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
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 = 0.064 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) 99.67 50.77 679.8 C C in carbyne.04 (.07) 800.00 C=C in cumulene.40 (.8) 54.99 8.0 644.0 S
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
FIG S. Conversion curves for carbyne and cumulene from MD (T MD ) to quantum-corrected (T QC ) temperatures. S4
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
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
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
From the PCFF force field used for carbyne, the anharmonic C C bond possesses a leading order harmonic spring constant of C = 99.67 kcal/mol/å, while the C C bond is quite harmonic with a spring constant of C = 800.00 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 = 54.99 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