Supplementary Information

Transcription

1 Supplementary Information Ballistic Thermal Transport in Carbyne and Cumulene with Micron-Scale Spectral Acoustic Phonon Mean Free Path Mingchao Wang and Shangchao Lin * Department of Mechanical Engineering, Materials Science and Engineering Program, FAMU- FSU College of Engineering, Florida State University, Tallahassee, Florida 0, USA * Corresponding author contact information: slin@eng.fsu.edu S

2 Table S. Force field parameters for covalent bonding interactions in carbyne and cumulene. The -4 non-bonded interactions between carbon atoms were modeled by the -6 Lennard- Jones potential with σ = 4.0 Å and ε = kcal/mol. The C-C bond and C=C bond were modeled using a 4th-order polynomial function (anharmonic), while the C C bond was modeled using the common quadratic function (harmonic) since it is very stiff. The equation for the anharmonic covalent bonding energy is: E(r) = K (r r 0 ) + K (r r 0 ) + K 4 (r r 0 ) 4. Experimentally interpolated bond lengths are shown in the parentheses. An equilibrium angle of 80 and harmonic spring constant of 00 kcal/mol was used for angular bending in both carbyne and cumulene. Bond Types r 0 (Å) K (kcal/mol/å ) K (kcal/mol/å ) K 4 (kcal/mol/å 4 ) C C in carbyne.50 (.80) C C in carbyne.04 (.07) C=C in cumulene.40 (.8) S

3 FIG S. Time evolutions of Green-Kubo MD simulation predicted thermal conductivity κ (left column, running average) and HFACF (right column, for the last ns) for (a) carbyne and (b) cumulene chains of various lengths, all at T MD = 00 K. S

4 FIG S. Conversion curves for carbyne and cumulene from MD (T MD ) to quantum-corrected (T QC ) temperatures. S4

5 FIG S. Phonon dispersion (top left), DOS (top right) and group velocity (bottom) of the carbyne chain under (a) 0% extension and (b) 0% compression. The computational method and color code are the same as in Fig. (a). S5

6 FIG S4. Phonon dispersion (top left), DOS (top right) and group velocity (bottom) of the cumulene chain under (a) 0% extension and (b) 0% compression. The computational method and color code are the same as in Fig. (b). S6

7 Analytical Lattice Dynamics (LD) Calculations The single carbyne chain can be modeled as a monoatomic spring consisting of beads of the same mass m but two alternating spring constants C (for C C) and C (for C C) as well as the corresponding equilibrium separations a and a (Fig. (a) in the main text). The classical Newton s second law of motion applied to this chain leads to the following equation for the displacement on particles n and n+, u n and u n+, from force balancing: u n m = C ( u ) ( ) n + u n C u n u n (S) t u t n + m = C n + n + n + n ( u u ) C ( u u ) (S) We propose the following complex wave equations as the general solutions to u n and u n+ : u n { i[ nk( a + a ) ωt] } = U exp (S) { i[ nk( a + a ) + ka ωt] } u n + = U exp (S4) Substituting Eqs. S and S4 into Eqs. S and S gives: which can be rearranged into: By canceling out U or U, we got: [ U exp( ika ) U ] C [ U U exp( ika )] mω U = C (S5) [ U exp( ika ) U ] C [ U U exp( ika )] mω U = C (S6) [ ( C + C )] U + [ C ( ika ) + C exp( ika )] U = 0 [ ( C + C )] U + [ C ( ika ) + C exp( ika )] U = 0 mω exp (S7) mω exp (S8) ( + C ) ω + C + C + C C = C + C + C C cos[ k( a )] 4 m m C + ω a (S9) which can be further simplified to: m ( + C ) ω + C C { cos[ k( a + a )]} = 0 4 ω m C (S0) which leads to the phonon dispersion relation: ( + C ) ± C + C + C C [ k ( a + a )] C cos ω = (S) m S7

8 From the PCFF force field used for carbyne, the anharmonic C C bond possesses a leading order harmonic spring constant of C = kcal/mol/å, while the C C bond is quite harmonic with a spring constant of C = kcal/mol/å. Other parameters can be obtained from the force field: a =.50 Å, a =.04 Å and m = g/mol. At the long wavelength limit (k = 0), the acoustic phonon dispersion is ω optical (k = 0) = 0 and the optical phonon dispersion becomes: ( ) ( C + C ) = = = ω m ω (S) optical k 0 C + C where (C + C ) is the effective spring constant. Similarly, at the first Brillouin zone (k = π/(a +a )), the acoustic and optical phonon dispersion become: π C ω acoustic k = = ωc a a + m = (S) π C ω optical k = = = ω C a a (S4) + m Similarly, a single cumulene chain can be approximated as a monoatomic spring of mass m, spring constant C (for C=C), and equilibrium separation a (Fig. (b) in the main text). The analytical solution for phonon dispersion of this chain is well-known as: C ka ω sin = ωc m = (S5) From the PCFF force field used for cumulene, the anharmonic C=C bond possesses a leading order harmonic spring constant of C = kcal/mol/å, equilibrium separation of a =.40 Å, and mass of m = g/mol. Supplemental References. Kastner, J. et al. Reductive preparation of carbyne with high yield. An in situ raman scattering study. Macromolecules 8, 44-5 (995). S8