Spatially periodic forcing

A number of interesting effects occur in spatially periodically forced pattern forming systems with a nonconserved order parameter, which have been investigated during recent years [60-73, 120], Here we focus on nearly unexplored effects of spatially periodic forcing in system with a conserved order parameter, as they occur in phase separating systems which are forced by spatial temperature modulations and where thermodiffusion plays a crucial role. 

Equation (53) describes the dynamics of phase separation in the presence of a spatially periodic forcing following a quench from the stable one-phase region (e < 0) to a reference temperature in the two-phase region (e > 0). In the following only the case of a symmetric quench with / dry = 0 as initial condition at t = 0 will be considered. 

Phase Separation of Polymer Blends Driven by Temporally and Spatially Periodic Forcing... 

It is a special layer formation susceptibility close to the phase transition the nematic is very sensitive to the spatially periodic molecular field, which induce the density wave with period 1. In order to study this phenomenon one is tempted to use an external spatially periodic force with the same period, but, at present, it is technically impossible. Therefore, we cannot find the Landau coefficient a above T va using some analogy with the Kerr or Cotton-Mouton effects. 

Such forced phase separating systems can be realized, for instance, in optical grating experiments on polymer blends with a spatially periodic light intensity 7(r,f) = Iocos(qx) [44, 87], Such a spatially periodic light intensity leads, together with (50), to the equation... 

For positive values of the control parameter , stationary, spatially periodic solutions y/s(x) = y/x(x I 2n/q) of (53) may be found with and without forcing. However, in the case of a vanishing forcing amplitude (a = 0) in (53), this equation has a i//-symmetry and one has a pitchfork bifurcation from the trivial solution l/r = 0 to finite amplitude periodic solutions as indicated in Fig. 19. In the unforced case, however, periodic solutions of (53) are unstable for any wave number q against infinitesimal perturbations that induce coarsening processes [114, 121],... 

Fig. 19 (a) The bifurcation diagram for spatially periodic solutions at a forcing wave number q = 0.5 and modulation amplitudes a = 0 dotted), a = 0.01 dashed), and a = 0.03 solid), (b)—(d) The three solutions yrs (x) over one period corresponding to the different branches of the bifurcation diagram for a = 0.03 and = 1. Figure from [122]. Reprinted with permission by Springer... 

As was shown before, if phase separation is forced by a stationary and spatially periodic temperature modulation, the coarsening dynamics is interrupted above some critical value of the forcing amplitude a and it is locked to the periodicity of the external forcing. However, if this forcing is pulled by a velocity v 0, the traveling periodic solutions of (61) exist only in a certain range of v depending on a. 

An underlying question remains in the quest for understanding the hydrophobic effect Can continuum models adequately describe this phenomenon Force measurements between mica surfaces suspended in KC1 solution clearly exhibit an oscillatory behavior with a spatial periodicity approximately the same as the diameter of a water molecule [17], suggesting a challenge to the use of continuum models. 

Figure 2 Impulsive stimulated scattering generation of material excitations with crossed beams, (a) Crossed optical beams interfere at the sample and produce a spatially periodic driving force on the material with wavelength A. (b) A probe beam incident upon the spatially periodic material response is partially diffracted, with the diffraction efficiency being proportional to the square of the amplitude of the material response.

Fig. 13.5. Mean burst size and dwell and burst duration as a function of [ATP], (a) The spatial periodicity observed in the average PWD for the low force data in Fig. 13.4b as a function of [ATP]. Error bars represent the standard deviation in a linear fit to the position of each of the peaks in the average PWD. (b) Mean dwell and burst durations for the low force data in Fig. 13.4b as a function of [ATP]. The mean dwell time before the 10-bp bnrsts circles) is well described by a Michaelis-Menten [ATP] dependence solid line). The average bnrst dnration squares) shows no apparent [ATP] dependence, with a mean of 10ms solid line). Modified from [74]...

The velocity of an individual phase front is uniquely determined by the properties of the medium and the forcing parameters. For wave trains, it additionally depends on the spatial period A of a train. Figure 7.2b shows dependences F(A) for two different values of the coefficient /3, obtained by numerical continuation of wave train solutions of equation (7.1) with n = 1. When (3 = 5.0, velocity V remains positive for all spatial periods. This means that both a solitary kink and any kink train in such a medium possess the right chirality. In contrast to this, kinks move at a positive velocity (and have the right chirality) only for sufficiently short spatial periods at /3 = 1.8. At a critical spatial period Ac, the propagation velocity of the train vanishes and F(A) <0 when A > Ac. Thus, solitary kinks and kink trains with large periods have the opposite left chirality in the latter case. 

In concentrated aqueous surfactant solutions, the sizes and shapes of the aggregates are also influenced by interaggregate forces. This leads to positionally ordered structures characterized by long-range orientational alignment and spatial periodicities that cannot be ascribed to spherical micelles [108]. Nevertheless, all three classical structural shapes—spheres, cylinders, and planes—are respectively revealed in hexagonal, discrete (globular) micellar. 

The surface induced layering produces an oscillatory periodic force, with period d = (4.2 0.1)nm, that corresponds to the inter-miceUes distance in the bulk. Compared to smectics (see Fig. 3.18), the force is spatially damped and only nine oscillations are clearly identified. It is therefore a structural force of pre-smectic origin, induced by the surface. The oscillations are not parabolic and the minima lay on a fully attractive baseline. 

Phase separation of polystyrene/poly(vinyl methyl ether) blends was induced under spatially and temporally periodic forcing conditions by taking advantage of either photodimerization of anthracene or photoisomerization of /rt//i .9 stilbene chemically labeled on polystyrene chains. Significant mode election processes driven by these reactions were experimentally observed and analyzed for both cases. These experimental results reveal a potential method of morphology control using periodic forcing conditions. 

We present here a study on phase separation of polymer mixtures forced with temporally and spatially periodic irradiation. The main purpose is to elucidate the contribution of the elastic stress associated with the changes in polymer structure generated by chemical reactions. In the temporal modulation experiments, phase separation was induced by periodic irradiation using ultraviolet (uv) light chopped with frequency varying between 1/200 to 100 Hz. 

Free-electron lasers (FELs) are essentially an outgrowth and extension of modem synchrotron light sources. The amplification and radiation are achieved by a beam of free electrons forced into transverse oscillations by a spatially periodic wiggler magnetostatic field, thereby emitting magnetic bremsstrahlung radiation in the for-... 

Intriguing connections to condensed-matter physics can be found when Feshbach molecules are trapped in optical lattices. In this spatially periodic environment, a Feshbach molecule can be used both as a well-controllable source of correlated atom pairs, and as an efficient tool to detect such pairs. A recent experiment [95] shows how Feshbach molecules, prepared in a three-dimensional lattice, are converted into repulsively bound pairs of atoms when forcing their dissociation through a Feshbach ramp. Such atom pairs stay together and jointly hop between different sites of the lattice, because the atoms repel each other. This counterintuitive behavior is due to the fact that the bandgap of the lattice does not provide any available states for taking up the interaction energy. 

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