3.1. Surface Beris–Edwards model for symmetric lipid bilayers

A hydrodynamic liquid crystal model for symmetric lipid bilayers can be obtained from the surface Beris–Edwards framework (Nitschke & Voigt Reference Nitschke and Voigt2025b ). We carry out this derivation in § 4.2 starting from the surface-conforming Beris–Edwards model (Nitschke & Voigt Reference Nitschke and Voigt2025b ) (§ 3.3), conveniently provided in Appendix D. This leads to the following assertion.

Find the material velocity field $ \boldsymbol{V}\in {T} {{\mathbb{R}}^3\vert _{\mathcal{S}}}$ , scalar field $ \beta \in {{T}^0 \mathcal{S}}$ and generalised pressure field $ \tilde {p}\in {{T}^0 \mathcal{S}}$ such that

(3.1a) \begin{align} & \rho \left ( \partial _t\boldsymbol{V} + (\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V})\left ( \boldsymbol{V} - \boldsymbol{V}_{\!\mathfrak{o}} \right ) \right ) = {\operatorname {Grad}}_{\textsf{C}}\tilde {p} + {\operatorname {Div}}_{\textsf{C}}\tilde {\boldsymbol{\varSigma }}\nonumber \\ \text{with }\quad \tilde {\boldsymbol{\varSigma }} &= \upsilon \left ( 1 + \frac {\xi }{2}\beta \right )^2 \left ( {\boldsymbol{I\!d}}_{\mathcal{S}}\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V} + ({\boldsymbol{I\!d}}_{\mathcal{S}}\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V})^T \right ) - \frac {3L}{2}\left ( \boldsymbol{\nabla} _{\!\textsf{C}}\beta \otimes \boldsymbol{\nabla} _{\!\textsf{C}}\beta + 3\beta ^2\mathcal{H}\boldsymbol{I\!I} \right ) \nonumber \\ &\quad + \frac {9}{2}\boldsymbol{\nu }\otimes \left ( \delta _{\mathfrak{m}}^{\varPhi } M \beta ^2 \boldsymbol{\nu }\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V}- L\beta \left ( \beta \boldsymbol{\nabla} _{\!\textsf{C}}\mathcal{H} + 2\boldsymbol{I\!I}\boldsymbol{\nabla} _{\!\textsf{C}}\beta \right ) \right ) ,\end{align}

(3.1b) \begin{align} & \left ( M + \frac {\upsilon \xi ^2}{2} \right ) \left ( \partial _t\beta + (\boldsymbol{\nabla} _{\!\textsf{C}}\beta )\left ( \boldsymbol{V} - \boldsymbol{V}_{\!\mathfrak{o}} \right ) \right ) \nonumber \\ &= L\left (\Delta _{\textsf{C}}\beta - 3\beta \left ( \mathcal{H}^2 - 2\mathcal{K} \right )\right ) -\left (2a + b\beta + 3c\beta ^2\right )\beta ,\end{align}

(3.1c) \begin{align} & {\operatorname {Div}}_{\textsf{C}}\boldsymbol{V} = 0 \end{align}

holds for $ \dot {\rho } = 0$ and given initial conditions for $ \boldsymbol{V}$ , $ \beta$ and mass density $ \rho \in {{T}^0 \mathcal{S}}$ .

Here $ \boldsymbol{V}_{\!\mathfrak{o}}\in {T}{{\mathbb{R}}^3\vert _{\mathcal{S}}}$ is the so-called observer velocity, which is arbitrary, not necessarily divergence free, and could be used as mesh velocity in a discretised problem for instance. We use $ \beta \in {{T}^0 \mathcal{S}}$ as a proxy for the scalar order parameter $ S\in {{T}^0 \mathcal{S}}$ . In fact, $ \beta$ is the eigenvalue of the corresponding Q-tensor field in the normal direction. Ultimately, $ \beta$ statistically quantifies the degree to which the lipid molecules are aligned along the normal direction: $ \beta = ({2}/{3})$ for the fully ordered state and $ \beta =0$ for the isotropic, i.e. disordered, state. In this process, the average molecular orientation is constant along the normal direction. The fluid stress tensor $\tilde {\boldsymbol{\varSigma }}\in {T} {{\mathbb{R}}^3\vert _{\mathcal{S}}} \otimes {{T}\! {\mathcal{S}}}$ is given modulo $ {\boldsymbol{I\!d}}_{\mathcal{S}}$ ; the resulting pressure gradient is, respectively, incorporated into the generalised pressure $ \tilde {p}$ . Material parameters are the isotropic viscosity $\upsilon$ , anisotropy coefficient $\xi$ , elastic parameter $L$ , immobility coefficient $M$ and thermotropic coefficients $a$ , $b$ and $c$ ; cf. table 3. The Kronecker delta $\delta _{\mathfrak{m}}^{\varPhi }$ acts as a switch that selects between the material model ( $\varPhi =\mathfrak{m}$ : $ \delta _{\mathfrak{m}}^{\varPhi } = 1$ ) and the Jaumann model ( $\varPhi =\mathcal{J}$ : $ \delta _{\mathfrak{m}}^{\varPhi } = 0$ ). This distinction is less relevant here than in the more general Beris–Edwards models (Nitschke & Voigt Reference Nitschke and Voigt2025b ). However, the resulting material immobility force accounts for changes of orientations of the lipids due to the motion of the surface with respect to the surrounding three-dimensional space. This may become relevant if one intends to include the mass, and hence, local torque, of the lipid molecules. The Jaumann model, on the other hand, is invariant under rigid-body rotations; see the discussion in Nitschke & Voigt (Reference Nitschke and Voigt2025b ). In any case, for slow and small lipid surfaces, the choice of model will hardly lead to any qualitative difference in the solution. The lack of a spontaneous curvature term reflects the presence of the up–down symmetry of the surface, something that the general surface Beris–Edwards model in Nitschke & Voigt (Reference Nitschke and Voigt2025b ) already exhibits by construction. As a consequence, the model is appropriate for symmetric lipid bilayers.

In Nitschke & Voigt (Reference Nitschke and Voigt2025a ) additional active mechanisms are considered in the derivation of the general surface Beris–Edwards model. These terms do not contribute in the present setting due to inextensibility and the orthogonal alignment of the lipid molecules with respect to the surface. As a consequence, even with these terms, the solutions exhibit a purely dissipative energy evolution.

The governing equations (3.1) can be transformed into a tangential calculus in the usual manner. The tangential and normal fluid forces on the right-hand side of (3.1a ) yield

(3.2) \begin{align} & {\boldsymbol{I\!d}}_{\mathcal{S}}\left ({\operatorname {Grad}}_{\textsf{C}}\tilde {p} + {\operatorname {Div}}_{\textsf{C}}\boldsymbol{\varSigma }\right ) \hspace {-7em} \nonumber \\ &\,= \boldsymbol{\nabla }\!\hat {p} +\upsilon \left (1+\frac {\xi }{2}\beta \right )^2 \left (\Delta \boldsymbol{v} + \mathcal{K}\boldsymbol{v} - 2\left ( \boldsymbol{I\!I} - \frac {\mathcal{H}}{2}{\boldsymbol{I\!d}}_{\mathcal{S}} \right )\boldsymbol{\nabla }v_{\bot } - v_{\bot }\boldsymbol{\nabla }\mathcal{H}\right )\nonumber \\ &\quad +\upsilon \xi \left (1+\frac {\xi }{2}\beta \right )\left ( \boldsymbol{\nabla }\boldsymbol{v} + \boldsymbol{\nabla} ^T\boldsymbol{v} - 2v_{\bot }\boldsymbol{I\!I} \right )\boldsymbol{\nabla }\!\beta -\frac {3L}{2} \left ( \Delta \beta - 3\beta \big ( \mathcal{H}^2 - 2\mathcal{K} \big ) \right ) \boldsymbol{\nabla }\!\beta \nonumber \\ &\quad +\frac {9 M}{2} \delta _{\mathfrak{m}}^{\varPhi } \beta ^2\left ( \mathcal{K}\boldsymbol{v} - \boldsymbol{I\!I}\left ( \boldsymbol{\nabla }v_{\bot } + \mathcal{H}\boldsymbol{v} \right ) \right ) ,\nonumber \\ & \boldsymbol{\nu }\left ({\operatorname {Grad}}_{\textsf{C}}\tilde {p} + {\operatorname {Div}}_{\textsf{C}}\boldsymbol{\varSigma }\right ) \hspace {-7em} \nonumber \\ &\,= \mathcal{H}\hat {p} +2\upsilon \left (1+\frac {\xi }{2}\beta \right )^2 \left ( \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla }\boldsymbol{v} - v_{\bot }\left ( \mathcal{H}^2 - \mathcal{K} \right ) \right ) -\frac {3L}{4}\left ( 14\boldsymbol{\nabla }\!\beta \boldsymbol{I\!I}\boldsymbol{\nabla }\!\beta - \mathcal{H}\left \| \boldsymbol{\nabla }\!\beta \right \|^2\right )\nonumber \\ &\quad -\frac {9L}{4}\beta \left ( 4\boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla} ^2\beta + 8\boldsymbol{\nabla }\mathcal{H}\boldsymbol{\nabla }\!\beta +\beta \left ( 2\Delta \mathcal{H} + \mathcal{H}\big(\mathcal{H}^2 - 4\mathcal{K}\big) \right )\right )\nonumber \\ &\quad +\frac {9 M}{2} \delta _{\mathfrak{m}}^{\varPhi } \beta \left ( \left ( \Delta v_{\bot } + \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla }\boldsymbol{v} + \boldsymbol{\nabla} _{\boldsymbol{v}}\mathcal{H} \right )\beta + 2\left ( \boldsymbol{\nabla }v_{\bot } + \boldsymbol{v}\boldsymbol{I\!I} \right )\boldsymbol{\nabla }\!\beta \right ) , \end{align}

where $\boldsymbol{V} = \boldsymbol{v} + v_{\bot }\boldsymbol{\nu }$ ; see § 4.2. For purely aesthetic reasons, we separated the elastic pressure contribution from the pressure $\tilde {p}$ again, i.e. $\tilde {p} = \hat {p} + ({3L}/{4}) ( \left \| \boldsymbol{\nabla }\!\beta \right \|^2+ 3\mathcal{H}^2 \beta ^2 )$ . The tangential and normal material accelerations forces on the left-hand side of (3.1a ) are

(3.3) \begin{align} \begin{aligned} \rho {\boldsymbol{a}} &= \rho {\boldsymbol{I\!d}}_{\mathcal{S}}\left ( \partial _t\boldsymbol{V} + (\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V})\left ( \boldsymbol{V} - \boldsymbol{V}_{\!\mathfrak{o}} \right ) \right )\\ &= \rho \left ((\partial _t v^i)\partial _i\boldsymbol{X}_{\!\mathfrak{o}} + \boldsymbol{\nabla} _{\boldsymbol{v}-\boldsymbol{v}_{\!\mathfrak{o}}}\boldsymbol{v} + \boldsymbol{\nabla} _{\boldsymbol{v}}\boldsymbol{v}_{\!\mathfrak{o}} - v_{\bot }\left ( \boldsymbol{\nabla }v_{\bot } + 2\boldsymbol{I\!I}\boldsymbol{v} \right )\right ) ,\\ \rho a_{\bot } &= \rho \boldsymbol{\nu }\left ( \partial _t\boldsymbol{V} + (\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V})\left ( \boldsymbol{V} - \boldsymbol{V}_{\!\mathfrak{o}} \right ) \right )\\ &= \rho \left (\partial _tv_{\bot } + \boldsymbol{\nabla} _{2\boldsymbol{v}-\boldsymbol{v}_{\!\mathfrak{o}}}v_{\bot } + \boldsymbol{I\!I}(\boldsymbol{v},\boldsymbol{v})\right ) , \end{aligned} \end{align}

where $\boldsymbol{V}_{\!\mathfrak{o}} = \boldsymbol{v}_{\!\mathfrak{o}} + v_{\bot }\boldsymbol{\nu }$ and $\{\partial _i\boldsymbol{X}_{\!\mathfrak{o}}\}$ is the observer frame; see table 5. The inextensibility constraint (3.4c ) becomes $ {\operatorname {div}}\boldsymbol{v} = v_{\bot }\mathcal{H}$ . The fully ordered case ( $\beta =({2}/{3})$ ) leads to a surface Navier–Stokes–Helfrich model and is discussed in § 3.3.

3.2. Hydrodynamic surface LH model for asymmetric lipid bilayers

The hydrodynamic surface LH model for asymmetric lipid bilayers is derived in Appendix B following the Lagrange-D’Alembert principle, and energetic considerations in § 4.3 and Nitschke & Voigt (Reference Nitschke and Voigt2025b ). The model can be written as follows:

Find the material velocity field $ \boldsymbol{V}\in {T}{{\mathbb{R}}^3\vert _{\mathcal{S}}}$ , scalar field $ \beta \in {{T}^0 \mathcal{S}}$ and generalised pressure field $ \tilde {p}\in {{T}^0 \mathcal{S}}$ such that

(3.4a) \begin{align} \rho \operatorname {D}^{\mathfrak{m}}_t\boldsymbol{V} &= {\operatorname {Grad}}_{\textsf{C}}\tilde {p} +{\operatorname {Div}}_{\textsf{C}}\left ( \tilde {\boldsymbol{\sigma }}_{\textit{NV}} + \tilde {\boldsymbol{\sigma }}_1 + \tilde {\boldsymbol{\sigma }}_0 + \boldsymbol{\nu }\otimes \left ( \delta _{\mathfrak{m}}^{\varPhi }\boldsymbol{\eta }_{\textit{IM}}^{\mathfrak{m}} + \boldsymbol{\eta }_0 + \boldsymbol{\eta }_{\bar {\kappa }}\right )\right ) ,\end{align}

(3.4b) \begin{align} \left ( M + \frac {\upsilon \xi ^2}{2} \right ) \dot {\beta } &= \omega _1 + \omega _0 + \omega _{\bar {\kappa }} + \omega _{\text{d}w}, \end{align}

(3.4c) \begin{align} {\operatorname {Div}}_{\textsf{C}}\boldsymbol{V} &= 0 \end{align}

holds for $ \dot {\rho } = 0$ and given initial conditions for $ \boldsymbol{V}$ , $ \beta$ and mass density $ \rho \in {{T}^0 \mathcal{S}}$ .

Table 1.

Quantities appearing in the fluid equation (3.4a ). Here $ \boldsymbol{V}_{\!\mathfrak{o}}\in {T}{{\mathbb{R}}^3\vert _{\mathcal{S}}}$ is the so-called observer velocity, which is arbitrary, not necessarily divergence free, and could be used as mesh velocity in a discretised problem for instance. Stress tensors $ \tilde {\boldsymbol{\sigma }}$ are written modulo $ {\boldsymbol{I\!d}}_{\mathcal{S}}$ , respectively the resulting pressure gradient, in comparison to the listed references, since the model (3.4) is inextensible. The use of $ \boldsymbol{\nabla} _{\!\textsf{C}}$ on $ {{T}^0 \mathcal{S}}$ instead of $ \boldsymbol{\nabla}$ is merely cosmetic; both are equivalent on scalar fields. Note that $ \boldsymbol{\eta }_{\textit{IM}}^{\mathfrak{m}}$ applies only within the material model.

Quantities for the fluid equation (3.4a ) are listed in table 1 and for the molecular equation (3.4b ) in table 2. Material parameters can be found in table 3. Again the stress tensors are given modulo $ {\boldsymbol{I\!d}}_{\mathcal{S}}$ ; the resulting pressure gradients are, respectively, incorporated into the generalised pressure $ \tilde {p}$ . Analogous to the surface Beris–Edwards model (3.1), activity, if additionally considered as in Nitschke & Voigt (Reference Nitschke and Voigt2025a ), has no effect, i.e. also the solutions of the hydrodynamic surface LH model exhibit a purely dissipative energy evolution as we can see in Appendix C. Many of the remarks made in § 3.1 remain valid in this context as well and are therefore not reiterated here, since (3.1) and (3.4) coincide for

(3.5) \begin{align} \begin{aligned} a &= \frac {2\hat {a}}{27} ,\quad b = -\frac {2\hat {a}}{3} - 27\varpi ,\quad c = \frac {2\hat {a}}{9} + \frac {27\varpi }{2} ,\\ \mathcal{H}_{0} &= \hat {\mathcal{H}}_0 = 0 ,\quad \kappa =\bar {\kappa } = 2L . \end{aligned} \end{align}

Table 2.

Quantities appearing in the molecular equation (3.4b ). Here $ \boldsymbol{V}_{\!\mathfrak{o}}\in {T}{{\mathbb{R}}^3\vert _{\mathcal{S}}}$ is the so-called observer velocity, which is arbitrary, not necessarily divergence free, and could be used as mesh velocity in a discretised problem for instance. The use of componentwise operators $ \boldsymbol{\nabla} _{\!\textsf{C}}, \Delta _{\textsf{C}}$ on $ {{T}^0 \mathcal{S}}$ instead of covariant $ \boldsymbol{\nabla }, \Delta$ is merely cosmetic; both are equivalent on scalar fields.

Table 3.

Material parameters for the LH model (3.4). Given domains are only necessary, but not sufficient, for solvability or physical plausibility. For instance, Nitschke & Voigt (Reference Nitschke and Voigt2025b ) suggest $ -3 \lt 2\xi \lt 3$ to maintain a positive definite lipid metric.

Again, we also formulate the model within a tangential calculus, including the use of covariant derivatives. For this purpose, we split the fluid equation (3.4a ) into its projective tangential and normal part, i.e.

(3.6) \begin{align} \rho {\boldsymbol{a}} &= \boldsymbol{\nabla }\!\bar {p} + \boldsymbol{f}_{\textit{NV}} + \delta _{\mathfrak{m}}^\varPhi \boldsymbol{f}_{\textit{IM}}^{\mathfrak{m}} + \boldsymbol{f}_1 + \boldsymbol{f}_0 + \boldsymbol{f}_{\bar {\kappa }} ,\nonumber \\ \rho a_{\bot } &= \bar {p}\mathcal{H} + f^{\bot }_{\textit{NV}} + \delta _{\mathfrak{m}}^\varPhi f_{\textit{IM}}^{\mathfrak{m},\bot } +f^{\bot }_{1} + f^{\bot }_{0} + f^{\bot }_{\bar {\kappa }}, \end{align}

where the tangential and normal fluid forces $ \boldsymbol{f}_{\textit{NV}}$ , $ f^{\bot }_{\textit{NV}}$  (B17), $ \boldsymbol{f}_{\textit{IM}}^{\mathfrak{m}}$ , $ f_{\textit{IM}}^{\mathfrak{m},\bot }$  (B21), $ \boldsymbol{f}_1$ , $ f^{\bot }_{1}$  (B4), $ \boldsymbol{f}_0$ , $ f^{\bot }_{0}$  (B5), (B6) and $ \boldsymbol{f}_{\bar {\kappa }}$ , $ f^{\bot }_{\bar {\kappa }}$  (B10) are given in Appendix B. It should be noted that

(3.7) \begin{align} \boldsymbol{f}_1 + \boldsymbol{f}_0 + \boldsymbol{f}_{\bar {\kappa }} &= -\frac {3}{2}\left ( \omega _1 + \omega _0 + \omega _{\bar {\kappa }} \right )\boldsymbol{\nabla }\!\beta \end{align}

holds, i.e. the tangential surface LH fluid force vanishes for constant $ \beta$ , as expected, since in this case the force is purely geometric and the associated normal force $ f^{\bot }_{1} + f^{\bot }_{0} + f^{\bot }_{\bar {\kappa }}$ reduces to a curvature-driven Helfrich force. Explicit representation of the tangential $ {\boldsymbol{a}} = {\boldsymbol{I\!d}}_{\mathcal{S}}\operatorname {D}^{\mathfrak{m}}_t\boldsymbol{V}$ and normal acceleration $ a_{\bot } = \boldsymbol{\nu }\operatorname {D}^{\mathfrak{m}}_t\boldsymbol{V}$ depending on an arbitrary observer frame can be found at the top of table 5, respectively (3.3). Even though this has no influence on the solution, the generalised pressure $ \bar {p}$ is different here than in (3.4a ). We have included in $ \bar {p}$ only the forces arising from the double-well potential (B11) and the activity (B23), both of which reduce to pure pressure gradients. The molecular equation (3.4b ) can essentially be used unchanged. In table 2 the subscript $ \textsf{C}$ may be omitted, as for scalar fields, the respective differential operators coincide with the covariant ones. Within the scalar rate $ \dot {\beta }$ the relation $ \boldsymbol{V} - \boldsymbol{V}_{\!\mathfrak{o}} = \boldsymbol{v} - \boldsymbol{v}_{\!\mathfrak{o}}$ already holds anyway, since all observers for the same moving surface have the same normal velocity. The inextensibility constraint (3.4c ) becomes $ {\operatorname {div}}\boldsymbol{v} = v_{\bot }\mathcal{H}$ . The fully ordered case ( $\beta = ({2}/{3})$ ) leads to a surface Navier–Stokes–Helfrich model and is discussed in § 3.3.

3.3. Surface Navier–Stokes–Helfrich models

In the vicinity of the ordered equilibrium, we may assume that $ \beta = ({2}/{3})$ . Incorporating this assumption into model (3.4), e.g. by means of a Lagrange multiplier formulation, eliminates the molecular equation (3.4b ) and yields the surface Navier–Stokes–Helfrich model: Find the material velocity field $ \boldsymbol{V}\in {T}{{\mathbb{R}}^3\vert _{\mathcal{S}}}$ and generalised pressure field $ \tilde {p}\in {{T}^0 \mathcal{S}}$ such that

(3.8a) \begin{align} \rho \operatorname {D}^{\mathfrak{m}}_t\boldsymbol{V} &= {\operatorname {Grad}}_{\textsf{C}}\tilde {p} +{\operatorname {Div}}_{\textsf{C}}\Big ( \tilde {\upsilon }\left ( {\boldsymbol{I\!d}}_{\mathcal{S}}\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V} + ({\boldsymbol{I\!d}}_{\mathcal{S}}\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V})^T \right ) - \kappa \left ( \mathcal{H} - \mathcal{H}_0 \right )\boldsymbol{I\!I}\nonumber \\ &\quad \hspace {0em}+\, \boldsymbol{\nu }\otimes \left ( 2M\delta _{\mathfrak{m}}^{\varPhi } \boldsymbol{\nu }\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{V} - \kappa \boldsymbol{\nabla} _{\!\textsf{C}}\mathcal{H} \right )\Big ),\end{align}

(3.8b) \begin{align} {\operatorname {Div}}_{\textsf{C}}\boldsymbol{V} &= 0 \end{align}

holds for $ \dot {\rho } = 0$ and given initial conditions for $ \boldsymbol{V}$ and mass density $ \rho \in {{T}^0 \mathcal{S}}$ .

For $\mathcal{H}_0 = 0$ , (3.8) also follows by assuming that $ \beta = ({2}/{3})$ in (3.1). Accordingly, the viscosity rescales as $ \tilde {\upsilon } = \upsilon (1+ ({\xi }/{3}))^2$ , although we may also simply set $ \xi =0$ without loss of generality, since the anisotropy coefficient no longer appears anywhere else, and thus, no actual anisotropic viscosity contributes to the solution. As expected, all Gaussian curvature-elastic terms vanish independently of $ \bar {\kappa }$ . It should be noted that if the assumption $ \beta = ({2}/{3})$ is justified by the fact that the immobility is small, i.e. $ M\approx 0$ , such that the molecular equation (3.4b ) relaxes on a much faster time scale than the fluid equation (3.4a ), then the immobility term in the fluid equation can obviously be neglected. In this case, the material ( $ \varPhi =\mathfrak{m}$ ) and Jaumann ( $ \varPhi =\mathcal{J}$ ) models are indistinguishable. Within the covariant differential calculus and the tangential-normal splitting of the fluid equations, the governing equations take the form

(3.9a) \begin{align} \rho {\boldsymbol{a}} &= \boldsymbol{\nabla }\!\bar {p} + \tilde {\upsilon } \left ( \Delta \boldsymbol{v} + \mathcal{K}\boldsymbol{v} - 2\left ( \boldsymbol{I\!I} - \frac {\mathcal{H}}{2}{\boldsymbol{I\!d}}_{\mathcal{S}} \right )\boldsymbol{\nabla }v_{\bot } \right )\nonumber \\ &\hspace {1em} + 2M\delta _{\mathfrak{m}}^{\varPhi }\left ( \mathcal{K}\boldsymbol{v} - \boldsymbol{I\!I}\left ( \boldsymbol{\nabla }v_{\bot } + \mathcal{H}\boldsymbol{v} \right ) \right )\!,\end{align}

(3.9b) \begin{align} \rho a_{\bot } &= \mathcal{H}\bar {p} + 2\tilde {\upsilon }\left ( \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla }\boldsymbol{v} - v_{\bot }\big ( \mathcal{H}^2 - \mathcal{K} \big ) \right ) + 2M\delta _{\mathfrak{m}}^{\varPhi }\left ( \Delta v_{\bot } + \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla }\boldsymbol{v} + \boldsymbol{\nabla} _{\boldsymbol{v}}\mathcal{H} \right )\nonumber \\ &\hspace {1em} -\kappa \left ( \Delta \mathcal{H} + \frac {1}{2}\left ( \mathcal{H}-\mathcal{H}_0 \right )\left ( \mathcal{H} \left ( \mathcal{H}+\mathcal{H}_0 \right )- 4\mathcal{K} \right ) \right )\! ,\end{align}

(3.9c) \begin{align} {\operatorname {div}}\boldsymbol{v} &= \mathcal{H}v_{\bot }. \end{align}

The associated Jaumann model ( $ \varPhi =\mathcal{J}$ ), respectively the neglected immobility ( $ M=0$ ), is consistent with Reuther et al. (Reference Reuther, Nitschke and Voigt2020), Krause & Voigt (Reference Krause and Voigt2023), Olshanskii (Reference Olshanskii2023), Sischka et al. (Reference Sischka, Nitschke and Voigt2025), Sauer (Reference Sauer2025) and with Arroyo & DeSimone (Reference Arroyo and DeSimone2009), Torres-Sánchez et al. (Reference Torres-Sánchez, Millán and Arroyo2019), Zhu et al. (Reference Zhu, Saintillan and Chern2025) in the Stokes limit. We would like to emphasise once again that the material model ( $ \varPhi =\mathfrak{m}$ , $ M \gt 0$ ) is, also in this setting, not invariant under rigid-body rotations. This lack of invariance may in fact be physically meaningful if a kind of rotational damping of the lipid molecules is assumed.

While the recovery of these known surface (Navier-)Stokes-Helfrich models for fluid deformable surfaces is appealing, their derivation differs. Brandner, Reusken & Schwering (Reference Brandner, Reusken and Schwering2022) summarises different derivations of the surface Navier–Stokes equations. Their extensions towards (3.8) or (3.9) in Arroyo & DeSimone (Reference Arroyo and DeSimone2009), Torres-Sánchez et al. (Reference Torres-Sánchez, Millán and Arroyo2019), Reuther et al. (Reference Reuther, Nitschke and Voigt2020) always consider an ad hoc additional Helfrich energy. The derivations in § 4 do not require such an approach, the Helfrich terms naturally follow from the underlying surface liquid crystal models, similarly to the original derivation of Helfrich (Reference Helfrich1971).

4.1. The Q-tensor ansatz

Essentially, we adopt a similar approach to that of Helfrich (Reference Helfrich1973). However, instead of considering a polar field normal to the surface ${\mathcal{S}}$ to describe a lipid bilayer, we employ an apolar field represented by an uniaxial Q-tensor field $ \boldsymbol{Q}\in {\operatorname {\mathcal{Q}}}{^2}{\mathbb{R}}^3\vert _{\mathcal{S}}$ with a normal eigenvector

(4.1) \begin{align} \boldsymbol{Q} &= S\left ( \boldsymbol{\nu }\otimes \boldsymbol{\nu } - \frac {1}{3}{\boldsymbol{I\!d}} \right ) = \beta \left ( \boldsymbol{\nu }\otimes \boldsymbol{\nu } - \frac {1}{2}{\boldsymbol{I\!d}}_{\mathcal{S}} \right )\!, \end{align}

where ${\boldsymbol{I\!d}} = {\boldsymbol{I\!d}}_{\mathcal{S}} + \boldsymbol{\nu }\otimes \boldsymbol{\nu }$ is valid and $ S= ({3}/{2})\beta \in {{T}^0 \mathcal{S}}$ is the scalar order field, respectively $ \beta = ({2}/{3}) S\in {{T}^0 \mathcal{S}}$ the eigenvalue field to the eigenvector field $ \boldsymbol{\nu }$ , i.e. it holds that $ \boldsymbol{Q}\boldsymbol{\nu }=\beta \boldsymbol{\nu }$ . With this ansatz, the lipid molecules are, on average, oriented orthogonally to the surface tangential planes and the amount of order is given by the scalar order $S$ as the weighted average of the molecular orientation (Mottram & Newton Reference Mottram and Newton2014) deviating from $\boldsymbol{\nu }$ . For convenience, we use $\beta$ instead of $S$ in this paper. Note that $\boldsymbol{Q}$ in (4.1) is surface conforming without any flat-degenerated part; see Nitschke & Voigt (Reference Nitschke and Voigt2025b ).

The advantage of modelling lipid layers as Q-tensor fields lies in the ability to describe molecular alignment as being, on average, normal to the surface, while also incorporating an additional degree of freedom for the statistical order. This approach allows us to build on established surface Beris–Edwards models to develop hydrodynamic descriptions of lipid bilayers, as discussed in § 4.2. A limitation of (4.1), however, is that it is an apolar ansatz, i.e. due to head–tail symmetry of the molecules we can only represent symmetric lipid bilayers a priori. The symmetry break required to obtain asymmetric lipid bilayers must occur elsewhere in the process. This is addressed in § 4.3.

4.2. Surface Beris–Edwards model for symmetric lipid bilayers

In this section we use the surface-conforming Beris–Edwards models in Nitschke & Voigt (Reference Nitschke and Voigt2025b ) as a starting point and incorporate a suitable constraint providing the ansatz (4.1). Since ansatz (4.1) as well as the surface-conforming models comprise up–down symmetry, i.e. the exterior and interior of the surface are indistinguishable in any way, we could only describe symmetric lipid bilayers, see figure 1, without introducing any additional symmetry-breaking mechanism. For the reader’s convenience, a summary of the surface Beris–Edwards model from Nitschke & Voigt (Reference Nitschke and Voigt2025b ) together with the active terms introduced in Nitschke & Voigt (Reference Nitschke and Voigt2025a ) are given in Appendix D.

A reasonable orthogonal decomposition of $ \boldsymbol{Q}\in {\operatorname {\mathcal{Q}}}{^2}{\mathbb{R}}^3\vert _{\mathcal{S}}$ is provided by

(4.2) \begin{align} \boldsymbol{Q} = \boldsymbol{q} + \boldsymbol{\eta }\otimes \boldsymbol{\nu } + \boldsymbol{\nu }\otimes \boldsymbol{\eta } + \beta \left ( \boldsymbol{\nu }\otimes \boldsymbol{\nu } - \frac {1}{2}{\boldsymbol{I\!d}}_{\mathcal{S}} \right )\!\text{,} \end{align}

where $ \boldsymbol{q}\in {\operatorname {\mathcal{Q}}}{^2}\mathcal{S}$ yields the flat-degenerated part, $ \boldsymbol{\eta }\in {{T}\! {\mathcal{S}}}$ the surface non-conforming part and $ \beta \in {{T}^0 \mathcal{S}}$ the uniaxial normal part. The surface-conforming constraint $ \boldsymbol{\eta }=\boldsymbol{0}$ is already incorporated in the surface-conforming Beris–Edwards model (D1). Therefore, we only need to consider a ‘no flat degeneracy’ ( $ \boldsymbol{q} = \boldsymbol{0}$ ) constraint for our purpose. We account for possible constraint forces in Appendix E and conclude that it suffices to omit the flat-degenerated molecular equation (D1c ) and simply substitute $\boldsymbol{q}=\boldsymbol{0}$ throughout the remaining model.

We first address the elastic contributions arising from the elastic energy functional

(4.3) \begin{align} \mathfrak{U}_{\textit{EL}} &= \frac {L}{2}\left \| \boldsymbol{\nabla} _{\textsf{C}}\boldsymbol{Q}\right \|_{\textrm{L}^2({T}^3 \mathbb{R}^3\vert _{\mathcal{S}})}^2 \!.\end{align}

The relevant terms in table 5 yield

(4.4) \begin{align} \boldsymbol{\sigma }_{\textit{EL}} &= -\frac {3}{2}L\!\left ( \boldsymbol{\nabla }\!\beta \otimes \boldsymbol{\nabla }\!\beta - \frac {\left \| \boldsymbol{\nabla }\!\beta \right \|}{2}^2 {\boldsymbol{I\!d}}_{\mathcal{S}} + 3\mathcal{H}\beta ^2\left ( \boldsymbol{I\!I} - \frac {\mathcal{H}}{2}{\boldsymbol{I\!d}}_{\mathcal{S}}\right ) \right )\!\text{,}\nonumber \\ \boldsymbol{\zeta }_{\textit{EL}} &= -\frac {3}{2}L\!\left ( 2\boldsymbol{I\!I}\boldsymbol{\nabla }\!\beta + \beta \boldsymbol{\nabla }\mathcal{H} \right )\!\text{,}\nonumber \\ \omega _{\textit{EL}} &= L\!\left ( \Delta \beta - 3\beta \big ( \mathcal{H}^2 - 2\mathcal{K} \big ) \right )\!. \end{align}

Note that the tangential elastic fluid force $ {\operatorname {div}}\boldsymbol{\sigma }_{\textit{EL}}$ combined with the elastic part of the tangential surface-conforming constraint force $ -3\beta \boldsymbol{I\!I}\boldsymbol{\zeta }_{\textit{EL}}$ results in

(4.5) \begin{align} {\operatorname {div}}\boldsymbol{\sigma }_{\textit{EL}} - 3\beta \boldsymbol{I\!I}\boldsymbol{\zeta }_{\textit{EL}} \hspace {-7em}\nonumber \\ &= -\frac {3}{2}L \Bigg ( \Delta \beta \boldsymbol{\nabla }\!\beta + \boldsymbol{\nabla} ^2\beta \boldsymbol{\nabla }\!\beta - \boldsymbol{\nabla} ^2\beta \boldsymbol{\nabla }\!\beta + 3\beta ^2\left ( \boldsymbol{I\!I}\boldsymbol{\nabla }\mathcal{H} - \frac {\mathcal{H}}{2}\boldsymbol{\nabla }\mathcal{H} \right ) \nonumber \\ &\hspace {1em} + 6\mathcal{H}\beta \left ( \boldsymbol{I\!I}\boldsymbol{\nabla }\!\beta - \frac {\mathcal{H}}{2}\boldsymbol{\nabla }\!\beta \right ) + \frac {3}{2}\mathcal{H}\beta ^2\boldsymbol{\nabla }\mathcal{H} -3\beta \left ( 2\mathcal{H}\boldsymbol{I\!I}\boldsymbol{\nabla }\!\beta - 2\mathcal{K}\boldsymbol{\nabla }\!\beta + \beta \boldsymbol{I\!I}\boldsymbol{\nabla }\mathcal{H} \right )\Bigg )\nonumber \\ &= -\frac {3}{2}L \left ( \Delta \beta - 3\beta \big ( \mathcal{H}^2 - 2\mathcal{K} \big ) \right ) \boldsymbol{\nabla }\!\beta = -\frac {3}{2} \omega _{\textit{EL}} \boldsymbol{\nabla }\!\beta, \end{align}

since $ {\operatorname {div}}\boldsymbol{I\!I}=\boldsymbol{\nabla }\mathcal{H}$ and $ \boldsymbol{I\!I}^2=\mathcal{H}\boldsymbol{I\!I} - \mathcal{K}{\boldsymbol{I\!d}}_{\mathcal{S}}$ is valid. The normal elastic fluid force $ \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\sigma }_{\textit{EL}}$ combined with the elastic part of the normal surface-conforming constraint force $ 3{\operatorname {div}}(\beta \boldsymbol{\zeta }_{\textit{EL}})$ becomes

(4.6) \begin{align} \boldsymbol{I\!I}\operatorname {:}\boldsymbol{\sigma }_{\textit{EL}} + 3{\operatorname {div}}(\beta \boldsymbol{\zeta }_{\textit{EL}}) &= -\frac {3}{4}L\Big ( 14\boldsymbol{\nabla }\!\beta \boldsymbol{I\!I}\boldsymbol{\nabla }\!\beta - \mathcal{H}\left \| { \boldsymbol{\nabla }\!\beta }\right \|^2\nonumber \\ &\hspace {0em} +3\beta \left ( 4\boldsymbol{I\!I}\operatorname {:}\boldsymbol{\nabla} ^2\beta + 8\boldsymbol{\nabla }\mathcal{H}\boldsymbol{\nabla }\!\beta +\beta \left ( 2\Delta \mathcal{H} + \mathcal{H}(\mathcal{H}^2 - 4\mathcal{K}) \right )\right )\Big ) . \end{align}

This implies that elasticity of symmetric lipid bilayers already comprises the bending force $ f^{\bot }_{{BE}}\boldsymbol{\nu }$ for surfaces with up–down symmetry and mean curvature-elastic moduli $ \kappa = ({9}/{2})L\beta ^2$ . Since the lipid molecules already define the surface geometry, we omit the lipid-independent bending force $f^{\bot }_{{BE}}$ and retain only the one arising from the elasticity in terms of the scalar order parameter. The molecular thermotropic force results in

(4.7) \begin{align} \omega _{{TH}} = -\big (2a + b\beta + 3c\beta ^2\big )\beta . \end{align}

Other thermotropic contributions have no influence on the model. The immobility terms yield

(4.8) \begin{align} \boldsymbol{\sigma }_{\textit{IM}}^{\varPhi }= \boldsymbol{0},\qquad &\boldsymbol{\zeta }_{\textit{IM}}^{\varPhi }= \begin{cases} \boldsymbol{0} &\text{for } \varPhi = \mathcal{J} \text{,}\\[3pt] \frac {3}{2}M \beta \left (\boldsymbol{\nabla }v_{\bot } + \boldsymbol{I\!I}\boldsymbol{v}\right ) &\text{for } \varPhi = \mathfrak{m} \text{;} \end{cases} \end{align}

see (B21) for the corresponding fluid forces. Nematic viscosity terms become

(4.9) \begin{align} \boldsymbol{\sigma }_{\textit{NV}}^0 + \xi \boldsymbol{\sigma }_{\textit{NV}}^1 + \xi ^2\boldsymbol{\sigma }_{\textit{NV}}^2 &= \tilde {\boldsymbol{\sigma }}_{\textit{NV}} + \frac {\upsilon \xi }{2}\dot {\beta }\left ( 1 + \frac {\xi \beta }{2} \right ){\boldsymbol{I\!d}}_{\mathcal{S}}, &\widetilde {\omega }^{2}_{\textit{NV}} &= 0 . \end{align}

Due to inextensibility, we only have to consider

(4.10) \begin{align} \tilde {\boldsymbol{\sigma }}_{\textit{NV}} &= \upsilon \left ( 1 - \frac {\xi \beta }{2} \right )^2 \big ( \boldsymbol{\nabla }\boldsymbol{v} + \boldsymbol{\nabla} ^T\boldsymbol{v} - 2v_{\bot }\boldsymbol{I\!I} \big ) \text{;} \end{align}

see (B17) for the corresponding fluid forces. Nematic activity does not contribute in a inextensible media, i.e.

(4.11) \begin{align} \boldsymbol{\sigma }_{\!{{NA}}} &= \boldsymbol{0} . \end{align}

Nevertheless, even though it does not affect the inextensible model, we note the presence of an active pressure $p_{{{AC}}} = \alpha _{\textit{IA}} - ({\alpha _{{{NA}}}}/{2})\beta$ , comprising a nematic part controlled by parameter $\alpha _{{{NA}}}\in {\mathbb{R}}$ and a geometric part controlled by the parameter $\alpha _{\textit{IA}}\in {\mathbb{R}}$ ; see Nitschke & Voigt (Reference Nitschke and Voigt2025a ). The corresponding pressure force therefore reads

(4.12) \begin{align} \boldsymbol{F}_{\!{{AC}}} &= {\operatorname {Div}}_{\textsf{C}}\left ( \left ( \alpha _{\textit{IA}} - \frac {\alpha _{{{NA}}}}{2}\beta \right ){\boldsymbol{I\!d}}_{\mathcal{S}} \right ) = \alpha _{\textit{IA}}{\operatorname {Grad}}_{\textsf{C}} 1 - \frac {\alpha _{{{NA}}}}{2} {\operatorname {Grad}}_{\textsf{C}}\beta \nonumber \\ &= - \frac {\alpha _{{{NA}}}}{2}\boldsymbol{\nabla }\!\beta + \left ( \alpha _{\textit{IA}} - \frac {\alpha _{{{NA}}}}{2}\beta \right )\mathcal{H}\boldsymbol{\nu } . \end{align}

Eventually, the surface Beris–Edwards model (D1), the ‘no flat degeneracy’ constraint, and rewriting all relevant terms in the $ {\mathbb{R}}^3$ calculus, yield the model (3.1) for symmetric lipid bilayers.

4.3. Hydrodynamic surface LH model for asymmetric lipid bilayers

In this section we derive the surface LH energy as a special case of a polarised surface Landau–De Gennes energy, which consists of a nematic elastic contribution, a thermotropic contribution, similar to Nitschke et al. (Reference Nitschke, Nestler, Praetorius, Löwen and Voigt2018, Reference Nitschke, Reuther and Voigt2020) and an up–down symmetry-breaking mechanism. The reduction to the lipid-molecule level is carried out using the ansatz (4.1). For this purpose, we identify all invariants for asymmetric lipid bilayers up to a certain order, similar to the procedure in Helfrich (Reference Helfrich1973) and Alexe-Ionescu (Reference Alexe-Ionescu1993). We then reformulate these into a form analogous to the Helfrich energy, which generalises it accordingly. The governing equations derived from this formulation and the Lagrange–D’Alembert principle for moving surfaces (Nitschke & Voigt Reference Nitschke and Voigt2025b ) are provided in Appendix B.

The componentwise surface derivative of (4.1) results in

(4.13) \begin{align} \boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q} &= \left ( \boldsymbol{\nu }\otimes \boldsymbol{\nu } - \frac {1}{2}{\boldsymbol{I\!d}}_{\mathcal{S}} \right ) \otimes \boldsymbol{\nabla }\!\beta -\frac {3}{2}\beta \left ( \boldsymbol{\nu }\otimes \boldsymbol{I\!I} + (\boldsymbol{\nu }\otimes \boldsymbol{I\!I})^{T_{(1\, 2)}}\right )\! , \end{align}

where $ [(\boldsymbol{\nu }\otimes \boldsymbol{I\!I})^{T_{(1\, 2)}}]_{ABC} = \nu _{B}I\!I_{AC}$ in arbitrary Euclidean coordinates. Both summands and all their transpositions are mutual orthogonal or symmetric, i.e. all scalar-valued quadratic invariants of (4.13) are given by $ \left \| \boldsymbol{\nabla }\!\beta \right \|_{{{T}\! {\mathcal{S}}}}^2$ and $ \left \| \beta \boldsymbol{I\!I} \right \|_{{{T}^2 \mathcal{S}}}^2= (\mathcal{H}^2-2\mathcal{K})\beta ^2$ or $ ({\operatorname {Tr}}\boldsymbol{I\!I})^2 \beta ^2 = \mathcal{H}^2 \beta ^2$ . Therefore, we could reduce the quadratic energy approach

(4.14) \begin{align} \int _{\mathcal{S}} l_{A_1,A_2,A_3,B_1,B_2,B_3}[\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q}]^{A_1,A_2,A_3}[\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q}]^{B_1,B_2,B_3} {\text {d}\mathcal{S}} \end{align}

to the elastic energy potential

(4.15) \begin{align} \tilde {\mathfrak{U}}_{{el}} := \frac {l_1}{2} \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}({T} \mathcal{S})}^2+ \frac {l_0}{2}\left \| \mathcal{H} \beta \right \|_{\textrm{L}^2({T}^0 \mathcal{S})}^2- \bar {l}_0\int _{\mathcal{S}}\mathcal{K}\beta ^2{\text {d}\mathcal{S}} \end{align}

with material parameters $ l_1 \ge 0$ and $ l_0 \ge \bar {l}_0 \ge 0$ . For instance, if we substitute ansatz (4.1) into the one-constant elastic energy potential (4.3), we obtain

(4.16) \begin{align} \mathfrak{U}_{\textit{EL}} &= \frac {3}{4}L\left ( \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2 + 3\left \| \mathcal{H} \beta \right \|_{\textrm{L}^2 ({T}^0 \mathcal{S})}^2{}{ }- 6\int _{\mathcal{S}}\mathcal{K}\beta ^2{\text {d}\mathcal{S}} \right )\!, \end{align}

i.e. the parameter relation would be $ l_1 = {3L}/{2}$ and $ l_0= \bar {l}_0 = {9L}/{2}$ . Other conceivable elastic energy potentials that are quadratic in $ \boldsymbol{Q}$ – for instance, those involving the commonly used elastic parameters $ L_1=L, L_2, L_3$ (Mori et al. Reference Mori, Gartland, Kelly and Bos1999; Mottram & Newton Reference Mottram and Newton2014) – yield energies with different proportions. (If we consider a constant extension in the normal direction of $ \boldsymbol{Q}$ , cf. Nitschke et al. (Reference Nitschke, Nestler, Praetorius, Löwen and Voigt2018), we obtain $ ({L_2}/{2})\| \mathrm{Div}_{\textsf {C}} \boldsymbol{Q} \|_{\textrm{L}^2({T}{\mathbb{R}}^3\vert _{\mathcal{S}})}^2 = ({L_2}/{8})( \| \boldsymbol{\nabla }\!\beta \|_{\textrm{L}^2({T} \mathcal{S})}^2+ 9\| \beta \mathcal{H} \|_{\textrm{L}^2 ({T}^0 \mathcal{S})}^2)$ and $({L_3}/{2}){} \langle {\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q},\boldsymbol{\nabla} _{\!\textsf{C}}^T\boldsymbol{Q}}\rangle _{\textrm{L}^2({T}^3\mathbb{R}^3\vert _{\mathcal{S}})} = ({L_3}/{8})( \| \boldsymbol{\nabla }\!\beta \|_{\textrm{L}^2({T} \mathcal{S})}^2 + 9\| \beta \mathcal{H} \|_{\textrm{L}^2({T}^0 \mathcal{S})}^2{}{ }- 18\int _{\mathcal{S}}\mathcal{K}{\text {d}\mathcal{S}} )$ for instance.) Note that (4.15) states Helfrich’s bending energy (Helfrich Reference Helfrich1973) for constant $ \beta$ and neglected spontaneous curvature, where the curvature-elastic moduli yield $ \kappa = l_0\beta ^2$ and $ \bar {\kappa } = \bar {l}_0\beta ^2$ .

In order to also incorporate spontaneous curvature effects we have to break the up–down symmetry of our approach. The elastic potential (4.15) cannot accomplish this on its own. In the field of liquid crystals, such a polarisation has already been derived via the piezoelectric effect (Meyer Reference Meyer1969; Helfrich Reference Helfrich1971). Essentially, this involves assuming an asymmetry in the physics of the nematic phase, which exhibits either splay and/or bend structures. Lipid bilayers, with average molecule directions perpendicular to the surface, may likewise exhibit a splay, since parallel alignment cannot be sustained on a curved surface, cf. figure 2, but no bending asymmetry, which would require molecules to possess, on average, a tangential component. Such a splay asymmetry can originate from a variety of mechanisms (Zimmerberg & Kozlov Reference Zimmerberg and Kozlov2005), for instance, if the inner layer consists of a different molecular composition than the outer one (London Reference London2019; Lorent et al. Reference Lorent, Levental, Ganesan, Rivera-Longsworth, Sezgin, Doktorova, Lyman and Levental2020; Bogdanov et al. Reference Bogdanov, Pyrshev, Yesylevskyy, Ryabichko, Boiko, Ivanchenko, Kiyamova, Guan, Ramseyer and Dowhan2020; Enoki & Heberle Reference Enoki and Heberle2023), if the molecular density differs between the two sides (Różycki & Lipowsky Reference Różycki and Lipowsky2015; Sreekumari & Lipowsky Reference Sreekumari and Lipowsky2018; Ghosh et al. Reference Ghosh, Satarifard, Grafmüller and Lipowsky2019; Sreekumari & Lipowsky Reference Sreekumari and Lipowsky2022; Lipowsky et al. Reference Lipowsky, Ghosh, Satarifard, Sreekumari, Zamaletdinov, Różycki, Miettinen and Grafmüller2023), or if binding of proteins, such as coat or scaffolds proteins, is included (Hu, Zhang & Weinan Reference Hu, Zhang and Weinan2007; Kirchhausen Reference Kirchhausen2012; Saleem et al. Reference Saleem, Morlot, Hohendahl, Manzi, Lenz and Roux2015; Kahraman, Langen & Haselwandter Reference Kahraman, Langen and Haselwandter2018; Mironov et al. Reference Mironov, Mironov, Derganc and Beznoussenko2020). This does not violate the apolarity of the average lipid orientations in the normal direction, but rather pertains to the physical conditions that a bilayer might inherently possess. In Govers & Vertogen (Reference Govers and Vertogen1984), Barbero et al. (Reference Barbero, Dozov, Palierne and Durand1986), Alexe-Ionescu (Reference Alexe-Ionescu1993), Mottram & Newton (Reference Mottram and Newton2014) flexoelectric polarisation in terms of Q tensors is already discussed. Considering the normal direction $ \boldsymbol{\nu }$ as the polarisation direction yields

(4.17) \begin{align} \boldsymbol{\nu }{\operatorname {Div}}_{\textsf{C}}\boldsymbol{Q} &= -\frac {3}{2}\mathcal{H}\beta , &\boldsymbol{\nu }(\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q})\operatorname {:}\boldsymbol{Q} &= \frac {3}{4}\mathcal{H}\beta ^2 , &\boldsymbol{\nu }\boldsymbol{Q}{\operatorname {Div}}_{\textsf{C}}\boldsymbol{Q} &= -\frac {3}{2}\mathcal{H}\beta ^2 , \end{align}

for the non-vanishing ( $ E_1,E_5,E_6$ ) terms in Alexe-Ionescu (Reference Alexe-Ionescu1993). As we see below, the thermotropic potential is a fourth-order polynomial in $ \beta$ and $ \mathcal{H}$ , albeit with vanishing coefficients for the mean curvature. Therefore, we extend the expansion of the polarisation up to third order, which, additionally, comprises the non-vanishing terms

(4.18) \begin{align} \boldsymbol{\nu }\boldsymbol{Q}(\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q})\operatorname {:}\boldsymbol{Q} &= \frac {3}{4}\mathcal{H}\beta ^3, &({\operatorname {Tr}}\boldsymbol{Q}^2)\boldsymbol{\nu }{\operatorname {Div}}_{\textsf{C}}\boldsymbol{Q} &= -\frac {9}{4}\mathcal{H}\beta ^3, \nonumber \\ \boldsymbol{\nu }(\boldsymbol{\nabla} _{\!\textsf{C}}\boldsymbol{Q})\operatorname {:}\boldsymbol{Q}^2 &= -\frac {3}{8}\mathcal{H}\beta ^3, &\boldsymbol{\nu }\boldsymbol{Q}^2{\operatorname {Div}}_{\textsf{C}}\boldsymbol{Q} &= -\frac {3}{2}\mathcal{H}\beta ^3. \end{align}

Therefore, the polarised elastic energy, up to forth polynomial order, respectively third order in $ \beta$ , is of the form

(4.19) \begin{align} \int _{\mathcal{S}} \left (k_1 + \frac {k_2}{2}\beta + \frac {k_3}{3}\beta ^2\right )\mathcal{H}\beta {\,{\textrm d}\mathcal{S}}, \end{align}

where $ k_1,k_2,k_3\in {\mathbb{R}}$ are the polarisation parameters. Order polarisation in terms of $\boldsymbol{\nabla }\!\beta$ is not included as a consequence. Eventually, we approach the polarised elastic energy by

(4.20) \begin{align} \mathfrak{U}_{{el}} &:= \frac {l_1}{2} \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2{}{ } + \int _{\mathcal{S}} \frac {l_0}{2} \mathcal{H}^2\beta ^2 - \bar {l}_0 \mathcal{K} \beta ^2 + \frac {k_3}{3}\mathcal{H}\beta ^3 + \frac {k_2}{2} \mathcal{H}\beta ^2 + k_1\mathcal{H}\beta {\text {d}\mathcal{S}} . \end{align}

Using (4.1) the thermotropic energy becomes

(4.21) \begin{align} \mathfrak{U}_{{TH}} &= \int _{\mathcal{S}} a {\operatorname {Tr}}\boldsymbol{Q}^2 + \frac {2b}{3}{\operatorname {Tr}}\boldsymbol{Q}^3 + c{\operatorname {Tr}}\boldsymbol{Q}^4 {\text {d}\mathcal{S}}\nonumber \\ &= \int _{\mathcal{S}} \frac {3a}{2} \beta ^2 + \frac {b}{2}\beta ^3 +\frac {9c}{8}\beta ^4{\text {d}\mathcal{S}} = \int _{\mathcal{S}} \frac {A}{2} \beta ^2 + \frac {B}{3}\beta ^3 + \frac {C}{4}\beta ^4{\text {d}\mathcal{S}}, \end{align}

i.e. $ A=3a$ , $ B = {3b}/{2}$ and $ C= {9c}/{2}$ holds for the thermotropic coefficients. Due to the zeroth-order derivative terms, the elastic and thermotropic energy should not be considered separately. The polarised surface Landau–de Gennes energy now reads

(4.22) \begin{align} \mathfrak{U}_{{LdG}} &:= \mathfrak{U}_{{el}} + \mathfrak{U}_{{TH}}\nonumber \\ &= \frac {l_1}{2} \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2 ({T} \mathcal{S})}^2\nonumber \\ &\quad + \int _{\mathcal{S}} k_1\mathcal{H}\beta + \frac {A + k_2 \mathcal{H} + l_0 \mathcal{H}^2 - 2\bar {l}_0\mathcal{K}}{2} \beta ^2 + \frac {B+k_3\mathcal{H}}{3}\beta ^3 + \frac {C}{4}\beta ^4 {\text {d}\mathcal{S}}. \end{align}

Note that this potential has a minimum for $ l_1 \ge 0$ , $ l_0 \ge \bar {l}_0 \ge 0$ , $ C\gt 0$ and $ k_3 \in ( -({3}/{2})\sqrt {C (2l_0 - \bar {l}_0 )}, ({3}/{2})\sqrt {C (2l_0 - \bar {l}_0 )} )$  (A8), see Appendix A.

The material parameters in (4.22) appear to be somewhat unintuitive, as they represent purely formal expansion parameters. Therefore, we aim to reformulate (4.22) in the form

(4.23) \begin{align} \mathfrak{U}_{{LH}} &:= \frac {3L}{4} \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2+ \int _{\mathcal{S}} \frac {f_{\kappa }(\beta )}{2}\big ( \mathcal{H} - f_{\mathcal{H}_{0},\hat {\mathcal{H}}_{0}}(\beta ) \big )^2 -f_{\bar {\kappa }}(\beta )\mathcal{K} + f_{\varpi ,\hat {a}}(\beta ) {\text {d}\mathcal{S}}, \end{align}

such that $f_{\kappa }$ , $f_{\mathcal{H}_{0},\hat {\mathcal{H}}_{0}}$ , $f_{\bar {\kappa }}$ and $f_{\varpi ,\hat {a}}$ are polynomials in $\beta$ , (4.22) is satisfied as generally as possible and $f_{\kappa } ( ({2}/{3}) ) = \kappa$ , $f_{\mathcal{H}_{0},\hat {\mathcal{H}}_{0}} ( ({2}/{3}) ) = \mathcal{H}_{0}$ , $f_{\bar {\kappa }} ( ({2}/{3}) ) = \bar {\kappa }$ , and $f_{\varpi ,\hat {a}} ( ({2}/{3}) ) = -\varpi$ (local minimum) hold at the fully ordered state. By providing a sufficient polynomial ansatz, incorporating the required conditions and comparing coefficients, we obtain the quadratic curvature-elastic polynomials $ f_{\kappa }(\beta ) = ({9}/{4})\kappa \beta ^2$ and $f_{\bar {\kappa }}(\beta ) = ({9}/{4})\bar {\kappa }\beta ^2$ , linear spontaneous curvature polynomial $ f_{\mathcal{H}_{0},\hat {\mathcal{H}}_{0}}(\beta ) = \hat {\mathcal{H}}_{0} + ({3}/{2}) ( \mathcal{H}_{0} - \hat {\mathcal{H}}_{0} )\beta$ and double-well potential

(4.24) \begin{align} f_{\varpi ,\hat {a}}(\beta ) &= \frac {9}{4}\beta ^2\left ( \frac {\hat {a}}{9}\left ( \frac {2}{3}-\beta \right )^2 - \frac {3\varpi }{2}\beta \left ( 4-\frac {9}{2}\beta \right ) \right ) \text{;} \end{align}

see figure 3. Similar linear spontaneous curvature polynomials have been considered to interpolate between spontaneous curvature values for coexisting, e.g. liquid-ordered and liquid-disordered, phases (Seifert Reference Seifert1993; Taniguchi Reference Taniguchi1996; Bachini et al. Reference Bachini, Krause, Nitschke and Voigt2023a ; Sischka & Voigt Reference Sischka and Voigt2025).

Figure 3.

Energy density plots of the double-well potential (4.24). In (a) plot we stipulate $ \hat {a}=0$ and choose some values for $ \varpi$ . Their additive inverse equal the minimum values at the fully ordered state $ \beta = ({2}/{3})$ . In (b) we use $ \varpi =1$ and specify some values for $ \hat {a}$ . Only $ \hat {a}=0$ yields an inflection point at the isotropic state $ \beta =0$ instead of a minimum for $ \hat {a} \gt 0$ .

As a consequence, the polarised surface Landau–de Gennes energy becomes

(4.25) \begin{align} \mathfrak{U}_{{LH}} &= \frac {3}{4} L \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2\\ &\quad + \int _{\mathcal{S}} \frac {9}{4}\beta ^2\left ( \frac {\kappa }{2} \left ( \mathcal{H} - \left (\hat {\mathcal{H}}_{0} + \frac {3}{2}\big ( \mathcal{H}_{0} - \hat {\mathcal{H}}_{0} \big )\beta \right ) \right )^2 - \bar {\kappa }\mathcal{K} \right ) + f_{\varpi ,\hat {a}}(\beta )\ {\text {d}\mathcal{S}},\nonumber \end{align}

and we name it surface LH energy. Alternative approaches beyond this resulting linear interpolation of the spontaneous curvature with respect to a scalar order are conceivable; see, e.g. Bartels et al. (Reference Bartels, Dolzmann, Nochetto and Raisch2012). Note that the surface LH energy (4.25) involves only seven material parameters (listed in table 3), instead of nine as in the polarised surface Landau–de Gennes energy (4.22). Its material parameters are uniquely determined as follows:

(4.26) \begin{align} \begin{aligned} A &= \frac {9\kappa }{4}\hat {\mathcal{H}}_{0}^2 + \frac {2}{9}\hat {a}, & k_1 &= 0 , & l_0 &= \frac {9\kappa }{4}, \\ B &= \frac {81\kappa }{8} \big ( \mathcal{H}_{0} - \hat {\mathcal{H}}_{0} \big )\hat {\mathcal{H}}_{0} - \frac {2\hat {a} + 81\varpi }{2}, & k_2 &= -\frac {9\kappa }{2} \hat {\mathcal{H}}_{0}, & \bar {l}_0 &= \frac {9\bar {\kappa }}{4} ,\\ C &= \frac {81\kappa }{8} \big ( \mathcal{H}_{0} - \hat {\mathcal{H}}_{0} \big )^2 + \frac { 4\hat {a} + 243\varpi }{4}, & k_3 &= -\frac {81\kappa }{8} \big ( \mathcal{H}_{0} - \hat {\mathcal{H}}_{0} \big ) , & l_1 &= \frac {3 L }{2}. \end{aligned} \end{align}

The restriction of $ k_3$ yields now the restriction (A10) of the double-well parameter $ \hat {a}$ and $ \varpi$ ; see Appendix A. It should be noted that the $C=0$ case is included here, although not discussed in Appendix A. Nevertheless, this yields $\mathcal{H}_{0} = \hat {\mathcal{H}}_{0}$ and $ \hat {a} = \varpi = 0$ , and thus,

(4.27) \begin{align} & \mathfrak{U}_{{LH}}\vert _{\mathcal{H}_{0} = \hat {\mathcal{H}}_{0}, \hat {a} = 0} \\ &= \frac {3}{4} L \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2 + \int _{\mathcal{S}} \frac {9}{4}\beta ^2\left ( \frac {\kappa }{2} \left ( \mathcal{H} - \mathcal{H}_{0} \right )^2 - \bar {\kappa }\mathcal{K} - \frac {3\varpi }{2}\beta \left ( 4-\frac {9}{2}\beta \right ) \right ) {\text {d}\mathcal{S}}, \nonumber \end{align}

which remains bounded below even for $\varpi = 0$ . The special case (4.27) essentially states that the spontaneous curvature $\mathcal{H}_{0} = \hat {\mathcal{H}}_{0}$ does not depend on the scale order $\beta$ , and that the isotropic state ( $\beta \equiv 0$ ) is possible but not stable due to $ \hat {a} = 0$ . By contrast, another special case occurs when no spontaneous curvature $\hat {\mathcal{H}}_{0} = 0$ is present in the isotropic state, i.e.

(4.28) \begin{align} & \mathfrak{U}_{{LH}}\vert _{\hat {\mathcal{H}}_{0}=0,\hat {a}=0} \\ & = \frac {3}{4} L \left \| \boldsymbol{\nabla }\!\beta \right \|_{\textrm{L}^2({T} \mathcal{S})}^2+ \int _{\mathcal{S}} \frac {9}{4}\beta ^2\left ( \frac {\kappa }{2} \left ( \mathcal{H} - \frac {3}{2} \mathcal{H}_{0}\beta \right )^2 - \bar {\kappa }\mathcal{K} - \frac {3\varpi }{2}\beta \left ( 4-\frac {9}{2}\beta \right ) \right ) {{\textrm d}\mathcal{S}} \text{;} \nonumber \end{align}

see figure 4. It should be noted that $ \hat {\mathcal{H}}_{0}$ and $ \hat {a}$ become increasingly insignificant near the ordered state ( $\beta \approx ({2}/{3})$ ) anyway.

Figure 4.

Energy density plots of the energy (4.28) ( $\hat {\mathcal{H}}_0=\hat {a}=0$ ) with $L=\bar {\kappa }=0$ . We stipulate $\kappa =5$ and $\mathcal{H}_{0}=\varpi =1$ . In panel (a) we fix the mean curvature $\mathcal{H}$ for some fixed values around $\mathcal{H}_{0}=1$ . Additional we add the $\kappa =0$ case, which does not depend on $\mathcal{H}$ and leads $\beta$ to the fully ordered state $ \beta = ({2}/{3})$ . Panel (b) shows the dependency with respect to $ \mathcal{H}$ for some fixed $\beta$ s.

The converse of material parameter relation (4.26) is not uniquely determined, due to the reduction of the amount of parameters. In table 4 we give an overview of these possible relations for the two special cases above. However, in order to address phase coexistence and account for the different properties of the liquid-ordered and liquid-disordered phases, as in Baumgart et al. (Reference Baumgart, Hess and Webb2003), these special cases are not sufficient and the full set of parameters, allowing for $\mathcal{H}_0 \neq \hat {\mathcal{H}}_0$ , are required.

Table 4.

Inverse of material parameter relations (4.26) for the special cases (4.27) (left column) and (4.28) (right column). The middle column holds for both cases. Since the number of material parameters is reduced from 9 to 5, four constraints on the parameters follow. The relations are not unique due to some of these constraints.

It should also be noted that the fully ordered regime ( $ \beta = ({2}/{3})$ ) yields the common Helfrich energy

(4.29) \begin{align} \mathfrak{U}_{{LH}}\vert _{\beta \equiv \frac {2}{3}} &= \int _{\mathcal{S}} \frac {\kappa }{2} \left ( \mathcal{H} - \mathcal{H}_0 \right )^2 - \bar {\kappa }\mathcal{K} - \varpi {\text {d}\mathcal{S}} \end{align}

with a constant offset ( $-\varpi$ ), which has not any impact on variations for inextensible materials.

The derivation of the governing equations from the LH energy (4.25) and additional dynamical contributions, such as linear momentum and viscous forces, is carried out directly via the Lagrange–d’Alembert principle in Appendix B.

Figure 5.

Evolution of the surface Beris–Edwards model (blue), the Navier–Stokes–Helfrich model with $\mathcal{H}_0 = 0$ (apricot), the hydrodynamic surface LH model (green) and the Navier–Stokes–Helfrich model with $\mathcal{H}_0 = -1.77$ (red). The parameters used are stated in Apppendix F. Shown is a double logarithmic plot of the time evolution of the Helfrich energy $\mathfrak{U}_H = \int _{\mathcal{S}} ({\kappa }/{2}) \mathcal{H}^2\,d\mathcal{S}$ , with $ {\kappa }/{2} = L = 0.1$ , together with snapshots for different levels of $\mathfrak{U}_H$ . Here $\mathfrak{U}_H=19.96$ corresponds to the initial condition and $\mathfrak{U}_H=5.03$ to the equilibrium configuration. In addition, we show intermediate states for $\mathfrak{U}_H=17, 11.5$ and $10$ for each model. The frames of the images follow the same colour coding as the plots. On the surface we show the value of the orientation field $\beta$ in colour, ranging from $\beta =0.35$ in light yellow to $\beta =0.67$ in dark orange. While the shape corresponds to the same Helfrich energy and, therefore, only slightly differ, their appearance in time differs between the models with a slower shape evolution for variable $\beta$ far away from the equilibrium state.