Crystalline membrane morphology beyond polyhedra

TL;DR

This study explores how magnetic interactions can induce and control complex, switchable morphologies in crystalline membranes, enabling shape manipulation for potential targeted delivery applications.

Contribution

It introduces a novel magnetoelastic membrane system where magnetic fields control membrane shape and symmetry, surpassing traditional polyhedral constraints.

Findings

Magnetic interactions induce concave membrane morphologies.

Symmetry decreases from 5-fold to 1-fold with increasing magnetic field.

Membrane shapes can be reversibly switched between convex and concave.

Abstract

Elastic crystalline membranes exhibit a buckling transition from sphere to polyhedron. However, their morphologies are restricted to convex polyhedra and are difficult to externally control. Here, we study morphological changes of closed crystalline membrane of super-paramagnetic particles. The competition of magnetic dipole-dipole interactions with the elasticity of this magnetoelastic membrane leads to concave morphologies. Interestingly, as the magnetic field strength increases, the symmetry of the buckled membrane decreases from 5-fold to 3-fold, to 2-fold and, finally, to 1-fold rotational symmetry. This gives the ability to switch the membrane morphology between convex and concave shapes with specific symmetry and provides promising applications for membrane shape control in the design of actuatable micro-containers for targeted delivery systems.

Equations

$$H_e=\sum _{e\in \mathbf{E}}\frac{1}{2}k{\left(\left|{\mathbf{r}}_1^e-{\mathbf{r}}_2^e\right|-l_0\right)}^2+\sum _{e\in \mathbf{E}}\frac{1}{2}\tilde{\kappa }{\left|{\mathbf{n}}_1^e-{\mathbf{n}}_2^e\right|}^2$$

$$H_{m}=-\frac{\mu_{0}}{4\pi}\sum_{\mathbf{r}_{i},\mathbf{r}_{j}\in\mathbf{V}}\frac{1}{\left|\mathbf{r}_{ij}\right|^{3}}\left[3\left(\mbox{\boldmath$\mu$}_{i}\cdot\mathbf{\hat{r}}_{ij}\right)\left(\mbox{\boldmath$\mu$}_{j}\cdot\mathbf{\hat{r}}_{ij}\right)-\mbox{\boldmath$\mu$}_{i}\cdot\mbox{\boldmath$\mu$}_{j}\right]$$

$$H_m\approx \left(\sum _{{\mathbf{r}}_i\in {\mathbf{V}}^{hex}}6+\sum _{{\mathbf{r}}_i\in {\mathbf{V}}^{pen}}5\right)\left({n_z^i}^2-\frac{1}{3}\right)\tilde{M}$$

$${\tilde{H}}_{em}\left[\{{\mathbf{r}}_i\};\gamma ,\Gamma \right]$$

$$={\tilde{H}}_e\left[\{{\mathbf{r}}_i\};\gamma \right]+{\tilde{H}}_m\left[\{{\mathbf{r}}_i\};\Gamma \right]$$

$$H_v=\Lambda {\left(\sum _k{\Omega }_k-V_{ref}\right)}^2$$

Full text

††preprint: APS/membrane-static

Polyhedra are of great interest of scientists, mathematicians and engineers. They emerge spontaneously in many fields of science. For example, single crystals take various polyhedra shapes, fullerenes adopt beautiful truncated icosahedron shapesKroto1985 , and bacterial micro-compartments are observed in multiple regular and irregular polyhedral shapesFan2010 .

Self-assembled crystalline membranes, like the shells of viruses, generally possess icosahedral symmetryZandi2004 . These icosahedral membrane shapes have been explained by homogeneous elasticity theoryPhysRevA.38.1005 ; PhysRevE.68.051910 . Furthermore, membranes with heterogeneous elasticity have been demonstrated to form various regular and irregular polyhedral shapesVernizzi4292 . Such polyhedral morphologies are formed by the competition between stretching energy and bending energy. Although it is possible to engineer membrane morphologies by arranging defects in closed membrane topologiesPhysRevLett.111.177801 , these morphologies cannot go beyond polyhedra.

Here, we explore the possibility to create new closed shell morphologies, other than polyhedra, in a controllable manner. For this purpose, we consider elastic membranes of super-paramagnetic particles because of the exceptional penetration of magnetic fields and bio-compatibility, which provide opportunities for biotechnology applications. Magnetoelastic materials form rich morphologiesKim2018 ; LumE6007 and can accomplish multimodal locomotionHu2018 as well as deformations that generate forces between surfacesBrisbois2019 when directed by magnetic fields. The versatility of magnetoelastic filaments, which consist of super-paramagnetic particles connected by elastic linkers, has also been demonstrated experimentallyDreyfus2005 ; Wang2011 ; Wei2016 and numericallyPhysRevE.95.052606 ; Vazquez-Montejo2017 .

Comparing with magnetoelastic filaments and open membranes, closed magnetoelastic membranes, which have additional topological constraints, are found here to generate specific symmetries due to the interplay between nonlinear elasticity and magnetic dipole-dipole interactions. By using molecular dynamics simulations, we find the minimum energy configurations of magnetoelastic membranes, which can be directly controlled by external magnetic fields.

As dictated by Euler’s polyhedron formula, we start by triangulating a spherical shell with twelve isolated 5-fold disclinations. The disclinations are positioned on the vertices of an inscribed icosahedron (Fig. 1) to minimize the interactions between themPhysRevB.62.8738 , as proposed by Caspar and KlugCaspar01011962 .

The elastic component of the Hamiltonian of a magnetoelastic membrane, following the discretization scheme of Nelson et alPhysRevA.38.1005 , is written as

[TABLE]

where $k$ is microscopic stretching constant and $\tilde{\kappa}$ is microscopic bending rigidity. The sum is over all $e$ elements of $\mathbf{E}$, which is the set of all edges; $\mathbf{r}_{1}^{e}$ and $\mathbf{r}_{2}^{e}$ are two vertices of the edge $e$; and $\mathbf{n}_{1}^{e}$ and $\mathbf{n}_{2}^{e}$ are normal vectors of the two adjacent triangles of the edge $e$; and $l_{0}$ is the equilibrium length. Note that the corresponding continuum limit of the above discretized Hamiltonian is mesh dependentGompper1996 . With the above described triangulation of a spherical shell, it has been shown that in the continuum limitPhysRevA.38.1005 ; Nelson2004 Young’s modulus $Y=\frac{2k}{\sqrt{3}}$, Poisson’s ratio $\nu=\frac{1}{3}$ and bending rigidity $\kappa=\frac{\tilde{\kappa}}{\sqrt{3}}$.

Incompressible membranes ($\nu=1/3$) of radius $R$ can be described by two parameters $Y$ and $\kappa$. Then, a single dimensionless parameter, $\gamma=\frac{YR^{2}}{\kappa}$, called the F$\mathrm{\ddot{o}}$ppl-von Kármán parameterLibai1998 , completely determines the buckling transition of the system. Nelson et alPhysRevE.68.051910 have shown that homogeneous elastic membranes undergo a spontaneous buckling transition from sphere to icosahedron when $\gamma>\gamma^{\ast}=154$, where $154$ is the value of $\gamma^{\ast}$ for a flat disk.

In our study, we place a small super-paramagnetic particle at each vertex. An external magnetic field induces a magnetic dipole on each vertex. Therefore, an additional term from magnetic dipole-dipole interactions is added into the Hamiltonian of the system:

[TABLE]

where $\mu_{0}$ is the magnetic permeability in vacuum, \mbox{\boldmath\mu}_{i} is the magnetic dipole moment at vertex $i$, $\mathbf{V}$ is the set of all vertices, $\mathbf{r}_{i}$ is the position vector of vertex $i$, $\mathbf{r}_{ij}=\mathbf{r}_{j}-\mathbf{r}_{i}$ and $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/\left|\mathbf{r}_{ij}\right|$ and the sum is over $i\neq j$.

The magnetic dipole-dipole interaction is long range and anisotropic. A useful simplified form which considers only nearest neighbor interactions in the inextensible limit111For details of derivation, please refer to SI. is derived and yields:

[TABLE]

where $\tilde{M}=\frac{1}{4}\frac{\mu_{0}}{4\pi}\frac{(3\mu)^{2}}{l_{0}^{3}}\frac{2}{3}$, $\mu$ is the induced magnetic dipole moment which assumes only one type of super-paramagnetic particles, $\mathbf{V}^{hex}$ is the set of vertices with 6 neighbors, $\mathbf{V}^{pen}$ is the set of vertices with 5 neighbors and $n_{z}^{i}$ is the z component of normal vector at vertex $i$.

$\tilde{M}$ gives the characteristic energy scale for each nearest neighbor pair of magnetic dipole-dipole interactions in the discretization limit. Similar to the case of elastic membranes, a magnetic modulus can be defined in the continuum limit as $M=8\sqrt{3}\frac{\tilde{M}}{l_{0}^{2}}$, and a dimensionless parameter, $\Gamma=\frac{MR^{2}}{\kappa}$, called magnetoelastic parameterPhysRevE.98.032603 , can be similarly defined. The magnetoelastic parameter $\Gamma$ characterizes the relative strength between magnetic energy and bending energy(see SI).

Therefore, the magnetoelastic membrane has one additional energy competition from magnetic dipole-dipole interactions, which is tunable via an external magnetic field. The total magnetoelastic energy of the membrane $H_{em}$ is the sum of elastic and magnetic energies, which divided by $\kappa$ gives the dimensionless form:

[TABLE]

where tilde indicates dimensionless quantites.

Besides magnetic and elastic contributions, a volume constraint is also imposed on the membranes to account for internal pressure. This internal pressure is necessary when the membrane is not penetrable, which is modeled as

[TABLE]

where $\Omega_{k}$ is the signed volume of the tetrahedron extended by k-th triangle on the membrane, $V_{ref}$ is the reference volume of the membrane and $\Lambda$ is the Lagrange multiplier which serves role of pressure. $V_{ref}$ is set as volume of the icosahedron after buckling and $\Lambda$ is set to a large enough value such that the membrane has additional rigidity from the volume constraint. The volume constraint is used to better capture effect from the environment surrounding the magnetoelastic membrane and eliminate possible crumpled statesRevModPhys.79.643 . Corresponding cases without the volume constraint are also explored, and their morphologies generally do not differ significantly from the cases with the volume constraint. Some crumpled states and collapsed states are observed in high field strength limit for the cases without the volume constraint(see SI for more discussions).

In the simulation, a shifted Lennard-Jones potential is also included for each pair of vertices to account for the exclude volume effect. Each vertex is assigned a point magnetic dipole moment. Stretching and bending are treated with a harmonic bond interaction and a harmonic dihedral interaction, respectively. Magnetic dipole-dipole interactions are calculated without a cutoff. The connectivity of the membrane is preserved during simulations. The external magnetic field is static along z direction. We assume super-paramagnetic particles respond to an external magnetic field instantaneously and ignore rotational degrees of freedom of each vertex because super-paramagnetic particles do not have spontaneous magnetization, which decouples magnetics and elasticity. Kinetic energy is also assigned to each vertex to give a fictitious temperature of the system. The simulations start at high temperature and are gradually annealed to find the minimum energy configuration of the system. This annealing process is repeated several times to ensure that the system is not trapped in local minima.

A collection of possible morphologies of magnetoelastic membranes obtained by systematically varying the two dimensionless parameters, the F$\mathrm{\ddot{o}}$ppl-von Kármán parameter $\gamma$ and the magnetoelastic parameter $\Gamma$, are shown in Fig. 2. Without magnetic dipole-dipole interactions ($\Gamma$=0), when $\gamma<\gamma^{\ast}$, the homogeneous elastic membrane tends to stay spherical (Fig. 2e) and when $\gamma>\gamma^{\ast}$ it buckles into an icosahedron (Fig. 2f) as expected in the conventional homogeneous elastic membranesPhysRevE.68.051910 .

At moderate strengths of the magnetic dipole-dipole interaction, as shown in second row of Fig. 2, the structures deform since the magnetic dipoles prefer to line up and stay closer to each other to minimize the magnetic energy. When the membrane is relatively soft ($\gamma<\gamma^{\ast}$), the membrane tends to elongate along the direction of the external magnetic field. However, this is opposed by elastic interactions since elasticity prefers the membrane to stay spherical, resulting in an ellipsoid like membrane morphology as shown in Fig. 2c.

When the membrane is relatively stiff ($\gamma>\gamma^{\ast}$), the membrane undergoes an elastically driven buckling transition. The interplay between nonlinear elasticity and magnetic dipole-dipole interactions distorts the icosahedron. The flat regions of the icosahedron bend inward to reduce the distance between magnetic dipoles and disclinations pair up, resulting in a star-like morphology with six ridges as shown in Fig. 2d. Unlike the conventional convex polyhedral morphologies of the purely elastic membranes, the magnetoelastic membranes develop concave regions.

Then, consider the case of strong magnetic dipole-dipole interactions, as shown in first row of Fig. 2. The elastic energy only becomes comparable with the magnetic energy until the membrane is highly deformed. In this regime, the competition between magnetic energy and elastic energy results in another new family of morphologies.

When the membrane is easily deformed ($\gamma<\gamma^{\ast}$), magnetic dipole-dipole interactions tend to elongate the membrane furthermore along the direction of external magnetic field in this high field strength regime. However, the elastic energy can no longer hold the membrane in a spherical or ellipsoidal shape. The membrane forms a cylindrical shape, as shown in Fig. 2a, to minimize the magnetic energy. Although the bending energy is high along edges of two end caps of the cylinder, the total energy decreases by lining up vertices on the side surface of the cylinder.

When the membrane is relatively rigid ($\gamma>\gamma^{\ast}$), the elastic energy tries to preserve the total surface area of the membrane since stretching is much more expensive than bending in this case. Meanwhile, the magnetic dipole-dipole interaction tries to reduce the total volume of the membrane to minimize the magnetic energy. This competition, combined with the nonlinearity introduced by the twelve disclinations, results in a star-like morphology with four ridges as shown in Fig. 2b. Note that the membrane in this case is highly bent inward, which reduces its total volume significantly and opens some possible applications as discussed later.

Among all these mentioned morphologies of the magnetoelastic membrane, the $\gamma\sim 1000$ cases are particularly interesting because this regime corresponds to a typical F$\mathrm{\ddot{o}}$ppl-von Kármán parameter of viral shellsPhysRevE.68.051910 . In this regime, where both nonlinear elasticity and magnetic dipole-dipole interactions are significant, we find that the magnetoelastic membrane tends to choose configurations that decrease symmetry with increasing external magnetic field strength. This point is illustrated by plotting the mean curvature and energy distribution of the membrane in spherical coordinates as shown in Fig. 3.

In the weak field strength limit, the membrane forms an icosahedron (Fig. 3a) with five-fold rotational symmetry around the z-axis as shown in Fig. 3b and 3c. In this limit, the elastic energy dominates and the magnetic energy is negligible (Fig. 3d). With a moderate external magnetic field strength, the membrane morphology has six ridges(Fig. 3e). However, the 12 isolated disclinations prefer to pair up and form ridges connecting each pair of disclinationsLobkovsky1995 . These disclinations pairs are arranged alternatively to maximize the mutual distance in order to reduce interactions between disclinationsPhysRevB.62.8738 and ridgesPhysRevE.55.1577 , as shown in Fig. 3f and 3g. Because of this alternative arrangement, the membrane with six ridges has only three-fold rotational symmetry around the z-axis, which is also reflected by magnetic energy distribution (Fig. 3h). If the field strength is further increased, the membrane starts to form morphologies with four ridges (Fig. 3i). In this regime, two pairs of disclinations break and there is a single disclination near each of the four concave regions as show in Fig. 3j and 3k. Therefore, the symmetry of the membrane reduces to two-fold rotational symmetry (Fig. 3j, 3k and 3l) around the z-axis. In the extremely high field strengths regime, the magnetic energy completely dominates and the membrane collapses and takes one-fold rotational symmetry(the collapsed state is not shown in Fig. 3).

A natural question is to ask why the four-fold rotational symmetry is missing among all these above mentioned morphologies. A qualitatively argument suggests that if the system was to form a morphology with four-fold rotational symmetry, the 12 disclinations must be divided into groups of 3 disclinations, which means pairs of disclinations must break up. A direct visualization of the four-fold rotational symmetry case is to image the two-fold rotational symmetry case without alternative arrangement of the disclinations. Since weak field strengths cannot break up disclinations pairs and strong field strengths prefer alternative arrangement of disclinations to lower the system energy, the four-fold rotational symmetry is missed in these morphologies.

When the volume constraint is removed, we find similar morphologies to those discussed above except in high magnetic field strengths, where they take crumpled or collapsed morphologies with one-fold rotational symmetry(see Fig.2 in SI).

In summary, crystalline magnetoelastic membranes exhibit concave morphologies beyond the conventional polyhedral shapes found in elastic membranes. Magnetic dipole-dipole interactions give an additional control parameter which is characterized by the magnetoelastic parameter $\Gamma$. Combining with the F$\mathrm{\ddot{o}}$ppl-von Kármán parameter $\gamma$ in the elastic membranes, these two dimensionless parameters provide guidelines for analyzing properties of crystalline magnetoelastic membranes. Importantly, since $\gamma$ is hard to change once a membrane is assembled, the magnetoelastic parameter, which can be easily manipulated by an external magnetic field, provides a way to tune the membrane morphology between convex shapes and concave shapes with specific symmetry.

Exciting applications, including reversible membrane shape control, design of micro-container, and targeted drug delivery, are expected for the closed crystalline magnetoelastic membranes. For example, since the volume to surface ratio of magnetoelastic membranes can be highly reduced by imposing an external magnetic field, the concentration inside can be much higher than that in the outside environment. This morphological change induced by the external magnetic field can facilitate release of cargoes. Therefore, the magnetoelastic membrane can be used as a container to carry and protect volatile or toxic molecules and release them in targeted region labelled by the external magnetic fields.