Fig. 6: Phase transition to the chiral state B* with respect to elastic anisotropy \(\kappa\) and Ericksen number \(\,\text{Er}\,\).

a \(B\) state appears in the phase diagram at large values of \(\kappa\) and at small values of \(\,\text{Er}\,\). The \({B}^{* }\) state is characterised by non-zero mid-channel tilt angle \({\theta }_{0}\) of the director. The phase border is fitted with the function \(\kappa =-A{(\text{Er}-C)}^{-\beta }+D\) with exponent \(\beta =1.36\), asymptote at \(\,\text{Er}\,=9.87\) and \(\kappa =0.97\). \({\text{Er}}_{D}=7.56\) indicates the value of Ericksen number at which \(D\) phase can be stabilised. b, c Cross sections of the phase diagram at \(\kappa =0.2\) and \(\,\text{Er}\,=50\), respectively, show a clear continuous phase transition between \(B\) and \({B}^{* }\) phase. In the vicinity of the transition region a power law reveals a critical exponent close to 0.5 (see insets). d Effect of intrinsic chirality on the mid-plane tilt angle \({\theta }_{0}\), obtained by numerical integration of Eq. (8). The symmetry of the pitchfork bifurcation at the second-order \(B\) to \({B}^{* }\) phase transition is broken by a chiral dopant into stable and metastable branches. In a chiral system, the tilt angle has an immediate onset at small non-zero flow rates instead of an abrupt onset at the transition flow rate. The dotted line indicates the solution for no intrinsic pitch, shown also in b.