Flow behavior of rod-like viruses in the nematic phase
M. P. Lettinga and J. K. G. Dhont
IFF, Institut Weiche Materie, Forschungszentrum J¨ulich,D-52425 J¨ulich, Germany Z. Dogic
Rowland Institute at Harvard, Cambridge, Massachusetts H. Wang
Department of physics, North Dakota state university, Fargo, North Dakota J. Vermant
Department of Chemical Engineering, Katholieke Universiteit Leuven, de Croylaan 46, B-3001 Leuven, Belgium
The nematic phase of rod-like particles shows a very rich dynamic behavior in shear flow. At low shear rate it was shown both theoretically [1, 2] and experimentally [3]
that the director, which describes the average orientation of the rods, undergoes a continuoustumbling motion in the plane described by the flow and the gradient in the flow. At higher shear rates a transition was predicted between this tumbling region and a regime where the di- rector iswaggingbetween two relatively small angles, and to flow alignment at even higher shear rates.[2] Since then an increasingly complex diagram of transitions between different kinds of oscillatory regimes has been found theo- retically. Meanwhile experimentalists confirmed the exis- tence of several of the predicted regimes by using rheolog- ical and birefringence measurements under shear. Typi- cal systems that where used for these studies were poly- meric liquid crystals like PBLG[3] and worm-like micelles [4].
Although the correspondence between theory and these experiments is rather good on a qualitative level, several features predicted by theory were not confirmed.
The main problem in the experimental studies is that the interpretation of the results is hindered by the fact that the viscosity is mostly determined by the inter-domain elasticity and not by the viscosity of the nematic phase it- self. The concentration dependence of the transitions be- tween tumbling and wagging, and between wagging and flow aligning could not be accessed because the textural contribution is dominating the response at higher concen- trations. Moreover, the period of the oscillatory response was strain scaling. It was assumed that this was due to an inherent length set by the size of the nematic domains, which emphasizes the influence of the textural stress.
Most of the theory developed is based on the Doi- Edwards theory describing the flow behavior of a homo- geneous ensemble of rods, and is not considering struc- tural effects. For zero-shear this theory extrapolates to the Onsager theory for hard rods, which predicts a first order transition between the isotropic and nematic phase
at a concentration of φ = DL4. However, most systems used in this field of research do not show a well defined phase transition and their equilibrium phase behavior can not be described by Onsager theory.
In this study we use fd viruses, which are inherently monodisperse, very thin (L = 880nm, D = 6.6nm) and stiff (P = 2.2µ) in a 20 mM Tris-HCl buffer at pH 8.2.
Their behavior in equilibrium is well described with On- sager theory, taking into account the semi-flexible nature of the rods.[5]. These authors also showed that the effec- tive diameter of the rods can be adjusted by tuning the ionic strength, sincefd is a poly-electrolyte.
The fact that this system is so well behaved in equi- librium prompted us to usefd also in the study of flow behavior of the nematic phase. It is expected that the response to shear flow of the nematic phase formed by these nearly ideal rods will be more dominated by the rods them selves, than by secondary effects like texture.
Thus, experiments on this system should provide us with a more rigorous test for the theory for a homogeneous system of rods, like the Doi-Edwards theory. It already proved to be a useful system for the study the behavior of the isotropic phase under flow [6].
We performed two types of experiments: stationary viscosity measurements and flow reversal experiments, where the direction of the shear flow is reversed att = 0, for a fixed shear rate. The measurements wee done using an ARES strain controlled rheometer (Rheometric Scientific, Piscataway NJ). A double couette geometry was used because of the very low viscosity of the samples under study.
The stationary viscosity as a function of the shear rate is depicted in Fig. 2 a. Here the viscosity is measured during 30 seconds for each shear rate. The main feature of the viscosity is that it is very low. Focussing on a single concentration, we observe shear slight thinning for the fd virus dispersion at low shear rates, but at intermediate shear rates the viscosity reaches a plateau. More striking is that the viscosity show a small peak before a stronger
2
FIG. 1:
Typical flow reversals for a concentration of 14.5 mg/ml showing the response in the tumbling regime at ˙γ = 10s−1 (solid line), the wagging regime at ˙γ= 55s−1 (dashed line), and the flow aligning at regime ˙γ= 80s−1(dotted line).
shear thinning region sets in.
For a range of shear rates we did a flow reversal ex- periment, especially focussing on the region where the viscosity has a local maximum. Typical flow reversals are depicted in fig 1. The response to the flow rever- sal is characterized by the dimensionless frequency of the oscillationsP =T·γ, and a damping constant˙ τd.
An overview of the behavior of the frequency, the damping constant, and viscosity of the liquid crystalline fd virus dispersions at several concentrations is given in Fig. 2 b and c. Interestingly there is an order of events: the period starts to increase at the shear rate where the first the viscosity reaches its maximum,
˙
γ(Pinc) = ˙γ(ηmax), and only then the damping time starts to increase to reach a maximum value at ˙γ(τd,max).
We interpret the results in terms of the tumbling to wag- ging, and wagging to flow-aligning transitions. The most clear signature for the transition to the Wagging state is the fact that the maximum in the viscosity is found, just before the damping constant reaches a maximum.
The increase in viscosity is generally due to a decrease of the ordering in the sample. Although it seems counter- intuitive that the ordering in a sample should decrease with increasing shear rate, Larson showed [2] that the director of the nematic phase passes a point during the tumbling process, where the flow is extensional, result- ing in a widening of the distribution. When the shear rate is sufficiently high, the widening of the distribution will cause an increase in the viscosity. At some point the shear rate is so high that the distribution will be ran- domized at that angle of extensional flow. As a result the sample will flow-aline, resulting in the wagging of the director between two angles at a different frequency
FIG. 2:
Stationary viscosity ηst (a), period of oscillation P =T ·γ˙ (b), and damping constantτd (c), as function of the shear rate at varying concentrations offd(given in mg/ml).
FIG. 3:
Dynamic phase diagram of the tumbling to wagging for ex- periments (solid points) giving the shear rate at which the different transitions occur, and theory (open points).
3 as the tumbling.
In Fig. 3 we plotted the three points, ˙γ(Pinc), ˙γ(ηmax), and ˙γ(τd,max), as a function of concentration. Following the argumentation that these p oints can be identified with the tumbling-to-wagging transition, we now have with Fig. 3 the first phase diagram of the non-equilibrium behavior of the nematic director. In this diagram we also plot the theoretical data solving the equation of motion of the orientational distribution function numerically. The experimental data can be compared semi-quantitatively by scaling the experimental concentration to the concen- tration at the isotropic to nematic transition with the theoretical value at this transition. The trends are the same, but the scaling of the shear rate to the Peclet num- ber still is unsatisfactory. Important features of the ex- perimental system, like the flexibility of the rods and the existence of nematic domains, still have to be incorpo-
rated in the theory. Nonetheless, the fact that we have been able to access the dynamic phase diagram of the nematic phase offd dispersions is a mayor step to a gain a full understanding of the dynamics of lyotropic liquid crystals.
[1] S. Hess, Z. Naturforsch.31, 1034 (1976).
[2] R. Larson, Macromolecules23, 3983 (1990).
[3] P. Moldenaers, H. Yanase, and J. Mewis, J. Rheol. 35, 1681 (1991).
[4] J.-F. Berret and D. Roux, J. Rheol.39, 725 (1995).
[5] J. Tang and S. Fraden, Liquid Crystals19, 459 (1995).
[6] C. Grafet al., J. Chem. Phys.98, 4921 (1993).
[7] L. M. Walker and N. J. Wagner, Macromolecules29, 2298 (1996).
