bn lammps parameters
TRANSCRIPT
h 46 � h 10 q Vol.46 No.10
2010 f 10 m h 1244—1249 ; ACTA METALLURGICA SINICA Oct. 2010 pp.1244–1249
i_SO2K6WfU8?q:JgRV ∗
svr tuw(6|�$<O.;℄.k, 6|� 150001)m h *SUZ�%�O*BanCÆ"m./Td��, b4 3 [ Tersoff 49ENBh"fa�J+S, Td3 C, BN? SiC [J-a�EY4:. 3�3��5p�vp?ld%�O�dLJw 3 �[J-a���Y�aQ�, �w 3 �[J-aY�%a�"3�w. |2�P: [J-a�Y� k Q���5p L asdzsd, 0wP�8 k ∝ L
α *�, ^}~#"E>�:^[a*B~>�; [J-a�Y�Qvp'�zrf; Q���%�O�asP, [J-a�Y�W�sdHrfa���4, T 3 �[J-a�Y�~%��+wOa%�O�"d; �d_m� 3 �[J-a�Y�JP[��Hq: C, BN ? SiC.�E5 nCÆ"m./, [J-, �Y�o℄=>IC O482.22 ab0ZN A an.C 0412−1961(2010)10−1244−06
MOLECULAR DYNAMICS SIMULATION ON THERMAL
CONDUCTIVITY OF ONE DIMENISON
NANOMATERIALS
GAO Yufei, MENG QingyuanSchool of Astronautics, Harbin Institute of Technology, Harbin 150001
Correspondent: MENG Qingyuan, professor, Tel: (0451)86414143, E-mail: [email protected] by National Natural Science Foundation of China (No.10772062)Manuscript received 2010–04–07, in revised form 2010–06–30
ABSTRACT The Non–equilibrium molecular dynamics (NEMD) simulation method which is basedon the linear response theory is applied to simulate the thermal conduction process of C, BN and SiCnanotubes. The three–body Tersoff potential is used to simulate the interactions among atoms. Theeffects of axial length, temperature and tensile strain on the axial thermal conductivity of the threekinds of nanotubes are investigated, and their thermal conductivities are compared and analyzed. Thesimulation results show that the axial thermal conductivity increases as the axial length increases, andexhibits a relationship k ∝ Lα that is in agreement with the solution of Boltzmann-Peierls phonontransport equation (B–P equation). It is found that the thermal conductivity of nanotube decreaseswith the increase of temperature. As the tensile strain increases, the thermal conductivity of nanotubesshow an slight increase first, and then decreases. But, the corresponding tensile strains at whichthe tendency of thermal conductivity of the three nanotubes changes are different. Under the sameconditions, the sequence of thermal conductivity from the biggest to the smallest is in the order ofcarbon nanotubes, boron nitride nanotubes and carbon silicon nanotubes.KEY WORDS non–equilibrium molecular dynamics simulation, nanotube, thermal conductivity
R�nj#℄Ab2L�u, j#gvnbW|>LDS��\K2[, e,�fb2Lq#vtQ, MGvnb�.+|Z�_�W�9. r��j#vn{k&,, )qvn&,5; 3&b�22LjC, MG)9* 1 $��/U� ��X 10772062:[���r : 2010–04–07, :['���r : 2010–06–30+wh� : �a�, ℄, 1984 g%, !3%DOI: 10.3724/SP.J.1037.2010.00164
1v�X�Z�b/�7AT[n\K>Lj#gvn� b)7. \K.,r?��?t)7�XZ�&b� [1,2]��ZA� eUi, y_Ffb 5,r#e)7xvnbF�&b�# �b�0��X4�R&bgp, MG =+�.
Mingo � Broido[3] 5~� Boltzmann ?n�;�V℄?n:F_\\K.b�ZA_t��6q�Æ�>Le}(-Fb+�, F�ye [4] xG�?%�#4m�,y��Y\K.>L�Pbk&+CDÆ/�y=.
h 10 q �`� : =sZI'5X$`` !l-.S 1245�Pop e [5] � Yu e [6] 5-64�4S� C \K.bZ�&b, /2_\q4w� C \K.b�ZA�N\x Wm−1K−1. UV[-6�Y\yD�)FY>q�Pe�Mb#^, Yy-6S_b C \K.b�ZA}3�$�pBQ (10—2×106 Wm−1K−1)[5−9], ����fb�Æ/�Æ�b-6}3. +T`O�Ueb��, Zhang � Li[10] 4�4�Gx C \K.�ZAbR�, YC4nV�G� V�Gb,i; Nanda e [11] 4�4� α– �a (PAO) �E�e9^,r\K)7byDF�x C \K.Z�&bbR�, }3�Q#ebyDF�x C \K.b�ZAX�Y(Tsgb#e,T;
Maruyama e [12] 4�4 C \K.�ZA_��6qb+�, ��R�6qbte, �ZAXtQb�5; Bi e[13] ToDV�#n/0��4�4K&i#x C \K.�ZAbR�, ��K&i#x C \K.�ZAXsg,T.w�r�, zx\K.Z�&bb4��9X q C\K.�, {x BN \K.� SiC \K.X�Y. P Hx\K.�0�/0&�b4�X4?kb73,Ux1xt1b�/-A;bb4�B�, {�/-A�Px� 5��\K.}(-F' �vnZ�&b [14]X�#Qb&;IJ, TFX�9x\K.b�/-A�P�#�fb4�.�y+T��qx�boDV�#n/04���4�4 C, BN � SiC \K.b�/-A�P, Z& P�x\K.Z�&bbR�; e,4�4��6q�wqx BN � SiC \K.Z�&bbR�, � 5~�Boltzmann–Peierls �;xUe}3�#46�.
1 D[Qe�yToD�#n/0�� [15] Ue4 10 10 �;C C, BN � SiC \K.b�FZ5;. Ue5; TV[2#/0l[,h�uu'b Tersoff y\5 [16]'OC i#gb�K,T. Terrsoff 5:Fb�\!0
��2rE =
1
2
∑
i
∑
j 6=i
Vij (1)
Vij = fC(rij)[fR(rij + bijfA(rij))] (2)
fC(r) =
1 : r < R − D
12 −
12 sin (π
2r−R
D ) : R − D < r < R + D
0 : r > R + D
(3)
fR(r) = Aexp(−λ1r) (4)
fA(r) = −Bexp(−λ2r) (5)
bij = (1 + βnζnij)
− 1
2n (6)
ζij =∑
k 6=i,j
fC(rik)g(θijk)exp[λm3 (rij − rik)m] (7)
g(θ) = rijk
(
1 +c2
d2−
c2
[d2 + (cos θ − cos θ0)2]
)
(8)0 , i, j � k ri#*=; E r�f'b2; Vij r�:i#gblb; rij r�:i#gbl6; fC r{v:F; fR ri#gj?b; fA ri#g�Nb; bij rl*-F; ζij rX�kuF; θijk rlu; R � D r{v5r-2; A � B ri�b; λ1 � λ2 r�-F; θ0 roDVlu; β \V�F; λ3 rOC{v!0b-F; n xPiS,��b?�F�; m � r r5:Fb(~�F;
g(θ) rlu�$:F; c �2uzqbR�; d OC�fxuqbA(;q.+T#eb-F�, Tersoff 5:F�-�x#egKg�K,TbOC. OC C \K.� SiC rlu\K.b Tersoff 5-F�-k`O�n Lammps b5:Fyn. V[ Lammps #�8OC N–B i#g�K,Tb 3 \ Tersoff 5-F, MG�y"� N–B b 2 \Tersoff 5-F [17] A7C 3 \ Tersoff 5-F, �� 1T2. TA7b5-FS_4 BN \K.b{k( , /1 1 N–B 3 [ Tersoff 4,E
Table 1 The parameters of three body Tersoff potential function of N–B
c d h n β λ2 B R D λ1 A
nm−1 eV nm nm nm−1 eV
BBB 0.526 1.587×10−3 0.5 3.99 1.6×10−6 2.18×10−5 43.13 0.2 0.01 2.348×10−5 40.05
BBN 0.526 1.587×10−3 0.5 3.99 1.6×10−6 2.18×10−5 43.13 0.2 0.01 2.348×10−5 40.05
BNB 0.526 1.587×10−3 0.5 3.99 1.6×10−6 1.51×10−4 89.07 0.2 0.01 1.634×10−4 82.71
BNN 0.526 1.587×10−3 0.5 3.99 1.6×10−6 1.51×10−4 89.07 0.2 0.01 1.634×10−4 82.71
NBB 17.8 5.9484 0 0.62 0.0193 1.51×10−4 89.07 0.2 0.01 1.634×10−4 82.71
NBN 17.8 5.9484 0 0.62 0.0193 1.51×10−4 89.07 0.2 0.01 1.634×10−4 82.71
NNB 17.8 5.9484 0 0.62 0.0193 0.2624 142.9 0.2 0.01 0.28252 133.0
NNN 17.8 5.9484 0 0.62 0.0193 0.2624 142.9 0.2 0.01 0.28252 133.0
Note: λ1 and λ2—lattice constant, A and B—cohesive energy, R and D—cutoff parameters, h—cosine of the energetically
optimal angle, β—bulk modulus, n—corrspond to simple cubic phase, d—determine the dependence on angle,
c—determine the strength of angular effect
�1246 ����-� h 46 �_ BN \K.bl6r 0.146 nm, _ BN \K.b-al6�zA, HQT�yA7b 3 \ Tersoff 5-F4�BN \K.6�#b.<"� C, BN � SiC \K.b}(Wi(ptA/( b�gj8, I 5=eS_U b{k( . =e5;r, <q\K.bA/( q 0 K �=e4×104 %, I�o(w, GI(w 100 K, GIo=e�eb%F, �\wq(\ 1600 K, *I�#kR5;'_\\K.b{k}(. *Æ_\ C \K.bl6r0.142 nm, BN \K.bl6r 0.146 nm, SiC \K.bl6r 0.18 nm, _ Tersoff–Brenner 5 [18] `O}3BzA, HQ.T Tersoff 5:F4�G 3 �\K.6�#b. C, BN � SiC \K.b}(U �i 1 T2.qUe5; , V[U b6q6XÆb, MG#,r�s&��`n, ~�i#bnn�;+Td/ – �~O� [19], W�%6,r 0.5 fs.�ZA`O+ToDV�#n/0��, q\K.56q��e�r N ��b, t � N=0 �r�)��, N=N/2 �r����, t\r~4btO� (�i2). G�?k%FY�)� nb*Qbi#��� n
^ 1 C, BN ? SiC [J-|'1HhFig.1 Scketchs of C (a), BN (b) and SiC (c) nanotubes
^ 2 nCUÆ"m./Td1HhFig.2 Schematic of non–equilibrium molecular dynamics
(NEMD) simulation
b*�bi#, �tO1xLq (?�|!��)�� nb*Qbi#bb2Q[���� nb*�bi#bb2), MGy8�3&4?�K)�\��b�=, {�fV[L�4)��3&4?�_�=����bw2, [6�Ff`_C�� 0bwq, �{�V 0bwq`O_wqXq, p}Af`_b�=V Fourier Z�k?J = −λ∇T , �`O_\\K.b�ZA [15,19].
2 D[FBl\M}�Ue5;�r 3 %: <, q NVT �& (Ue5; �< N– i#F2�V – \W�T– wq#�b�&) �Ue 1×105 step, TLq�k�.�fbwqN\d��; I, q NVE �& (Ue5; �< N– i#F2�V – \W�E– �fb2#�b�&) �b+n#1×106 step, .�f=eN\oD�V; *Iq NVE �&�n# 2×106 step, 5tO)�i#.�f_&!7?kbw2, �f`�f3&b�=��Pbw2, *I 5�=_wqXqb��~_\K.b�ZA.qUe5; ��b2tOAy�-F�yR��Ue}3b��q, {{��?%4���6Su,g_bb2tOIF{#6G%tOb2bQ��k�Ue}3b��q. Ue}3�Qb2tOIFlm�, _\b}3�2�pl�, U5[m�bb2tOqZwqb�&�P, qo� Fourier Z�k? [20]. �Q2Ue6�, Z.q 1200 K b�w�, 10 step tO?Ib23&bwq�PAr�&�P, { 5 step ?Iq3&Q�b�&�P. MGq�ybUe b2tOmA,rr 10 step tO?I.
2.1 pd4;<TPAX7L9k ��6qx\K.���ZAX��bR� [3], MG�y��`O4 20, 30, 40, 50, 60, 80 � 100 nm b10 10 �;C C, BN � SiC \K.q 300 K ,b�ZA, `O}3�i 3 T2.Vi 3 �k, 3 �\K.b���ZA�Rt��6qbte{te, {teb�5�ojN. y����TXÆ>L�P [20] ℄F�8. XÆ�fbX�%V;��f6qQX��+�:
1
leff∝
1
l∞+
4
LZ(9)0 , leff rXÆ�fbX�%V;, l∞ r}Æ�fb%V;, LZ rU b6q.q2#/0 )7b�ZA κ ��2r
κ =1
3cvl (10)
c =3
2kBn (11)
h 10 q �`� : =sZI'5X$`` !l-.S 1247�
20 40 60 80 1000
50
100
150
200
250
300
350
400
450
Th
erm
al c
ondu
ctiv
ity, W
(/mK
)
L, nm
SiC BN C C from ref.[13] C from ref.[4]
^ 3 [J-�Y�^��5pa*�Fig.3 The relationship between nanotube’s thermal con-
ductivity and axial length0 , l r$#o�%V;, v r$#Lq, c rSu\Wb��, n r$#F, kB r Boltzmann 4F.MG, &A0 (9—11) �_1
κ∝
2
kBnv
( 1
l∞+
4
LZ
)
(12)V0 (12) ��R�U 6q LZ btQ, U b�ZA κ :�otQ, {{tQbLA�oj�, _�yUeT_}3��. 5~� Boltzmann–Peierls $#F?�; (B–P�;)[3] �FS_#��b+C�. B–P $#F?�;��,−vp
dnp
dx= ∂cnp (13)0 , vp r$#Lq, np r$#�$:F, n r$#F, np � n ����2r
np = n + n(n + 1)g(x) (14)
n ≡ (eh̄ω/kBT− 1)−1 (15)q�wqXq�, B–P �; YMb?��,�&�I
∂cnp/(n(n + 1)) =∑
dir(p)
(n′− n′′)ϑp′′
p,p′(g′′− g′ − g)+
1
2
∑
rev(p)
(1 + n′ + n′′)ϑp′,p′′
p (g′′ + g′ − g) (16)
t dir(p) � rev(p) R�$#b��5;, �$:
p+p′↔ p′′ � p↔ p′+p′′ 1�5;. p, p′ � p′′ r��5; �&,Tb$#; ϑp′′
p,p′ � ϑp′,p′′
p ��rOC�C1���5;,gb�+2; g, g′ � g′′ ��r p, p′ �p′′ xPbOC$#�$bRF�.?t)7b�ZA� 5�$:F�2r
κ =1
S
∑
α
∫
vph̄ωpgpnp(np + 1)dq (17)0 , h̄ r Planck 4F, ωp r$#mA, q r F,
S r\K.{MMW. OCW�k$#�$bRF� gprgp ≈
h̄ω
kBT 2vpτp
∆T
L(18)0 , ∆T r\K.1sbwq2, L r\K.��6q.�fb=e,g��2r
τ−1p =
∑
dir(p)
(n′−n′′)ϑp′′
p,p′+∑
rev(p)
(1+n′+n′′)ϑp′,p′′
p +2vp
L
(19)0 , n′, n′′ ��r$# p′ � p′′ bF2.qgmu, ��>$#?I��, ��I_\τ−1p ≈
dir∑
pp′p′′
(n′−n′′)ωω′ω′′+
1
2
rev∑
pp′p′′
(1+n′+n′′)ωω′ω′′+a
L
(20)t a r42; ω, ω′ � ω′′ xP p, p′ � p′′ bmA. Wω →0 ,
n′− n′′
∝ 1/ω′− 1/ω′′
≈ ω/(ω′ω′′) (21)
1+n′+n′′∝ 1+1/ω′+1/ω′′
≈ 1+ω/(ω′ω′′) ≈ ω/(ω′ω′′)
(22)_\κ ≈
∫
(q2 + a/L)−1dq ∝ L1/2 (23)�>?&�$#�&}I��, ��IÆrκ ≈
∫
(q3/2 + a/L)−1dq ∝ L1/3 (24)&��, 5�I~� B–P �;, _\�ZA_ Lα7~�+�, { α �r�[ 0.5 b�. �yUe_\b}3�eAr Lα ��!0, �`��b��[ 0.25—0.45�g, _+C}3��. y [4] � [13] `O_\, 20,
80 � 100 nm b (10 10)C \K.q 300 K wq�b�ZA��r 247, 405 � 456 W/mK(�OOr�G�*V0gg� 0.34 nm ,xPb}3), _�y}3zAbx<, �i 3 T2.
2.2 `;<TPAX7L9k r4�wqx\K.Z�&bbR�, �yUe4 3� 10 10 �;C \K.q#ewq� (100—1200 K)b�FZ5;, �f`��_\ 3 �\K.�ZA_wqb+��i 4. Vi 4 ��, \K.b�ZA_wq7��, tR�wqb(�{sg, 3 �\K.b�ZA�XG0?. y����T$#��+C℄F�8.
�1248 ����-� h 46 �
0 200 400 600 800 1000 12000
50
100
150
200
250
300
350
400
450
T, K
Ther
mal
con
duct
ivity
, W/(m
K)
SiC BN C C from ref.[13]
^ 4 [J-�Y�^vpa*�Fig.4 The relationship between nanotube’s thermal con-
ductivity and temperatureqgwu, Uewqj�[ Debye wq θD, "�Einstein b+C [21] �_\
κ ∝ eθD
αT (25)q�wu, UebwqQ[ Debye wq, h̄ω <<
kBT , MG$#F n D(n =
1
eh̄ω
kBT − 1≈
kBT
h̄ω(26)
V0 (26) ��, R�wqb(�, $#F n :tQ, �{te4$#l�b�A, j�4$#bo�%V;. V0(9) ��, $#bo�%V;_)7b�ZA7~�, MG_\
κ ∝1
T(27)}A0 (25—27) Ue}3�i 4 T2. y [13]!C, qwq��r 300, 600 � 800 K ,, C \K.b�ZA��r 247, 148 � 120 W/(m·K)(�OOr�G�*V0gg�,xPb}3), _�y}3�zA.
2.3 pdHYj/<TPAX7L9k x\K.)e��& q��\K.����bi#j$, i#b{nQMG=\R�, �{R�\$#��, K{��\K.bZ�&b. r44���& P�x\K.Z�&bbR�,�y,r 20 nmb 10 10�;C \K.r4�x�, qwqr 300 K ,, ��x\K.1s*�?0�*�?0i#)e���b& uB,G)em?IuB<�f=e?k%F, G���uBo>eoLArG 5000 steps )e 0.0025 nm buB. �5Ue�f`��_\ 3 �\K.���ZA_��& P�b+��i 5. Vi 5 �F��R���& P�btQ, 3 �\K.b�ZA�XtQIj�b�5,U 3 x�M;q#e, R�P�bte, C \K.b�ZA*�/j�, BN \K.I�, SiC \K.pI�.
0.00 0.02 0.04 0.06 0.08 0.100
20406080
100120140160180200220240
Ther
mal
con
duct
ivity
, W/(m
K)
Axial strain
SiC BN C
^ 5 [J-���Y�^��%�O�a*�Fig.5 The relationship between nanotube’s thermal con-
ductivity and axial strainq�/zu, )ebP�x�, R�P�bte, \K.����b lÆ& , Z7li#gb,T/te, �i#=\����bD�tQ, MGi#b{nj�, ��b$#j�, $#bo�%V;te, "�0(10) ��Gzu\K.b�ZAte; R�P�bb+te, ����b&�lM&/5Q{v9, 3&4vl��u"�uMen,>qbM&�!, y�oj�4i#q����=\bD�, MGqGzui#{ne�, ��$#te, $#o�%V;j�, "�0 (10) ��Gzu\K.b�ZAj�.
3 �\K. lbl6bj*r: C–C l <B–Nl <C–Si l. R�& P�bte, ltbll�GÆ&v, MG 3 �\K.3&vl��u"�uMen,>qbM&�!bG*AIr: C \K., BN \K., SiC\K.. TFV�yb$#��i+�Fjv_\, R�P�bte, C \K.b�ZA*j�, BN \K.I�, SiC \K.b�ZA*IC�j�.&A 3 �\K.bUe}3��, �e`n�, 3 �\K.bZ�&br: C \K. >BN \K. >SiC \K.. y6Mr��bZ�&b�k[i#gb�K,T�i#�2eMK. ?�|!�, i#g�K,Tlz, i#b�2lQ, ��bZ�&b�l2 [22]. �yUeb 3�\K.bM0lz�j*r: C–Cl >B–Nl >Si–Cl, lzbM0lxi#bD�lQ, yqZ$#��j�, $#o�%V;tQ, *ÆZ�ZAtQ.
3 FM�yPToDV�#n/0��Ue4 C, BN �SiC 3 �\K.b�FZ5;. Ue}3�Q:
(1) \K.b���ZAR���6qbte{te, 1xQ9 κ ∝ Lα +�.
(2) \K.b���ZARwqbte{j�, Uj��5�o�E.
h 10 q �`� : =sZI'5X$`` !l-.S 1249�(3)\K.b���ZAR���& P�bte9teIj�b�5, U 3 �\K.b�M�q#e.
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