At equilibrium, Bloch walls are chiral interfaces between domains with different magnetization. Far from equilibrium, a set of forced oscillators can exhibit walls between states with different phases. In this Letter, we show that when these walls become chiral, they move with a velocity simply related to their chirality. This surprising behavior is a straightforward consequence of nonvariational effects, which are typical of nonequilibrium systems.PACS numbers: 47.10.+g, 05.45.+b, 47.20.Ky Domain walls exhibit phase transitions, where Ising walls become Bloch walls,' and which have been related to spontaneous breaking of chirality, 2,3 in the frame of an anisotropic X-Y model. In this Letter, we investigate the nonequilibrium analog of these transitions. Namely, we consider the parametric forcing of an assembly of self-oscillators. In such a system, "phase anisotropy," induced by the forcing, provides a natural counterpart to crystalline anisotropy which allows the existence of walls. We show that spontaneous breaking of chirality is accompanied by the motion of chiral interfaces. A simple analysis allows us to describe this behavior. Since this motion is due to nonvariational effects, it is generic to nonequilibrium systems. The occurrence of such effects has been recently emphasized, 4 " 8 and here is another manifestation of these phenomena. In conclusion, we suggest an experiment where these phenomena are likely to be observed.We consider a one-dimensional system which undergoes a spatially homogeneous Hopf bifurcation, 9 with temporal frequency coo. Such a system can be seen as a continuous assembly of oscillators, which auto-oscillate above the Hopf bifurcation threshold. Now, we parametrically force this system at twice its natural frequency too. Let A be the complex order parameter which measures the amplitude of oscillations. It is assumed to vary slowly in space and time and obeys the following Ginzburg-Landau equation:where p measures the distance from the oscillatory instability threshold, v is the detuning parameter, V stands for d/dx, and y > 0 is the forcing amplitude.When the real parameters v, a, and j3 vanish, Eq. (1) can be cast into a variational form:bt 8Awhere 7 -J {-p CY 2 + v 2 ) + \vx\ 2 + |vr| 2 + HX 2 +Y 2 ) 2 -r (X 2 -Y 2 )}dx and A =X + iY. The quantity J turns out to be the free energy of the X-Y model in the presence of weak anisotropy whose amplitude is controlled by 7. In this limit, Eq.(1) exhibits spontaneous breaking of chirality. 2 Namely, Eq. (2) possesses stationary kinklike solutions 1,10 which connect stable homogeneous solutions (X=± VAI + 7, F=0). They read X I = ±Jp + ytanh{[{(p + y)] {/2 x}, r,=0, A-fl = ±v^T7tanh(V27x), Y B -±

Biquadratic coupling between W O order parameters Q , and Q2 can lead to transitionsbetweenaphase withonlyoneof thehvoorderparametersactive(phase1orII) andamivedphaseIIIinwhich bothorderparametenaresymmetry breaking. Domain walls for positive coupling energies are chiral under thermodynamic conditions close to the transition points 1-111 and 11-111. Annihilation of chirality in the wall leads to wall widening and wall bifurcations. The widths of the walls are different for the hvo order parameters leading to greater lattice distortions on either side of the wall than in the centre of the wall,

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AbslrncL Structural relaxations near surfaces and inlerfaces are analysed in a simple, generic model with firs-, second-and lhird-layer interaclions. The relaxations have u p n e n l i a l envelopes with lhree lypes of stmuctural distonion: (i) ferrodislartive, (ii) anlifemcdislorlivc, and (iii) mcdulaled (incommensurable). meir stability conditions in the field of conlml paramelen and their relationship with stmcl~ral phase transilions is derived. A lricriliral p i n t is found far three-layer interaclions and a lransition to an incommensurate phase. All phase transitions to the ferrcdislortive phase are first-order in our model.The theory is applied to the analysis of lalliee relations near internal interfaces in ionic polytypic materials (e.g. Pbll). Diffuse x-ray scatrering and the shifl of diffraction angles are lhe typical fingerprinls for such relaxations. The relevant structure factors are calculated. mere is tentative agreemenl between lhe calculated and observed diffraction profiles.

Most equimolecular mixtures of enantiomers, i.e., tively few (there are ca. 250 inventoried cases1) are conracemates, crystallize as racemic compounds, while reíaglomerates, i.e., cases which permit spontaneous resolution.