Virtual bending method to calculate bending rigidity, saddle-splay modulus, and spontaneous curvature of thin fluid membrane

A method to calculate the bending rigidity $\kappa$, saddle-splay modulus $\bar{\kappa}$, and spontaneous curvature $C_0$ of a fluid membrane is proposed. Virtual work for the bending deformations into cylindrical and spherical shapes is calculated for a flat membrane. This method does not require a force decomposition, unlike the existing stress-profile method. The first derivative of the deformation gives $\kappa C_0$ and is a discrete form of the first moment of the stress profile. The second derivatives give $\kappa$ and $\bar{\kappa}$, and include the variance terms of the first derivatives, which are not accounted for in the stress-profile method. This method is examined for a solvent-free meshless membrane model and a dissipative-particle-dynamics two-bead amphiphilic molecular model. It is concluded that $\kappa$ and $\bar{\kappa}$ of a thin membrane can be accurately calculated, whereas for a thick membrane or one with an explicit solvent, a further extension to include the volume-fluctuation effects is required for an accurate estimation. The amplitude of the volume-fluctuation effects can be evaluated using the parameter dependence in the present method.