Periodic shear dynamic viscosity

The complex viscosity is called the dynamic viscosity, and the loss of energy viscosity is the real viscosity, while the in-phase or elastic component is the imaginary 

In everything we have done so far, we have drawn many graphs of modu-lus/loss against temperature/time/frequency, but nowhere did we have a specific equation for these curves. We shall now derive formulae for compliance and loss against time and frequency.The equations for temperature are more complex, but can be obtained quite simply from those for frequency by applying a frequency-temperature relationship such as the Arrhenius equation. 

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Oscillatory shear experiments using, for example, cone-and-plate devices constitute the third main group of viscometric techniques. These techniques enable the complex dynamic viscosity rj ) to be measured as a function of the angular velocity (cu). The fundamental equations are presented in section 6.2 (eqs (6.22H6.27)). Another arrangement is two rotating parallel excentric discs by which the melt is subjected to periodic sinusoidal deformation. 

A few theoretical and computational studies have already addressed in some detail the problem of viscosity in ILs.[136] However, a complete microscopic theory of viscosity is currently not available. It is a challenging task to accurately compute the viscosity of a complex system by means of simulation methods. For a system with high viscosity, it is extremely difficult to reach the hydrodynamic limit (zero wave number) where the experimental data is observed. This is because, in order to reach this limit, a very large simulation box is required. Traditional simulation methods normally used for shear viscosity of fluids fall into two categories (a) the evaluation of the transverse-current autocorrelation function (TCAC) through equilibrium molecular dynamics (HMD) trajectories and (b) non-equilibrium molecular dynamics (NEMD) simulations that impose a periodic perturbation. [137] In recent work, Hess[138] compared most of the above methods by performing simulations of Lermard-Jones and water system. They concluded that the NEMD method using a periodic shear perturbation can be the best option. 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

Real world materials are not simple liquids or solids but are complex systems that can exhibit both liquid-like and solid-like behavior. This mixed response is known as viscoelasticity. Often the apparent dominance of elasticity or viscosity in a sample will be affected by the temperature or the time period of testing. Flow tests can derive viscosity values for complex fluids, but they shed light upon an elastic response only if a measure is made of normal stresses generated during shear. Creep tests can derive the contribution of elasticity in a sample response, and such tests are used in conjunction with dynamic testing to quantity viscoelastic behavior. 

This section draws heavily from two good books Colloidal Dispersions by Russel, Seville, and Schowalter [31] and Colloidal Hydrodynamics by Van de Ven [32] and a review paper by Jeffiey and Acrivos [33]. Concentrated suspensions exhibit rheological behavior which are time dependent. Time dependent rheological behavior is called thixotropy. This is because a particular shear rate creates a dynamic structure that is different than the structure of a suspension at rest. If a particular shear rate is imposed for a long period of time, a steady state stress can be measured, as shown in Figure 12.10 [34]. The time constant for structure reorganization is several times the shear rate, y, in flow reversal experiments [34] and depends on the volume fraction of solids. The viscosities discussed in Sections 12.42.2 to 12.42.9 are always the steady shear viscosity and not the transient ones. 

The evidence which points to the role of extensional viscosity is partly direct and partly circumstantial. In the case of laminar flows, the role is obvious in flows through porous media or around small bodies, regions of high extensional deformation are easily identified and strain rates in these zones are consistent with the rates necessary for high extensional viscosity. In turbulent flows altered by polymer addition, principally boundary layers and jets, the regions most affected by the polymer are zones which are dominated by shear but periodically subjected to significant extensional motion hence extensional viscosity is linked with the effect. However, in these turbulent flows, simultaneous measurements of the hydro-dynamic effect and the fluid property have not yet been made and thus a direct relationship has not been established. 

In the method of nonequilibrium molecular dynamics (NEMD), transport processes are usually driven by boundary conditions. For example, the calculation of shear viscosity is based on the Lees-Edwards flow-adapted sliding brick periodic boundary conditions (PBCs) (Panel 4 or their equivalent Lagrangian-rhomboid... 

An alternative NEMD method has been developed that is much simpler to implement than is the SLLOD method, particularly for charged systems such as ionic liquids. The method is called reverse nonequilibrium molecular dynamics (RNEMD) and was first developed as a means for computing thermal conductivity but has also been applied to viscosity. It differs from conventional equilibrium and nonequilibrium methods where the cause is an imposed shear rate and the measured effect is a momentum flux/stress. RNEMD does the opposite it imposes the difficult to compute quantity (the momentum flux or stress) and measures the easy to compute property (the shear rate or velocity profile). The method is very simple to implement because it only requires periodic swapping of momenta between atoms at different positions in the box. These swaps set up a velocity profile in the system (i.e., a shear rate). By tracking the frequency and amount of momentum... 

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