16.2 Elasticity 309
16.2.5 Microscopic Expressions for Elasticity Coefficients.
Consider a system ofNparticles, located at the positionsri,i= 1 , 2 ,...Nwithin
a volumeV. The interaction potential is denoted byΦ=Φ(r^1 ,r^2 ,...,rN). When
the potential is pairwise additive, one hasΦ =
∑
i<jφ
ij=∑
i<jφ(r
ij), where
rij=ri−rj, andφ=φ(r)is the binary interaction potential. In thermal equilibrium,
at the temperatureT, the configurational part of the free energyFpotis given by
βFpot=−lnZpot, Zpot=
∫
exp[−βΦ]dr{N},β=
1
kbT
,
whereZpotis the configurational partition integral,dr{N}is the 3N-dimensional
volume element. The difference between the free energy, where the position vectors
are displaced according torνi→rνi+rμiuμνand the original free energy,δFpot=
Vppotμνuμνyields the expression
Vppotμν=
〈
Φμν
〉
,Φμν=
∑
i
rμi∂νiΦ=−
∑
i
rμIFνi,∂νi=
∂
∂rνi
, (16.29)
for the potential contribution to the pressure tensor. HereFνiis the force acting on
particlei. In the absence of external forces, the total force vanishes:
∑
iF
i
ν=0. The
bracket〈···〉indicates the configurational canonical average
〈...〉=Zpot−^1
∫
...exp[−βΦ]dr{N}.
The change of the pressure tensor (16.29) under a deformationrjλ→rλj+rκjuκλ
yields a relation between the potential contributionσμνof the stress tensor and
the deformation tensor, and consequently a microscopic expression for the elastic
moduli. Starting from
σμν=−
(
p
pot,def
μν −p
pot, 0
μν
)
=−δp
pot
μν=δ
(
V−^1 〈Φμν〉
)
,
whereppotμν,defandppotμν,^0 are the pressure tensors in the strained and in the unstrained
states, and usingσμν=Gμν,λκuλκ, one obtains
VGμν,λκ=〈Φμν,λκ〉 0 +Vppotδμν,δλκ−β
(
〈ΦμνΦλκ〉 0 −〈Φμν〉 0 〈Φλκ〉 0
)
.
(16.30)
The subscript 0 in〈...〉 0 indicates the average in the undeformed state. The first term
viz.
Φμν,λκ=
∑
j
∑
i
rμi∂νirλj∂κjΦ (16.31)