310 16 Constitutive Relations
represents the deformation-induced variation ofΦμν. The second term on the right
hand side of (16.30) stems from the volume changeδV, sinceδV/V =uλλand
−〈Φμν〉 0 =Vppotδμν, whereppotis the potential contribution to the pressureP.
For pairwise additive interaction, (16.31) reduces to
Φμν,λκ=∑
i<jφijμν,λκ,φμν,λκij =φμν,λκ(rij), φμν,λκ(r)=rμ∂νrλ∂κφ(r).(16.32)
As before, the abbreviationrij=ri−rjis used, andφ(r)is the pair potential. For
spherical particles whereφ=φ(r), withr=|r|, holds true, one has
φμν=rμrνφ′,φμν,λκ(r)=(rμrκδνλ+rνrλδμκ)r−^1 φ′+rμrνrλrκr−^1 (r−^1 φ′)′.
(16.33)The prime denotes the differentiation with respect tor.
TheBorn-Greenexpression for the elastic modulus tensor [100, 101], corresponds
to the first and second terms of (16.30), when the total interaction potential is the sum
of pair potentials. The remaining terms with the factorβare referred to asfluctuation
contributions. In solids, the fluctuation parts of the elastic properties are small at low
temperatures [102]. They are of crucial importance, however, for the fact that the
low frequency shear modulusof a liquid vanishes whereas it has a finite value for a
solid [103]. This underlies the fundamental difference in the mechanical behavior of
a solid and a liquid.
The Born-Green part of the orientationaly averaged shear modulus is
GBG=
1
15 V
〈
∑
i>j(
r^3 (r−^1 φ′)′)ij〉0−ppot=1
15 V
〈
∑
i>j(
r−^2 (r^4 φ′)′)ij〉0.
(16.34)
The shear modulusGBGcan be expressed in terms of an integral over the pair
correlation functiongaccording to
GBG=
1
30
n^2 kBT∫
r−^2 (r^4 φ′)′g(r)d^3 r. (16.35)Heren=N/Vis the number density. The quantityGBGis also calledhigh frequency
shear modulus. It is not only well defined in solid state, but also for liquids, where it
reflects a rigidity, observable on a short time scale only. The pair correlation function
can be written asg(r)=χ(r)exp[−βφ(r)]. The quantityχapproaches 1 for small
densities, corresponding to a dilute gas. Notice thatGBGis not zero even in this limit.
The total shear modulusG=GBG+Gfluct, on the other hand, vanishes in the liquid
state and the more in a gas. This cancellation ofGBGby the fluctuation partGfluctis