1.10.1 Next-Nearest Neighbors and the Approach to Isostaticity
Note that for all of the lattices we are considering, adding the NNN springs will sta- bilize them into a solid, providing a finite energy for all of the possible relative particle displacements. This can be seen from the Maxwell counting argument as well as from
direct calculations.
Hyper-isostatic lattices are stable with NN springs, by definition. However, for lattices at or below isostaticity, soft deformations are stabilized by adding NNN springs. Thus, we take all NN springs to have the spring constant k, while NNN springs have the spring constant k′. Importantly, as k′/k → 0, we recover back the soft modes, so that this limit allows us to continuously approach the isostatic transition. Note that for generic springs, a hyper-isostatic system will be under a state of self-stress. However, instead we pick all the spring rest lengths so that the springs are at rest without external stresses.
There are several other means of approaching the isostatic transition by introducing additional terms in the Hamiltonian, thus adding more constraints to the total degrees of freedom of the system.
1.10.2 States of Stress
One means of stabilizing the soft modes of an isostatic system is to put it under uniform tension, i.e. a negative pressure. For a stable isotropic solid, to lowest order the pressure
simply modifies the elastic moduli as λ → λ+p and µ → µ −p. Extending this to
lattice models, we must linearize the spring potential in Eq. 1.30 around some spring length|r◦j−r◦i| ≡Rijo 6=aij. While the Hamiltonian of unstressed springs in Eq. 1.31 was
proportional toV′′= ∂2∂RV(oR2o) =kij, here a term proportional toV′=
∂V(Ro) ∂Ro =kij(Roij−aij) is added, ∆H= 1 2 V′(Roij) Roij (ui−uj)·(I− (r◦j −r◦i) Roij ⊗ r◦j −r◦i) Roij )·(ui−uj) (1.41)
However, this expresses the Hamiltonian in terms of the compression ratio of the springs,
Rijo = Λaij, while we want to know the relation to the pressure p. To do so, we solve the
equation of force balance on each spring to obtain
p= 1 2Λd−1v 0 X hi,ji V′(Roij)Roij (1.42)
Note that it is possible to generate unstable lattices which resist uniform pressure. These lattices would have soft deformations, but a finite bulk modulus. This must arise due to local force balance, as seen in the honeycomb, square and kagome lattices.
1.10.3 Bending Stiffness
Besides NNN bonds and pressure terms, we could imagine stepping away from isostatic- ity by breaking the isotropic nature of particle interactions. As already mentioned, this seems most relevant for the case of semi-flexible polymer gels, where the bending rigidity of the rod-like polymers gives more force constraints at each cross-link.
Besides the radial force terms, there would be a force which restores bond angles to their initial values. Another possibility for lattices with inversion symmetry around some site is to consider the angle between opposite bonds. Similarly to the terms already mentioned, we can consider quadratic terms arising from the deformation of the bond angles.
We know that
cosθij,ik=
(rj −ri)·(rk−ri) |rj−ri||rk−ri|
=eij ·eik (1.43)
where eij = (|rrjj−−rrii)| is the unit vector along the bond connecting particles i and j. Since
θij,ik will have a preferred valuesθ◦, we can expand around it in powers ofδθij,ik and u on
the left and right sides. For the left side,
cosθij,ik= cosθ◦−sinθ◦δθ− 1
2cosθ◦δθ
2 (1.44)
Note that the last term is only relevant if sinθ◦ = 0, which for a lattice is equivalent to
θ◦=π. Here, cosθ◦ = (r ◦ j−r◦i)·(r◦k−r◦i) R◦ ijR◦ik =e ◦ ij ·e◦ik.
The right side of Eq. 1.43 can be expanded in the displacementsu. We restrict ourselves to two dimensions and define
δukij = 1 R◦ ij e◦ij ·(uj−ui) (1.45) δu⊥ij = 1 R◦ ij fij◦ ·(uj−ui) (1.46)
wheref =e⊥ is the unique 2D vector such that|f|= 1, f·e= 0 ande×f >0. The lowest order terms are
−sinθ◦δθ=e◦ik·δu⊥ij+e◦ij ·δu⊥ik (1.47) Note that forθ◦=π these terms vanish, and we must expand to next order. For this angle we can use the identityeik=−eij to simply the expression,
1 2δθ 2= 1 2(δu ⊥ ij +δu⊥ik)2 (1.48)
Overall, the energy contribution from the bending of the angles becomes
∆H= b
2
X
θij,ik
(δθij,ik)2 (1.49)
These terms have some similarity to the terms of the formf⊗f =I−e⊗efrom adding a uniform tension to the system, and serve the same stabilizing role. Thus, in addition to introducing NNN bonds and taking the spring constant k′ →0+, we can instead make use of these terms, taking p→0− and b→0+. These different means of stabilization allow us to expand the range of the models to particles with different interactions, as well as note universality of certain commonly observed features.