Listofsymbols
a Sidelengthofacubicsimulationbox
Meannearest-neighborseparation
a Accelerationvector(thesecond-ordertimederivativeofa positionvector)
Edgevectorofaunitcell
apre
Predictedaccelerationvector
acor Correctedaccelerationvector
aGE
aSE
A
AIn
Latticeconstantofgraphene(2.46A ˚ )
Latticeconstantofsilicene(3.98A ˚ )
Aphysicalquantity
Amaterial-dependentconstant
Amplitudeofsurfaceroughness
Contactarea
Cross-sectionarea
Surfacearea
Areaoftheinterfacebetweenagraphenemonolayeranda silicenemonolayer
A(RN)ConfigurationalcomponentoftheHelmholtzfreeenergy
A hi
A hiblk
A hisim
AðRNn ; tÞ
bij
b
Truemeanvalueofaphysicalquantity A
Time-averagedvalueofaphysicalquantity A obtainedina blockperiodofanMDsimulation
Time-averagedvalueofaphysicalquantity A obtainedfroman MDsimulation
Amplitudefactorofanuclearwavefunction
BondorderintheREBOpotential
Third-ordertimederivativeofapositionvector
Edgevectorofaunitcell
B Aphysicalquantity
Bijk
Aconstantthatdenotesinteractionstrength
BOij
Bondorderbetweenthe ithandthe jthatom
c Afittingparameter
c Edgevectorofaunitcell
cIi
Linearcoefficientforthemappingfunctionofthecoordinatesof the Ithcoarse-grainedsites
c(t)Autocorrelationfunctionofhydrogenbonding
C Amaterial-dependentconstant
CGE
CSE
CV
Constantvolumeheatcapacityofagraphenenanosheet
Constantvolumeheatcapacityofasilicenenanosheet
Isochoricheatcapacity
d Meangrainsize
Thicknessofananochannel
D
Diffusionconstantofagaseousorliquidsystem
Diameterofthenanowire
Interlayerdistance
Amaterial-dependentconstant
Dc Diffusioncoefficient
D0
DimerbindingenergyintheTersoffpotentialenergyfunction
De PotentialenergywelldepthoftheMorsepotentialenergy function
D(ω )Phonondensityofstates
e Elementaryelectroncharge(1.60217662 3 10 19 C)
ePot i Potentialenergyoftheatom i
E,Ea Electricfieldintensity
E0 Originalenergyofawavepacket
E Electricfieldstrengthvector
Ei Electricfieldvectorattheposition r i
ΔE Interfacialbondingstrengthofananocomposite
Ea Adsorptionenergy
Activationenergy
Eangle
Eavg
Ebond
Potentialenergyrelatedtothebendingofabondangleinthe ReaxFFforcefield
Averagekineticenergyofagroupofatoms
Potentialenergyassociatedwiththebondingbetweentwo atomsintheReaxFFforcefield
Ecoh
Ecut
ECoul
Ee ðt0 Þ
Egap
EH-bond
Ematrix
Enanofillers
Eover
Erep
Cohesiveenergyoftheinterface
Highestkineticenergypertainingtoanexpansionofawave functionintermsofanelectronicorbitalset
Potentialenergycorrespondingtoelectrostaticinteractions
Phasefactorofthecompletewavefunctionofanatomicsystem
Energydifferencebetweenthelowestunoccupiedorbitaland thehighestoccupiedorbital
Cohesiveenergyduetohydrogenbonding
Energyofthematrixofananocomposite
Energyofthenanofillersofananocomposite
Energypenaltytermthatpreventstheovercoordinationof atomsintheReaxFFforcefield
Potentialenergyofanatominteractingwitharepulsivewall
Est Springpotentialenergyfunction
Especific System-specificenergytermsintheReaxFFforcefield
Esystem
Etot
Etr
Etors
EvdW
TotalenergyofamolecularsystemintheReaxFFforcefield
Totalenergyofthenanocomposite
Transmittedenergyofawavepacket
Potentialenergyrelatedtothetwistofadihedralbondanglein theReaxFFforcefield
PotentialenergycorrespondingtononbondingvanderWaals interactionsintheReaxFFforcefield
f NumberofdegreesoffreedomofanMDsystem
f Resultantatomisticforceactingonanatom
fA(rij)AttractiveatomicinteractionfunctionoftheTersoffpotential
fc(rij)Cutofffunctionfornearest-neighborinteractions
fd Drivingforce
fFVF
fij
Freevolumefraction
Interatomicforceimposedbythe jthatominthewallonthe ith atominthefluid
fi(r ri)Radialdistributionoftheremainingchargeoutsidethecoreof the ithatom
fiw Forceexertedonthe ithfluidparticlebythechannelwall
fp
Forcearisingfromthepressuredifferenceattheboundaryof the“pump”region
Listofsymbols
fR(rij)RepulsiveatomicinteractionsoftheTersoffpotential
fw
fz
Forceexertedonachannelwallbythefluidparticles
Netelectrostaticforceaveragedoverallthewatermolecules
Fi(RN)Forceactingonthe ithparticle
FI
Fij
Fijk
Forceactingonthe Ithcoarse-grainedsite
Forceimposedontheatom i duetoitsinteractionwithatom j
Forceontheatom i duetothethree-bodyinteractionsamong atoms i, j,and k
Fj(ijk)Three-bodyforcetermforthecalculationofheatflux
F[ni(ri)]Functionofthelocalelectrondensity ni(ri)attheatomicsite ri
g Gradientofthepotentialenergy
g(r)Radialdistributionfunction
g(θijk)BondangledistributionfunctionoftheTersoffpotentialenergy function
gref MN ðRÞ
gkMN ðRÞ
Targetradialdistributionfunction
Pairedradialdistributionfunctionfortheeffectivepotential obtainedfromthe kthIBIiterationstep
gμVT MN ðRIJ Þ Radialdistributionfunctioninthe μVTensemble
G Interfacialthermalconductance
G+
Interfacialthermalconductanceofthesystemcorrespondingto theforwardheatfluxdirection
G Interfacialthermalconductanceofthesystemcorrespondingto theoppositeheatfluxdirection
G Metrictensorgivenby G=hhT,with h=(a, b, c)T
GN MN
GkMN ðRÞ
Gref MN ðRÞ
GVMN
Kirkwood-Buffintegralinthethermodynamiclimit
PlateauvalueoftherunningKirkwood Buffintegralafterthe kthIBIiterationstep
PlateauvalueoftherunningKirkwood Buffintegralcalculated fromthereferenceall-atomsystem
Kirkwood Buffintegraloftheradialdistributionfunction betweenspecies M and N
h ReducedPlanckconstant(1.054572 3 10 34 Js 1)
h
Amatrixdefinedbytheedgevectors a, b,and c ofaunitcell
H Separationdistance
Hamiltonianofasimpleone-dimensionalharmonicoscillator
H
Hij
Hessianmatrixofpotentialenergy
Strengthofthestericrepulsionforthetwo-bodytermofthe Vashishtapotentialenergyfunction
He PartialHamiltonian
H2
Interlayerdistancebetweenthetwographenelayersineachelectrode
H(RN , PN)Hamiltonianofanatomicsystem
H2;adi ðp2 ; x2 Þ Hamiltonianofparticle2inanadiabatictwo-particlesystem
J Heatfluxvector
J+
Heatfluxintheforwarddirection(fromsilicenetographene)
J Heatfluxinthereservedirection(fromgraphenetosilicene)
Jx(y,t)Momentumofthefluxdensity
Jμ Componentoftheheatfluxvector J inthe μ direction
k Forceconstantintheharmonicfunction
ks Springconstantofanoscillator
Springconstantofasimpleone-dimensionalharmonicoscillator
k0 Wavenumber
k0 Wavevector
kB
kb i
Boltzmannconstant(1.380649 3 10 23 JK 1)
Material-dependentconstantsthatdeterminetheinteraction strengthofbondstretching
kθ i Material-dependentconstantsthatdeterminetheinteraction strengthofbondanglebending
kχ Forceconstant
K Evaporationconstant
K2 Kineticenergyofparticle2
K(t)Instantaneouskineticenergy
Kfict Fictitiouselectronic“kineticenergy” KnKnudsennumber
Kst
Stochastickineticenergydefinedinthe Bussi Donadio Parrinellothermostat
l LengthoftheMFPofthephonons
l Setofvariationallydeterminedparameters
lb IntrinsiclengthofthephononMFPofthebulkmaterial
li,0
Referencevalueofbondlength
Bondlength
ly Lengthofthesimulationboxinthe y-direction
L LengthofanMDsimulationsystem
L Second-ordertensorormatrixofthedeformationrate
Lp
m
me
Periodlength
Massofasimpleone-dimensionalharmonicoscillator
Massofanelectron(9.109383 3 10 31 kg)
mi Massofthe ithatomorparticle
Massofthe ithnucleus
mw
minx1 V ðx1 ; x2 Þ
M
MI
M N R ðr n Þ
M N P ðpn Þ
M RI ðr n Þ
M PI ðpn Þ
Fictitiousmassofthewall
Thelowestenergystateofthesystem
Massofanucleus
Fictitiousmasshavingtheextradegreeoffreedom Q
Massofthe Ithnucleus
Mappingfunctionofthecoordinatesbetweenthesitesofa coarse-grainedsystemandthecorrespondingatomsina referenceall-atomsystem
Mappingfunctionofthemomentabetweenthesitesofacoarsegrainedsystemandthecorrespondingatomsinareferenceallatomsystem
Mappingfunctionofthepositionofthe Ithsiteinacoarsegrainedsystem
Mappingfunctionofthemomentaofthe Ithsiteinacoarsegrainedsystem
n Empiricalfittingparameter
Localelectrondensity
AdjustingparameterintheTersoffpotentialenergyfunction
ne Phononoccupationnumber
n(r ’)Electrondensitydistribution
nb
nk
nt
Numberofblocksfortheblockaveragemethod
Numberofatomsinthe kthslab
Totalnumberofconfigurationssampledduringtheentire courseofanall-atomMDsimulation
N Numberofatoms
Averagenumberofwatermoleculesinananochannel
Na
NA
Nc
Ne
Numberofatoms
Avogadroconstant(6.022140 3 1023 mol 1)
Numberofatomsinacell
Numberofelectrons
Nf Numberofatomsinthe“pump”region
Ni,nei
Nm
Nn
Numberofneighborsoftheatom i
Numberofmolecules
Numberofnuclei
Ns Numberofatomsontheboxsurface
Nt Totalatomnumber
Ntransfer
Nw
N0
N1
minx1 V ðx1 ; x2 Þ
Totalnumberofvelocityexchangeeventsthatoccurovertime t
Numberofatomsintheconstraintwall
HalfofthetotalnumberofCatomsinagraphenenanosheet
NumberofHatomsononesideofagraphenenanosheet
Minimizedpotentialenergy V ðx1 ; x2 Þ withrespectto x1 fora fixedvalueof x2
p Momentumvectorofanatom
Dipolemomentofawatermolecule
pi
pbo
p elc
p n
pNe
p nul
pp ðpn Þ
prp ðr n ; pn Þ
pr ðr n Þ
Momentumvectorofthe ithatom
Empiricalparameter
Momentumvectorsoftheelectrons
Momentumvectorsoftheatoms
Collectionofthemomentumvectors p1 ; ; pN e noofanMD simulationsystemcontaining Ne electrons
Momentumvectorsofthenuclei
Probabilitydensityoftheatomicmomentaofanall-atom systemdescribedbymomentumvectors pn ¼ p1 ; ; pn
Probabilitydensityofthedynamicstatesofanall-atomsystem describedbypositionvectors r n ¼ r 1 ; ; r n fg andmomentum vectors pn ¼ p1 ; ; pn
Probabilitydensityoftheatomicpositionsofanall-atomsystem describedbypositionvectors r n ¼ r 1 ; ; r n fg
p(t)Instantaneousmomentumofa1Doscillator
p0
Initialmomentumofa1Doscillator
pðθjik ; θ0 jik Þ
FunctionforthebendingofthebondangleoftheVashishta potential
P Pressure
Pα
Padi ðx2 Þ
Padi ðxÞ
Pext
TotalmomentumofanMDsysteminthe α-direction,with α being x, y,or z
Probabilitydistributionfunctionof x2 intheadiabaticlimit
Probabilitydistributionfunctionof x intheadiabaticlimit
Externalpressure
Pf Pressureexertedonthefluctuatingwall
Pin
PNn
PN
PP ðPN Þ
Inletpressure
Collectionofthemomentumvectors P1 ; ...; PNn ofanMD simulationsystemcontaining Nn atoms
GeneralizedspatialmomentaofalltheparticlesinanMD simulationsystem
Probabilitydensityofthesitemomentaofacoarse-grained systemdescribedbymomentumvectors PN ¼ P1 ; ; PN fg
P(q)Normalizedprobabilitydistributionofaspecificdegreeof freedom q
pR ðRN Þ
PRP ðRN ; PN Þ
PR ðRN Þ
Referenceprobabilitydensityofacoarse-grainedsystem determinedbyitscorrespondingall-atomtrajectories
Probabilitydensityofthedynamicstatesofacoarse-grained systemdescribedbypositionvectors RN ¼ R1 ; ; RN fg and momentumvectors PN ¼ P1 ; ; PN fg
Probabilitydensityofthesitepositionsofacoarse-grained systemdescribedbypositionvectors RN ¼ R1 ; ; RN fg
P(t)Instantaneouspressure
P(x)
Pout
Gaussiandistributionofarandomvariable x
Outletpressure
q Chargeofaparticle
qi
Partialchargeofthe ithatom
Q Aphysicalquantity
Chargepossessedbyacoreintheshellmodel
Fictitiousmassfortheadditionaldegreeoffreedomintroduced intheNose ´ Hooverthermostat
FictitiousdegreeoffreedomintroducedintheAndersen barostat
Waterflux
QHP
EstimatedflowratebasedontheHPequation
Q(RN ,PN)AphysicalquantityofanMDsystemdescribedbythe generalizedspatialcoordinates RN andspatialmomenta PN
Q0
Estimationofaquantity Q forabulkmaterial
r Distancebetweentwoatoms
_ r Thefirst-orderderivativeofthepositionvectorwithrespectto time
r € Thesecond-orderderivativeofthepositionvectorwithrespect totime
r0
rcm
rcut
r elc
Distancecorrespondingtotheminimumpotentialenergy
Equilibriumdistance
Equilibriumbondlengths
Positionofthecenterofmass
Cutoffdistanceforapotentialenergyfunction
Positionvectorsoftheelectrons
ri Positionvectorofthe ithatom
r i ðΔtÞ
ri,c
r ij
rij
ri,s
ri,α
Positionofthe ithatomwithoutconsideringthebondconstraint betweentheatoms i and j
Positionofthe ithcorefortheshellmodel
Displacementvectorfromtheatom i toatom j
Distancebetweentheatoms i and j
Positionofthe ithshellfortheshellmodel
Componentof ri inthe α-direction
ri(t)Time-dependentpositionofthe ithatom
rm
EmpiricalparameterfortheLJpotentialfunction
rnei NeighborlistdistancedefinedfortheVerletneighborlist method
r Ne Collectionofthepositionvectors r 1 ; ; r N e ofanMD simulationsystemcontaining Ne electrons
r n Spatialcoordinatesofthe n atomsinthedetailedatomistic system
rn,i Configurationofthe ithall-atomMDsimulationsystem
r nul
r4s
Positionvectorsofthenuclei
Decaylengthofthetwo-bodytermoftheVashishtapotential
rN Radiusoftheinnerboundaryoftheambientregion
Rc Criticaldistanceofanatomwithinwhichitsclosestneighbors caninteractwithit
RI
RIJ
RI
RMSD
RN
ΔRN
RNn
Idealgasconstant(8.314462JK 1 mol 1)
Relativepositionvectorofthe Ithand Jthcoarse-grainedsite
Positionofthe Ithcoarse-grainedsite
Meansquaredisplacement
GeneralizedspatialcoordinatesofalltheparticlesinanMD simulationsystem
Atomicdisplacement
Collectionofthepositionvectors R1 ; ; RNn ofanMD simulationsystemcontaining Nn atoms
RyRydbergconstant(1Ry=13.605693eV)
s Anenhancementfactor
Afictitiousdegreeoffreedom
si Arowvectorwiththecomponents(li ,mi ,and ni )
s(k) Directionvector
Smap r n Averageentropyresultingfromthedegeneraciescausedby configurationmapping
Srel Relativeentropy
SðRNn ; tÞ
Phasefactorforthenuclearwavefunction
t Time
Δt Timestep
Δtmax
ThemaximumvalueoftheAIMDtimestep
tsignal TimeatwhichtheHFACFpossessesamagnitudeequivalentto thatof W
T Temperature
ΔT Temperaturedifference
r T Temperaturegradient
Tc Criticaltemperature
Td
Telec
Highthresholddisplacementenergy
“Fictitious”temperatureoftheelectrons
Tf Tiltfactorthatdenotestheanglebetweenthe xz-planeandthe yz-plane
T 0 f
Initialtiltfactorat t=t0 thatdenotestheanglebetweenthe xzplaneandthe yz-plane
TG
T 0 G
Temperatureofthegraphenemonolayer
Initialtemperatureofthegraphenemonolayer
Ti Totalnuclearchargeofthe ithatom
ΔTIn
TMD
Tnucl
Temperaturedropataninterface
CalculatedtemperaturebasedonMDsimulation
Temperatureofthenuclei
TN m (zk)Instantaneoustemperatureofthe kthslabatthe(N m)thtime step
Tr(.)Traceofamatrix
TS Temperatureofthesilicenemonolayer
T 0 S
Initialtemperatureofthesilicenemonolayer
T(t)Instantaneoustemperature
T ðzk Þ M
@T =@z
Temperatureofthe kthslabaveragedoverthefinal M time stepsofthesimulation
Averagedvalueofthetemperaturegradientinthe z-direction overtime
ui Displacementvectorofthe ithatom
ui Timederivativeofthedisplacementvectoroftheatom i
u(rn)Atomisticinteractionpotentialenergy
Ubond-bending
UBuck
Ucross
UCG
UCoul , Uel
UES
Bondbendingpotentialcomponent
Buckinghampotential
Crosspotentialfunctiontermsforthecouplingsamongbond stretching,bondanglebending,andbondtorsion
EffectivepotentialsoftheCGMDmodel
PotentialenergyduetoCoulombicinteractions
Potentialenergyduetoelectrostaticinteractions
ΔU KBI ðRÞ KBI IBIrampcorrectionterm
UMorse
U 0 MN ðRÞ
Morsepotentialfunction
Initialguessofthenonbondedtwo-bodypotentialofaCGMD model
UMN(R)Nonbondedtwo-bodypotentialofaCGMDmodel
U k MN ðRÞ Effectivepotentialobtainedfromthe kthIBIiterationstep
Unb PotentialenergyduetononbondedvanderWaalinteractionin theMMforcefield
ΔUP ðRÞ Linearpressurecorrectionterm
U(RN)Potentialenergyfunction
U’(RN)Gradientof U(RN)
URigid ðRN Þ PotentialenergyofanMDsimulationsystemwitharigidbond constraintimposed
U SR(rij)Potentialenergyduetotherepulsionoftheelectronshells
Utot Potentialenergyfunction
Uχ
Out-of-planebendinginteractionsthatarecloselyrelatedtothe bondtorsionterm
U2(rij)Two-bodytermoftheVashishtapotential
U3(rij, rik, rjk)Three-bodytermoftheVashishtapotential
v Velocityofanatom
Rotationalfrequency
v Averagevelocityofthewatermoleculesinthenanochannel
vC Velocityofthecenterofmassofalltheparticlesintheheat sourceortheheatsink
vcom Velocityofthecenterofmassoftheatomsinastrip
vi,α Componentofthevelocityofthe ithatominthe α-direction
vi Velocityvectorofthe ithatom
vij
Relativevelocityofthetwopoints(oratoms) i and j
vj(t)Velocityoftheatom j attime t
vmax
Maximumvalueoftheadvectionvelocity
vmax hi Timeaverageofthehighestatomicvelocity
vp Groupvelocityofthephonons
Velocityoftheincidentwave
vs Velocityofthesurfacewaveonthesolid
vx x-componentofthelocalaveragestreamvelocity
V Volumeofasystem
V(x1, x2)Potentialofasimpleone-dimensionalchainmodelconsisting thetwoparticles1and2
V(r)Electron nucleiinteractionpotentialenergy
V(rij)Two-bodypotentialthataccountsfortheinteractionsamongthe nuclei
V2(rij)Two-bodypotentialenergyfunction
V3(ri, ri, rk)Three-bodypotentialenergyfunction
V E e ðRNn Þ
Effectivepotentialenergycontributedbytheelectrons
Vfree Totalfreevolume
Voccupy
Totalvolumeoccupiedbythenanofillersandthematrixofthe composite
Vvacuum Sumofthefreevolumesthatexceedacriticalsize
VH(r)Hartreepotentialenergy
Vm Volumeofthecomposite
Vne ðr Ne ; RNn Þ
Operatorforthetotalpotentialenergyofthenucleiand electrons
Vn Energybarriersforbondtorsion
VREBO
REBOtermoftheAIREBOpotentialenergyfunction
VLJ Lennard Jonespotentialenergyfunction
VTorsion
Vw
PotentialenergycontributedbybondrotationsfortheAIREBO potentialenergyfunction
Potentialenergycontributedbytheinteractionbetweenthe ith particleandthewall
W(τ )Amplitudeoftheautocorrelationfunction
wA Waveamplitudeofawavepacket
x0
x1;adi ðx1 ð
Initialpositionofa1Doscillator
Valueof x1 attime t when x1 evolvesadiabaticallyunderthe initialconditionsof x1(0)and v1(0)with x2 beingfixedatits initialvalue
x(t)Instantaneouspositionofa1Doscillatorattime t
xi(t0)Coordinateofatom i at t=t0 inthe x-direction
yi y-componentofthepositionofthecenterofmassofwater molecule
yi(t0)Coordinateoftheatom i at t=t0 inthe y-direction
Z Effectiveconfigurationpartitionfunction
ZI Atomicnumberofthe Ithnucleus
Z1 ðx2 ; β 1 Þ Effectiveconfigurationpartitionfunctionof x2
Zi,eff
Effectivenuclearchargeofthe ithion
zi Coordinateofthe ithatominthedirectionofheattransport
zk Positionofthecenterofthe kthslabofatoms
z0 Centerofthewavepacket
zðr n;i Þ
Partitionfunctionsofthe ithall-atomconfiguration r n;i
zðRN ;i Þ
Listofsymbols
Partitionfunctionsofthe ithcoarse-grainedconfiguration RN ;i
αi Electronicpolarizabilityofthe ithion
Anglesatthecoreofthenanowireforthe ithwedge
ρ Amaterial-dependentconstant
ρ y; t
Massdensityattheposition y andtime t
ρ(RN , PN)ProbabilitydensitydistributionofthephasesofanMDsystem describedbythegeneralizedspatialcoordinatesRN andspatial momentaPN
ρi
Electrondensityatthesiteofthe ithatom
ρij(rij)Electrondensityofthe jthatomatthesiteofthe ithatom
ρN
ρN
NumberdensityoftheatomsinanMDsystem
Numberdensityofcoarse-grainedparticlesofthe N type
θ Generalangle
Misorientationangle
Rayleighangle
θD
Debyetemperatureofthesystem
θi Bondangle
θi,0
θijk
θ0 jik
θ hi
Referencevalueof θi
Anglebetweentwosuccessivebondvectors
Equilibriumbondangle
Time-averageddipoleorientationangleofthewatermolecules
ε Depthofthepotentialenergywell
ScalingfactorintheSWpotentialenergyfunction
Effectivespringconstantofasimpleharmonicspringpotential energyfunction
Rectificationratio
εAB PotentialenergywelldepthoftheLennard Jonespotentialfor thetwoatomtypesAandB
εcoh
Cohesiveenergydensity
εi Eigenvaluesoftheunoccupiedorbitals
εPot i Potentialenergytermofthesiteenergy
εi(k0)Polarizationparameter
εi(t)Siteenergyofthe ithatomattime t
εfw Potentialenergywell(orbindingenergy)oftheLJpotential
εj Eigenvaluesoftheoccupiedorbitals
_ εi Atomicstrainratetensor
εxy Shearstrainrate
ε0
Δε
Vacuumdielectricconstant(8.854 3 10 12 Fm 1)
Amountofheatinputintotheheatsource
ξ Widthofthewavepacket
ξ i Material-dependentconstant
NoisetermintheLangevinequationthatimposesastochastic forceonthe ithatom
ζ EffectivecoordinationnumberoftheatomsfortheTersoff potentialenergyfunction
Thermalrelaxationtime
Velocityrescalingfactor
ζ ij Effectivecoordinationnumberoftheatoms
ς i ScaledcoordinatevectorfortheAndersenbarostat
π i Momentumthatconjugatesthescaledcoordinatevectorforthe Andersenbarostat
Π Momentumthatconjugatesthedegreeoffreedom Q inthe Andersenbarostat
ηi Coefficientthatdeterminesthemagnitudeofthestochasticforce imposedonthe ithatomfortheLangevinthermostat
ηij ExponentofthestericrepulsionfortheVashishtapotential energyfunction
μ MeanofaGaussiandistribution
ReducedmassoftheatomfortheMorsepotential
Fictitiousmassparameter(intheunitofeVs2)
υ Constantthatdenotesthenonzerocollisionfrequencyforthe Andersenthermostat
δ αβ Kroneckerdelta
σ Criticalseparationdistanceforanegativepotentialenergy
StandarddeviationofaGaussiandistribution
Zero-crossingdistancesoftheLJpotential
σ AB
Zero-crossingdistanceoftheLennard Jonespotentialforthe twoatomtypesAandB
σ 2 VarianceoftheGaussiandistribution
σ virial(r)Virialstressofavolumeelement
λ Positivecoefficientforthesteepestdescentmethod
Thermalconductivity
Time-dependentLagrangemultiplier
λ+
Thermalconductivityintheforwarddirection
λ Thermalconductivityintheoppositedirection
λμv Componentofthethermalconductivitytensor λ
λ Thermalconductivitytensor
λGE
λH
Thermalconductivityofthegraphenesheet
Thermalconductivityofthehybridgraphene/silicene monolayer
λk Lagrangemultipliersforbondconstraints
λ(k)
λSE
Acoefficient
Thermalconductivityofthesilicenesheet
λ0 Predefinedvalueof λ(k)
λ3 CutoffparameterintheTersoffpotentialenergyfunction
λmax N
Upperlimitofbulkthermalconductivity
Λ deBrogliethermalwavelength
Λij
Lagrangemultipliersintroducedtodynamicallysatisfythe condition ψi jψj DE ¼ δ ij
γ FittingconstantfortheSWpotentialenergyfunction
γ i Positiveatomicfrictioncoefficient
φ Ageneralangle
φ0 Torsionanglesatpointsofminimumpotentialenergy
ω Vibrationalfrequencyofabond
Frequencyofmotionofaparticle
Strengthofawedgedisclination
Angularfrequency
Phononfrequency
ωc Criticaldisclinationstrength
ωD
Debyefrequencythatcorrespondstothehighestphonon frequencyallowedintheDebyemodel
ωe Vibrationalfrequencyoftheelectrons
ωmax
e Highestpossibleelectronicfrequency
ωmin
e Lowestpossibleelectronicfrequency
ω n Vibrationfrequencyofthenuclei
ω max n Highestphononmodefrequency
ΞðRNn Þ Wavefunctionofthenucleiinasystem
Φðr Ne ;RNn Þ Wavefunctionoftheelectronsinasystem
Φðr Ne ; RNn ; tÞ Time-dependentwavefunctionthatdependsonboththe nuclearandelectronicdegreesoffreedom
Φ(ri rj)Puretwo-bodypotentialformodelingnuclei
Φi Rc Setofatomswithinasphereofradius Rc withtheatom i atits center
ΨðRNn ; r Ne Þ Totalwavefunctionofanatomicsystem
Ψðr Ne ; tÞ Time-dependentwavefunctionoftheelectronsinasystem
χðRNn ; tÞ Time-dependentwavefunctionofthenucleiinasystem
Ψ Wavefunction
Ψ i One-particleorbitalsubjectedtotheorthonormalconstraint
Ψ 0 Potentialenergyofelectronsatthegroundstate
Ψ 1 Potentialenergyofelectronsatthefirstexcitedstate
α Electronicpolarizabilityofanion
β AdjustingparameterintheTersoffpotentialenergyfunction Isothermalcompressibilityofamaterial
χ Dihedralangle Nuclearwavefunction
χ2 Objectiveresidualfunction
τ b Timestepsfortheblockaveragemethod Intrinsicrelaxationtimeofthecorrespondingsingle-crystalbulk material
τ g Relaxationtimeundertheeffectofgrainboundaryscattering
τ s Relaxationtimeundertheeffectofsurfacescattering
τ im Relaxationtimeinthepresenceofimpurities
τ m UpperintegrallimitintheGreen Kuboexpression
τ m;max Largestnumericallymeaningfulupperlimitfortheintegralin theGreen Kuboexpression
τ P Couplingparameterthatdetermineshowcloselyasystemis coupledwithabarostat
τ T CouplingparameteroftheBerendsenthermostat
Δτ Timeintervalbetweentwosuccessivecollisionsforagiven atom
Σ Reciprocalcoincidencesitedensity
γ AtomicfrictioncoefficientoftheLangevinthermostat Interfacialadhesionenergy
hi Averageofaphysicalquantityoverallthepossiblephasesofan MDsystem
j hi Innerproductoftwofunctions
Ω Avolumeelement
Systemvolume
Ω i Volumesurroundingtheatom i
Ωmap ðM N R ðr n;i ÞÞ
Degreeofdegeneracyof RN ;i
Γ Transmissioncoefficient
Dyadicproductoftwovectors
