The distribution of bond lengths in (V 3+ O 6 ) polyhedra shows a maximum between 1.98 and 2.04 Å, and limits of 1.88 and 2.16 Å, respectively. The bond lengths in (V 4+ O n ) and (V 5+ O n ) (n ) 5, 6) polyhedra show distinct populations which allow us to define the following types of bonds: (1a) vanadyl bonds in (V 4+ O n ) polyhedra, shorter than 1.74 Å; (1b) vanadyl bonds in (V 5+ O 5 ) polyhedra, shorter than 1.76 Å; (1c) vanadyl bonds in (V 5+ O 6 ) polyhedra, shorter than 1.74 Å; (2a) equatorial bonds in (V 4+ O n ) polyhedra, in the range 1.90 to 2.12 Å; (2b) equatorial bonds in (V 5+ O 5 ) polyhedra, longer than 1.76 Å; (2c) equatorial bonds in (V 5+ O 6 ) polyhedra with one vanadyl bond, in the range 1.74 to 2.10 Å; (2d) equatorial bonds in (V 5+ O 6 ) polyhedra with two vanadyl bonds, in the range 1.80 to 2.00 Å; (3a) trans bonds in (V 4+ O 6 ) polyhedra, longer than 2.10 Å; (3b) trans bonds in (V 5+ O 6 ) polyhedra with one vanadyl bond, longer than 2.15 Å; (3c) trans bonds in (V 5+ O 6 ) polyhedra with two vanadyl bonds, longer than 2.025 Å. The average equatorial bond length in (V 4+ O n ) and (V 5+ O n ) polyhedra can be used to calculate the mean valence state of V in mixed-valent structures. We define characteristic bond valences for vanadyl, equatorial, and trans bonds in different coordinations and examine which binary linkages are possible and which linkages occur in minerals and synthetic compounds. Here, V 5+ -O-V 5+ , V 5+ -O-V 4+ , and V 4+ -O-V 4+ linkages between vanadyl-trans and equatorial-equatorial bonds occur often in synthetic compounds, whereas the corresponding V 4+ -O-V 4+ linkages are rare in minerals.
We propose a simple model to analyze the traffic of droplets in microfluidic "dual networks." Such functional networks which consist of two types of channels, namely, those accessible or forbidden to droplets, often display a complex behavior characteristic of dynamical systems. By focusing on three recently proposed configurations, we offer an explanation for their remarkable behavior. Additionally, the model allows us to predict the behavior in different parameter regimes. A verification will clarify fundamental issues, such as the network symmetry, the role of the driving conditions, and of the occurrence of reversible behavior. The model lends itself to a fast numerical implementation, thus can help designing devices, identifying parameter windows where the behavior is sufficiently robust for a device to be practically useful, and exploring new functionalities.
The overall traffic of droplets in a network of microfluidic channels is strongly influenced by the liquid properties of the moving droplets. In particular, the effective hydrodynamic resistance of individual droplets plays a key role in their global behavior. Here we propose two simple and low-cost experimental methods for measuring this parameter by analyzing the dynamics of a regular sequence of droplets injected into an "asymmetric loop" network. The choice of a droplet taking either route through the loop is influenced by the presence of previous droplets that modulate the hydrodynamic resistance of the branches they are sitting in. We propose to extract the effective resistance of a droplet from easily observable time series, namely, from the choices the droplets make at junctions and from the interdroplet distances. This becomes possible when utilizing a recently proposed theoretical model based on a number of simplifying assumptions. Here we present several sets of measurements of the hydrodynamic resistance of droplets, expressed in terms of a "resistance length." The aim is twofold: ͑1͒ to reveal its dependence on a number of parameters, such as the viscosity, the volume of droplets, their velocity as well as the spacing between them. At the same time ͑2͒, by using a standard measurement technique, we compare the limitations of the proposed methods. As an important result of this comparison, we obtain the range of validity of the simplifying assumptions made in the theoretical model.
Molecules that only differ by their chirality, so called enantiomers, often possess different properties with respect to their biological function. Therefore, the separation of enantiomers presents a prominent challenge in molecular biology and belongs to the "Holy Grail" of organic chemistry. We suggest a new separation technique for chiral molecules that is based on the transport properties in a microfluidic flow with spatially variable vorticity. Because of their size the thermal fluctuating motion of the molecules must be taken into account. These fluctuations play a decisive role in the proposed separation mechanism.
The flow profile in a capillary gap and the pumping efficiency of an acoustic micropump employing surface acoustic waves is investigated both experimentally and theoretically. Ultrasonic surface waves on a piezoelectric substrate strongly couple to a thin liquid layer and generate a quadrupolar streaming pattern within the fluid. We use fluorescence correlation spectroscopy and fluorescence microscopy as complementary tools to investigate the resulting flow profile. The velocity was found to depend on the applied power approximately linearly and to decrease with the inverse third power of the distance from the ultrasound generator on the chip. The found properties reveal acoustic streaming as a promising tool for the controlled agitation during microarray hybridization.
We formulate a model of self-propelled hard disks whose dynamics is governed by mutually coupled vectors for velocity and body orientation. Numerical integration at low densities reveals that the expected transition from isotropic to aligned collective motion is present. However, the transition at the Landau mean-field level is strongly first-order, while it is continuous in the Vicsek model. We show that this difference is rooted in the completely opposite effect that individual scattering events have on alignment. We argue that such differences will generically hold for systems of self-propelled particles with repulsive short-ranged interactions. We obtain these results by matching the numerical results to the framework of Boltzmann theory, based on the statistics of binary scattering properties, always assuming that the system is small enough to stay homogeneous. We further show that the presence of noise in the dynamics can change the nature of the transition from discontinuous to continuous.
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