PSI - Issue 32

O.B. Naimark et al. / Procedia Structural Integrity 32 (2021) 144–151 Author name / Structural Integrity Procedia 00 (2019) 000–000

145

2

thermodynamic properties [Baggioli et al. (2020)]. The GMS emerge in the Maxwell–Frenkel approach to liquid viscoelasticity as the gaps in the dispersion relations (DR) leading to continuously change from the energy to momentum space. DR has been emerging in different areas of physics as the gap in momentum to study the interplay between propagation and dissipation effects. Seminal idea by Frenkel (1947) was that liquid particles oscillate as in solids for some time and then diffusively move to neighboring quasi-equilibrium positions. It was introduced the average time τ between diffusive jumps and predicted the existence in liquids shear modes at time shorter than τ , or above the Frenkel frequency F ω : F F τ ω ω 1 > = . Frenkel qualified the solid state as a correct starting point of liquid description and proposed the definition of viscosity using the operator d dt A F τ = + 1 , where G F η τ = is the relaxation time in the Maxwell equation

G η σ ε 1 = +

d

d

σ

,

(1)

dt

dt

where ε is the shear strain, σ is the shear stress. It follows from (1) that the hydrodynamic viscous flow is equivalent to the substitution of G by the operator ( ) 1 1 − = − M G A in the momentum conservation law that has the form of the Navier–Stokes equation for the velocity v

2

2

v

v

t v

∂

∂ ∂

∂ ∂

,

(2)

η

ρτ

ρ

+

=

F

2

2

x

t

∂

(

( i k x

) ) t

the dispersion equation has the form

Representing the solution (2) as

= exp 0

v v

ω −

0 F 2 + − = i c k G τ ω ω leading to the complex frequency 2 2

1

i

.

(3)

2 2

c k

ω

= − ±

−

2 F

2

4

τ

τ

F

( ) 2 F τ

( ) τ 1 2 > ck has the real part of

2 2 1 4 −

c k

ω

=

It follows from (3) that for

and the solution of (2) is

(

) ( ) . exp i t ω

The gap in k -space emerges in the liquid transverse spectrum for

g k k > , where

exp 2

v

t

=∝ −

τ

F

( ) F 1 2 τ c k g = .The k -gap characterizes the propagating shear modes. The value of k g is interpreted as an order parameter quantifying the rigidity transition. Microscopically, the gap in k -space can be related to a finite propagation length of shear waves in a liquid: F τ с gives the shear wave propagation length, or liquid elasticity length el d : F τ d c el = . This definition of the k -gap means that a liquid can be considered as ensemble of dynamical regions of characteristic size F τ c , where the solid-like response appears as the shear waves with the wavelength λ . The propagation range of shear waves d in solid-like elastic regime, F d τω λ ≈ ⋅ , where 1 > F τω . It follows from 1 > F τω . The gap develops in k -space but not in the frequency space. It is the consequence of a difference between a local nature of a relaxation event (particle jump envisaged by Frenkel originally) and an extended character of a wave. In liquids a propagating wave does not require all particles to be solid-like. Instead, the wave propagation is affected only by particles jumping at a distance , where the wave front has reached, disrupting the wave continuity and dissipating the wave. This proposal corresponds to the Frenkel definition of a liquid state as a state that retains the structural features of a solid state and, as a consequence, a corresponding set of internal (thermodynamic) variables. These statements are confirmed by Bazaron et al. (1990) by measurements of relaxation spectra in a shear simple liquid flow accompanying the superposition of shear oscillations in a liquid at frequency 10 5 Hz , when the existence of F d τω λ ≈ ⋅ that the propagation length is much longer than the wavelength λ in the regime