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Microphysical derivation of the Canham–Helfrich free-energy density

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Abstract

The Canham–Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham–Helfrich free-energy density. Our derivation (1) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (2) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.

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Notes

  1. See (26) and (27) for the definitions of the mean \(H\) and Gaussian \(K\) curvatures.

References

  • Canham PB (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theor Biol 26:61–81

    Article  Google Scholar 

  • Cosserat E, Cosserat F (1909) Théorie des Corps Deformables. Herman et fils, Paris

    Google Scholar 

  • Döbereiner HG, Selchow O, Lipowsky R (1999) Spontaneous curvature of fluid vesicles induced by trans-bilayer sugar asymmetry. Eur Biophys J 28:174–178

    Article  Google Scholar 

  • Föppl A (1907) Vorlesungen über technische Mechanik, Bd. 5, Die wichtigsten Lehren der höheren Elastizitätstheorie. Teubner, Leipzig

    Google Scholar 

  • Germain S (1821) Recherches sur la Théorie des Surfaces Élastique. Huzard-Courcier, Paris

    Google Scholar 

  • Gurtin ME, Fried E, Anand L (2010) The Mechanics and Thermodynamics of Continua. Cambridge University Press, New York

    Book  Google Scholar 

  • Helfrich W (1973) Elastic properties of lipid bilayers: Theory and possible experiments. Zeitschrift für Naturforschung 28c:693–703

    Google Scholar 

  • Keller JB, Merchant GJ (1991) Flexural rigidity of a liquid surface. J Stat Phys 63:1039–1051

    Article  MathSciNet  Google Scholar 

  • Lasic DD (1988) The mechanism of liposome formation. A review. Biochem J 256:1–11

    Google Scholar 

  • Lipowsky R (1990) Shape fluctuations and critical phenomena. In: van Beijeren H (ed) Fundamental Problems in Statistical Mechanics VII. North-Holland, Amsterdam, pp 139–170

    Google Scholar 

  • Ljunggren S, Eriksson JC (1985) Comments on the origin of the curvature elasticity of vesicle bilayers. J Colloid Interface Sci 107:138–145

    Google Scholar 

  • Luisi PL, Walade P (2000) Giant Vesicles. Wiley, Chichester

    Google Scholar 

  • Maïer W, Saupe A (1958) Eine einfache molekulare Theorie des nematischen Kristallinflüssigen Zustands. Zeitschrift für Naturforschg 13a:564–566

    Google Scholar 

  • Maleki M, Seguin B, Fried E (2012) Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature. Biomech Model Mechanobiol. doi:10.1007/s10237-012-0459-7

  • McMahon HT, Gallop JL (2005) Membrane curvature and mechanisms of dynamic cell membrane remodelling. Nature 438:590–596

    Article  Google Scholar 

  • Paunov VN, Sandler SI, Kaler EW (2000) A simple molecular model for the spontaneous curvature and the bending constants of nonionic surfactant monolayers at the oil/water interface. Langmuir 16:8917–8925

    Article  Google Scholar 

  • Poisson SD (1812) Mémoire sur les surfaces élastiques. Mémoire de Classe des Sciences Mathématiques et Physiques de I’Institut de France 2nd pt, pp 167–225

  • Sackmman E (1994) The seventh Datta Lecture. Membrane bending energy concept of vesicle- and cell-shapes and shape-transitions. FEBS Lett 346:3–16

    Article  Google Scholar 

  • Safran SA (1994) Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. Addison-Wesley, Reading

    Google Scholar 

  • Seguin B, Fried E (2012) Statistical foundations of liquid-crystal theory I. Discrete systems of rod-like molecules. Arch Ration Mech Anal 206:1039–1072

    Article  MATH  MathSciNet  Google Scholar 

  • Seifert U (1997) Configurations of fluid membranes and vesicles. Adv Phys 46:13–137

    Article  Google Scholar 

  • Steigmann DJ (1999) Fluid films with curvature elasticity. Arch Ration Mech Anal 150:127–152

    Article  MATH  MathSciNet  Google Scholar 

  • von Kámán Th (1910) Festigkeitsprobleme im Maschinenbau. In: Klein F, Müller C (eds) Encyklopädia der mathematischen Wissenschaften IV/4. Teubner, Berlin, pp 311–385

    Google Scholar 

  • Winterhalter M, Helfrich W (1988) Effect of surface charge on the curvature elasticity of membranes. J Phys Chem 92:6865–6867

    Google Scholar 

  • Yuan H (2010) A solvent-free coarse-grained model for biological and biomimetic fluid membranes. Ph. D. Thesis, Pennsylvania State University

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Acknowledgments

We thank Mohsen Maleki for very fruitful discussions.

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Correspondence to Eliot Fried.

Appendix

Appendix

In these appendices we present the necessary linear algebra and differential geometry necessary to carry out the expansions needed in Sect. 2. We also include some details of the expansion.

1.1 A.1 Linear algebra

Here we recall some basic properties of tensors. Let \(\mathcal{V }\) be a finite-dimensional vector space equipped with an inner-product. For the purposes of this work, a tensor \(\mathbf{T}\) of order \(m\) is a multilinear mapping that takes in \(m\) vectors and gives a real number. A vector can be viewed as a first-order tensor and vice versa. A linear mapping from \(\mathcal{V }\) to itself can be viewed as a second-order tensor and vice versa; we denote the set of all such linear mappings by \({\mathop {\text{ Lin}}\,}(\mathcal{V },\mathcal{V })\). Other such statements regarding higher-order tensors can also be made. For example, a fourth-order tensor may be viewed as a bilinear mapping from \(\mathcal{V }\) to the space of second-order tensors and vice versa.

Given a tensor \(\mathbf{T}\) of order \(m\) and a tensor \(\mathbf{S}\) of order \(n\), it is possible to form their tensor product \(\mathbf{T}\otimes \mathbf{S}\), which is a tensor of order \(m+n\) defined by

$$\begin{aligned}&\mathbf{T}\otimes \mathbf{S}(\mathbf{v}_1,\ldots ,\mathbf{v}_m,\mathbf{v}_{m+1},\ldots ,\mathbf{v}_{m+n})\nonumber \\&\quad =\mathbf{T}(\mathbf{v}_1,\ldots ,\mathbf{v}_m)\mathbf{S}(\mathbf{v}_{m+1},\ldots ,\mathbf{v}_{m+n})\quad \mathrm{for \ all}\ \ \mathbf{v}_i\in \mathcal{V },\ i\in \{1,\ldots ,m+n\}.\quad \quad \quad \end{aligned}$$
(20)

The tensor product is associative, meaning that \((\mathbf{T}\otimes \mathbf{S})\otimes \mathbf{R}=\mathbf{T}\otimes (\mathbf{S}\otimes \mathbf{R})\) and, thus, the use of parenthesis in this context may be dropped.

Just as it is possible to take the inner-product of two vectors, it is possible to take the inner-product of two tensors of the same order. The details of how this can be done in general will not be spelled out here. However, there are two useful identities that we will need regarding the inner-product of tensors. Namely, for any vector \(\mathbf{z}\) and second-order tensor \(\mathbf{L}\), the following hold:

$$\begin{aligned} (\mathbf{z}\otimes \mathbf{z})\cdot \mathbf{L}&= \mathbf{z}\cdot \mathbf{L}\mathbf{z}\end{aligned}$$
(21)
$$\begin{aligned} (\mathbf{z}\otimes \mathbf{z}\otimes \mathbf{z}\otimes \mathbf{z})\cdot (\mathbf{L}\otimes \mathbf{L})&= (\mathbf{z}\cdot \mathbf{L}\mathbf{z})^2 \end{aligned}$$
(22)

Assume the dimension of \(\mathcal{V }\) is \(2\). The following lemma, which can be established using polar coordinates and the integral gradient theorem, involving second- and fourth-order tensors will also be needed. Here, \(\mathbf{1}\) denotes the identity mapping on \(\mathcal{V }\) and, thus, is a second-order tensor.

Lemma 1

Let \(g:\mathbb P \longrightarrow \mathbb R \) be a continuous function. Then

$$\begin{aligned} \int \limits _{|\mathbf{z}|\le \ell }g(|\mathbf{z}|)\mathbf{z}\otimes \mathbf{z}d a_{\mathbf{z}}&= \left(\pi \int \limits _0^\ell g(r)r^3\,d r\right) \mathbf{1},\end{aligned}$$
(23)
$$\begin{aligned} \int \limits _{|\mathbf{z}|\le \ell }g(|\mathbf{z}|)\mathbf{z}\otimes \mathbf{z}\otimes \mathbf{z}\otimes \mathbf{z}d a_{\mathbf{z}}&= \left(\frac{3\pi }{4}\int \limits _0^\ell g(r)r^5 d r\right)\mathbf {1}\otimes \mathbf {1}. \end{aligned}$$
(24)

1.2 A.2 Geometry of surfaces

Consider a surface \(\mathcal{\mathcal S }\) in a three-dimensional Euclidean point space \(\mathcal{\mathcal E }\) with associated vector space \(\mathcal{V }\). Given a point \(y\) in \(\mathcal{\mathcal S }\), let \(\mathcal{\mathcal T }_y\mathcal{\mathcal S }\) denote the tangent space of \(\mathcal{\mathcal S }\) at \(y\). Let \(\mathbf{n}\) denote a mapping that determines a unit-normal to the surface at each point. Given a mapping \(h:\mathcal{\mathcal S }\longrightarrow \mathcal{\mathcal W }\) defined on the surface that takes values in some vector space \(\mathcal{\mathcal W }\), the surface gradient \(\nabla ^\mathcal{\mathcal S }h\) of \(h\) can be defined by

$$\begin{aligned} \nabla ^\mathcal{\mathcal S }_x h:=\nabla _x h^e (\mathbf 1 -\mathbf{n}(x)\otimes \mathbf{n}(x))\quad \mathrm{for \ all}\ \ x\in \mathcal{\mathcal S }, \end{aligned}$$
(25)

where \(h^e\) is an extension of \(h\) to a neighborhood of \(x\) and \(\nabla _x h^e\) is the classical three-dimensional gradient of this extension at \(x\). It can be shown that the definition of the surface gradient is independent of the extension used on the right-hand side of (25).

Of particular interest is the opposite \(\mathbf{L}:=-\nabla ^\mathcal{\mathcal S }\mathbf{n}\) of the surface gradient \(\nabla ^\mathcal{\mathcal S }\mathbf{n}\) of \(\mathbf{n}\)—called the curvature tensor, which is a second-order tensor field defined on \(\mathcal{\mathcal S }\). The curvature tensor is symmetric and has two scalar invariants: the mean curvature \(H\) and Gaussian curvature \(K\) defined by

$$\begin{aligned} H:&= \frac{1}{2}\mathrm{tr}\,\mathbf{L},\end{aligned}$$
(26)
$$\begin{aligned} K:&= \frac{1}{2}[(\mathrm{tr}\,\mathbf{L})^2-\mathrm{tr}(\mathbf{L}^2)]. \end{aligned}$$
(27)

If \(\kappa _1\) and \(\kappa _2\) are the two nontrivial eigenvalues of \(\mathbf{L}\), often called the principle curvatures, then

$$\begin{aligned} H&= \frac{1}{2}(\kappa _1+\kappa _2),\end{aligned}$$
(28)
$$\begin{aligned} K&= \kappa _1\kappa _2. \end{aligned}$$
(29)

A useful identity involving the curvature tensor is provided by the following result.

Lemma 2

For \(\mathbf{z}\) in \(\mathcal{V }\) and \(x\) in \(\mathcal{\mathcal S }\),

$$\begin{aligned} \mathbf{n}(x)\cdot (\nabla _x^\mathcal{\mathcal S }\mathbf{L}\mathbf{z})\mathbf{z}=|\mathbf{L}(x)\mathbf{z}|^2. \end{aligned}$$
(30)

Proof

Using the definition and symmetry of \(\mathbf{L}\), we have

$$\begin{aligned} \mathbf{n}(x)\cdot \mathbf{L}(x)\mathbf{z}=0\quad \mathrm{for \ all}\ \ x\in \mathcal{\mathcal S }. \end{aligned}$$
(31)

Taking the surface gradient of both sides of (31) in the direction \(\mathbf{z}\), using the product rule, and evaluating the result at \(x\) yields

$$\begin{aligned} -\mathbf{L}(x)\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z}+\mathbf{n}(x)\cdot (\nabla _x^\mathcal{\mathcal S }\mathbf{L}\mathbf{z})\mathbf{z}=0, \end{aligned}$$
(32)

which is (30).

Fig. 2
figure 2

Depiction of a local curving for a surface \(\mathcal{\mathcal S }\) at \(x\)

From here on fix a point \(x\) in \(\mathcal{\mathcal S }\). It is possible to parameterize a small neighborhood of \(\mathcal{\mathcal S }\) at \(x\) using a local curving \(f\). Put \(\mathcal{\mathcal T }:=T_x\mathcal{\mathcal S }\) and \(\mathcal{\mathcal N }:=(T_x\mathcal{\mathcal S })^\perp \), and let \(\mathcal{\mathcal U }\) be an open neighborhood of \(\mathbf 0 \) in \(\mathcal{V }\). A local curving \(f:\mathcal{\mathcal U }\longrightarrow \mathcal{\mathcal E }\) of \(\mathcal{\mathcal S }\) at \(x\) has the following properties (Fig. 2):

  1. 1.

    \(f(\mathbf 0 )=x\),

  2. 2.

    \(f(\mathbf{u})\in \mathcal{\mathcal S }\) for all \(\mathbf{u}\in \mathcal{\mathcal T }\cap \mathcal{\mathcal U }\),

  3. 3.

    \(f(\mathbf{u})-(x+\mathbf{u})\in \mathcal{\mathcal N }\) for all \(\mathbf{u}\in \mathcal{\mathcal T }\cap \mathcal{\mathcal U }\),

  4. 4.

    \(f(\mathbf{u}+\mathbf{w})=f(\mathbf{u})+\mathbf{w}\) for all \(\mathbf{u}\in \mathcal{\mathcal T }\cap \mathcal{\mathcal U }\) and \(\mathbf{w}\in \mathcal{\mathcal N }\cap \mathcal{\mathcal U }\),

  5. 5.

    \(\nabla _\mathbf{0 } f=\mathbf 1 \).

The gradients of \(f\) at \(\mathbf 0 \) describe the local shape of the surface at \(x\). Consider the mappings

$$\begin{aligned}&\mathbf{F}:\mathcal{\mathcal U }\longrightarrow {\mathop {\text{ Lin}}\,}(\mathcal{V },\mathcal{V }),\end{aligned}$$
(33)
$$\begin{aligned}&{\varvec{\Lambda }}:\mathcal{V }\times \mathcal{V }\longrightarrow \mathcal{V },\end{aligned}$$
(34)
$$\begin{aligned}&{\varvec{\Gamma }}:\mathcal{V }\times \mathcal{V }\longrightarrow {\mathop {\text{ Lin}}\,}(\mathcal{V },\mathcal{V }), \end{aligned}$$
(35)

defined by

$$\begin{aligned} \mathbf{F}:&= \nabla f,\end{aligned}$$
(36)
$$\begin{aligned} {\varvec{\Lambda }}(\mathbf{u},\mathbf{v}):&= [(\nabla _\mathbf{0 }\mathbf{F})\mathbf{u}]\mathbf{v}\quad \mathrm{for \ all}\ \ \mathbf{u},\mathbf{v}\in \mathcal{V },\end{aligned}$$
(37)
$$\begin{aligned} {\varvec{\Gamma }}(\mathbf{u},\mathbf{v})\mathbf{w}:&= ([(\nabla _\mathbf{0 }\nabla \mathbf{F})\mathbf{u}]\mathbf{v})\mathbf{w}\quad \mathrm{for \ all}\ \ \mathbf{u},\mathbf{v},\mathbf{w}\in \mathcal{V }. \end{aligned}$$
(38)

The mapping \(\mathbf{F}\) is a second-order tensor field while the quantities \({\varvec{\Lambda }}\) and \({\varvec{\Gamma }}\) can be viewed as third- and fourth-order tensors, respectively. Using Item 3, it can be shown that

$$\begin{aligned} {\varvec{\Lambda }}(\mathbf{z}_1,\mathbf{z}_2)\in \mathcal{\mathcal N }\quad \mathrm{and}\quad {\varvec{\Gamma }}(\mathbf{z}_1,\mathbf{z}_2)\mathbf{z}_3\in \mathcal{\mathcal N }\quad \mathrm{for \ all}\ \ \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3\in \mathcal{\mathcal T }; \end{aligned}$$
(39)

further, using Item 4, it can be shown that

$$\begin{aligned} {\varvec{\Lambda }}(\mathbf{z}_1,\mathbf{n})=\mathbf 0 \quad \mathrm{and}\quad {\varvec{\Gamma }}(\mathbf{z}_1,\mathbf{z}_2)\mathbf{n}=\mathbf 0 \quad \mathrm{for \ all}\ \ \mathbf{z}_1,\mathbf{z}_2\in \mathcal{\mathcal T }. \end{aligned}$$
(40)

It follows from (39)–(40) that

$$\begin{aligned} \mathrm{tr}\, {\varvec{\Lambda }}\mathbf{z}_1=0\quad \mathrm{and}\quad \mathrm{tr}({\varvec{\Gamma }}(\mathbf{z}_1,\mathbf{z}_2))=0\quad \mathrm{for \ all}\ \ \mathbf{z}_1,\mathbf{z}_2\in \mathcal{\mathcal T }. \end{aligned}$$
(41)

The curvature tensor \(\mathbf{L}(x)\) is related to \({\varvec{\Lambda }}\) by

$$\begin{aligned} {\varvec{\Lambda }}(\mathbf{z}_1,\mathbf{z}_2)=\mathbf{n}(\mathbf{z}_1\cdot \mathbf{L}(x)\mathbf{z}_2)\quad \mathrm{for \ all}\ \ \mathbf{z}_1,\mathbf{z}_2\in \mathcal{\mathcal T }. \end{aligned}$$
(42)

It follows from (40) and (42) that

$$\begin{aligned} {\varvec{\Lambda }}\mathbf{z}=\mathbf{n}\otimes (\mathbf{L}(x)\mathbf{z})\quad \mathrm{for \ all}\ \ \mathbf{z}\in \mathcal{\mathcal T }. \end{aligned}$$
(43)

1.3 A.3 Useful approximations

In this subsection we delineate the key expansions that will be needed in the asymptotic expansion of the energy. The three basic expansions, which follow from the results stated in Appendix A.2, are

$$\begin{aligned} f(\epsilon \mathbf{z})&= x+\epsilon \mathbf{z}+\frac{\epsilon ^2}{2}(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})\mathbf{n}(x)+\text{ o}(\epsilon ^2),\end{aligned}$$
(44)
$$\begin{aligned} \mathbf{n}(f(\epsilon \mathbf{z}))&= \mathbf{n}-\epsilon \mathbf{L}(x)\mathbf{z}-\frac{\epsilon ^2}{2}[(\nabla _x^\mathcal{\mathcal S }\mathbf{L})\mathbf{z}]\mathbf{z}+\text{ o}(\epsilon ^2),\end{aligned}$$
(45)
$$\begin{aligned} \mathbf{F}(\epsilon \mathbf{z})&= \mathbf 1 +\epsilon \mathbf{n}(x)\otimes \mathbf{L}(x)\mathbf{z}+\frac{\epsilon ^2}{2}{\varvec{\Gamma }}(\mathbf{z},\mathbf{z})+\text{ o}(\epsilon ^2). \end{aligned}$$
(46)

Using (30) and (44)–(45), it transpires that

$$\begin{aligned} \epsilon ^{-2}|x-f(\epsilon \mathbf{z})|^2&= |\mathbf{z}|^2+\frac{\epsilon ^2}{4}(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})^2+\text{ o}(\epsilon ^2),\end{aligned}$$
(47)
$$\begin{aligned} (x-f(\epsilon \mathbf{z}))\cdot \mathbf{n}(x)&= -\frac{\epsilon ^2}{2}(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})+\text{ o}(\epsilon ^2),\end{aligned}$$
(48)
$$\begin{aligned} (x-f(\epsilon \mathbf{z}))\cdot \mathbf{n}(f(\epsilon \mathbf{z}))&= \frac{\epsilon ^2}{2}(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})+\text{ o}(\epsilon ^2),\end{aligned}$$
(49)
$$\begin{aligned} \mathbf{n}(f(\epsilon \mathbf{z}))\cdot \mathbf{n}(x)&= 1-\frac{\epsilon ^2}{2}|\mathbf{L}(x)\mathbf{z}|^2+\text{ o}(\epsilon ^2). \end{aligned}$$
(50)

It can be shown that the cofactor \(\mathbf{F}^\text{ c}\) of \(\mathbf{F}\) is given by (Gurtin et al. 2010)

$$\begin{aligned} \mathbf{F}^\text{ c}:=\left[\mathbf{F}^2-(\mathrm{tr}\,\mathbf{F})\mathbf{F}+\frac{1}{2}[(\mathrm{tr}\,\mathbf{F})^2-\mathrm{tr}(\mathbf{F}^2)]\mathbf 1 \right]^{\top }. \end{aligned}$$
(51)

Using (30), (41), (45)–(46), and (51), it follows that

$$\begin{aligned} \mathbf{n}(f(\epsilon \mathbf{z}))\cdot \mathbf{F}^\text{ c}(\epsilon \mathbf{z})\mathbf{n}(x)=1+\frac{\epsilon ^2}{2}|\mathbf{L}(x)\mathbf{z}|^2+\text{ o}(\epsilon ^2). \end{aligned}$$
(52)

Given a scalar-valued function \(W\) defined only on the surface, if we write

$$\begin{aligned} W^{\prime }(x):=\nabla _x^\mathcal{\mathcal S }W\quad \mathrm{and}\quad W^{\prime \prime }(x):=\nabla _x^\mathcal{\mathcal S }\nabla ^\mathcal{\mathcal S }W, \end{aligned}$$
(53)

then

$$\begin{aligned} W(f(\epsilon \mathbf{z}))=W(x)+\epsilon W^{\prime }(x)\mathbf{z}+\frac{\epsilon ^2}{2}\mathbf{z}\cdot W^{\prime \prime }(x)\mathbf{z}+\text{ o}(\epsilon ^2). \end{aligned}$$
(54)

1.4 A.4 Expansion of the energy

Here we use the results of the previous two subsections to perform the expansion of (11) in powers of \(\epsilon \). Since the domains of integration of the integrals (7)–(10) depend on \(\epsilon \) through \(d\), this cannot presently be achieved. To proceed, we introduce a change of variables that transfers this dependence to the integrand. To achieve this, we parameterize \(\mathcal{\mathcal S }_d(x)\) using the notion of a local curving \(f\) introduced in Appendix A.2, which is made possible by (3). In particular, it is convenient to introduce the set

$$\begin{aligned} \mathcal{\mathcal T }_\ell (0):=\{\mathbf{z}\in T_x\mathcal{\mathcal S }:|\mathbf{z}|\le \ell \}, \end{aligned}$$
(55)

where \(T_x\mathcal{\mathcal S }\) denotes the tangent space of \(\mathcal{\mathcal S }\) at \(x\). Focusing on (7) and using using the abbreviation (36), the change of variables \(y=f(\epsilon \mathbf{z})\) yields

$$\begin{aligned} \psi _{11}(x)&=\epsilon ^{-2}\int \limits _{\mathcal{\mathcal T }_\ell (0)}\varPhi _{11}^d (x-f(\epsilon \mathbf{z}),\mathbf{n}(x),\mathbf{n}(f(\epsilon \mathbf{z})))W_{1}(x)W_{1}(f(\epsilon \mathbf{z}))\nonumber \\&\quad \quad \qquad \qquad \cdot \mathbf{n}(f(\epsilon \mathbf{z}))\cdot \mathbf{F}^\text{ c}(\epsilon \mathbf{z})\mathbf{n}(x)\,\text{ d}a_{\mathbf{z}}. \end{aligned}$$
(56)

Let us focus our attention on the quantity

$$\begin{aligned}&\varPhi _{11}^d(x-f(\epsilon \mathbf{z}),\mathbf{n}(x),\mathbf{n}(f(\epsilon \mathbf{z}))\nonumber \\&\quad =\phi _{11}(\epsilon ^{-2}|x-f(\epsilon \mathbf{z})|^2,(x-f(\epsilon \mathbf{z}))\cdot \mathbf{n}(x), (x-f(\epsilon \mathbf{z}))\cdot \mathbf{n}(f(\epsilon \mathbf{z})),\mathbf{n}(x)\nonumber \\&\quad \quad \cdot \mathbf{n}(f(\epsilon \mathbf{z}))). \end{aligned}$$
(57)

For all \(0\le s\le \ell \), put

$$\begin{aligned} \phi _{11,0}(s):=\phi _{11}(s^2,0,0,1), \end{aligned}$$
(58)

and

$$\begin{aligned} \phi _{11,k}(s):=\frac{\phi _{11}(\xi _1,\xi _2,\xi _3,\xi _4)}{\partial \xi _k} \bigg |_{(\xi _1,\xi _2,\xi _3,\xi _4)=(s^2,0,0,1)}, \quad k\in \{1,2,3,4\}. \end{aligned}$$
(59)

Using the expansions (47)–(50) and the notation (58)–(59), we have

$$\begin{aligned}&\varPhi _{11}^d(x-f(\epsilon \mathbf{z}),\mathbf{n}(x),\mathbf{n}(f(\epsilon \mathbf{z}))\nonumber \\&\quad \quad =\psi _{11,0}(|\mathbf{z}|)+\frac{\epsilon ^2}{2}\left(\frac{1}{2}\phi _{11,1}(|\mathbf{z}|)(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})^2-\phi _{11,2}(|\mathbf{z}|)(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})\right.\nonumber \\&\qquad \quad +\left.\phi _{11,3}(|\mathbf{z}|)(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})+\phi _{11,4}(|\mathbf{z}|)|\mathbf{L}(x)\mathbf{z}|^2\right)+\text{ o}(\epsilon ^2). \end{aligned}$$
(60)

where \(\mathbf{L}=-\nabla ^{\mathcal{\mathcal S }}\mathbf{n}\) denotes the curvature tensor of \(\mathcal{\mathcal S }\) (see Appendix A.2). Combining this expansion with (52) and using the expansion (54) for \(W_1\) yields an expansion for the right-hand side of (56):

$$\begin{aligned} \phi _{11}(x)&= \int \limits _{T_x\mathcal{\mathcal S }_1(0)}\Big (\phi _{11,0}(|\mathbf{z}|)W^2_1(x) +\frac{\epsilon ^2}{2}W_1(x)\big [W_1(x)(\phi _{11,0}(|\mathbf{z}|)\nonumber \\&-\phi _{11,4}(|\mathbf{z}|))|\mathbf{L}(x)\mathbf{z}|^2\!+\phi _{11,0}(|\mathbf{z}|)\mathbf{z}\cdot W^{\prime \prime }_1(x)\mathbf{z}+\!\frac{1}{2}W_1(x)\psi _{11,1}(|\mathbf{z}|)(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})^2\nonumber \\&+W_1(x)(\phi _{11,3}(|\mathbf{z}|)-\psi _{11,2}(|\mathbf{z}|))(\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})\big ]\Big )\,\text{ d}a_{\mathbf{z}}+\text{ o}(\epsilon ^2). \end{aligned}$$
(61)

Similarly, the right-hand sides of (8)–(10) can be expanded in powers of \(\epsilon \), using notation similar to that of (58)–(59). Putting these expansions together, we obtain

$$\begin{aligned} \psi (x)&= \int \limits _{T_\ell (0)}[\phi _{11,0}(|\mathbf{z}|)W^2_1(x)+\phi _{22,0}(|\mathbf{z}|)W^2_2(x)+2\phi _{12,0}(|\mathbf{z}|)W_1(x)W_2(x)]\,\text{ d}a_{\mathbf{z}}\nonumber \\&+\frac{\epsilon ^2}{2}\int \limits _{T_\ell (0)}[W_1^2(x)(\phi _{11,3}(r)-\phi _{11,2}(r))-W_2^2(x)(\phi _{22,3}(r)\nonumber \\&\quad \qquad \qquad -\phi _{22,2}(r))] (\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z}) \,\text{ d}a_{\mathbf{z}}\nonumber \\&+\frac{\epsilon ^2}{2}\int \limits _{T_\ell (0)}[W_1^2(x)(\phi _{11,0}(|\mathbf{z}|)-\phi _{11,4}(|\mathbf{z}|))+W^2_2(x)(\phi _{22,0}(|\mathbf{z}|)\nonumber \\&\qquad \quad \qquad -\phi _{22,4}(|\mathbf{z}|))\!+\!2W_1(x)W_2(x)(\phi _{12,0}(|\mathbf{z}|)\!+\!\phi _{12,4}(|\mathbf{z}|))]|\mathbf{L}(x)\mathbf{z}|^2\,\text{ d}a_{\mathbf{z}}\nonumber \\&+\frac{\epsilon ^2}{2}\int \limits _{T_\ell (0)}[W_1^2(x)\phi _{11,1}(|\mathbf{z}|)+W^2_2(x)\phi _{22,1}(|\mathbf{z}|)\nonumber \\&\qquad \qquad \quad +2W_1(x)W_2(x)\phi _{12,1}(|\mathbf{z}|)](\mathbf{z}\cdot \mathbf{L}(x)\mathbf{z})^2\,\text{ d}a_{\mathbf{z}}\nonumber \\&+\frac{\epsilon ^2}{2}\int \limits _{T_\ell (0)}[(W_1(x)\phi _{11,0}(|\mathbf{z}|)+W_2(x)\phi _{12,0}(|\mathbf{z}|)) (\mathbf{z}\cdot W^{\prime \prime }_1(x)\mathbf{z})\nonumber \\&\qquad \qquad \quad +(W_2(x)\psi _{22,0}+W_1(x)\phi _{12,0}(|\mathbf{z}|))(\mathbf{z}\cdot W^{\prime \prime }_2(x)\mathbf{z})]\,\text{ d}a_{\mathbf{z}}. \end{aligned}$$
(62)

Notice that the first term on the right-hand side of (62) is independent of \(\epsilon \), while the others are proportional to \(\epsilon ^2\). Of the terms involving \(\epsilon \), the first is linear is \(\mathbf{L}\) and the second and third are quadratic in \(\mathbf{L}\). The last term in (62) is independent of \(\mathbf{L}\). Using the identities (21)–(24), the abbreviations

$$\begin{aligned} \psi _0(x):&= 2\pi \int \limits _0^\ell [\phi _{11,0}(r)W^2_1(x)+\phi _{22,0}(r)W^2_2(x)+2\phi _{12,0}(r)W_1(x)W_2(x)]r\,\text{ d}r,\quad \quad \quad \end{aligned}$$
(63)
$$\begin{aligned} A(x):&= \frac{\pi }{2}\int \limits _0^\ell [W_1^2(x)(\phi _{11,3}(r)-\phi _{11,2}(r))-W_2^2(x)(\phi _{22,3}(r)-\phi _{22,2}(r))]r^3\,\text{ d}r,\quad \quad \end{aligned}$$
(64)
$$\begin{aligned} B(x):&= \frac{\pi }{2}\int \limits _0^\ell [W_1^2(x)(\phi _{11,0}(r)-\phi _{11,4}(r))+W^2_2(x)(\phi _{22,0}(r)-\phi _{22,4}(r))\nonumber \\&\qquad \quad +2W_1(x)W_2(x)(\phi _{12,0}(r)+\phi _{12,4}(r))]r^3\,\text{ d}r,\end{aligned}$$
(65)
$$\begin{aligned} C(x):&= \frac{3\pi }{16}\int \limits _0^\ell [W_1^2(x)\phi _{11,1}(r)+W^2_2(x)\phi _{22,1}(r)+2W_1(x)W_2(x)\phi _{12,1}(r)]r^5\, dr,\quad \quad \end{aligned}$$
(66)
$$\begin{aligned} D(x):&= \frac{\pi }{2}\int \limits _0^\ell [(W_1(x)\phi _{11,0}(r)+W_2(x)\phi _{12,0}(r))\mathrm{tr} (W^{\prime \prime }_1(x))\nonumber \\&\qquad \quad +(W_2(x)\phi _{22,0}+W_1(x)\phi _{12,0}(r))\mathrm{tr} (W^{\prime \prime }_2(x))]r^3\,\text{ d}r, \end{aligned}$$
(67)

and, motivated by the grouping in (62), we find (on suppressing explicit dependence on \(x\)) that

$$\begin{aligned} \psi =\psi _0+[ A\,\text{ tr}\, \mathbf{L}+B\,\text{ tr}(\mathbf{L}^2)+C(\text{ tr}\, \mathbf{L})^2+D]\epsilon ^2+\text{ o}(\epsilon ^2). \end{aligned}$$
(68)

Further, using (26)–(27) and the additional abbreviations

$$\begin{aligned} \kappa :&= 8\epsilon ^2(B+C),\end{aligned}$$
(69)
$$\begin{aligned} \bar{\kappa }:&= -2\epsilon ^2B,\end{aligned}$$
(70)
$$\begin{aligned} H_\circ :&= -\frac{A}{4(B+C)},\end{aligned}$$
(71)
$$\begin{aligned} K_\circ :&= \frac{4D(B+C)-A^2}{8B(B+C)}, \end{aligned}$$
(72)

allows us to write the expansion of the right-hand side of (11) asFootnote 1

$$\begin{aligned} \psi =\psi _0+\frac{1}{2}\kappa (H-H_\circ )^2+\bar{\kappa }(K-K_\circ ), \end{aligned}$$
(73)

where the correction of \(\text{ o}(\epsilon ^2)\) has been neglected.

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Seguin, B., Fried, E. Microphysical derivation of the Canham–Helfrich free-energy density. J. Math. Biol. 68, 647–665 (2014). https://doi.org/10.1007/s00285-013-0647-9

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