Data associated with the paper "Structural parameters and flexoelectric properties of twist-bend nematics according to Shamid's theory"

1 Data associated with the paper “Structural parameters and flexoelectric properties of twistbend nematics according to Shamid’s theory” Michał Szmigielski, Mariola Buczkowska Institute of Physics, Lodz University of Technology, ul. Wólczańska 217/221, 93-005 Łódź, Poland Corresponding author e-mail: [email protected] Content of files 1. The file “Dateset_1.csv” contains the results of estimation of unknown material parameters (i.e. 𝜇, 𝜆 and 𝜅) from Shamid’s model. 2. The file “Dataset_2.csv” contains the data illustrating how the pitch 𝑝, the heliconical angle 𝜃, the polarisation parameter 𝑃 and the flexoelectric coefficient 𝑒33 of twist-bend nematics depend on the values of material parameters (𝐾22, 𝐾33, 𝜇, 𝜈, 𝜆 and 𝜅) from Shamid’s model. 3. In the file “Dataset_3.csv” the values of 𝑝, 𝜃 and 𝑃 calculated from various approximate formulae are compared with those obtained numerically. 4. The file “Dataset_4.csv” presents the example data obtained in the procedure of the transition from Shamid’s model to Dozov’s one. 5. The file “Dataset_5.csv” presents the example data obtained in the procedure of the transition from Dozov’s model to Shamid’s one. Additional comments on the file “Dataset_3.csv” 1. Explanation of the table header: • p_numerical — the value of the pitch 𝑝 obtained in numerical calculations; • θ_numerical — the value of the heliconical angle 𝜃 obtained in numerical calculations; • P_numerical — the value of the polarisation coefficient 𝑃 obtained in numerical calculations; • P_Parsouzi — the value of the polarisation coefficient 𝑃 calculated from Parsouzi’s formulae: 𝑃=−3𝜆2(𝜅𝐾22)12 2𝐾33 2𝜈+√9𝜆4𝜅𝐾22 4𝐾33 4𝜈2+1𝜈(𝜆2 𝐾33−𝜇) for 𝜈>0 or 𝑃=𝐾33(𝜆2−𝜇𝐾33) 3𝜆2(𝜅𝐾22)12 for 𝜈=0; • θ_Parsouzi — the value of the heliconical angle 𝜃 calculated from Parsouzi’s formula: 𝜃=arcsin√−𝜅𝑃2 𝐾22+√𝜅𝑃2 𝐾22(1+𝜅𝑃2 𝐾22); • p_Parsouzi — the value of the pitch 𝑝 calculated from Parsouzi’s formula: 𝑝=2𝜋(𝜅𝑃2+𝐾33sin2𝜃cos2𝜃+𝐾22sin4𝜃) 𝜆𝑃sin𝜃cos𝜃; 2 • θ_Parsouzi_P_num — the value of the heliconical angle 𝜃 calculated from Parsouzi’s formula (assuming the value of the polarisation parameter 𝑃 obtained numerically); • p_Parsouzi_P_num — the value of the pitch 𝑝 calculated from Parsouzi’s formula (assuming the value of the polarisation parameter 𝑃 obtained numerically); • P_Shamid — the value of the polarisation coefficient 𝑃 calculated from Shamid’s formula: 𝑃=𝐾33 2 4𝜆2(𝜆2 𝐾33−𝜇)√3 2𝐾22𝜅; • θ _Shamid — the value of the heliconical angle 𝜃 calculated from Shamid’s formula: 𝜃=arcsin(𝐾33 2𝜆√𝜆2−𝜇𝐾33 𝐾22𝐾33 ); • p_Shamid — the value of the pitch 𝑝 calculated from Shamid’s formula: 𝑝= 4𝜋 √3 2𝜅(𝜆2 𝐾33−𝜇); • P_alternative — the value of the polarisation coefficient 𝑃 calculated from the alternative formula derived by us: 𝑃=𝐾33√9𝐾22𝜅𝜆4+4(𝜆2−𝜇𝐾33)[4(𝐾33−2𝐾22)𝜅𝜆2+𝜈𝐾33 3]−3𝐾33𝜆2√𝜅𝐾22 2[4(𝐾33−2𝐾22)𝜅𝜆2+𝜈𝐾33 3]; • θ_Parsouzi_P_alternative — the value of the heliconical angle 𝜃 calculated from Parsouzi’s formula (assuming the value of the polarisation parameter 𝑃 calculated from the alternative formula); • p_Parsouzi_P_alternative — the value of the pitch 𝑝 calculated from Parsouzi’s formula (assuming the value of the polarisation parameter 𝑃 calculated from the alternative formula); • δ — the quantity given by the formula: 𝛿=4(𝐾33−2𝐾22)𝜅𝜆2+𝜈𝐾33 3+9𝐾22𝜅𝜆4 4(𝜆2−𝜇𝐾33). 2. When 𝛿≥0, the alternative formula for the polarisation parameter 𝑃 can be used. 3. The percentage differences are calculated as follows: ΔP_Parsouzi = [(P_Parsouzi – P_numerical) / P_numerical]∙100%, Δθ_Parsouzi = [(θ_Parsouzi – θ_numerical) / θ _numerical]∙100%, Δp_Parsouzi = [(p_Parsouzi – p_numerical) / p_numerical]∙100%, Δθ_Parsouzi_P_num = [(θ_Parsouzi_P_num – θ_numerical) / θ _numerical]∙100%, Δp_Parsouzi_P_num = [(p_Parsouzi_P_num – p_numerical) / p_numerical]∙100%, ΔP_Shamid = [(P_Shamid – P_numerical) / P_numerical]∙100%, Δθ_Shamid = [(θ_Shamid – θ_numerical) / θ _numerical]∙100%, 3 Δp_Shamid = [(p_Shamid – p_numerical) / p_numerical]∙100%, ΔP_alternative = [(P_alternative – P_numerical) / P_numerical]∙100%, Δθ_Parsouzi_P_alternative = [(θ_Parsouzi_P_alternative – θ_numerical) / θ _numerical]∙100%, Δp_Parsouzi_P_alternative = [(p_Parsouzi_P_alternative – p_numerical) / p_numerical]∙100%. 4. The percentage differences inform by what percentage the value of 𝑝, 𝜃 or 𝑃 calculated from the formulae is larger (or smaller when the difference is negative) than that obtained numerically. 5. Parsouzi’s formulae can be found in [1]. 6. Shamid’s formulae can be found in [2]. 7. The annotation “not computable” means that the formula cannot be used for a given set of material parameters for mathematical reasons (i.e. the expression under the square root becomes negative or sin𝜃 exceeds unity). References [1] Z. Parsouzi et al., Fluctuation modes of a twist-bend nematic liquid crystal, Phys Rev X 6, 021041 (2016). [2] S. M. Shamid, S. Dhakal, and J. V. Selinger, Statistical mechanics of bend flexoelectricity and the twist-bend phase in bent-core liquid crystals, Phys Rev E 87, 052503 (2013).