  The purpose of this report is presentation of the main modifications of the
standard Kibble-Zurek formalism caused by the existence of the unperfections in
the system. We know that the distribution of kinks created during a second
order phase transition in pure systems is determined solely by the correlation
length at freeze-out time. The correlation length at that instant of time
intuitively describes the size of the defect and therefore the number density
of defects is limited by the possibility of holding kinks in a unit volume. On
the other hand if the system is populated by the impurities then kinks emerge
mainly in knots of the force distribution which correspond to extremes of the
impurity potential i.e. positions of imperfections. The purpose of this report
is to show that, due to existence of the strong gradients of the impurity
potential, kinks can be created mainly in the close vicinity of the impurities.
It seems that this simple mechanism can be responsible for occurrence, in the
number density formula, the additional length scale describing the impurity
distribution. We know that in pure systems, as a consequence of kink-antikink
annihilation, the number density of kinks decrease in time. In contradiction to
pure systems, kinks produced in the systems populated by impurities could be
confined by the impurity centers and therefore they may not disappear from the
system and may remain above the level established by thermal nucleation of
pairs.

Soft Condensed Matter