Theory-Group

Annual Report 2006

Non-Newtonian Complex Plasma Fluids

A. Ivlev, R. Kompaneets, H. Höfner, H. Thomas, C. Räth, G. Morfill

Rheological and structural properties of complex plasma fluids were studied. First, the shear viscosity was measured using the PK-4 setup. The shear flow was induced either by an inhomogeneous gas flow or with laser radiation. Complex plasmas exhibited substantial shear-thinning and shear-thickening in a broad range of shear rates. Second, the formation of a string fluid phase in electrorheological complex plasmas was investigated, by applying an external AC electric field and measuring the induced structural anisotropy. The observed isotropic-to-string phase transition is believed to be of second order, and thus is a good candidate to study kinetics of critical phenomena.

One of the remarkable features of complex plasmas is that although they are intrinsically multiphase systems, the rate of momentum exchange through collisions between the charged microparticles can significantly exceed the coupling to the background neutral gas. Therefore fluid complex plasmas can act as an essentially single-fluid system. This opens up an unique opportunity to investigate generic phenomena occurring in strongly coupled media at the kinetic level, and to get insights into microscopic properties of hydrodynamics as well as thermodynamics of fluids, which is considered as one of the outstanding problems in fundamental physics.

Fluids exhibit a rich variety of rheological properties. Along with conventional Newtonian fluids (e.g., air, water, gasoline) that have constant viscosity, there exists a broad class of non-Newtonian fluids with their rheology strongly depending on the flow conditions (e.g., fluids with shear-thinning and shear-thickening effects, viscoelastic fluids). Classic examples encountered in everyday life include paint, ketchup, gelatine, etc. Other fluids such as molten polymers and slurries are of considerable technological importance.

Another remarkable class of non-Newtonian fluids are the so-called electro- or magnetorheological (ER/MR) fluids. The interparticle interaction in such fluids and, hence, their rheology is determined by external electromagnetic fields: At low fields they may be ''normal'' fluids, but above a critical field, at low shear stresses ER/MR fluids behave like solids, and at stresses, greater than a ''yield stress'' they flow with enhanced viscosity. ER/ MR fluids have potential use in industrial applications as, e.g., hydraulic valves, clutches, brakes. Moreover, for colloidal ER/MR fluids, the possibility to tune the crystal structure of these suspensions by an external field makes them appealing for display production and photonic applications.

Our recent experiments demonstrated that complex plasmas are very well suited to study the kinetics of ''microscopic'' processes that govern the non-Newtonian behavior. We investigated shear flows in complex plasma fluids (by applying stresses of different magnitudes and measuring the dependence of the viscosity on the local shear rate), and studied the formation of a string fluid phase in ER complex plasmas (by applying an external ac electric field and measuring the induced structural anisotropy).

Fig. 1: Qualitative dependence of the viscosity ν and the shear stress σ = ν γ on the velocity shear rate γ. Three ranges of γ are indicated, corresponding to the Newtonian (constant-viscosity, I), shear-thinning (II), and shear-thickening (III) regimes. The ν(γ) dependence may have an anomalous N-shaped profile (dashed line).

We developed a theoretical model of the non-Newtonian viscosity of complex plasmas. The model is based on the assumption of local balance between viscous heat deposition and neutral friction sink, which yields a local relation between the shear rate γ and the kinetic temperature T of the microparticles. Along with the known dependence of the kinematic viscosity ν on T, this gives a parametric dependence for ν(γ). According to the model, there are three distinct regimes shown schematically in Fig. 1: If γ is small enough, the temperature remains constant and the viscosity does not depend on γ, the shear stress σ = ν γ has a linear scaling (regime I). At larger γ, the temperature starts increasing and the behavior becomes non-Newtonian: ν(γ) falls off showing shear-thinning (regime II). Note that in some cases the behavior can become anomalous, dσ/dγ < 0 (dashed line). As γ grows further, ν(γ) and hence σ(γ) increase again (shear-thickening, regime III). The transition from regime II to III is due to the interplay between the so-called 'potential' and 'kinetic' parts in the ν(T) dependence, determining the viscosity at low and high temperatures, respectively.

We performed a series of shear flow experiments using the PK-4 prototype setup (Fig. 2, right side). The particles formed a very elongated 3D cloud (of length 3 - 10 cm and diameter of 4 - 6 mm) that was oriented along the axis of the discharge tube. The shear flow (along the axis) was induced by two different methods: (i) we used the gas flow in the tube (the velocity of the gas has a parabolic profile and hence exerts shear stresses due to coupling to microparticles), or (ii) applied laser radiation (along the tube axis). The gas-induced flow turned out to have an one-dimensional plane topology with zero net flux (resembling the Couette flow between parallel plates), the laser- induced flow, on the other hand, had cylindrical symmetry (similar to the Poiseuille flow in a tube).

We performed a two-parametric fit of the measured velocity profile with solutions of the Navier-Stokes equation for (i) our non-Newtonian viscosity model and (ii) the conventional constant- viscosity case. The non-Newtonian viscosity yields an extended region of quasi-linear profile and thus provides good agreement. In contrast, for the constant-viscosity model the curve is of fairly different shape, which allows a qualitative distinction from the non-Newtonian case. Quantitatively, the distinction becomes evident by chi square-fitting. The theoretical fit allows us to retrieve an explicit dependence of the viscosity on γ with well- pronounced shear-thinning and -thickening.

For the laser-induced flows, the cylindrically symmetric velocity profiles were measured for different values of the laser power P and normalized to the maximum velocity at the center v_max, as shown in Fig. 3. Qualitatively one can see that the normalized profiles seem to be similar for different P, and v_max grows linearly with P. This, along with the assumption that the laser force on a particle is proportional to P, suggests that - if hydrodynamics still works  the effective viscosity should be constant. Such a hypothesis is well confirmed by the corresponding fit shown in Fig. 3.

Fig. 3: Experimental velocity profiles (symbols) of the laser-induced shear flow measured for different laser power P. The profiles are normalized by the maximum velocity at the center of the beam v_max. The beam boundaries are indicated by the vertical dashed lines. The solid line represents the theoretical fit (for P = 16.5 W) with constant viscosity. The inset shows the dependence of v_max on P.

The range of shear rates achieved in our experiments with the gas-induced flow was sufficiently broad to reveal a non-Newtonian behavior of 3D complex plasmas, accompanied by substantial shear-thinning and shear- thickening effects. At the same time, the shear was small enough to expect hydrodynamics to be applicable. In contrast, the inhomogeneity of the flow induced by the laser was very high  the velocity changed significantly at the scale of the interparticle distance. Thus, a combination of both methods of the flow generation made it possible to cover the entire range of γ up to the edge where complex plasmas cannot be considered as a continuous medium. Nevertheless, the formal hydrodynamic description of ''extreme'' laser-induced shear flows gives reasonable results. Apparently, the transport properties in this case are no longer local and in equilibrium, and further investigations of the mechanisms governing the momentum exchange in such flows would be highly desirable.

The derived magnitude of the kinematic viscosity in the Newtonian regime is ~100 mm²/s, which is about the viscosity of, e.g., glycerin. The observed shear-thinning, however, diminishes the viscosity by factor of 10, making it close to the viscosity of air at atmospheric pressure. Hence, in terms of viscosity the fluid complex plasmas are very similar to ordinary classic fluids.

Electrorheological complex plasmas were investigated with the plasma crystal experiment PK-3 Plus under microgravity conditions on board the ISS. We employed a modulation with an external electric field at 100 Hz (well above the eigenfrequency of the microparticles) that caused a periodic polarization of the plasma cloud around each microparticle, so that the resulting interaction between the particles was identical to that in conventional dipolar (Stockmayer) fluids. Such electrorheological fluids are famous because of their exceptionally diversified phase diagram that includes a number of second-order structural phase transitions. Thus, electrorheological complex plasmas may allow us to investigate at the kinetic level critical phenomena accompanying such transitions.

The ''string'' formation in gaseous ER complex plasmas is one of the candidates for the second-order phase transition: Theory predicts that for a given thermodynamic state there exists a critical amplitude of the electric field above which the isotropic gaseous phase loses the thermodynamic stability and particles form lines oriented along the field. Experiments seem to confirm the theory: At low fields the microparticles form an isotropic gaseous-like phase shown in Fig. 4a, whereas above a certain threshold an ordering of the particles along the field emerges. At sufficiently large fields the string fluid can be observed by naked eye - vertically, particles form clear strings (see Fig. 4b), but horizontally there is no apparent order. The magnitude of the critical field for the transition is in good agreement with the theoretical prediction.

Fig. 4: Isotropic-to-string phase transition observed in experiments with electrorheological complex plasmas. The dipolar interaction between microparticles is induced by an external vertical AC electric field with a tunable amplitude at 100 Hz. Particles are visualized by a vertical laser sheet of about 200 µm thickness. At small fields the particles form an isotropic gas-like phase (a), at larger fields the ''string fluid'' phase with vertical order emerges (b).

Further kinetic investigations of 3D fluid complex plasmas will require the development of a convenient technique for individual particle tracking. In particular, such technique is essential for reliable temperature measurements in the fluid phase, and its development will be one of the main challenges for our future work.