Elastic Deformations of Spherical Core-Shell Systems Under an Equatorial Load

Solid sphere-like core-shell systems containing an inner part, the core, of elastic properties different from a surrounding outer part, the shell, are encountered in various contexts on different length scales. On large macroscopic scales, many stars, planets and moons can be approximated by a core and a shell of different elasticity [1]. Jelly sweets covered by a solid layer represent a popular example of not only mechanical or haptic but also culinary experience. Conversely, on the mesoscopic colloidal scale and even down to the nanoscale, there are numerous soft matter systems involving core-shell particles. These can be prepared in various ways [2, 3] as spherical colloidal particles with a polymer coating [4‐6], as micelles [7] or as polymer networks with different crosslinking degrees in the inner and outer part [8‐10]. Their controlled fabrication is not only pivotal for applications (such as microreactors [11, 12], targeted drug delivery [13, 14] or smart elastic materials [15, 16]). They also serve as model systems to tailor effective repulsive square-shoulder potentials [17‐27] and to understand fundamental questions of statistical mechanics such as freezing and glass formation [4, 28‐30].

Our focus in this work is laid on the coupled elastic deformation of inner and outer part, that is core and shell, respectively. We address spherical elastic systems when exposed to a radially oriented force line density along the equatorial circumference of the shell. This setup is motivated by the elasticity problem underlying colloidal core-shell microgel particles that are adsorbed to the interface between two immiscible fluids. At their common contact line, the two fluids pull on the shell of the microgel particle approximately in a radially outward direction in a symmetric setup [31‐33]. In many of such interfacial situations, the wetting properties of the surface of a material are crucial for adsorption. A core-shell system provides an appropriate opportunity to adjust by a shell these surface wetting properties to the current need. At the same time, the elastic properties of the core under the influence of an interface are explored. Moreover, the particles may be density matched or functionalized, for example integrating magnetic behaviour, by the selected core material [34‐36]. On macroscopic toy model systems, the force densities can be applied by hand, while on even larger, global scales atmospheric effects may lead to equatorially located line-like force densities on planets. An example is the thin area of low atmospheric pressure located around the equator of the earth in the inter-tropical convergence zone. In view of these different systems and situations, the imposed equatorial line force density can either be oriented radially inward (compressive) or radially outward (tensile). In the mathematical treatment this difference is represented by an inversion of the sign of the load.

In this paper, we study the underlying elasticity problem. We present a general continuum theory to compute and predict the shape change of an elastic core-shell system when loaded by an equatorial ring of line force density. Importantly not only the shell deforms, but also the inner core, and the two deformations are coupled to each other by the overall architecture. Through this coupling, the core can influence or even determine the type of deformation of the shell, although the load is applied from outside to the shell, not to the core. We analyse the resulting change of shape in detail, as a function of the relative size of core and shell, different mechanical stiffness of core and shell, as well as their compressibility. In particular, we include the possibility of an elastic auxetic response [37‐42]. The latter is characterized by a negative Poisson ratio, i.e. when stretched along one axis the system expands along the perpendicular axes. Materials exhibiting corresponding elastic properties have been identified, constructed and analysed [43‐46]. In particular it is interesting to consider core and shell materials with different Poisson ratios, as their competition can result in qualitatively different modes of deformation. Our study links to previously investigated geometries, particularly spherical one-component systems [47] or hollow capsules [48] as special cases. Moreover, our additional predictions can be verified by experiments on different scales.

Within linear elasticity theory, small deformations of elastic materials are described. The position \(\mathbf{r}\) of a material element can be mapped to its position \(\mathbf{r}'\) in the deformed state by adding the displacement vector \(\mathbf{u}\). The displacement field \(\mathbf{u}\left (\mathbf{r}\right )\) in the bulk satisfies the homogeneous Navier-Cauchy equations [49] with \(-1<\nu \leq 1/2\) denoting the Poisson ratio of the elastic substance in three-dimensional situations [50]. Materials with \(\nu =1/2\) are incompressible, while those with negative Poisson ratio are referred to as auxetic materials [50]. The latter, when stretched along a certain axis, expand along the lateral directions (instead of undergoing lateral contraction). We ignore any force acting on the bulk, for example gravity. Consequently, in the bulk, the right-hand side of Eq. (1) is set equal to zero.

Furthermore, linear elasticity theory for homogeneous isotropic materials dictates the stress-strain relation [50] Equation (2) describes the relationship between the strain tensor \(\underline{\boldsymbol{\varepsilon}}\left (\mathbf{r}\right )=\bigl ( \nabla \mathbf{u}\left (\mathbf{r}\right )+ \left (\nabla \mathbf{u} \left (\mathbf{r}\right )\right )^{\text{T}}\bigr)/2\) as the symmetrized gradient of the displacement field \(\mathbf{u}\left (\mathbf{r}\right )\) and the symmetric Cauchy stress tensor \(\underline{\boldsymbol{\sigma}}\) (we mark second-rank tensors and matrices by an underscore). \(E\) is the Young modulus of the elastic material and \(\underline{\mathbf{I}}\) is the unit matrix. The Young modulus \(E\) and the Poisson ratio \(\nu \) are sufficient to quantify the properties of a homogeneous isotropic elastic material.

The boundary conditions at the surface of the elastic shell are Here, \(\mathbf{n}\) describes the outward normal unit vector of the surface and \(\delta \left (\theta -\frac{\pi}{2}\right )/R_{\text{s}}\), with \(\delta \) the Dirac delta function, sets the location of the line at which the loading force line density of amplitude \(\lambda \) is acting on the core-shell system. We use spherical coordinates so that \(\theta =\frac{\pi}{2}\) specifies the equator.

Since we are describing a core-shell material, different elastic properties and radii are attributed to the core and to the shell, see Fig. 1. The core (green) is assigned the radius \(R_{\text{c}}\), the Young modulus \(E_{\text{c}}\), and the Poisson ratio \(\nu _{\text{c}}\). The shell (red) is defined by the outer radius \(R_{\text{s}}\), the Young modulus \(E_{\text{s}}\), and the Poisson ratio \(\nu _{\text{s}}\). According to Eq. (3), \(\lambda >0\) marks the amplitude of a line density of force pointing radially outward along the equator of the outer surface of the shell.

The system is characterized by the following five dimensionless parameters. First, the ratio \(\lambda /E_{\text{s}}R_{\text{s}}\) of the loading force line density on the surface to the Young modulus of the shell describes the relative strength of the load magnitude and is proportional to the amplitude of deformation. The second parameter is the ratio of Young moduli \(E_{\text{c}}/E_{\text{s}}\) of the core to the shell and in addition, the two dimensionless Poisson ratios \(\nu _{\text{c}}\) and \(\nu _{\text{s}}\) of core and shell, respectively, enter the elasticity theory. The fifth parameter is the size ratio \(R_{\text{c}}/R_{\text{s}}\) of the core to the shell.

In spherical coordinates, the position vector \(\mathbf{r}\) transforms from the unloaded configuration to the loaded configuration as \(\mathbf{r}'=\mathbf{r}+u_{r}\mathbf{e}_{r}+u_{\theta}\mathbf{e}_{\theta}\), with \(u_{r}\) the radial and \(u_{\theta}\) the polar component of the displacement field. \(\mathbf{e}_{r}\) and \(\mathbf{e}_{\theta}\) denote the radial and polar unit vector, respectively. Due to the special axial symmetry of the problem, the azimuthal component of the displacement field, \(u_{\phi}\), is zero. Concerning the homogeneous Navier-Cauchy equations Eq. (1) and the stress-strain relation Eq. (2) recast in spherical coordinates, where in our case the azimuthal dependence vanishes, see the Supporting Information (SI).

For both core and shell we solve Eq. (1) by separation into a series expansion of the polar dependence in terms of Legendre polynomials \(P_{n}\left (\cos \theta \right )\) and associated \(r\)-dependent prefactors \(\left (r=\left |\mathbf{r}\right |\right )\) [51]. We distinguish by superscripts c and s the solutions for core and shell, respectively. More precisely, the solutions [47, 52, 53] of the Navier-Cauchy equations (1) split into a radial component \(u^{(c)}_{r}\left (\mathbf{r}\right )\) and a polar component \(u^{(c)}_{\theta}\left (\mathbf{r}\right )\) for the core and take the form The solutions for the shell additionally contain terms inverse in the radial distance from the origin

As boundary conditions, we use that the traction vectors at the interface of core and shell (at radius \(R_{c}\)) must be equal Requiring strict elastic no-slip coupling, also the deformations at the interface must be equal Since the Legendre polynomials form a complete orthogonal set, the Dirac delta function in Eq. (3) can be expanded in Legendre polynomials

Due to the assumed mirror symmetry with respect to the equatorial plane, all odd series expansion components in the core and shell solution in Eqs. (4)-(7) vanish. Therefore we can write for the radial displacement with \(i=c\) for the core and \(i=s\) for the shell, respectively. Here, the first component \(u^{(i)}_{r,0}(r)\) describes the overall volume change. We note that this term will vanish for \(\nu _{i} \rightarrow 1/2\) and remains as the only component for \(\nu _{i} \rightarrow -1\). The second component gives the first correction to a spherical shape. A positive prefactor \(u_{r,2}^{(i)}(r)\) describes a relative prolate deformation while \(u_{r,2}^{(i)}(r)<0\) implies a relative oblate deformation. It is in fact the latter case of an oblate deformation which we expect when the core-shell particle is pulled outwards at the equator \((\lambda >0)\).

The solutions for the displacements of the core and the shell diverge in response to the Dirac delta function at the equator on the surface of the shell, see the boundary condition Eq. (3). Yet, each mode of deformation is only excited to a finite degree by the Dirac delta function, see Eq. (10). Therefore, the second components \({u_{r,2}^{(c)}(r)}\) and \({u_{r,2}^{(s)}(r)}\) for the core and the shell remain finite and \({u_{r,2}^{(s)}(r)}\) is even finite at the surface of the shell. We shall use them as parameters to characterise the relative oblate or prolate deformation of the core and the shell shape.

For convenience, we evaluate these second components at the core and shell radii and normalize them with the corresponding unloaded radii of the core and the shell, respectively. Hence, we use subsequently \({u_{r,2}^{(c)}}/{R_{c}}\equiv {u_{r,2}^{(c)}(R_{c})}/{R_{c}}\) and \({u_{r,2}^{(s)}}/{R_{s}}\equiv {u_{r,2}^{(s)}(R_{s})}/{R_{s}}\) as dimensionless measures for the shape of the core and the shell.

We have analysed in detail the deformational response of an elastic core-shell system to a radially oriented force line density acting along the outside equatorial line. Natural extensions of our considerations include the following.

First, the axially symmetric situation that we addressed could be generalized to systems exposed to line densities that are modulated along the circumference. Moreover, the effect of surface force densities applied in patches or distributed over the whole surface area could be analysed, instead of pure force line densities. In a further step, the imposed distortions may not only be imposed from outside, but could additionally result from internal active or actuation centers. Obvious candidates for corresponding actuatable cores are given by magnetic gels [54, 55]. For these types of systems, magnetically induced deformations have already been analysed by linear elasticity theory in the case of one-component elastic spheres [56‐58].

The considered geometry of loading can effectively be realised in experiments on the mesoscale by exposing core-shell microgel particles to the interface between two immiscible fluids acting on the elastic system [32, 33]. There, interfacial tension radially pulls on the equatorial circumference along the common contact line in a symmetric setup. Yet, our description can be applied to any system on any scale that can be characterized by continuum elasticity theory. For example, macroscopic elastic core-shell spheres could be generated as toy models using soft transparent elastic shells on an elastic core. The line of loading force could then simply be imposed by tying a cord around the equator of these macroscopic core-shell spheres and tightening it. In this setup, the direction of the force is inverted as well. However, this in our evaluation simply means that all directions of displacement are inverted. Such macroscopic approaches may support the involvement of auxetic components [37‐42]. Depending on the materials at hand, this strategy may facilitate the experimental confirmation of our results, possibly by direct visual inspection.