1 by 44

22

212

1 LDLD ππ= . Hence, the draw (stretching) ratio in Step 1

can be estimated to be 2

2 12

1 2

6 ~ 16L DL D

= ≈ . In Step 2, the final length of the nanofiber L3

has been successfully drawn in the range of 1,000-10,000 μm, thus giving a draw ratio of

3

2

10 ~ 50LL

≈ . The overall draw ratio in the two steps is thus 3 3 2

1 2 1

60 ~ 800L L LL L L

= ⋅ ≈ for

ultradrawn nanofibers. Such a large range in draw ratio is indicative of our ability to

control the geometrical parameters of our nanofiber samples; though certainly, higher

draw ratios are more difficult to achieve.

The overall draw ratios for three samples presented in the paper are estimated to

be: ~ 410 for sample 1, ~ 270 for sample 2 and ~ 160 for sample 3. Based on the




accuracies of length and diameter measurements of polymer fibers during the drawing

process, we estimated that the uncertainty in the draw ratios is around 10 %.

3. Thermal conductivity measurement system

The thermal conductivity measurement system can be represented by a three-

junction thermal circuit with two thermal resistances corresponding to the polyethylene

nanofiber and the AFM bi-material cantilever, as shown in Fig. S3. The two quantities

that are varied during the measurement are the laser power absorbed by the cantilever QB

and the temperature of the thermocouple T

B

A. The specific thermal resistance of the

nanofiber is ~10 K/W, while typical thermal contact resistance between two solids is

usually 10 -10 K/W range. In addition, we have also used silver epoxy to join the

nanofiber and the thermocouple, thus increasing the contact area. Hence, the thermal

contact resistance at the two ends of the nanofiber is neglected in the following analysis.

Note that if there is thermal contact resistance, the fiber thermal conductivity will be

higher than reported values. The nanofiber is assumed to be uniformly cylindrical along

its length. Although irregularities at the ends are possible, they are neglected in this

analysis. Our SEM images of the nanofiber justify this assumption. All the measurements

are done under high vacuum (< 50 μTorr) and therefore heat convection from air is

negligible. Based on the low emissivity (~ 0.1) of the microfiber , we estimate that the

emissivity of the nanofiber is smaller than 0.1 according to Rayleigh scattering theory

where emissivity is proportional to volume. Due to the very low emissivity (< 0.1) of the

nanofiber and the small temperature difference between the nanofiber and the

surrounding, the radiation loss from the fiber is estimated to be < 1 nW, which is

9

7 8 1

2,3




negligibly small compared to the heat conduction (~ 100 nW) through the nanofiber. As

an approximation, we also assumed that the heat input from the laser can be modeled as

an input at point B. But, in reality, the distance between the cantilever tip and the laser

spot on the cantilever was estimated to be ~ 20 μm. Since the cantilever is very thermally

conductive, we neglect the thermal resistance between the cantilever tip and the laser spot

in the heat transfer model. This assumption will be justified in subsequent sections.

The two-step process developed in our paper can be understood as follows. In the

first step, we calibrate the bending of the cantilever by varying the laser power. In the

second step, we measure the heat transfer via the nanofiber by varying the thermocouple

temperature. These two steps are further explained below.

Step 1: Calibration via Varying of Laser Power

At steady state, the heat balance of the three junctions at point B in Fig. S3 is,

, (1) CAB QQQ +=

where QB is the laser energy absorbed by the cantilever. QB

C

)

A is heat conduction through

the nanofiber and QC the heat conduction through the cantilever. Since the thermal

conductance of the AFM cantilever GC (~ 10 μW/K) is around three orders of magnitude

larger than the thermal conductance of the nanofiber GF (~10 nW/K), the heat conduction

through the nanofiber (~ 100 nW) can be ignored compared to the change of the

absorbed power (~ 10 μW in Fig. 3 (c)) at point B. Equation (1) can then be written

in differential form as follows,

AQΔ

BQΔ

, (2) BQ QΔ = Δ

In Fig. 3 (a), we can obtain the bending1 2(P PB −Δ when the absorbed power on the

cantilever tip changes from P1 to P2. Since the bending signal measured by the




photodiode is effectively a representation of the cantilever temperature, which is

proportional to the heat transfer through the cantilever, , it can also be shown that, CQ

1 2 1 2( ) 1 (C P P P PQ )Bα−Δ = ⋅Δ − , (3)

where α1 is a proportionality constant, determined by the properties and dimensions of

the cantilever.4 From Eqs. (2) and (3), we have,

1 2

1 2

(1

( )

B P P

P P

QB

α −

−

Δ=

Δ) , (4)

Step 2: Variation of Thermocouple Temperature

The second step measures the heat transfer changes through the nanofiber when

the temperature of the thermocouple is changed. The laser input to the cantilever is held

constant. In Fig. 3 b, when the temperature of the thermocouple TA changes from TA1 to

TA2, the heat fluxes through the nanofiber can be expressed as,

) , (5) ( 11 ABFA TTGQ −=

) , (6) ( 22 ABFA TTGQ −=

where GF is the thermal conductance of the nanofiber. Since GC is much larger than GF,

the temperature of cantilever tip TB is assumed as a constant during the temperature

change of the thermocouple. The heat conduction through the nanofiber in the experiment

is estimated to be ~100 nW which corresponds to a ~ 10 K temperature change on T

B

2

-2B.

This justifies our assumption. Thus, subtracting Eq. (6) from Eq. (5), we have,

1 2( ) 1A AA T T F A AQ G T−Δ = − Δ −

)

, (7)

where can be obtained by the thermocouple measurement. In Fig. 3b, we can also

obtain the bending of the cantilever caused by the temperature change from T

1 2A AT −Δ

1 2( A AT TB −Δ A1




to TA2. Similarly, we can relate the heat flux through the cantilever and the bending of the

cantilever by,

1 2 1 2( ) 2 (A A A AC T T T TQ B )α−Δ = ⋅Δ −

)

, (8)

where is the change of the heat flux through the cantilever due to the

temperature change from T

1 2( A AC T TQ −Δ

A1 to TA2, and α2 is a proportionality constant when changing

the temperature of the thermocouple.

In Step 2, the absorbed laser power does not change (BQ 0=Δ BQ ). This leads to

the following,

, (9) CA QQ Δ−=Δ

Thus, based on Eqs. (8) and (9), we have,

1 2 1 2 1 2( ) ( ) 2 (A A A A A AA T T C T T T TQ Q B )α− −Δ = −Δ = − ⋅Δ − , (10)

From Eqs. (7) and (10), the proportionality constant α2 can be expressed as,

1 2

1 22

( )A A

A AF

T T

TGB

α −

−

Δ= ⋅

Δ, (11)

In these two cases, the proportionality constants α1 and α2 are equal, which will be

proved in the next sections. In terms of Eqs. (4) and (11), the conductance of the

nanofiber can be expressed by,

1 2 1 2

1 2

( ) (

1 2 ( )

//

A A

B P P P PF

A A T T

Q BG

T B−

− −

Δ Δ=Δ Δ

)− , (12)

where )()( 2121PPPPB BQ −−

ΔΔ and )()21( 21 AA TTA BT −− ΔΔ can be obtained from the slopes of

Figs.3 c and d, respectively. The thermal conductivity of the fiber can then be found by

assuming an one-dimensional heat transfer along the fiber,




2

4d

LGK FF π= , (13)

where L and d are the length and the diameter of the nanofiber, respectively. It should be

emphasized that though it may appear that the fiber conductance is independent of the

cantilever conductance in Eq. (12), the effect of the cantilever conductance is actually

embedded within the measured bending signals.

From the above discussion, we see that the differential heat transfer through the

nanofiber is directly measured from the change of the bending in the cantilever, since the

initial bending of the cantilever is well recorded by the photodetector [Figs.3 a and b in

the manuscript]. Although the nanofiber thermal conductance is much smaller than that

of the cantilever, our technique allows direct measurement of very small heat transfer

through the nanofiber.

4. Temperature distribution along the cantilever due to laser heating

During the experiment, the area of the laser spot (~ 20 μm × 20 μm) on the

cantilever is observed, while our model assumes heating occurs at one point. In this

section, we will show that for the experimental conditions, the finite laser spot size has a

negligible effect. Using general beam theory, the deflection of a bi-material strip with

different thermal expansion coefficients can be solved from the following differential

equation ,

( )021

21122

2

)()(6 TxTKtttt

dxZd

−+

−= γγ, (14)

where Z(x) is the vertical deflection at a location x, γ is the thermal expansion coefficient,

K is a constant defined by the thickness ratio and the Young’s modulus of the layers, t is




the thickness of the layers (t1= 650 nm, t2= 70 nm), T(x) is the temperature distribution

along the cantilever, T0 is the reference temperature at zero deflection. The boundary

condition for the differential equation is Z(l) = 0, and dZ(l)/dx = 0 since the base of the

cantilever is fixed. Subscripts 1 and 2 refer to the two layers of the strip: “1” for Si3N4

and “2” for Au. Thus, the temperature distribution of (T(x)-T0) must be determined to

solve the deflection Z(x).

Based on equation (14), it can be observed that variations in heating input will be

reflected by the temperature distribution across the cantilever. To show the validity of our

assumption that all the incident heat by the laser is absorbed at a point, we will consider

two cases: (1) the assumed case where all the heat is absorbed at the end and (2) a finite

distribution of heat across some length, l1, from the tip. These cases are displayed in Figs.

S4 (a) and (b). Note that this model is only one-dimensional. Because the relative area

that the incident heat impinges upon the cantilever is a smaller proportion with respect to

the surface area of one side of the cantilever, the current model can be assumed to

provide a conservative estimate.

For case 1, the temperature distribution follows the previously published

distribution,

0 11 1 2 2

( ( ) ) ( )( )

PT x T l xw k t k t

− = −+

, (15)

where P is the total heat incident on the cantilever, l is the length of the cantilever, w is

the effective width of the cantilever, and ki is the thermal conductivity for layer i.

For case 2, the temperature distribution can be described by a piece wise

distribution where (T(l)-T0) = 0 and continuity is assumed at l1,




[ ]

21

11 1 2 2 1

0 2

11 1 2 2

- - ;0( ) 2 2

( ( ) - )- ;

( )

lP xl x l w k t k t l

T x TP l x l x l

w k t k t

⎧ ⎡ ⎤≤ ≤⎪ ⎢ ⎥+⎪ ⎣ ⎦= ⎨

⎪ ≤ ≤⎪ +⎩

, (16)

The temperature distribution for this case is shown in Fig. S4 (c), where it is assumed l =

200 µm and l1 = 20 µm. It can be observed that a distinct parabolic region exists near the

tip of the cantilever, as expected. But overall, it can be qualitatively observed that this

distribution is approximately linear.

Given these temperature distributions, integration of Equation (14) will yield a tip

displacement Z(0), which can be compared between both cases to test how close the real

case can be approximated by the assumed case. For case 1, integration using (15) will

yield,

31(0)Z Cl= , (17)

Similarly, for case 2, integration using Eq. (16) yields,

3 31

2 (0) 624 6l lZ C

⎛ ⎞= − +⎜

⎝ ⎠⎟ , (18)

where,

1 22 1 2

2 1 1 2

( )( )

t t PCt K w k t k t

γ γ += −

+ 2

Upon substitution of known values, the tip displacements become Z1(0) = 8E6·C

and Z2(0) = 7.998E6·C. As observed, the assumed case where all incident heat is incident

at the tip only differs by the more realistic case by 0.002E6 m3 or 0.025 % uncertainty.

Hence, based on this analysis, the assumption that all the heat is incident at a point is

validated for approximation.




5. Proportionality constants α1 and α2

In calibrating the cantilever bending, the laser spot inputs heat at a point to the

cantilever at some distance x1 (~20 µm) from the tip. The calibrated bending constant is

α1. In the second step of the experiment, heat is transferred at the tip-nanofiber junction

(~x=0) and the corresponding bending constant is α2. In both cases, the bending of the

cantilever is measured at x = x1, where the laser spot is located. We will show that α1 is

equal to α2.

For case 1, when the laser power is delivered to the cantilever, as shown in Fig.

S5 (a), the temperature profile at steady state is,

GP

lxTxT 1

0 1)( ⎟⎠⎞

⎜⎝⎛ −=− , for lxx ≤≤1 , (19)

where l is the effective length of the cantilever (l = 200 μm), G is the effective thermal

conductance of the cantilever and P1 is the absorbed power. The temperature is constant

for because of the adiabatic boundary condition at10 xx ≤≤ 0=x . By solving Eq.(14)

using Eq. (19), we have,

( ) 13

21

2112 )()( Pxl

KGttttxZ −

+−−= γγ , for lxx ≤≤1

Thus, the deflection measured at x = x1 (the location of the laser spot) is,

( ) 1113

121

21121 )()( PPxl

KGttttxZ αγγ =−

+−−= , (20)

For case 2, when the heat is conducted through the nanofiber, as shown in Fig. S5

(b), the temperature profile at steady state is,




GP

lxTxT 2

0 1)( ⎟⎠⎞

⎜⎝⎛ −=− , for lx ≤≤0 , (21)

By solving Eq. (14) using Eq.(21), we have,

( ) 23

21

2112 )()( Pxl

KGttttxZ −

+−−= γγ , for lx ≤≤0

Thus, the deflection measured at x = x1 is,

( ) 2223

121

21121 )()( PPxl

KGttttxZ αγγ =−

+−−= , (22)

As can be observed, the two proportionality factors, α1 and α2, are equal from Eqs.

(20) and (22),

( )31 21 2 2 1 1

1 2

( ) t t l xt t KG

α α γ γ += = − − −

6. Radiation heat transfer between the cantilever and the heated needle

Upon completion of the measurements, both varying the absorbed power of the

cantilever and the temperature of the thermocouple, the motorized control stage is used

(100 nm step resolution) to move the needle with the thermocouple backwards until

breaking the nanofiber. Then, the needle is moved forward to the original location where

both measurements were previously conducted and the bending of the cantilever is

measured again by varying the temperature of the thermocouple. The observed bending

signal without the nanofiber is caused only by the radiation transfer between the

cantilever and the heated needle (Fig. S6). The maximum influence of thermal radiation

is ~ 25 %. The deflection signals in Fig. S6 and Fig. 3 b are not in the same range due to

the thermal drift of the cantilever. In Fig. 3 d, we have normalized the deflection signals




in Fig. S6 to ambient temperature (T1) and subtracted them from the deflection signals in

Fig. 3 b. When varying the absorbed power by the cantilever in Fig. 3 a, no correction of

radiation is needed because the conductance of the cantilever is much larger.

7. Laser power absorbed by the cantilever

The output of the photodiode is converted into an X or Y signal corresponding to

the position of the laser spot and a sum signal proportional to the incident laser power. A

power meter (Newport, Model: 818-UV) is used to measure the radiant power in the

incident beam (1.809 mW, 650 nm wavelength), the reflected beam (1.222 mW) and the

strayed beam (0.427 mW). The strayed beam is defined as the beam passing through the

unblocked area of the cantilever and is measured behind the cantilever perpendicular to

the laser incidence direction. We also put the power meter very close to the cantilever at

different locations around the cantilever to measure the scattered light from the cantilever.

The scattered light intensity was measured to be negligibly small because the cantilever is

very thin and flat. Thus, the absorbed power by the cantilever is calculated to be 0.160

mW by (1.809 mW-1.222 mW-0.427 mW). Due to the strayed light (0.427 mW), the

light power incident on the cantilever is 1.382 mW by (1.809 mW-0.427 mW). Thus, the

absorptivity of the cantilever (gold film) is calculated to be 0.116 by the ratio of 0.160

mW and 1.382 mW, which is consistent with the literature value. The accuracy of each

power measurement is ~ 0.002 mW which is obtained by multiple measurements. From

uncertainty propagation, that would yield an overall error on the absorbed power of

~0.004 mW. The uncertainty of the absorbed power by the cantilever is thus ~ 2.5 %.

During the experiments, it is not the incident light that is measured but the reflected light.




The ratio of absorbed to reflected light is 0.131. These numbers can vary depending on

the shape of the cantilever and optical arrangement. In Fig. S7 (a), the photodiode sum

signal of the reflected light is plotted as a function of the reflected laser power. The linear

relationship between them corresponds to a slope of 0.6434 mW/V. The absorption of the

cantilever can be calculated from the photodiode sum signal as 0.131 × 0.6434 ×

(photodiode sum signal). In Fig. S7 (b), we also plot the deflection signal as a function of

sum signal.

8. Uncertainty analysis

The total uncertainty incurred during measurement is a composition of multiple

uncertainties propagating throughout the experiment. The general formulation for KF is,

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⋅==

2

122

44

BTBQ

dL

dLG

KA

B

FF ππ

From uncertainty propagation rules, the total uncertainty in KF is,

( ) ( ) ( )

2

2

2

2

1

1222

2

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

BTBT

e

BQBQ

e

dde

LLe

KKe

A

A

B

B

F

F , (23)

where the uncertainties of the nanofiber length and diameter are ~ 3% and ~ 8 %,

respectively. Therefore, to completely determine the total uncertainty, the uncertainty of

each bending measurement must be derived.

In the first measurement where the incident laser power is varied, only the

deflection signal and sum signal of the photodector were directly measured. Hence, a




conversion is needed to change the sum signal term into absorbed power. This can be

accomplished as follows,

(reflectedabsorbedB

reflected

PPQ SumP Sum

⎛ ⎞ ⎛ ⎞Δ = ⋅ ⋅ Δ⎜ ⎟ ⎜ ⎟⎜ ⎟ Δ⎝ ⎠⎝ ⎠

) , (24)

where this expression is intentionally written as the product of three terms to utilize

previous measurements. The ratio of absorbed to reflected power can be calculated by the

power meter data, as previously discussed, and the ratio of the reflected power to the sum

signal can be obtained by previous data shown in Fig. S7 (a). Thus, the ratio of the

absorbed power to the deflection signal is,

2 2

reflectedabsorbedB

reflected

PPQ SumB P Sum B

⎛ ⎞ ⎛ ⎞⎛ ⎞Δ Δ= ⋅ ⋅⎜ ⎟ ⎜⎜ ⎟⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎟Δ, (25)

where the ratio of the sum signal to the bending signal reflects what was actually

measured. The estimated error for the absorbed power and the reflected power is thus

0.004 mW and 0.002 mW with averages of 0.160 mW and 1.222 mW, respectively,

( ) ( )0025.0

222.1002.02

160.0004.02 2222

=⎟⎠⎞

⎜⎝⎛ ⋅

+⎟⎠⎞

⎜⎝⎛ ⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

reflected

reflected

absorbed

absorbed

reflected

absorbed

reflected

absorbed

PPe

PPe

PPPP

e

In the measurement relating the reflected power to the sum signal (Fig. S7 (a)),

the standard deviation and range of the reflected power are 0.004 mW and 0.18 mW.

Likewise, for the sum signal, they are 0.4 mV and 0.28 V. Note that the reflected power

in this instance is different than what was previously used since this quantity is

contingent on the sum signal.




( ) ( )

2

2 2 2 22 0.004 2 0.0004 0.00190.18 0.28

reflected

reflected

reflected reflected

Pe e P e SumSum

P P SumSum

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎛ ⎞ Δ⎛ ⎞Δ ⋅ ⋅⎛ ⎞ ⎛ ⎞⎝ ⎠⎜ ⎟ ⎜ ⎟= + = + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ Δ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎜ ⎟Δ⎝ ⎠

Similarly, in the measurement relating the deflection signal of the cantilever to the

sum signal (Fig. S7 (b)), the standard deviation and range of the deflection signal is 2.21

mV and 823.3 mV. For the sum signal, they are 0.4 mV and 0.125 V. Again, the sum

signal considered in this case is different from what was previously used.

( ) ( )

2

2 2 2 222

2

2

2 0.0004 2 2.21 6.98 50.125 823.3

Sumee Sum e BB

ESum Sum BB

⎛ ⎞⎛ ⎞Δ⎜ ⎟⎜ ⎟ Δ Δ⎛ ⎞ ⎛ ⎞Δ ⋅ ⋅⎛ ⎞ ⎛ ⎞⎝ ⎠⎜ ⎟ = + = + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Δ Δ Δ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟Δ⎝ ⎠

−

The total uncertainty of equation (24) is thus,

22 22

2 2

22

absorbed reflectedB

reflected

B absorbed reflected

reflected

P PQ Sumee eePB BSumQ P P Sum

BB P Sum

⎛ ⎞⎛ ⎞⎛ ⎞ ⎛⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞Δ Δ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟Δ ΔΔ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜⎜ ⎟= + +⎜ ⎟⎜ ⎟ ⎜⎜ ⎟Δ Δ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜ ΔΔ Δ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝⎝ ⎠

⎞⎟⎟⎟⎟⎟⎠

2

2

2

0.0025 0.0019 6.98 5 0.00447

B

B

QeB

EQB

⎛ ⎞⎛ ⎞Δ⎜ ⎟⎜ ⎟Δ⎝ ⎠⎜ ⎟→ = + + − =⎜ ⎟Δ⎜ ⎟⎜ ⎟Δ⎝ ⎠

, (26)

In the second measurement where the temperature of the heated needle is varied,

the temperature is directly measured by an attached thermocouple with a resolution of 0.1

K, as shown in Fig. 3d in the manuscript. The standard deviation of the deflection signal

is ~ 0.6 mV. Because the deflection signal in Fig. 3b is found by subtracting the radiation

signal, caused by the heated needle, from the total signal, an additional error is incurred.




Thus the final standard deviation of the deflection signal in Fig. 3(b) is ~ 2 0.6⋅ mV. The

range of the temperature and deflection considered, is 40 K and 7 mV, respectively, from

Fig. 3d. Note that it was assumed the error was normally distributed, hence two times of

standard deviations were taken for a 95% confidence interval. This will be assumed for

all subsequent parameters. This results in the following uncertainty,

( ) ( )

2

22 2 222

2

2

2 0.1 2 2 0.6 0.05940 7

A

A

A A

Tee T e BB

T T BB

⎛ ⎞⎛ ⎞Δ⎜ ⎟⎜ ⎟ ⎛ ⎞Δ Δ⎛ ⎞ ⎛ ⎞Δ ⋅ ⋅ ⋅⎛ ⎞⎝ ⎠⎜ ⎟ = + = + =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Δ Δ Δ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎜ ⎟Δ⎝ ⎠

,(27)

Upon substitution of the uncertainties in (26) and (27) into (23),

( ) ( ) ( ) %8.27278.000447.0059.008.0203.0 22 ==++⋅+=⎟⎟⎠

⎞⎜⎜⎝

⎛

F

F

KKe

The values of thermal conductivities of three samples in the main text were averaged

based on two or three repeated measurements on one individual sample, and the largest

error among the repeated measurements on each sample was chosen as the error of the

sample.

9. Scanning electron microscope (SEM) images of nanofiber samples

Figures S8 (a), (b) & (c) show the measured diameters of samples 1, 2 and 3 by

SEM, respectively. The uniformity of the samples is evaluated by measuring the diameter

at ~ 10 different locations (with varying intervals 2 - 20 μm) along one sample. The

standard deviation of the diameter measurements is ~ 12 nm except at the end which is

connected with the cantilever.




10. Molecular Dynamics Simulations

Equilibrium molecular dynamics simulations were conducted, using the adaptive

intermolecular reactive empirical bond order (AIREBO) potential, which was

implemented in LAMMPS – a parallel molecular dynamics software package developed

at Sandia national labs. A time step of 0.25 fs was used and the simulations were run for

10 ns. Periodic boundary conditions were applied in all directions and the minimum

energy unit cell dimensions were 7.1 Å × 5.05 Å × 2.58 Å, which are within 4% of the

values obtained experimentally5. All simulations were initialized with each atom at its

equilibrium (minimum energy) lattice site and random velocities corresponding to room

temperature. In contrast to our previous work6, we have used the more widely employed

temperature definition, which is based on equipartition of the kinetic and potential

energy7. The size of the systems ranged from 6 × 5 × 40 unit cells (7200 atoms) to 8 × 6

× 40 unit cells (11,520 atoms); however, no size dependence was observed in the thermal

conductivity results. The simulations were run in parallel on processor grids ranging from

30-48 processors, using Sandia computing facilities. Each simulation required

approximately 20 days of computational time to complete. Other details associated with

computing the heat flux can be found in our previous works8, which are based on the

operator originally derived by Hardy9.

Figure 4a shows the full normalized heat flux autocorrelation function for a

simulation of a single polyethylene chain, which was calculated up to t = 5 ns. The inset

shows the first 35 ps in greater detail and provides a better view of the strong symmetric

oscillatory contributions close to ~ 50 THz. These oscillations correspond to

contributions from the C-C optical phonon bending motions ~ 50 THz, which do not




contribute significantly to the thermal conductivity, because of their symmetry about zero.

Figure 4b, however, shows the normalized heat flux autocorrelation function, when these

oscillations are suppressed by averaging over 0.025 ps intervals. This is done to highlight

the asymmetric features of the function which decay over a longer time scale and assist in

delineating the difference between the single chain and bulk crystalline behaviors.

References

1. Shi, L. & Majumdar, A. J. Heat Transfer 124, 329 (2002). 2. Marla, V. T., Shambaugh, R. L. & Papavassiliou, D. V. Ind. Eng. Chem. Res. 46, 336 (2007).3. Fujikura, Y., Suzuki, T. & Matsumoto, M. J. Appl. Polym. Sci. 27, 1293 (1982). 4. Shen, S., Narayanaswamy, A., Goh, S. & Chen, G. Appl. Phys. Lett. 92, 63509 (2008). 5. Sperling, L. H., Introduction to Physical Polymer Science (Wiley-Interscience, New Jersey, 2006)6. Henry, A. & Chen, G. Phys. Rev. Lett. 101, 235502 (2008).7. Turney, J. E. McGaughey, A. J. H. & Amon, C. H. Phys. Rev. B, 79, 224305 (2009).8. Henry, A. & Chen, G. J. Theoretical and Computational Nanosci. 5, 2, 141 (2008). 9. Hardy, R. J. Phys. Rev. 132, 168 (1963).

Figure Captions:

Figure S1 Schematic diagram of experimental setup to fabricate ultradrawn

nanofibers.

Figure S2 Two-step drawing process for fabricating the nanofibers.




Figure S3 Three-junction thermal circuit model for analyzing heat transfer in the

experiment.

Figure S4 (a) Case 1: Total heat at tip, (b) Case 2: Heat distribution at tip, (c)

temperature distribution in Case 2.

Figure S5 (a) Case 1: configuration and temperature distribution, (b) Case 2:

configuration and temperature distribution.

Figure S6 Experimental data of the radiation heat transfer between the heated

needle and the AFM cantilever.

Figure S7 (a) Measured reflected power versus the sum signal of photodiode, (b)

Deflection signal versus the sum signal of photodiode.

Figure S8 SEM images of measured nanofiber samples.



supplementary information doi: 10.1038/nnano.2010.27

Heater 2

Heater 1

Gel Nanofiber

Tungsten tip

Microscope

Motor

FIG1



supplementary informationdoi: 10.1038/nnano.2010.27

FIG2




TB

A CB

TA TC

Nanofiber AFMCantilever

QC

QB

QA

FIG3




FIG4




x=l

T -T0

x=l

T -T0

x =x1

T0

Vacuum

Power, P2

x =l x =0

Laser beam

T0

Vacuum

Power, P1

x =l x =0

x =x1

FIG5




0 500 1000 1500−4874

−4872

−4870

−4868

−4866

−4864

−4862

−4860

Time (arb. units)

Def

lect

ion

Sign

al (m

V)T1

T3T4

T1T2

FIG6




1.3 1.35 1.4−4600

−4500

−4400

−4300

−4200

−4100

−4000

−3900

−3800

−3700

Sum Signal (V)

Def

lect

ion

Sign

al (m

V)

Experimental dataLinear fitting

1.5 1.6 1.7 1.8

1

1.05

1.1

1.15

Sum Signal (V)

Pow

er (m

W)

Experimental datay=0.6434x+0.001

FIG7




c

a b

Sample 1 Sample 2

Sample 3

FIG8


Polyethylene Nanofibers with Very High Thermal Conductivities · the nanofiber and QC the heat...

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