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We have investigated non-equilibrium properties of proper uniaxialSn$_2$P$_2$(Se$_x$S$_{1-x}$)$_6$ ferroelectrics with the Type II incommensuratephase above Lifshitz point $x_{\rm LP} \sim 0.28$. We measured dielectricsusceptibility with cooling and heating rate ranging 0.002-0.1~K/min, andhigh-resolution ultrasound experiments and hypersound Brillouin scattering. Forsamples with $x \geqslant 0.28$ clear anomalies were observed at incommensuratesecond order transition ($T_i$) and at first order lock-in transition ($T_c$)in the regime of very slow cooling rate, whereas the intermediate IC phase isnot observed when the rate is faster then 0.1~K/min. In general, increasing thecooling rate leads to smearing the anomaly at $T_c$. We explain this effect interms of Kibble-Zurek model for non-equilibrium second order phase transitions.In the ferroelectrics with strongly nonlinear local potential cooling ratedefines concentration of domain walls and their size: domain width decreaseswhen cooling rate increases. At certain conditions the size of domain iscomparable to the incommensurate phase modulation period, which lies inmicrometer scale in the vicinity of Lifshitz point and leads to pinning of themodulation period by domain wall.

Observation of Kibble-Zurek behavior near the Lifshitz point in ferroelectrics with incommensurate phase