21.4 Formulation of Multifield Perturbation
403
hydrogen bond (O:H–O) in water ice. Molecular undercoordination shortens the H–
O covalent bond but lengthens the O:H nonbond because of the coupling between
electron pairs on adjacent O
2− [4].
The local bond average (LBA) represents the true Fourier transformation of the
phonon relaxation dynamics, which sorts the constituent bonds according to their
force constants or vibrational frequencies. In contrast to the volume partition approximation focusing on the value of a quantity in the partitioned volume, the LBA
approach connects the deviation of the quantity from its known bulk value under an
applied external stimulus. The volume partition approximation therefore describes
only the local representative atomic bonds, disregarding the manner of distribution
and the number of bonds. In the absence of phase transitions, the nature and total
number of bonds keep unchanged. The LBA applies to all specimens of interest:
crystalline, non-crystalline, solid and liquid, and those with or without defects or
impurities. One can therefore focus on the performance of the representative bond,
or the average of all bonds, toward the bond-phonon-property cooperativity.
A typical example is the graphene and carbon nanotube (CNT) that follows the
BOLS prediction. Using the diamond C–C bond length of 0.154 and 0.142 nm for
graphite, one can readily derive the effective CN for the bulk graphite as z g = 5.335
from the bond contraction coefficient C z . For the C atom in the bulk diamond, the
effective CN is 12 instead of 4 because the diamond structure is an interlock of two
fcc unit cells. Given the atom cohesive energy in diamond, 7.37 eV [18], and the bond
nature index m = 2.56, one can derive, from the E z = C
−m
z E b relation, the single
C–C bond energy in the diamond is E b = 7.37/12 = 0.615 eV and it is E 3 = 1.039 eV
in the monolayer graphene of z = 3. The cohesive energy per atom in graphene is
3.11 eV/atom.
Theoretical reproduction of the elastic modulus enhancement [19–21], melting
temperature depression of the SWCNT [19, 22], and the C 1s core level shift of the
graphene edge, graphene interior, graphite, and diamond [23], confirmed consistently
that the C–C bond at the graphene edge contracts by 30% from 0.154 to 0.107 nm
with a 152% bond energy gain [19, 20]. The bond contraction and polarization dictate
the ribbon width dependence of the band gap expansion of GNR [24], and the DiracFermi polaritons generation and hydrogenation [25]. The C–C bond between the
3-coordinated atoms in GNR contracts by 18.5% to 0.125 nm with a 68% increase
of bond energy [20]. The Young’s modulus of the SWCNT was determined to be
2.595 TPa with respect to the bulk modulus of 865 GPa. The effective wall thickness
of the SWCNT is determined to be 0.142 nm instead of the layer spacing 0.34 nm.
It has been found [26] that breaking a C–C bond of the 2-coordinated carbon atom
near the vacancy requires 7.50 eV per bond that is 32% higher than the energy
(5.67 eV/bond) required for breaking one bond of a 3-coordinated carbon atom in
a suspended graphene. This fact further evidences for the BOLS-LBA prediction of
the shorter and stronger bonds between undercoordinated atoms.
403
hydrogen bond (O:H–O) in water ice. Molecular undercoordination shortens the H–
O covalent bond but lengthens the O:H nonbond because of the coupling between
electron pairs on adjacent O
2− [4].
The local bond average (LBA) represents the true Fourier transformation of the
phonon relaxation dynamics, which sorts the constituent bonds according to their
force constants or vibrational frequencies. In contrast to the volume partition approximation focusing on the value of a quantity in the partitioned volume, the LBA
approach connects the deviation of the quantity from its known bulk value under an
applied external stimulus. The volume partition approximation therefore describes
only the local representative atomic bonds, disregarding the manner of distribution
and the number of bonds. In the absence of phase transitions, the nature and total
number of bonds keep unchanged. The LBA applies to all specimens of interest:
crystalline, non-crystalline, solid and liquid, and those with or without defects or
impurities. One can therefore focus on the performance of the representative bond,
or the average of all bonds, toward the bond-phonon-property cooperativity.
A typical example is the graphene and carbon nanotube (CNT) that follows the
BOLS prediction. Using the diamond C–C bond length of 0.154 and 0.142 nm for
graphite, one can readily derive the effective CN for the bulk graphite as z g = 5.335
from the bond contraction coefficient C z . For the C atom in the bulk diamond, the
effective CN is 12 instead of 4 because the diamond structure is an interlock of two
fcc unit cells. Given the atom cohesive energy in diamond, 7.37 eV [18], and the bond
nature index m = 2.56, one can derive, from the E z = C
−m
z E b relation, the single
C–C bond energy in the diamond is E b = 7.37/12 = 0.615 eV and it is E 3 = 1.039 eV
in the monolayer graphene of z = 3. The cohesive energy per atom in graphene is
3.11 eV/atom.
Theoretical reproduction of the elastic modulus enhancement [19–21], melting
temperature depression of the SWCNT [19, 22], and the C 1s core level shift of the
graphene edge, graphene interior, graphite, and diamond [23], confirmed consistently
that the C–C bond at the graphene edge contracts by 30% from 0.154 to 0.107 nm
with a 152% bond energy gain [19, 20]. The bond contraction and polarization dictate
the ribbon width dependence of the band gap expansion of GNR [24], and the DiracFermi polaritons generation and hydrogenation [25]. The C–C bond between the
3-coordinated atoms in GNR contracts by 18.5% to 0.125 nm with a 68% increase
of bond energy [20]. The Young’s modulus of the SWCNT was determined to be
2.595 TPa with respect to the bulk modulus of 865 GPa. The effective wall thickness
of the SWCNT is determined to be 0.142 nm instead of the layer spacing 0.34 nm.
It has been found [26] that breaking a C–C bond of the 2-coordinated carbon atom
near the vacancy requires 7.50 eV per bond that is 32% higher than the energy
(5.67 eV/bond) required for breaking one bond of a 3-coordinated carbon atom in
a suspended graphene. This fact further evidences for the BOLS-LBA prediction of
the shorter and stronger bonds between undercoordinated atoms.