Supporting information for relative importance of
grain boundaries and size effects in thermal
conductivity of nanocrystalline materials
Huicong Donga, Bin Wen a 1, Roderick Melnikb
a State
Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao
066004, China
b M2NeT
Lab, Wilfrid Laurier University, Waterloo,75 University Ave. West, Ontario, Canada N2L 3C5
(October 15, 2014)
1
Authors to whom any correspondence should be addressed
E-mail address: wenbin@ysu.edu.cn (Bin Wen), Tel: 086-335-8568761
1
Discussion 1: Simulation method (Non-Equilibrium Molecular Dynamics
method)
Non-Equilibrium Molecular Dynamics (NEMD) method
1
is used to impose a
heat flow across the interface. The thermal conductivity can be calculated from the
heat flux and temperature gradient by using the Fourier’s law2:
J  kdT / dx
(1)
where k is the thermal conductivity, dT / dx is the temperature gradient, and J is
the heat flux, which is given by the sum of exchanged energy per unit time and area,
J
1
m 2
2
(vhot  vcold
) ,

2tA transfer 2
(2)
where t is the simulation time, A is the cross-sectional area of the simulation
system perpendicular to the direction of the heat flux, m is the atom mass, vhot and
vcold denote the velocities of the hottest and coldest atoms to be exchanged at each
step, and the factor 2 arises from the periodicity.
2
Discussion 2: NEMD simulation of single crystal diamond thermal conductivity
The method used for single crystal diamond thermal conductivity calculation is
the same as the method for nanocrystalline diamond thermal conductivity. The atomic
structure models of single crystal diamond, as well as the temperature profile obtained,
are shown in Fig. S1. As shown in Fig. S1 (a), in this model, the hot region is located
in the middle, and the cold regions are located at the ends. To avoid the edge effect 3,
the middle section of the temperature profile in Fig. S1 (b) has been chosen in order to
obtain temperature gradient, which can be fit with a linear function.
Due to the previous reports about the effect of modeling length scales on single
crystal thermal conductivity3, in our work, models with length scales ranging from 40
nm to 200 nm are created in three directions, [100], [111], [110]. The calculated
results indicate that single crystal diamond thermal conductivity increases with the
increasing modeling length scale4, and it differs in three directions1, probably due to
different arrangement of atoms. Following the previous analysis3,5,6, we expect the
inverse of thermal conductivity to be proportional to the inverse of the modeling
length scale, which is indeed the case as shown in Fig. S2. From the extrapolation of
this plot to infinite sample length (inverse of model length tends to 0), we estimate
that the thermal conductivities along [100], [111], and [110] equal to 1021 W/mK,
1225 W/mK, and 1548 W/mK, respectively. The results obtained in this work are in
agreement with previously reported experimental and computational results 1,7.
3
Discussion 3: NEMD simulation of the Kaptiza thermal resistance
A single diamond crystal is characterized by four covalent bonds pointing along
the [111] direction. When observing the main planes in the diamond lattice, we will
find that GBs on the (111) plane have three bonds back into the crystal and one across
the interface; GBs on the (110) plane have one bond pointing back into the crystal,
two parallel to the interface, and one pointing across the interface; while GBs on the
(001) plane have two bonds pointing back into the crystal and two pointing across the
interface. Therefore, a high-angle twist boundary on the (001) plane can be considered
as being representative of GBs in crystalline and nanocrystalline diamond8.
In this work, twist GBs are created by the coincidence site lattice (CSL) method9,
which is a geometrical construction based on the geometry of the lattice10. A twist
grain boundary can be easily constructed by rotating part of a crystal around a plane
direction, as shown in Fig. S3 (a), Part A is rotated around Z direction by a certain
angle (  ), after that there appears a grain boundary between Part A and Part B. Here
the twist angles should be carefully designed so as to avoid unexpected boundaries10.
After rotation, a few lattice sites in Part A coincide with the equal number of lattice
sites in Part B (the red circles in Fig. S3 (b)), and these coincident lattice sites are the
so-called Coincidence Site Lattice (CSL). CSL is characterized by  , which is defined
as the ratio of CSL and crystal unit cell volumes11. In order to get a more accurate
value for the Kaptiza thermal resistance, GBs with different twist angles are needed.
Considering the twofold symmetry axis in the [001] direction in diamond lattice8,
we only need to consider twist angles from 0 to 90 . In our simulation, nine
different twist angles ranging from 0 to 90 are chosen.
NEMD has also been applied in the Kaptiza thermal resistance calculation. The
4
atomic structure model of twist grain boundary used in calculation as well as the
temperature profile obtained, are shown in Fig. S4. As can be seen in Fig. S4 (a), in
this model, the hot region is located at the quarter of the total length, the cold region is
located at the three quarters of that, and a grain boundary is located in the middle, thus
we can observe a jump in the temperature across the grain boundary as shown in Fig.
S4 (b). The Kaptiza thermal resistance is defined as the ratio of the temperature jump
T across a grain boundary to the heat flux J passing through it,
RK  T / J .
(3)
Usually a relaxation process is necessary for the twist angles initially generated
because of high energy of the atoms which are very close to each other.
It has been known that the model length scale has effect on the single crystal
diamond thermal conductivity. In this paper, we also consider this effect on the
Kaptiza thermal resistance. A high-angle twist boundary has been formed by a
  73.74 twist rotation which is known the so-called   25 boundary. Then,
simulation models for   25 GBs with lengths ranging from 20 nm to 100 nm have
been created. The results for T are listed in Fig. S5. It can be seen that the
variation in the jump of temperature at the GBs is very small, although the modeling
length scales are different. Therefore, the effect of modeling length on the Kaptiza
thermal resistance of diamond is insignificant.
Finally, we note that in this work, the same thermal current has been used. Based
on Eq. (3), from the temperature jump T and the thermal current, the Kaptiza
thermal resistance can be obtained. Fig. S6 shows the calculated thermal resistance of
GBs with different twist angles. In order to obtain a more accurate RK, the average
5
value ( 0.143 109 m2 KW 1 ) has chosen, which conforms well with the results from
Ref 12.
6
Discussion 4: Comparison between experimental and our theoretical ETC for
naocrystalline silicon at 300 K
To get theoretical ETC from our theoretical model, the following parameters
have been used. Single crystal thermal conductivity13 is 140Wm 1 K 1 , single crystal
mean free path13 is 40 nm and Kaptiza resistance12 is 0.4 109 m2 KW 1 . For
comparison, three groups’ experimental values14-16 have been plotted with our
theoretical one, and they are shown in Fig. S7. As can be seen that our theoretical
values agree well with Wang’s experimental results14, but there are some deviations
with Bux
16
and Maldovan’s
15
experimental values. It is worth noted that Bux’s
experimental ETC with less grain size is larger than Maldovan’s ETC with larger
grain size, this result is obviously not reasonable. Hence, the deviations between our
theoretical ETC with Bux16 and Maldovan’s15 experimental values may be attributed
to the experimental error.
7
Discussion 5: Relative importance of GBs and size effect on ETC for
nanocrystalline silicon at 300 K
On the basis of our theoretical model and parameters used in Discussion 4,
intragranular thermal resistance ( Rint ra ), intergranular thermal resistance ( RK / d ), and
total thermal resistance ( R pol ) for nanocrystalline silicon at 300 K have also been
calculated and plotted in Fig. S8 (a). As shown in Fig. S8 (a), with the increasing
grain size, the intragranular thermal resistance Rint ra and intergranular thermal
resistance RK / d both decrease. When the grain size is large enough (or 1/d tends to
0), the intragranular thermal resistance approaches to the single crystal thermal
resistance, while the intergranular thermal resistance tends to zero. These results
demonstrate that with increasing grain size, both the effect of GBs and the size effects
of thermal properties become weaker, which conforms to the trend of nanocrystalline
diamond at 300 K and nanocrystalline silicon at 500 K.
The relative importance of GBs and size effects on ETC of nanocrystalline
silicon at 300 K has also been studied by calculating the ratio of intragranular thermal
resistance, as well as the intergranular thermal resistance, to the total nanocryatlline
silicon thermal resistance ( Rint ra / R pol and ( RK / d ) / R pol ). As shown in Fig. S8 (b),
there is a crossover at d  3nm , which is different from that of nanocrystalline
diamond at 300 K and nanocrystalline silicon at 500 K due to different materials and
temperatures. When the grain size is less than 3 nm, ( RK / d ) / R pol is larger
than Rint ra / R pol , when the grain size is larger than 3 nm, Rint ra / R pol is larger
than ( RK / d ) / R pol , and with increasing grain size, the value of Rint ra / R pol is
8
increasing, while the value of ( RK / d ) / R pol is decreasing. It indicates that with
increasing grain size, size effects become stronger than the GBs effect on ETC for
nanocrystalline silicon at 300 K, and this trend is the same as that of nanocrystalline
silicon at 500 K and nanocrystalline diamond at 300 K. As shown in Fig. S8 (b), when
the grain size is less than 10 nm, Rint ra / R pol is less than 0.58, and ( RK / d ) / R pol is
larger than 0.42, the effect of GBs and size effects are both significant for ETC. When
the grain size is 100 nm, Rint ra / R pol is 0.8, and ( RK / d ) / R pol is 0.2, size effects
become much stronger than the effect of GBs on ETC, and both effects still play very
important roles on ETC. When grain size is 1000 nm, Rint ra / R pol is 0.96, and
( RK / d ) / R pol is 0.04, size effects are still relatively stronger than the effect of GBs on
ETC. When the grain size is 10000 nm, Rint ra / R pol is 0.99, and ( RK / d ) / R pol is only
0.01, at that time, both effects have almost no effect on ETC, and ETC of
nanocrystalline silicon becomes very close to the single crystal thermal conductivity.
Analysis of relative importance of size effects and GBs on ETC shows that besides
GBs, size effects also play a very important role on ETC.
9
References for Supplementary Information
1
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2
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3
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14
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16
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10
Hot region
Cold region
(a)
390
Temperature (K)
360
330
300
270
240
210
0
20
40
60
80
100
120
Position (nm)
(b)
Figure S1. Schematic diagram of NEMD approach for calculating single crystal
diamond thermal conductivity. (a) Atomic structure model of single crystal diamond.
(b) Temperature profile along the heat flow direction.
11
along [100]
0.0024
0.0016
0.0008
0.0000
0.000
(a)
0.005
0.010
0.015
0.020
Inverse of model length (nm-1)
0.025
Inverse of thermal conductivity (mK/W)
0.0032
Inverse of thermal conductivity (mK/W)
Inverse of thermal conductivity (mK/W)
0.0040
0.0040
along [111]
0.0032
0.0024
0.0016
0.0008
0.0000
0.000
(b)
0.006
0.012
0.018
0.024
0.030
Inverse of model length (nm-1)
0.0040
along [110]
0.0032
0.0024
0.0016
0.0008
0.0000
0.000
(c)
0.006
0.012
0.018
0.024
0.030
Inverse of sample length (nm-1)
Figure S2. The linear relationship between the inverse of the thermal conductivity
and the inverse of modeling length scale of single crystal diamond along three
crystallographic directions: (a) [100], (b) [111] and (c) [110].
12
Part A
Grain boundary
Part B
Z
Y
X
(a)
θ
Y
X
(b)
Figure S3. A coincident site lattice formed from two simple cubic lattices rotated by
 around Z direction. (a) The main view. (b) The top view. The red circles are the
so-called coincident lattice sites (CLS). A CLS is characterized by  , which is defined
as the volume ratio of the unit cell of the CSL to that of the original crystal lattice or
reciprocal density of coinciding sites.
13


G
B
(a)
345
Temperature (K)
330
315
T
300
285
270
255
0
10
20
30
40
50
60
Position (nm)
(b)
Figure S4. A schematic diagram of the simulation system for calculating the Kaptiza
thermal resistance of crystalline and nanocrystalline diamond in the framework of
NEMD approach. (a) Simulation model for calculating the Kaptiza thermal resistance,
the hot region is located at the quarter of the total length, the cold region is located at
the three quarters of that length, and a grain boundary is located in the middle. (b)
Temperature profile along the heat flow direction.
14
A jump in temperature (K)
25
20
15
10
5
0
20
40
60
80
100
Length of simulation model (nm)
Figure S5. A jump in temperature at a grain boundary for different modeling length
scales.
15
Kaptiza thermal resistance(10-9m2KW-1)
0.40
0.32
0.24
0.16
0.08
0.00
0
16
32
48
64
80
Angle of rotation (degree)
Figure S6. Calculated Kaptiza thermal resistances of [100] tilt GBs in diamond.
16
Thermal conductivity (Wm-1K-1)
1000
Wang's experimental results[14]
Bux's experimental results[16]
Maldovan's experimental results[15]
K0 for silicon at 300K[13]
140
100
10
theoretical ETC for Si at 300 K
1
1
10
100
1000
Grain size (nm)
10000
Figure S7. Grain-size dependence of thermal conductivity for nanocrystalline silicon
(300 K).
17
Thermal resistance (m2KW-1)
0.10
Theoretical Rpol
0.08
Theoretical Rintra
0.06
Theoretical RK/d
Rbulk for Si at 300 K[13]
0.04
0.02
0.007
0.00
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Inverse of grain size (nm-1)
(a)
Relative importance of GBs and
thermal properties size effect on ETC
1.0
Theoretical Rintra/Rpol
0.8
Theoretical (R /d)/Rpol
K
0.6
0.4
10nm
100nm
(RK/d)/Rpol
0.58
0.8
1000nm 10000nm
0.96
0.99
Rintra/Rpol
0.42
0.2
0.04
0.01
0.2
0.0
1
10
100
1000
Grain size (nm)
10000
(b)
Figure S8. Relative importance of GBs and size effects of thermal properties on ETC
of nanocrystalline silicon (300 K). (a). Grain-size dependence of thermal resistance
for nanocrystalline silicon (300 K). (b) Relative importance of GBs and size effects of
thermal properties on ETC in nanocrystalline silicon (300 K).
18