Effective Topological Charge Cancelation Mechanism

Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions. The topological origin makes their key features independent of systems’ microscopic details; therefore TDs display many universalities. Because of their strong impact on numerous material properties and their significant role in several technological applications it is of strong interest to find simple and robust mechanisms controlling the positioning and local number of TDs. We present a numerical study of TDs within effectively two dimensional closed soft films exhibiting in-plane orientational ordering. Popular examples of such class of systems are liquid crystalline shells and various biological membranes. We introduce the Effective Topological Charge Cancellation mechanism controlling localised positional assembling tendency of TDs and the formation of pairs {defect, antidefect} on curved surfaces and/or presence of relevant “impurities” (e.g. nanoparticles). For this purpose, we define an effective topological charge Δmeff consisting of real, virtual and smeared curvature topological charges within a surface patch Δς identified by the typical spatially averaged local Gaussian curvature K. We demonstrate a strong tendency enforcing Δmeff → 0 on surfaces composed of Δς exhibiting significantly different values of spatially averaged K. For Δmeff ≠ 0 we estimate a critical depinning threshold to form pairs {defect, antidefect} using the electrostatic analogy.


Intrinsic and extrinsic elastic contributions
Below, we present the minimal model with which we demonstrate the key difference between the intrinsic and the extrinsic elastic contributions in describing orientational ordering on curved surfaces. Afterwards, we illustrate the typical impact of the extrinsic curvature contribution on nematic structures within nematic ellipsoidal shells, considering the tensor order parameter Q (Eq. (2)) and the free energy contributions introduced in Eq. (5) and Eq. (6).

Minimal model
In general, an elastic free energy penalty of an elastic medium living within a 2D curved surface consists of the so-called intrinsic and extrinsic contributions [25][26][27]. To illustrate their role, we consider the simplest possible elastic free energy term f e = k |∇ s n| 2 . (S1) Here, n is a unit vector field lying within a surface, k is a positive elastic constant, and ∇ s stands for the surface gradient [28]. The latter is related to the conventional 3D gradient operator via ∇ s = (I − v ⊗ v)∇. We express n in the surface principal curvature direction frame {e 1 ,e 2 } as n = e 1 cos θ + e 2 sin θ.
where θ is the angle between the unit vector field n and the first principal curvature direction e 1 . The local surface curvature C, seen by the molecule aligned along n, can be expressed by the Euler relation as: C = C 1 cos 2 θ +C 2 sin 2 θ. It follows ∇ s n = cos θ∇ s e 1 − sin θe 1 ⊗ ∇ s θ + sin θ∇ s e 2 + cos θe 2 ⊗ ∇ s θ.
Taking into account [28] where κ q1 (κ q2 ) are geodesic curvatures along e 1 (e 2 ), we obtain The quantity is the so called spin connection and it holds [26,29] K = |∇ × A|. The first contribution in Eq. (S6) is referred to as the intrinsic term. It enforces the spatially non-homogeneous orientation of n if K = 0, which introduces a geometric frustration into the system. It derives from the incompatibility of parallel and straight directions on surfaces with Gaussian curvature. Note that using the covariant derivative approach in expressing elastic free energy contributions yields only intrinsic-type contributions. The covariant derivative is defined via parallel transport. While parallel transported n experiences minimal distortions, it holds [28] If n is parallel transported along a closed path, then the difference in orientation of n before and after such transport reveals a geometric frustration. Taking into account Eq. (S3) on the left side of Eq. (S8) and considering ∇ s v =C 1 e 1 ⊗ e 1 + C 2 e 2 ⊗ e 2 one obtains ∇ s θ= −A. Therefore, the intrinsic term in Eq. (S6) is minimised if n is parallel transported. Such a configuration represents a local ground state, exhibiting minimal possible elastic free energy distortions. The second term in Eq. (S6) is referred to as the extrinsic contribution. Its influence is reminiscent of an external orientational field enforcing its orientation to n. Expressing the term in the principal curvature frame yields n · C 2 n =C 2 1 cos 2 θ + C 2 2 sin 2 θ.
One sees that for k > 0 this term is minimised if n is aligned along the principal direction exhibiting lower curvature. The energy term (Eq. (S9)) is closely related to the energy term f H , commonly used to study biological membranes [30], thin plates [31,32] with anisotropic properties, and the orientational ordering of anisotropic components in biological membranes [7,11,33] where K 1 and K 2 are (positive) constants. The mismatch tensor is defined as M = RC m R −1 − C, where the tensor C m describes the intrinsic curvature of the inclusion. The rotational matrix R describes the rotation of the system by angle θ, which is the angle of rotation of the membrane element relative to the first principal direction e 1 . In the case of non-curved rod-like molecule the energy term (Eq. (S10)) is minimised if the molecule is aligned along the principal direction exhibiting lower curvature, which is also true for the energy term in Eq. (S9). Therefore, the intrinsic elastic component is associated with variations of n living in 2D curved space. On the other hand, the extrinsic elastic component tells how n is embedded in 3D space. The difference between these contributions is well visible in an infinitely long cylinder of radius R. The intrinsic elastic penalty equals to zero for n, pointing either along the symmetry axis or at right angles with it. To show this, we use parametrisation defined in Eq. (S2), where e 1 is aligned along the symmetry axis. It holds κ q1 =κ q2 = A = ∇ s θ = 0 for both orientations, and consequently the intrinsic term equals zero (see Eq. (S6)). On the contrary, the extrinsic term is different for these orientations due to different values of principal curvatures; namely, C 1 = 0 and C 2 = 1/R. Consequently, it forces n to align along e 1 .

Impact of extrinsic term
In the core of the paper, we calculated TDs in oblate and prolate shells in the presence of different number of NPs in the absence of extrinsic term. The positions of NPs are depicted in Figure S1 on cases of spherical shells, where we superimpose n and spatial variations in λ. Figure S1. Nematic ordering on spherical shells. The panels (a), (b), (c) and (d) show cases with no, one, two and three nanoparticles, which are denoted by "NP". Each NP effectively acts as a topological defect, bearing m = 1. Superimposed are λ/λ 0 and the vector field n spatial variations. Configurations were calculated for a/b = 1, The impact of the orientation of n on the extrinsic free energy contribution is depicted in Figures S2. In Figure  S2a, we plot g e = f e /(k e λ 2 0 ) as the function of zenith angle v for prolate and oblate shells for either n along e 1 (meridians, full lines) or n along e 2 (parallels, thin lines). The plots which show extreme cases and variation for the arbitrary orientation of n are presented in Figure S2b (a prolate shell) and Figure S2c (an oblate shell). The extrinsic contribution effectively acts as an external field which for k e > 0 favours the orientation of n along lines, exhibiting minimal curvature. One sees that in both geometries the extrinsic field is absent at v = 0 and v = π, where C 1 = C 2 . In between, the field is different from zero. In prolate shells, it prefers an alignment along meridians and is in general significant for all values of v ∈ [0, π] for a large enough value of k e . On the contrary, in oblate geometries it enforces the alignment along parallels. In this case, the extrinsic field tends to be localised at the equatorial region, which progressively narrows on increasing the ratio b/a. Of our interest is the impact of a relatively strong extrinsic field (e.g., k e /k i = 1) on the number and position of TDs in ETCC limit structures. In prolate shells, structures are qualitatively similar, because the extrinsic field does not introduce any additional frustrations. However, in oblate structures the extrinsic field can strongly modify patterns, because it tends to expel TDs from the equatorial region. This is depicted in Figures S3. The corresponding textures for k e = 0 are plotted in the column (b) of Figure 3.

Electrostatic analogy
Here, we sketch key steps in developing electrostatic analogy which enables us to estimate the threshold condition to form pairs {defect,antidefects}. Furthermore, we schematically visualise the depinning process for two qualitatively different geometries.

Critical condition
We first consider a flat liquid crystalline film in the Cartesian coordinates (x, y), and the free energy, which is given by Eq. (6). We neglect spatial variations in λ, set k e = 0, and express the director field using Eq. Nanoparticles are labelled by NP. Nematic ordering was calculated for: a/ξ = 3.5, k = k e . localised at (x i , y i ), can be expressed as Here, c is a constant and m i is the winding number. We consider a pair consisting of {defect,antidefect}={m 1 = m, m 2 = −m} placed at (x 1 = −ρ/2, y 1 = 0) and (x 2 = ρ/2, y 2 = 0), respectively. The orientational pattern of the resulting structure is determined by θ = θ 1 + θ 1 . The spatial integral of the corresponding free energy yields the interaction potential between TDs separated for a distance ρ. Here k F = kλ 2 0 and ρ c ∼ ξ is the cut-off radius estimating a typical defect's core size. In our model notations it roughly holds The magnitude of the corresponding attractive force per unit length is then f int = 2πm 2 k F /ρ. We define the elastic electric field via f int ≡ mE e . Therefore, a topological defect bearing topological charge m creates an elastic field of strength E e = 2πk F m ρ . (S14) Next, we estimate energies needed to form a pair {m = 1 2 , m = − 1 2 } of TDs at line K = 0 which then "fall" on relevant capacitor plates. We express the total free energy cost for this process as ∆F = ∆F cond + ∆F work + ∆F gain . (S15) Here, ∆F cond describes the free energy costs to form a pair {1/2, −1/2} of TDs, ∆F work corresponds to the work needed to separate the newly born pair, and ∆F gain describes the free energy gain due to the "fall" of TDs within the capacitor's elastic electric field. We suppose that E(∆m eff ) is strong enough to trigger off TDs at an arbitrary point along the line where K = 0. The corresponding penalty is roughly given by the condensation free energy cost The work needed to pull apart the new-born defects with charges m = ± 1 2 from the initial separation ξ to the final separation ρ 2 − ρ 1 overcoming their mutual attraction is equal to Finally, the energetic gain is estimated by The critical condition to form a stable pair is estimated by ∆F = 0, yielding Eq. (13).

Depinning event
We next schematically illustrate events above and below the critical depinning threshold for two qualitatively different geometries, emphasising the role of the effective topological charge. We consider (i) dumb-bell and (ii) spherocylindrical shells. In the latter case we also introduce a NP acting as a "virtual" TD bearing a virtual topological charge ∆m v = 1. Examples of a "real" and "virtual" charge are schematically depicted in Figure S4a and Figure  S4b, respectively. Figure S4. Schematic presentation of (a) "real" and (b) "virtual" topological defect, bearing m = 1.
We first consider configurational changes in geometry, depicted in Fig. 1c. Let us assume that the structure is mirror symmetric (the up and down parts of the dumb-bell are the same) and that it possesses four m = 1/2 defects as predicted by Poincaré-Hopff and Gauss Bonnet theorem. For a better visualisation of estimates let us assume that the dumb-bell structure consists of nearly two spherical parts connected by a narrow neck. Note that for a closed sphere surface ζ it holds ∆m K (ζ) = − 1 2π ζ Kd 2 r = −2. We focus, henceforth, to the upper part of the dumb-bell and consider patches ∆ζ + and ∆ζ defined in Fig. 1c. Below the depinning threshold the two m = 1/2 are assembled in the region exhibiting positive Gaussian curvature because this region effectively acts as a smeared negative curvature topological charge. The resulting configuration is depicted in Figure S5a. The effective topological charge in the upper ∆ζ patch equals to ∆m eff (∆ζ ) = ∆m+∆m v +∆m K ∼ 1+0−2 = −1, where the used approximation is ∆m K (∆ζ ) ∼ −2. Furthermore, according to the Poincaré-Hopff and Gauss Bonnet theorem it holds ∆m K (∆ζ ) + ∆m K (∆ζ + ) = −1, consequently ∆m K (∆ζ + ) ∼ 1 and ∆m eff (∆ζ + ) = ∆m + ∆m v + ∆m K ∼ 0 + 0 + 1 = 1. The upper dumb-bell part is therefore roughly equivalent to the capacitor shown in Figure 1d, where the plates at ρ = ρ 1 and ρ = ρ 2 bear charges ∆m eff (∆ζ + ) ∼ 1 and ∆m eff (∆ζ − ) ∼ −1, respectively. If two pairs {defect,antidefect} are created, then the capacitor plates are partially discharged, as shown in Figure S5b. The complete discharging of the capacitor, corresponding to the ETCC limit structure, requires a formation of four pairs {defect,antidefect}. Figure S5. Schematic illustration of equilibrium configurations above (b,d) and below (a,c) depinning threshold. We consider dumb-bell (a,b) and sphero-cylindrical (c,d) shells. Nanoparticles act as "virtual" topological defects bearing a virtual topological charge ∆m v = 1.
Next, we treat a qualitatively different case, similar to the one depicted in Figure S1b. For illustration purpose we consider a sphero-cylinder, see Figure S5c, where one NP enforcing m = 1 is present. Below the depinning threshold such configuration possesses two m = 1/2 defects to fulfil Eq. (10). As in the case above, we limit to the upper part of the structure, assuming that the lower part is mirror symmetrical. To make the derivation more transparent we set that the spherical patch is represented by ∆ζ − and the cylindrical part by ∆ζ + . Consequently, it holds ∆m K (∆ζ + ) = 0, ∆m K (∆ζ ) = −1 (due to Eq. (9)), and m = 1/2 defect is attracted to the ∆ζ − patch. The effective topological charge within patches ∆ζ − (∆ζ + ) equals ∆m (−) eff = ∆m + ∆m v + ∆m K = 1/2 + 0 − 1 = −1/2 (∆m (+) eff = ∆m + ∆m v + ∆m K ∼ 0 + 1/2 + 0 = 1/2 ). Here, we assumed that a half of the central NP bearing m = 1 contributes to each half of the sphero-cylinder. To cancel ∆m eff in the patches, two pairs {defect,antidefect}={m = 1/2, m = −1/2} must be formed. The two antidefects are needed to screen the central "virtual" charge enforced by NP. On the other hand, defects are moved toward poles. The corresponding structure is "neutral", i.e. represents the ETCC limit structure shown in Figure S5d.