Concentrated Gaussian curvature in curved creases of actuated spiral nematic solids
CConcentrated Gaussian curvature in curved creases ofactuated spiral nematic solids
Fan Feng , Daniel Duffy , John S. Biggins , and Mark Warner Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom * [email protected] February 10, 2021
Abstract
Liquid crystal elastomers/glasses are active materials that can have significant metricchange upon stimulation. The local metric change is determined by its director pattern thatdescribes the ordering direction and hence the direction of contraction. We study logarithmicspiral patterns on flat sheets that evolve into cones on deformation, with Gaussian curva-ture localized at tips. Such surfaces, Gaussian flat except at their tips, can be combined togive compound surfaces with GC concentrated in lines. We characterize all possible metric-compatible interfaces between two spiral patterns, specifically where the same metric changeoccurs on each side. They are classified as hyperbolic-type, elliptic-type, concentric spiral,and continuous-director interfaces. Upon the cone deformations and additional isometries,the actuated interfaces form creases bearing non-vanishing concentrated Gaussian curvature,which is formulated analytically for all cases and simulated numerically for some examples.Analytical calculations and the simulations agree qualitatively well. Furthermore, the relax-ation of Gaussian-curved creases is discussed and cantilevers with Gaussian curvature-enhancedstrength are proposed. Taken together, our results provide new insights in the study of curvedcreases, lines bearing Gaussian curvature, and their mechanics arising in actuated liquid crystalelastomers/glasses, and other related active systems.
Thin and slender structures find wide applications in engineering, ranging from small-scale medicalstents [28] to large-scale space structures [38] by forming desired surfaces. The geometry-material-mechanics coupling in these structures has attracted interest in many areas, including the bucklingof shells, the crumpling of paper, and the design of active surfaces etc. [3]. Gaussian curvature(GC), defined as the product of the two principal curvatures, is an intrinsic geometrical quantitythat plays a crucial role in determining the mechanical behaviour of an elastic surface; for example,the developability of a surface without stretch. Therefore, a method for designing a desired GC isurgently required. GC can be distributed smoothly over surfaces, leading to smooth, curved surfaces1 a r X i v : . [ c ond - m a t . s o f t ] F e b n space. It is well-known from Gauss’s Theorema Egregium [22] that isometric deformations areincapable of changing the GC. Deformations that do change GC are hence quite rigid compared withpure bending. Thus, effective non-isometric deformations induced in active materials [29, 32, 33, 35,15], and in origami/kirigami constructions [14, 6, 7, 9] etc. are employed to achieve non-trivial GC,any blocking of which leads to strong actuation. In this paper, we study active materials, namelyliquid crystal elastomers (LCEs) and glasses [53], and examine the GC induced on actuation.LCEs can have significant shape change determined by their ordering direction, the director n .The shape change can be driven by heat, light, or solvent. The director field is patterned on atwo dimensional (2D) sheet. Upon stimulation, LC solid sheets contract by a factor λ < λ − ν , where ν is the optothermal Poisson ratio . The deformation isnon-isometric and generically induces non-trivial GC [29, 32, 35] and metric change locally. Non-uniform director fields result in non-uniform metric change throughout the pattern, and therefore thecoupled mechanical behavior becomes sophisticated. This metric-driven paradigm has been termed“metric mechanics” [51] and can be used to design targeted, three-dimensional shapes actuatedfrom a prescribed 2D (usually smooth) director pattern [29, 1, 35, 52, 2, 23]. For example, acircular pattern and its actuated state, cones [Fig. 1(a)], have been used to design heavy liftersthat experimentally can lift thousands of times their own mass through strokes that are hundredsof times their own thickness [50, 54]. Fundamentally, the strength of the lifter is enhanced by thetips of cones, which are point defects of concentrated GC preventing the lifter from being flattened.Beyond smooth director patterns and point defects, we are interested in “interfaces” formedat curves separating two patterns. To form a geometrically-consistent deformed configuration withinterfaces, the deformations on the two sides have to be different, but obey the so-called geometricalcompatibility (or rank-1 compatibility [5]) condition to ensure the continuity of the configurationacross the interface. The study of compatible interfaces can be traced back to martensitic phasetransformations in crystals with straight-line interfaces [5, 45], more recently in helical structureswith curved interfaces [21, 19], and straight-fold/curved-fold origami [17, 47, 10, 11, 24, 41]. Ex-amples in LCEs include the straight-line interface between piecewise constant director patterns[30, 40, 39] and the hyperbolic/elliptical interfaces between circular patterns [31, 18] (see Figs. 1(b)-(d)). These examples of non-isometric origami have interfaces that are geometrically consistent,before and after deformation. For example, in Figs. 1(b–d), the actuated interfaces under the con-ical deformations from the different patterns that meet are identical and fit together at the foldwithout the need for further (isometric) shape change. We name this particular scenario the conicalconsistency .However, the actuated state can be further deformed isometrically. Like in classical, isometriccurved-fold origami [20, 13, 4, 8], the additional isometric deformation preserves the GC and thegeodesic curvature distributed along the deformed interface. This constraint links the deformedinterface and the surfaces on two sides via the crease’s opening angle which can be calculated fromthe total curvature and geodesic curvature of the interface. Usually these additional isometricdeformations are sophisticated and can not be determined analytically, but they do occur in softmaterials such as paper and LCEs. Allowing such arbitrary additional isometric deformations, metric compatibility is employed. That is, the metric changes at the interface induced by the twosides need to be identical. As a result, the deformed interfaces from the two sides have the samelocal arc length. Interfaces that satisfy the metric compatibility and have the same deformed localarc length are named metric-compatible interfaces, which are the focus of this paper.In this paper, we move beyond the above conically-consistent, non-isometric origami in Fig. 12igure 1: The circular director patterns (top), the theoretical actuated configurations (middle),and the simulated configurations (bottom). (a) A circular pattern and its actuated state, a cone.(b) Straight-line interface between two circular patterns that bisects the line joining their centres.The actuated cones have equal height and parallel cone axes. (c) Hyperbolic interface between twocircular patterns. The interface is not bisecting and hence the actuated cones have different heights,but parallel cone axes. (d) Elliptical interface between two circular patterns. The actuated coneshave nonparallel cone axes. The deformed interfaces are geometrically consistent, that is, they havethe same shape under either conical deformation.and study both the metric-compatible interfaces and the post-actuation isometries in LCEs. Inparticular, we study logarithmic spiral director patterns, which cause sheets to evolve into cones in3D under so-called cone deformations . Similar to those from circular patterns, cones actuated fromspiral patterns have a point source of positive GC localized at the tip, with a value of 2 π (1 − sin ϕ )calculated from the angular deficit, where ϕ is the cone angle [18]. We are interested in GCconcentrated in activated tips and curved creases, in systems that are otherwise Gaussian flat andconvenient to solve. In contrast with that of circular patterns, the cone deformation of spiralpatterns contains a non-uniform rotation term that causes conical inconsistency of the deformedinterface. The metric-compatible interfaces between spiral patterns are also more complicated,while in circular patterns the interfaces are simply hyperbolae or ellipses [18]. Fortunately, onecan employ elliptic coordinates to derive the analytical forms for all possible metric-compatibleinterfaces between two spiral patterns. We characterize these interfaces into four types: hyperbolic-type, elliptic-type, continuous-director, and concentric spiral interfaces. These interfaces genericallycarry non-trivial GC, fundamentally because the deformation from 2D to 3D is non-isometric. TheGC distributed along a curve parallel or perpendicular to the director in a general director pattern3s given in a recent work [15], by applying the Gauss-Bonnet theorem locally to a loop containingthe curve. Here we employ the same approach and further derive formulae for the GC concentratedalong a metric-compatible interface between two director patterns.As yet we have no theoretical tools to compute the post-actuation isometries that can resolve theconical inconsistency for actuated spiral patterns, so we instead present simulations that providethis resolution. Our numerical approach models a thin patterned LCE sheet as a 2D surface,assigns it an elastic energy containing both stretch and bend terms, and minimizes that energyto find an equilibrium state. The simulated post-actuation deformations are not exact isometries,but we quantify the degree of non-isometry in Section 3 and find it to be negligibly small, apartfrom localised ‘blunting’ deformations that regularize any singular curvature encoded in the directorpattern.The paper is organized as follows. In Section 2, we study isolated logarithmic spiral patternsand derive the explicit cone deformations that map the reference domain to actuated cones. InSection 3, all possible metric-compatible interfaces between two spiral patterns are characterized.The disconnected actuated states under the separate cone deformations, and the connected, con-tinuous simulated configurations are presented. In Section 4, we discuss the conical inconsistencyunder cone deformations and argue that additional isometries, which preserve the GC, are neededto form continuous, actuated configurations. In Section 5, we provide three methods for computingthe analytical GC concentrated along the actuated interface. Simulations and analytical results arecompared qualitatively for two examples, and agree well. In the discussion section, we propose adesign of LCE cantilevers with their strength potentially enhanced by concentrated GC, and dis-cuss the relaxation of non-isometric folds. Our findings provide insights into curved creases, lines ofconcentrated GC, and the coupled mechanics of spiral director patterns and of anticipated generalpatterns in active systems. Let { e , e , e } be the orthonormal basis of the Cartesian coordinates. Consider a general directorfield n ( x, y ) = n ( x, y ) e + n ( x, y ) e at the position ( x, y ) ∈ D ⊂ R satisfying | n ( x, y ) | = 1,where D is the domain of the director field. Logarithmic spiral director patterns are in the classof circularly symmetric patterns – the GC on actuation of such patterns has been investigatedin [29, 1, 35] who proved that spiral patterns deform into cones. It is convenient to use the polarcoordinates ( r, θ ) to denote the position ( x, y ) in Cartesian coordinates, i.e., ( x, y ) = ( r cos θ, r sin θ ).The components n ( x, y ) and n ( x, y ) for spiral pattern have the forms n ( x, y ) = ¯ n ( r, θ ) = cos( θ + α ) ,n ( x, y ) = ¯ n ( r, θ ) = sin( θ + α ) , (1) n ( r, θ ) = cos( θ + α ) e + sin( θ + α ) e , (2)where α ∈ ( − π/ , ∪ (0 , π/
2) is a constant angle between the tangent of the spiral passing through( x, y ) and the radial direction (see Fig. 2(a)). Here we carefully choose the domain of α to excludetwo degenerate cases: the radial pattern with α = 0 and the circular pattern with α = ± π/
2. Thesetwo degenerate cases make some of our formulae not well-defined and thus will be treated separately.4igure 2: (a) A logarithmic spiral curve r ( θ ) = r exp (cid:0) θ tan α (cid:1) in red with r = 0 . α = π/ . α . (b) A logarithmic director pattern in R with radius R . (c) A cone evolved from the patternedreference state has the slant height ˜ ρR , base radius ρR , and cone angle ϕ determined by the conedeformation Eq. (13). The coefficients ˜ ρ and ρ are given in the text. (d) The proto-radius Γ(red) is deformed to the actual radius Γ (cid:48) (blue) upon actuation. The actual radius is illustrated byprojecting onto R . Without loss of generality, we assume the r = r circle remains unrotated buthas a rescaled radius after actuation.The circular pattern has been extensively studied in [35, 34, 18], and will also be discussed in thispaper, as a special case of the spiral pattern. For smaller α , and in particular α = 0 (radial director),actuation is to an anti-cone where circumferences are long relative to their radii and ruffled shapeswith negative localised GC result [29].The spiral pattern satisfying Eq. (1) is constructed in the following way. Firstly the single spiral[Fig. 2(a)] is of the form r ( θ ) = r exp (cid:18) θ tan α (cid:19) , (3)where r defines the reference radius for θ = 0. The director on the spiral is defined as its unittangent. One can easily confirm that the director satisfies Eq. (2) by this definition. The spiraldirector pattern D (initially assumed to be infinite) in Fig. 2(b) is constructed by applying acontinuous rotation to a single radius, say, D = (cid:26) r exp (cid:18) θ tan α (cid:19) R ( ξ ) e r : ξ ∈ [0 , π ) , θ ∈ R (cid:27) , (4)where R ( . ) = cos( . )( e ⊗ e + e ⊗ e ) + sin( . )( − e ⊗ e + e ⊗ e ) is a rotation in R , and e r = cos θ e + sin θ e is the radial direction. The domain D is well defined in the sense that eachpoint in R corresponds to one and only one pair of ( ξ, θ ) in the domain defined in (4). To achievea finite domain of director, for example a circular plate with radius R > r , one has to truncate θ by θ ∈ [0 , θ ] for tan α > θ ∈ [ θ , ∞ ) for tan α <
0, where θ = tan α ln( R/r ). The spiral pattern with radius R illustrated in Fig. 2(b) will deform into a cone in Fig. 2(c) or ananticone [29] upon stimulation, depending on the material parameters and the spiral constant α . It is5hown in [30, 29, 35, 34] that the spiral pattern results in a cone or an anti-cone depending on whetherthe spiral constant α is greater than or less than the threshold angle α c = arcsin (cid:16)(cid:113) λ − ν − λ − ν λ − ν − λ (cid:17) .The spontaneous deformation gradient associated with the director n has the symmetric form U n = λ n ⊗ n + λ − ν n ⊥ ⊗ n ⊥ , (5)where n ⊥ = R ( π/ n is perpendicular to n in the plane of the sheet and the tensor component( n ⊗ n ) ij = n i n j . Under U n , the pattern has a contraction λ < n and an elongation λ − ν along n ⊥ with the optothermal Poisson ratio ν .We focus on the cone case here and revisit the metric tensor argument in [35] to derive an explicitdeformation that maps the material point on the reference director pattern to the correspondingpoint on the actuated cone.Recalling the argument in [35], the components of the metric tensor a for an actuated spiralpattern in polar coordinates are a rr = λ + ( λ − ν − λ ) sin α,a rθ = a θr = − r ( λ − ν − λ ) sin(2 α ) ,a θθ = r [ λ − ν − ( λ − ν − λ ) sin α ] . (6)We define proto-radii Γ in Fig. 2(d) on the reference domain as curves that deform to be actual radiiΓ (cid:48) after actuation [34]. The proto-radii are indicators of how much rotation takes place on actuation,and it is rotation that gives the difficulty in fitting the actuated cones together at their interface.By circular symmetry, the deformation takes circles into circles. Let Γ have the parametric form( r ( s ) , θ ( s )) in polar coordinates. Then the tangents of Γ and circles have to obey t · a · t = 0 , (7)where t = (0 ,
1) is the tangent to a circle and t = (d r/ d s, d θ/ d s ) is the tangent to Γ. SolvingEq. (7), one can obtain the expression of the proto-radii Γ as r ( θ ) = r exp (cid:18) θb (cid:19) , (8)where b = (1 − λ ν ) ) cot αλ ν ) +cot α . Without loss of generality, we assume the circle r = r is unrotatedafter actuation. Then the material point x in Fig. 2(c) is deformed to x (cid:48) with a relative rotation − b ln (cid:16) rr (cid:17) , where r = | x | .As illustrated in Fig. 2(c), the deformed cone has a rescaled base radius ρr and a rescaled slantheight ˜ ρr . The coefficients ρ and ˜ ρ are ρ = (cid:113) λ − ν − ( λ − ν − λ ) sin α, ˜ ρ = λ − ν (cid:112) λ − ν − ( λ − ν − λ ) sin α . (9)These scaling coefficients can be calculated by the metric change along the proto-radii and thecircumference: 6. The circumference of the deformed cone, which accordingly defines the in-space radius, is ρ πR = (cid:90) π (cid:112) a θθ ( R )d θ = 2 πR (cid:113) λ − ν − ( λ − ν − λ ) sin α ; (10)2. The proto-radii emanating from the origin give radii upon actuation of in-material length˜ ρR = (cid:90) R (cid:112) a rr + 2 a rθ d r/ d θ + a θθ (d θ/ d r ) d r = λ − ν (cid:112) λ − ν − ( λ − ν − λ ) sin α R. (11)The cone angle ϕ is then given by the ratio of radii as ϕ = arcsin( ρ/ ˜ ρ ) = arcsin[ λ − − ν − ( λ − − ν − λ ν ) sin α ] . (12)According to the previous discussion, we next define the cone deformation that encodes therotation of the in-material point and maps the reference pattern to the corresponding cone surface.For a spiral director pattern occupying the reference region D = { ( r cos θ, r sin θ ) : r ∈ (0 , R ) , θ ∈ (0 , π ] } with the spiral constant α , we define the cone deformation y α : D → R as y α ( r ) = ρr (cid:20) R e ( − b ln rr ) e r − cot ϕ e (cid:21) , r ∈ D , (13)where R is the radius of the reference domain (see Fig. 2(b)), r = ( r cos θ, r sin θ ), e r = cos θ e +sin θ e , ρ = √ λ sin α + λ − ν cos α , b = (1 − λ ν ) ) cot αλ ν ) +cot α , R e ( . ) = cos( . )( e ⊗ e + e ⊗ e ) +sin( . )( − e ⊗ e + e ⊗ e ) + e ⊗ e is a rotation with axis e , and the cone angle ϕ is given by (12).For circular patterns with α = ± π/
2, the cone deformation degenerates to y c ( r ) = ρr ( e r − cot ϕ e ) , (14)where ρ = λ and ϕ = arcsin( λ ν ), with no rotation ( b = 0), by substituting α = ± π/ Now we consider two director patterns separated by an interface. Generally, the actuated interfacesfrom the two patterns are intrinsically different – the local metric changes of the interface from twopatterns are not equal and therefore no isometry can stitch these two patterned pieces together atcommon interface isometrically achieved from the original two, deformed interfaces. Incompatibil-ity in metric (for example metrics induced by swelling [27]) usually leads to complex shape changecoupled with mechanics, which is beyond the scope of this paper. To construct a non-isometricorigami, we focus our attention on a metric-compatible interface that has the same stretches alongits length induced by actuating the two patterns. A canonical example is the interface betweencircular director patterns [Fig. 1]. In [18], it is found that the metric-compatible interface evolvesinto a conically consistent interface in 3D, and then the actuated configuration is simply the combi-nation of cones with the same cone angles, and no rotations are involved. Since the metric changeinduced by inhomogeneous director patterns is non-trivial, the actuated interface carries non-trivial,concentrated GC. 7ooking beyond circular patterns, we now study the metric-compatible interfaces between twospiral patterns. The metric compatibility condition, as discussed in [18], ensures that the interfacesarising from actuation according to the two respective patterns have the same metric locally. Themetric compatibility derived from the metric change has the following equivalent forms, using thedeformation gradient of Eq. (5): | U n t | = | U n t | ⇔ t · a · t = t · a · t ⇔ ( n · t ) = ( n · t ) , (15)where t is the unit tangent to the interface, n , n are directors on two sides of the interface, and a i = U T n i U n i is the metric tensor for i = 1 ,
2. Now we explicitly and exhaustively characterize all possiblemetric-compatible interfaces between two spiral patterns, by solving the metric compatibility (15)for n and n belonging to two spiral patterns. Suppose two spiral patterns centered at c = ( − c, c = ( c,
0) are given by D = (cid:26) r exp (cid:18) θ tan α (cid:19) R ( ξ ) e r − c e : ξ ∈ [0 , π ) , θ ∈ R (cid:27) , D = (cid:26) r exp (cid:18) θ tan α (cid:19) R ( ξ ) e r + c e : ξ ∈ [0 , π ) , θ ∈ R (cid:27) . (16)Let γ ( s ) be a parametric form of the curved interface separating D and D . The unit tangent isthen t ( s ) = γ (cid:48) ( s ) / | γ (cid:48) ( s ) | . In the following we drop explicitly displaying s -dependence to abbreviatenotations, and t is not generally a constant vector.We will prove that the solution to Eq. (15) essentially has four situations: • Hyperbolic-type interface. If c (cid:54) = 0 and n (cid:54) = ± n , one solution to (15) is t · n = t · n . Inthis case, the tangent t bisects the two directors n and n . • Elliptic-type interface.
The other solution for the case c (cid:54) = 0 and n (cid:54) = ± n is t · n = − t · n .In this case, the tangent t is perpendicular to the bisector of n and n . • Continuous-director interface.
The directors on two sides of the interface are parallel, i.e., n = ± n . Then Eq. (15) holds trivially. • Concentric spiral interface. If c = 0, the two spiral patterns have the same center. Wefind that the metric-compatible interface between them is a logarithmic spiral with the samecenter, but different spiral constant α , or the interface degenerates to a radial line or a circle.These four types of interfaces are named according to their geometrical features on the referencedomain which will be discussed in detail in the following subsections. Before heading to theoreticalresults and simulations, we elucidate some settings throughout the paper: • The material constants λ = 0 . ν = 0 . • For simulations, the spiral patterns have the centers at ( − ,
0) and (2 , • The boundary of the reference domain is a circle with radius R = 8 if not additionally noted.8 The left/right actuated configuration (e.g. the yellow conical configurations in Fig. 4) isobtained by cutting along the interface and then applying the corresponding cone deformationto the left/right pattern separately.Our simulations were conducted with Morphoshell, a bespoke ‘active-shell’ code. For full detailssee Ref. [15]. The numerical approach models a thin sheet of incompressible Neo-Hookean elastomeras a 2D surface, assigns an elastic energy to that surface, and minimises that energy to find equilib-rium configurations. The energy penalises both stretch (deviations from the ‘programmed’ metricin Eq. (6)), and bend. Taking the reference-state thickness of the sheet to be h , the stretch andbend energies are ∝ h and ∝ h respectively, as is usual for a thin shell. Thus, for suitably small h the stretch energy dominates, and the activated surface approaches an isometry of the programmedmetric.The programmed metrics considered in this work generically encode singular curvature at spiralcenters and along interfaces. Thus at any non-zero h there are localised regions around these featureswhich are significantly non-isometric, and exhibit only finite curvature, thereby relieving a singularbend energy at the cost of some stretch energy. This effect is the cause of the visible ‘blunting’ of allsimulated interfaces and cone tips. In the h → (cid:15) (cid:46) .
003 in allsimulations, confirming that our simulated surfaces are indeed near-isometries of the programmedmetric. This in turn provides strong evidence that even when the two activated surfaces formedon either side of a metric-compatible interface are conically inconsistent, they can nonetheless be‘fitted together’ isometrically.Finally, we note that some of the simulated surfaces, although entirely physical, may only belocal energy minima because the surfaces are multistable in general.
We employ elliptic coordinates to calculate the hyperbolic-type and elliptic-type interfaces. Asillustrated in Fig. 3(a), two spirals Γ ∈ D and Γ ∈ D from the two patterns intersect at thepoint x = ( x ( u, v ) , y ( u, v )) = c (cosh u cos v, sinh u sin v ) , (17)where u ≥ v ∈ [0 , π ), and two spiral centers locate at ( − c,
0) and ( c, u form ellipses and the curves of constant v form hyperbolae, as shown in Fig. 3(a). For n (cid:54) = ± n and c (cid:54) = 0, metric-compatible interfaces are either hyperbolic-type, I , with tangent t bisecting n and n , or elliptic-type, I , with tangent t (cid:48) bisecting − n and n . Following the geometry inFig. 3(a), elliptical-hyperbolic systems have the properties: r = x − c , | r | = c (cos v + cosh u ) , n = R ( α ) r | r | , r = x − c , | r | = c ( − cos v + cosh u ) , n = R ( α ) r | r | , u = ∂ x ∂u = c (cos v sinh u, sin v cosh u ) , v = ∂ x ∂v = c ( − sin v cosh u, cos v sinh u ) . (18)9igure 3: (a) Metric-compatible interfaces (red) between two spiral patterns in elliptic coordinates.Two spirals Γ and Γ centered at c = ( − c,
0) and c = ( c,
0) intersect at x . The directors at x are n and n . Two interfaces I and I passing through x have tangents t bisecting n and n ,and t (cid:48) bisecting − n and n . The vectors u and v (blue) are the unit tangents along the u -lineand v -line that pass through x . The vectors r and r point from the spiral centers c and c to x . These vectors are given explicitly in Eq. (18). (b) Illustration of angles between the vectors. n i is obtained by rotating r i / | r i | counterclockwise by an angle α i , for i = 1 ,
2. By the property ofelliptical coordinates (a proof is given in the text), the vector u bisects r and r with the angle β .The angle ξ between t and u is ξ = α + α (see text).Assume the interface is described by f ( x ( u, v ) , y ( u, v )) = 0. Differentiating f yields( t ⊥ · u )d u + ( t ⊥ · v )d v = 0 , (19)where t ⊥ = ( ∂f∂x , ∂f∂y ) / | ( ∂f∂x , ∂f∂y ) | is the unit normal to the interface. The unit tangent is given by t = ( − ∂f∂y , ∂f∂x ) / | ( − ∂f∂y , ∂f∂x ) | satisfying t · t ⊥ = 0. We discuss compatibility solutions t · n = t · n and t · n = − t · n for hyperbolic-type and elliptic-type interfaces respectively . Hyperbolic-type interface.
For n · t = n · t and n (cid:54) = ± n , we have two solutions for t : t = ± n + n | n + n | , and choose the “+” solution t = n + n | n + n | . The “-” solution inverts the tangent t to − t and gives the same curve. Then t ⊥ = − n + n |− n + n | . Substituting t ⊥ in terms of n and n into (19), bydirect calculation we have the differential equation for u and v :d v d u = tan α + α , (20) Changing α to α ± π or α to α ± π will invert one of the directors and result in the exchange of hyperbolic-typeand elliptic-type interface, while preserving the two spiral patterns. The previous bisector of the directors is thenchanged to the orthogonal dual of the bisector of the new directors. We use this fact in Figs. 4(c)(d) to unify thefeatures of the interfaces in the same column. v ( u ) = u tan α + α v . (21)Eq. (20) can also be obtained by a more intuitive geometrical approach. In elliptic coordinates, u bisects r and r as seen by checking u · r / | r | = u · r / | r | . We assume the angle between r and u is β , as illustrated in Fig. 3(b). Since t bisects n and n , then the angle between t and n is(2 β − α + α ) /
2. The angle ξ between t and u is ξ = (2 β − α + α ) / α − β = ( α + α ) / u and v holds.Finally, the interface x ( u ) = ( x ( u ) , y ( u )) is of the form (cid:40) x ( u ) = c cosh u cos (cid:0) u tan α + α + v (cid:1) y ( u ) = c sinh u sin (cid:0) u tan α + α + v (cid:1) . (22)Notice that ( x (0) , y (0)) = c (cos v , v is determined by the intersection between theinterface and the horizontal axis. Elliptic-type interface.
For n · t (cid:48) = − n · t (cid:48) and n (cid:54) = ± n , we have two solutions for t (cid:48) : t (cid:48) = ± − n + n |− n + n | . Again, we choose the “+” solution, i.e., t (cid:48) = − n + n |− n + n | . Following exactly the samecalculations in hyperbolic-type interface case, or simply replacing α + α in Eq. (20) by α + α + π/ u d v = − tan α + α u ( v ) = − v tan α + α u . (24)Thus the interface x ( v ) = ( x ( v ) , y ( v )) is of the form (cid:40) x ( v ) = c cosh (cid:0) − v tan α + α + u (cid:1) cos vy ( v ) = c sinh (cid:0) − v tan α + α + u (cid:1) sin v . (25)Notice that ( x (0) , y (0)) = c (cosh u ,
0) and thus u is determined by the intersection between theinterface and the horizontal axis. Examples of hyperbolic-type and elliptic-type interfaces.
The appearances of the inter-faces with analytical forms (22) and (25) highly depend on the parameters α , α , u , v . Despitehaving analytical forms, generically the interfaces are sophisticated curves with no symmetries. No-tice that hyperbolic-type and elliptic-type interfaces exist in pairs. Here we provide two generalcases in Figs. 4(a)(b), and highlight some examples with special α and α in Figs. 4(c)-(f). Fig. 4(a)shows the hyperbolic-type interface between a spiral pattern with α = 5 π/
12 and a circular patternwith α = − π/
2. One can clearly see that the tangent of the interface bisects the directors wherethey meet. Fig. 4(b) shows the elliptic-type interface between these two patterns. Figs. 4(c)(d) il-lustrate two examples of the two types of interfaces that have α = α = α . Recalling the formulaeEqs. (22) and (25), we have (cid:40) x ( u ) = c cosh u cos ( u tan α + v ) y ( u ) = c sinh u sin ( u tan α + v ) (26)11igure 4: Examples of hyperbolic-type interfaces and their elliptic-type pairs. (a)-(f) Top left:reference domains. Bottom left: analytical separated cone parts. Top right and bottom right: twoviews of the simulated configurations obtained by the simulation. The parameters are providedwith the figures. 12or hyperbolic-type interface passing through ( x (0) , y (0)) = (0 ,
0) when v = π/
2, and (cid:40) x ( v ) = c cosh ( − v tan α + u ) cos vy ( v ) = c sinh ( − v tan α + u ) sin v (27)for elliptic-type interface passing through ( x ( π/ , y ( π/ ,
0) when u = π tan α . Examiningthese two forms, we find that the two interfaces have 180 degree rotation symmetry, as illustratedin Figs. 4(c)(d).Figs. 4(e)(f) show interfaces between two patterns with equal and opposite spirals, α = − α ,and hence also opposite twists. Notice that for these two cases, the parametric forms of interfacesin Eqs. (22) and (25) degenerate to a hyperbolic interface (cid:40) x ( u ) = c cosh u cos v y ( u ) = c sinh u sin v (28)and an elliptical interface (cid:40) x ( v ) = c cosh u cos vy ( v ) = c sinh u sin v . (29)Furthermore, the hyperbolic interface degenerates to a straight-line when v = π/ Interfaces in circular patterns.
Now we briefly discuss the metric-compatible interfacesbetween two circular patterns. Notice that the analysis for spiral patterns in elliptical coordinatesstill holds for circular patterns − we only need to replace α and α by ± π/
2. Without loss ofgenerality, we choose α = π/ α = − π/
2. The other choices of π/ − π/ v and u in (28) and (29). For more details of the actuated states and complextopographies on combining circular patterns, we refer to a previous work [18]. A trivial “interface” between two spiral patterns is the so-called continuous-director interface. Thatis, the directors n and n are parallel across the interface, and a director field integral curve (blue,see Fig. 5a) suffers no kink upon crossing the interface. The metric compatibility condition holdstrivially by substituting n = ± n into Eq. (15). Establishing the coordinate system in Fig. 3, thedirectors n and n at position x = ( c cosh u cos v, c sinh u sin v ) again have the forms n ( x ) = R ( α ) x − c | x − c | , n ( x ) = R ( α ) x − c | x − c | , (30)13igure 5: (a)-(b) Top left: reference domains. Bottom left: analytical separated cone parts. Topright and bottom right: two views of the actuated configurations obtained by the simulation. Themetric-compatible interface between two spiral patterns is (a) the horizontal axis for α = α =1 . α = 1 .
173 and α = − . c = ( − c, c = ( c,
0) and R ( . ) = cos( . )( e ⊗ e + e ⊗ e ) + sin( . )( − e ⊗ e + e ⊗ e ) is arotation in R . Recalling | n ( x ) | = | n ( x ) | = 1, we have n ( x ) = ± n ( x ) ⇔ ( n ( x ) · e )( n ( x ) · e ) = ( n ( x ) · e )( n ( x ) · e ) , (31)the latter condition expressing equal tangents of the angles the directors make with e and eliminatesthe modulus denominators of Eq. (30). Substituting the expression of x and Eq. (30), a directcalculation of Eq. (31) yieldssin( α − α )(sin v − sinh u ) + 2 cos( α − α ) sin v sinh u = 0 . (32)We have the following two cases depending on ( α − α ): • α = α or α − α = ± π . Eq. (32) implies sin v sinh u = 0 = y , then the interface is thehorizontal axis, as depicted in Fig. 5(a). We parameterize the interface by x ( u ) = ( u, , u ∈ R . (33) • α (cid:54) = α and α − α (cid:54) = ± π . Eq. (32) is equivalent to x + (cid:18) y − c tan( α − α ) (cid:19) = c sin ( α − α ) (34)by substituting ( x, y ) = ( c cosh u cos v, c sinh u sin v ). Clearly, the interface (34) is a circle withthe radius c/ | sin( α − α ) | and center (0 , c/ tan( α − α )), as depicted in Fig. 5(b). The circlebecomes x + y = c when α − α = ± π/
2. We can parameterize the interface by x ( θ ) = (cid:18) , c tan( α − α ) (cid:19) + c | sin( α − α ) | (cos θ, sin θ ) , θ ∈ [0 , π ) . (35)The degeneracy α − α = ± π of (32) only occurs for circular patterns since the constant α forspiral patterns is not equal to ± π/ − π/ , − α c ) ∪ ( α c , π/
2) for forming cones.14 .3 Interfaces between concentric spirals
Figure 6: (a)-(d) Examples of concentric spiral interfaces. Top left: reference domains. Top bottom:analytical separated cone parts. Top right and bottom right: actuated configurations obtained bythe simulation. (a)-(b) The concentric spiral interface between a spiral pattern with α = 1 and acircular pattern with α = π/
2. The spiral interface has the spiral parameter (a) ¯ α = ( α + α ) / α = ( α + α ) / π/
2. The orange spiral interface is constructed by rotating the red one by π . (c)-(d) The radial line interface (c) and the circular interface (d) between two spiral patterns with α = 1 . α = − .
2. The actuated state with the circular interface is geometrically consistentunder the cone deformations, with the two domains having the opposite relative rotations indicatedby the radial dashed line in black before and after actuation.We now consider the interfaces between two spiral patterns centered at the same position (0 , c = 0. The elliptical coordinates are not appropriate for this situation, so we write the directors n = (cos( θ + α ) , sin( θ + α )) and n = (cos( θ + α ) , sin( θ + α )) at the position ( r, θ ) in polarcoordinates. Assuming α (cid:54) = α (the case α = α is trivial), the tangent t has two solutions t = ± n + n | n + n | ⇔ t = ± (cid:18) cos (cid:18) α + α θ (cid:19) , sin (cid:18) α + α θ (cid:19)(cid:19) (36)15nd t = ± n − n | n − n | ⇔ t = ± (cid:18) cos (cid:18) α + α + π θ (cid:19) , sin (cid:18) α + α + π θ (cid:19)(cid:19) (37)which correspond to the two spirals with parameters α = ¯ α = ( α + α ) / α = ( α + α ) / π/ α + α (cid:54) = 0 , ± π , and where ¯ α = ( α + α ) / r ( θ ) = r exp (cid:18) θ tan ¯ α (cid:19) , (38)corresponding to the interface in Fig. 6(a) or the form r ( θ ) = r exp (cid:18) θ tan( ¯ α + π/ (cid:19) (39)corresponding to the interface in Fig. 6(b). Here we construct two interfaces in each of Figs. 6(a)(b),so that the domain is divided into two subdomains occupied by the two spiral patterns (one is acircular pattern for the particular example) respectively.The idea can be simply generalized to n concentric spiral patterns with the spiral constants α i , i = 1 , , . . . , n . Then the interface between the i -th pattern and the ( i +1)-th pattern is a logarithmicspiral with the spiral constant ( α i + α i +1 ) / α i + α i +1 ) / π/
2, for i = 1 , , . . . , n and we define α n +1 = α . However, interfacial spirals will generically intersect, dividing the domain into regionsthat would actuate into facets in the target space. However, we see relaxation of boundaries in thesimulations of Fig. 6(a), and so these facets would presumably also relax.For the cases α + α = 0 , ± π , the interface degenerates to a radial line [Fig. 6(c)] parameterizedas x ( r ) = ( r cos ξ, r sin ξ ) , r ∈ [0 , R ] , (40)where ξ ∈ [0 , π ) is a constant angle denoting the radial direction and R is the radius of the referencedomain, or degenerates to a circle with the same center [Fig. 6(d)] parameterized as x ( θ ) = (¯ r cos θ, ¯ r sin θ ) , θ ∈ [0 , π ) , (41)where 0 < ¯ r < R is the radius of the interface. As illustrated in Fig. 6(d), the actuated state isa cone, which is conically consistent. The two domains have the opposite rotations relative to theinterface upon actuation, indicated by the actuated radial dashed line in black. Summary.
By considering the situations c = 0 or c (cid:54) = 0, n = ± n or n (cid:54) = ± n , we havecharacterized all possible metric-compatible interfaces between two spiral patterns: hyperbolic-type, elliptic-type, concentric spiral and continuous-director interfaces. We term the interface with n (cid:54) = ± n the twinning case , and the interface with n = ± n the continuous-director case . Namely,the hyperbolic-type, elliptic-type and concentric interfaces belong to the twinning case, and the twocontinuous-director interfaces in Fig. 5 belong to the continuous-director case. In the previous discussion, we have systematically found all possible metric-compatible interfacesbetween two spiral patterns, by solving the metric compatibility condition (15). We assume that:16. The two patterns evolve into cones so that we can use the explicit cone deformations (13) tostudy the deformed interfaces; 2. The director n ( n ) on the left (right) side of the interface isassociated with the left (right) pattern when c (cid:54) = 0. In this subsection, we generalize the method ofconstructing patterns in several natural ways, extending it to more complex intersections of circlesand spirals, and thereby show that the class of metric-compatible interfaces is even richer.First of all, the method is also applicable to spiral/radial patterns that evolve into anticones.Fig. 7(b) illustrates a hyperbolic-type interface between a spiral pattern with α = 5 π/
12 and aradial pattern with α = 0. The spiral pattern evolves into a cone with positive GC localized atthe tip, whereas the radial pattern evolves into an anticone with negative GC localized at the tip.The simulation in Fig. 7 illustrates the feature of the combination of cone-like and anticone-likestructures.Figure 7: Top row: reference domains. Middle and bottom rows: two views of the actuated con-figurations obtained by the simulation. (a) The hyperbolic-type interface between a spiral patternwith α = 5 π/
12 and a circular pattern with α = − π/
2. (b) The hyperbolic-type interface betweena spiral pattern with α = 5 π/
12 and a radial pattern with α = 0. (c) The complement of thepattern in (a). (d) The orthogonal dual of the pattern in (a). (e) The complement of the orthogonaldual. The reference domains of the simulations for (c) and (d) are the smaller circular domains inthe dashed circles for the sake of clarity.Next, we can construct new metric-compatible patterns by taking the complements of the existingones. For example, the complement of the original pattern in Fig. 7(a) is the pattern in Fig. 7(c)constructed by combining the hidden parts of spiral/circular patterns on the other side of theinterface. The original pattern satisfies the metric compatibility condition (15), i.e., | n · t | = | n · t | .This implies that the complement pattern is also metric-compatible at the same interface, because17he director of the complement at the interface becomes n on the left side and n on the rightside, then the condition n · t = n · t still holds. This construction relies on the smoothness ofeach pattern across the interface. It is worthwhile mentioning that the virtual centers of the spiralpattern and the circular pattern in the complement are located at the same places as those realcenters in the original pattern. In the original pattern, these centers deform into the tips of cones,which are point defects of Gaussian curvature. These point defects are found to play a crucialrole in the mechanical behaviour of thin sheets [46, 50, 55]. By removing the centers, we anticipatecomplements will have significantly different mechanical properties, as illustrated by the comparisonbetween the simulations in Fig. 7(a) and (c).Furthermore, another way of constructing a different metric-compatible pattern is taking the orthogonal dual of the original one. The orthogonal dual of the original pattern with director field n ( x ) is the pattern with director field n ⊥ ( x ), where n ⊥ ( x ) = R ( π/ n ( x ) is perpendicular to n ( x ).For example, the orthogonal dual of a circular pattern is the radial pattern with director pointingalong the radial direction. Upon actuation, the orthogonal dual deforms to an anticone, whereasthe circular pattern deforms to a cone [29, 15]. A metric-compatible interface between two patternsis still metric-compatible after taking the orthogonal duals by the fact that | n · t | = | n · t | ⇔ | n ⊥ · t | = | n ⊥ · t | . (42)Fig. 7(c) shows the orthogonal dual of the original pattern in Fig. 7(a), with its complement inFig. 7(d).Figure 8: (a) A 2 × α , α , α , α ) =(1 . , − . , . , − .
25) separated by straight-line interfaces. (b) An irregular 2 × α , α , α , α ) = (1 . , − . , π/ , π/ α , α , α , α ) = ( − . , . , , − × − the interface intersections are hard to predict and might break thetopology of the pattern easily because of the highly irregular shapes of the interfaces. An exampleof the mixture of hyperbolic-type and elliptic-type interfaces is given in Fig. 8(c). The topographiesevolved from these large spiral patterns are also expected to be more complex because of conicalinconsistency. Despite obeying the local metric change induced by the director, the deformation that maps a generaldirector pattern to its actuated state is not unique. Consequently, the resultant deformed interfacebetween two patterns is not unique. This is because the metric change only determines the firstfundamental form of a surface, whereas the second fundamental form is still unknown. For spiral andcircular patterns, the simplest deformations satisfying the metric change are the cone deformations(13) and (14). In [18], it is proved that the shapes of the actuated, metric-compatible interfacesobtaining under the cone deformations between circular patterns are the same – the interfaces areconically consistent, since the two pieces of cone can be fitted together without further isometriesfrom conical to make them match. To be specific, let x ( u ) denote the metric-compatible interfaceand c , c be the centers of two circular patterns. Then, there exists a constant rotation R ∈ SO (3)and a constant translation h ∈ R such that Ry c ( x ( u ) − c ) + h = y c ( x ( u ) − c ), where y c is thecone deformation for circular patterns defined by Eq. (14).However, generically, the deformed metric-compatible interfaces y α and y α between spiralpatterns are not geometrically consistent under their cone deformations, unless the spiral constantsare | α | = | α | = π/
2, i.e., the two spiral patterns degenerate to circular patterns. These phenomenacan be seen in the examples of geometrically consistent interfaces in circular patterns [Fig. 1] andgeometrically inconsistent interfaces in spiral patterns [Figs. 4, 5 and 6]. The problem is thatdeformations of spirals carry rotations, see after Eq. (8) and after (13), and that these rotationsstemming from each side of the metric-compatible line rotate its elements differently on actuationinto a crease. An example is in Fig. 4(a) where the boundary evolved under the circular patternsuffers no rotations, whereas that under the spiral suffers rotations dependent upon the distancefrom the spiral center to the metric-compatible line in the reference state. Explicitly, supposea hyperbolic metric-compatible interface is x ( u ) = ( x ( u ) , y ( u )), calculated from Eq. (22) (andequivalently an elliptic metric-compatible interface x ( v ) = ( x ( v ) , y ( v )), calculated from Eq. (25)).The centers of the two patterns are c = ( − c,
0) and c = ( c,
0) with c >
0. Recalling the conedeformation y α ( r ) in Eqs. (13) and (14), the deformed interfaces y ( u ) and y ( u ) mapped by thetwo cone deformations from two sides yield y ( u ) = y α ( x ( u ) − c ) , y ( u ) = y α ( x ( u ) − c ) . (43)For the particular example in Fig. 4(a), we have the parameters α = 5 π/
12 and α = − π/
2. Wereplace y − π/ ( r ) by the cone deformation y c ( r ) for circular patterns. The cone deformations’ y , ( r )rotation angles − b , ln ( r , ( u ) /r , (0)) (recall Eq. (13), with α (cid:54) = ± π/
2) are functions of r , ( u ),19hich generically ( r ( u ) (cid:54) = r ( u )) leads to different rotations along the interface driven from twosides. As a result, the shapes of the deformed interfaces y ( u ) and y ( u ) are generically different.Since bend energy scales as ∼ h and stretch energy scales as ∼ h , with h the thickness of thereference LCE sheet, the actuated configuration tends to follow the metric condition as h →
0, thatis, avoiding stretch from the actuated shape. Under this circumstance, it is physically favorable tohave two further isometric deformations g and g mapping the two interfaces to an identical one g ( u ), i.e., g ( u ) := g ( y ( u )) = g ( y ( u )) . (44)The additional isometries g and g preserve the geodesic curvatures of y ( u ) and y ( u ), andtherefore preserve the concentrated GC discussed in the next section. Since geodesic curvature and GC are intrinsic quantities preserved by isometric deformations, wecan analytically calculate the distributed GC along the deformed interface resulting from the conedeformations without considering the additional isometries. We then compare our analytical GCwith that of simulations, and see numerically that the simulation yields a configuration satisfyingthe metric change induced by the director. In other words, the isometries g and g that match theinterfaces together exist.We employ three distinct approaches for analytically calculating the GC concentrated along thedeformed interface. The first two approaches are based on the typical differential geometry methoddealing with the geodesic curvature on the actuated states (cone surface) or the unrolled 2D flatstate. The third approach considers the metric change directly from the reference director fieldsand then applies Liouville’s formula [37, 15] to calculate the geodesic curvature and the GC. Thethree approaches are illustrated by the sketch of circular patterns in Fig. 9, but the ideas apply tothe general spiral patterns.
The concentrated GC along the deformed interface is the sum between two geodesic curvatures fromtwo sides. This argument is consistent with considering the Gauss-Bonnet theorem along a loopcontaining the interface with the width of the loop tending to zero. By choosing the appropriateGauss-Bonnet loop such as that in Fig. 9(b), the Gauss-Bonnet theorem for the actuated domain M within the loop is (cid:90) M K A d A A + (cid:73) ∂M κ g d s A + Σ β i = 2 πχ ( M ) , (45)where κ g is the geodesic curvature along the loop ∂M , the β i are the turning angles, and χ ( M ) isthe Euler characteristic of M . Taking the limit of the width of the loop to zero, the Gauss-Bonnettheorem for the actuated interface yields the integrated GCΩ = (cid:90) K A d A A = (cid:90) ( k g − k g )d l A (46)after canceling the Euler characteristic term and the turning angle term (both are 2 π ), and neglect-ing the geodesic integral along the infinitesimal sections of the contour transverse to the interface.20igure 9: (a) Straight-line interface (red) between two circular patterns. The unit tangent t of theinterface is expressed as t = cos φ n + sin φ n ⊥ , where n is the director and n · n ⊥ = 0. (b) y ( u ) is theactuated interface deformed from the left circular pattern. The white loop is the Gauss-Bonnet loopcomposed of two paths along the interface connected by infinitesimal geodesic caps that traversethe interface orthogonally. l A is the arc length of the actuated interface and A A is the area enclosedwithin the loop. The dashed lines are where the cuts are performed to unroll the cones onto the2D plane in (c). (c) The unrolled states of two separated domains. ˜ y ( u ) is the interface of the leftdomain after the unrolling process.Here Ω is the integrated GC, K A is the distributed GC within the loop, k g and k g are the geodesiccurvatures of the actuated interfaces y ( u ) and y ( u ) from the left side and the right side, A A is thearea within the Gauss-Bonnet loop, and l A is the arclength of the actuated interface. The actuatedarclength l A can be defined by l A ( u ) = (cid:40)(cid:82) u | y (cid:48) ( u ) | d u, if u ≥ − (cid:82) u | y (cid:48) ( u ) | d u, if u < l A (0) = 0 as the reference point. Since l A ( u ) is monotonically increasing, we canfind its inverse as u = ¯ u ( l A ). The arclength l A derived from y ( u ) is consistent with that from y ( u )by the metric-compatibility condition | y (cid:48) ( u ) | = | y (cid:48) ( u ) | . This equivalence allowed us to transform (cid:72) ∂M κ g d s A of (45), a loop integral, to (cid:82) ( k g − k g )d l A of (46), a line integral along the crease. Wethen define dΩ / d l A as the concentrated GC per unit actuated arclength.However, it is usually convenient to write the geometrical quantities in terms of the parametricparameter u of the reference metric-compatible interface. Consequently, the integrated GC, Ω, interms of u is Ω = (cid:90) K A d A A = (cid:90) ( k g − k g ) d l A d u d u, (48)where d l A d u = | y (cid:48) ( u ) | = | y (cid:48) ( u ) | is the Jacobian. Following the standard method in differentialgeometry [37], κ g and κ g can be calculated by κ g ( u ) = N ( u ) · ( y (cid:48) ( u ) × y (cid:48)(cid:48) ( u )) | y (cid:48) ( u ) | , κ g ( u ) = N ( u ) · ( y (cid:48) ( u ) × y (cid:48)(cid:48) ( u )) | y (cid:48) ( u ) | , (49)where N i ( u ) = cos ϕ i ( I − e ⊗ e ) y ( u ) | ( I − e ⊗ e ) y ( u ) | + sin ϕ i e (50)21s the unit normal to the cone surface at y i ( u ) in which ϕ i is the cone angle, for i = 1 ,
2. Then theconcentrated GC in terms of the parametric reference state parameter u isdΩd u = ( κ g ( u ) − κ g ( u )) d l A d u , (51)and accordingly the concentrated GC in terms of the actuated arc length isdΩd l A = κ g (¯ u ( l A )) − κ g (¯ u ( l A )) . (52) A cone has a center of concentrated Gaussian curvature, i.e. the cone surface is Gaussian flat ev-erywhere except for the tip. We can then cut a cone along its slant flank and unroll it onto the 2Dplane isometrically with the angular deficit 2 π (1 − sin ϕ ), where ϕ is the cone angle. Accordingly,the actuated interface is then mapped to a curve on the 2D plane. As an example, we perform theunrolling process separately for the two actuated cones in Fig. 9(b) yielding the unrolled configu-rations in Fig. 9(c) with their interface boundaries. This procedure will simplify the calculation ofgeodesic curvature − one can calculate the total curvature of the 2D curve instead to obtain theequivalent geodesic curvature. The approach essentially uses the basic differential geometry ideathat the geodesic curvature is preserved by isometric deformations (here the unrolling process).Let ( x, y, z ) denote a point on the cone surface, which is parameterized by ( r, θ ) with the form x = r sin ϕ cos θ, y = r sin ϕ sin θ, z = − r cos ϕ, (53)where θ ∈ [ − π, π ] and ϕ is the cone angle. The cone surface is obtained by applying the conedeformation y α i on the reference domain D i . The deformed interface, as described above, hasthe parametric form y i ( u ). The cut-and-unroll process takes the angle θ in Eq. (53) to θ sin ϕ by considering the angular deficit induced by the unrolling with the cone angle ϕ . Then thepoint ( x, y, z ) parameterized by ( r, θ ) in Eq. (53) is mapped to ( r cos( θ sin ϕ ) , r sin( θ sin ϕ )) in 2D.Specifically, for the actuated interface y ( u ) in Eq. (43), the unrolled 2D curve is then (cid:40) ˜ x ( u ) = | y ( u ) | cos[ θ ( u ) sin ϕ ]˜ y ( u ) = | y ( u ) | sin[ θ ( u ) sin ϕ ] (54)by substituting ( x, y, z ) = y ( u ), where θ ( u ) = sign( y ( u ) · e ) arccos (cid:104) y ( u ) · e | y ( u ) | sin ϕ (cid:105) ∈ [ − π, π ] is theangle between the point on y ( u ) and e after projecting onto { e , e } plane. The unrolled interface˜ y ( u ) is then given by ˜ y ( u ) = (˜ x ( u ) , ˜ y ( u )). This unrolling process is isometric and thus preserves thegeodesic curvature. Following the standard method in differential geometry, the geodesic curvature κ g of the curve (˜ x ( u ) , ˜ y ( u )) on the 2D plane is the same as the total curvature, i.e., κ g ( u ) = ˜ x (cid:48) ˜ y (cid:48)(cid:48) − ˜ y (cid:48) ˜ x (cid:48)(cid:48) (˜ x (cid:48) + ˜ y (cid:48) ) / . (55)Then we can apply the same approach to reparameterize κ g ( u ) in terms of l A , the arclength of theactuated interface. The same approach applies to calculating κ g ( u ), and then the calculation ofthe concentrated GC follows via (51). 22 .3 Concentrated GC calculated from the reference director fields GC is an intrinsic geometrical quantity that can be derived from the metric change induced byactuation determined by a (reference state) director distribution. Thus the director pattern deter-mines the GC distribution over the entire actuated configuration, including point and curve defects[15]. A useful method for deriving the geodesic curvature of curve defects after actuation is to useLiouville’s formula [37]. Specifically, given an orthonormal coordinate system ( u, v ) with the met-ric tensor diag( E, G ), the geodesic curvature of a reference state, arc-length parameterized curve( u ( l ) , v ( l )) is κ g = φ (cid:48) − √ EG ( u (cid:48) E v − v (cid:48) G u ) , (56)where φ is the angle between between the unit tangent t of the curve and the u − line.Here we employ Liouville’s formula to derive the geodesic curvature and the Gaussian curvatureconcentrated along the metric-compatible interfaces for both the twinning cases (hyperbolic-type,elliptic-type and concentric spiral interfaces) and the continuous-director cases. Suppose the arclength parameterized curve r ( l ) = ( x ( l ) , y ( l )) is defined on the reference domain of a directorpattern. The tangent t of r ( l ) is simply t = r (cid:48) ( l ). It is natural to take the director n and itsperpendicular vector n ⊥ as the u and v basis respectively and then φ is the angle rotating n to t counterclockwise (see Fig. 9(a)). Following Liouville’s formula (56) and the metric tensor (see [36])associated with r ( l ), the geodesic curvature of r ( l ) after actuation, derived in [15], is κ g = λ − ν ( λ cos φ + λ − ν sin φ ) / d φ d l + λ − − ν s sin φ + λ ν b cos φ (cid:112) λ cos φ + λ − ν sin φ , (57)where φ , obtained by t = cos φ n + sin φ n ⊥ , is the angle between the curve tangent t and the director n , while b and s are the 2D bend and splay [36] defined by b = ∇ × n = n · ∇ ψ, s = ∇ · n = n ⊥ · ∇ ψ, (58)recalling n = cos ψ e + sin ψ e . For the spiral pattern, we have ψ = θ + α (Eq. (2)). Applying thegradient tensor in polar coordinates, we have b = sin α/r, s = cos α/r, (59)where r is the distance between the spiral center and the point we consider.We employ the formula (57) for geodesic curvature to obtain the concentrated GC along theactuated interface. Notice that the metric tensor a = U T n U n is invariant under the transformation n → − n . Thus the resulting metric change and the induced geodesic curvature (57) are alsoinvariant. This can be seen by rewriting Eq. (57) equivalently in terms of the bend, splay, tangentvectors and the metric tensor: κ g = λ − ν ( t · a · t ) / d φ d l + λ ν b · t ⊥ − λ − (1+ ν ) s · t ⊥ √ t · a · t , (60) We use the standard notation ( u, v ) in differential geometry as the basis. It should be distinguished from theelliptic coordinates in Sect. 3.1. t ⊥ = R ( π/ t is the unit vector perpendicular to t . We define the bend vector b and thesplay vector s in Eq. (60) as b = b n ⊥ , s = s n , (61)which are invariant under n → − n since the transformation induces a minus sign in b , s , n and n ⊥ .Following Eq. (48), the integrated GC is of the form( Twinning case: ) Ω = (cid:90) K A d A = (cid:90) (cid:18) λ − ν t · a · t φ (cid:48) ( l ) + ( λ ν ∆ b ⊥ − λ − (1+ ν ) ∆ s ⊥ ) (cid:19) d l, (62)when the directors on the two sides of the interface are not parallel, i.e., n (cid:54) = ± n . Here we haveused the facts that φ (cid:48) ( l ) = d φ d l = − d φ d l and d l A / d l = √ t · a · t = √ t · a · t = √ t · a · t given bymetric compatibility when n (cid:54) = ± n . In Eq. (62), we also define the perpendicular jumps of thebend and splay vectors as∆ b ⊥ = ( b − b ) · t ⊥ and ∆ s ⊥ = ( s − s ) · t ⊥ . (63)The quantities with subscript 1 or 2 are associated with the pattern on the left or right side ofthe interface. For the continuous-director interface n = ± n , such as the examples in Fig. 5, thefirst term in (60) for the two patterns on the two sides of the interface will be canceled by the fact d φ d l = d φ d l . Then the integrated GC along a line is given by the integral of the perpendicular jumpsof the bend and splay vectors:( Continuous-director case: ) Ω = (cid:90) K A d A = (cid:90) ( λ ν ∆ b ⊥ − λ − (1+ ν ) ∆ s ⊥ )d l. (64)The two examples in Fig. 5 have non-vanishing jumps of the bend and splay vectors across theinterfaces by the formulae (59), (61) and (64), therefore, the resultant actuated interfaces bearnon-zero GC for both cases. The equations (62) and (64) for the distributed GC along lines donot contain any defect-like contributions, e.g. from spiral centres. For the twinning exampleswe have, the interfaces in any event do not pass through the spiral centres. However, for thecontinuous-director cases (Fig. 5), the spiral centres are on the interface, but their contributionsare not contained in Eq. (64).We comment here about the three distinct methods of calculating the concentrated GC along acrease. The first two methods are intrinsically equivalent in applying differential geometry to theexplicit deformation that maps the reference domain on 2D to a 3D surface. The spiral patternwe study in this paper induces the explicit cone deformation so that the first two methods areapplicable. But these two methods can not handle the concentrated GC along a crease on a generaldirector pattern, because the explicit deformation is unknown. Then the formulae (62) and (64)derived from the metrics are necessary. More importantly, the third method provides a possible wayof considering a general crease and its concentrated GC on a general director pattern. We reservethis point as a future work. Remark.
The formulae for the geodesic curvature and the concentrated Gaussian curvature arederived for LCE systems, which have a contraction factor λ along the director and an elongationfactor λ − ν along the perpendicular direction to the director. But these formulae for concentratedGC can be generalized to apply to any 2D contraction/elongation system, such as the inflationsystems [42, 44, 43], swelling [49], and kirigami [25], by replacing λ and λ ν with the metric-changingfactors λ (cid:107) and λ ⊥ of the particular system. One should also notice that the interface has to bemetric-compatible in these systems. 24 .4 Concentrated GC for all types of metric-compatible interfaces We derive the analytical forms of the concentrated GC for all types of the metric-compatible in-terfaces by the third method. We have confirmed that these formulae agree with those computedfrom the other methods. For hyperbolic-type and elliptic-type interfaces, we only provide the con-densed but explicit formulae. For continuous-director interfaces and concentric spiral interfaces, theformulae are concise and obviously reveal the features of the GC.
Hyperbolic-type and elliptic-type interfaces.
Recalling the hyperbolic-type interface x ( u ) in(22) and the GC (62) for the twinning case, the once(transverse)-integrated concentrated GC interms of u is dΩd u = 2 λ − ν t · a · t d φ d u + ( λ ν ∆ b ⊥ − λ − − ν ∆ s ⊥ ) d l d u , (65)where t ( u ) = x (cid:48) ( u ) / | x (cid:48) ( u ) | , φ ( u ) = sign( n × t · e ) arccos( n · t ) , d l d u = | x (cid:48) ( u ) | , ∆ b ⊥ = (cid:18) sin α | r | n ⊥ − sin α | r | n ⊥ (cid:19) · t ⊥ , ∆ s ⊥ = (cid:18) cos α | r | n − cos α | r | n (cid:19) · t ⊥ , r i = x ( u ) − c i , n i = R ( α i ) r i | r i | , n ⊥ i = R ( π/ n i , t ⊥ = R ( π/ t , a = λ n ⊗ n + λ − ν n ⊥ ⊗ n ⊥ , (66)where i = 1 , R ( . ) is a rotationin 2D. The definition of φ ( u ) in (66) has encoded the counterclockwise rotation from n to t . Yetthe formula (65) is generically complicated, the special choices of α i and x ( u ) can simplify theexpression. We provide one example in the next section showing the concise formula explicitlyfor the straight-line interface between two equal and opposite spiral patterns. The calculation forelliptic-type interfaces follows naturally by replacing x ( u ) with x ( v ) given in (25). Continuous-director interfaces.
For the horizontal interface (33) with α = α or α − α = ± π ,we employ Eq. (11) to calculate the concentrated GC. Recalling the formulae (59) and (61) for thebend and splay vectors, we have b · t ⊥ = sin α cos αu + c , b · t ⊥ = sin α cos αu − c , s · t ⊥ = sin α cos αu + c , s · t ⊥ = sin α cos αu − c . (67)Substituting the bend and splay vectors into Eq. (64) yields the analytical form of the concentratedGC as dΩd u = ( λ ν − λ − − ν ) c sin(2 α ) c − u . (68)For the circular interface (35) with α (cid:54) = α and α − α (cid:54) = ± π , we have the parametric form of theinterface x ( θ ) = (¯ r cos θ, c/ tan( α − α ) + ¯ r sin θ ) (69)where ¯ r = c/ | sin( α − α ) | . We obtain the quantities for the GC in a way similar to (66), but with θ as the parametric parameter. By direct calculations, we have the concentrated GC unpon θ asdΩd θ = − ( λ ν − λ − − ν ) cos( α + α − θ ) sin( α − α )2 σ cos( α − α ) + 2 sin θ , (70)25here σ = sign(sin( α − α )). The minus sign in front of the formula arises because the spiralpattern with α ( α ) is on the left (right) side of the interface, not the other way which is commonin the hyperbolic-type interface case in Fig. 4. It should be noted that, quite counter-intuitively,the continuous director fields are also possible to result in non-vanishing concentrated GC alonglines, due to the jumps in splay and bend. Concentric spiral interface.
Recall that the concentric spiral interface is given by r ( θ ) = r exp (cid:18) θ tan α (cid:19) (71)in polar coordinates, where α = ¯ α = ( α + α ) / α = ¯ α + π/
2, and α ( α ) is the spiral parameterfor the pattern on the left (right) side of the interface satisfying α + α (cid:54) = 0 , ± π . See eqn. (38) andfigure 6. The “side” of the interface is consistent with the interface tangent t in terms of θ , i.e., t = (cid:2) r exp (cid:0) θ tan α (cid:1) (cos θ, sin θ ) (cid:3) (cid:48) / (d l/ d θ ). For example, the vector e × t points to the left side ofthe interface. The angle φ between the interface tangent and the director is constant, so the firstterm of (62) vanishes. Prior to calculating the GC, we haved l d θ = r ( θ ) | sin α | , ∆ b ⊥ = ∆ s ⊥ = ± sin( α − α ) cos αr ( θ ) , (72)where “+” corresponds to the case α = ¯ α = ( α + α ) / − ” corresponds to the case α =( α + α ) / π/
2. Then the concentrated GC in terms of θ isdΩd θ = ± ( λ ν − λ − − ν ) sin( α − α ) cos α | sin α | . (73)The formula reveals that the concentrated GC of the spiral interface is a non-zero constant in-dependent of θ . Although the length of a log spiral from −∞ → θ is finite ( l = r ( θ ) sec α ), theGC integrated along the crease diverges since the integrated GC per unit length along the creasediverges as dΩ / d l = ± ( λ ν − λ − − ν ) sin( α − α ) /l on approaching the spiral centre where l = 0,the length l in the problem determining the scale of geodesic curvature.For the case α + α = 0 , ± π , the radial line interface x ( r ) = ( r cos ξ, r sin ξ ) , r ∈ [0 , R ] inFig. 5(c), has the concentrated GCdΩd r = ( λ ν − λ − − ν ) sin(2 α ) r (74)by substituting ∆ b ⊥ = ∆ s ⊥ = cos α sin α r − cos α sin α r . (75)Again, α is chosen as the parameter for the spiral pattern on the left side of the interface. Thecircular interface in Fig. 5(d) bears no GC since it is on the developable cone surface. We provide two examples of the analytical and simulated concentrated GC along creases for thetwinning case and the continuous-director case. The twinning case in Fig. 10 consists of two spiral26atterns with equal and opposite α s, that is α = − α . The metric-compatible interface is astraight line passing through the bisector of the two patterns. The continuous-director case consistsof two spiral patterns with α = α . The interface is a horizontal line passing through the twospiral centers. The analytical formulae and plots of the concentrated GC for the two examples areobtained according to Eqs. (62) and (64). The results of simulations with the code Morphoshellqualitatively agree with the analytical solutions. We establish the coordinates in Fig. 10(a). The centers of the two spiral patterns with α = α = − α are located at ( − c,
0) and ( c, − d to d . The parameters we choose are α = arctan( λ − − ν ), d = 2 √
15 and c = 2,where the materials constants are λ = 0 . ν = 0 .
5. This spiral constant α leads to a cone withcone angle ϕ = π/
2, that is, the actuated configuration is in 2D with no cone tip, as illustrated inFig. 10(a). We are thus able to dis-aggregate the localise tip and the crease contributions to theGC.The quantities required in Eq. (57) for the left pattern are φ = π/ − α − arctan( y/c ) , b = sin α/ (cid:112) c + y , s = cos α/ (cid:112) c + y . (76)The arclength l of the reference interface is simply y . Substituting φ , b and s into (57), one canobtain the geodesic curvature κ g for the left pattern. A similar argument applies to the geodesiccurvature κ g of the right pattern. In fact, transforming α → − α and φ → − φ in (57) for theright pattern, we realize that κ g = − κ g . To calculate the concentrated GC per unit referencestate length, d Ω / d y , by (48) and (62), one has to multiply the geodesic curvature in (60) by theJacobian (cid:112) λ cos φ + λ − ν sin φ . Omitting calculation details, we provide the theoretical resultfor the concentrated GC for the given example asdΩd y = 2 λ − − ν cos α sin φ + λ ν sin α cos φ (cid:112) c + y − c ( c + y )( λ − − ν sin φ + λ ν cos φ )= 2( λ − − ν s sin φ + λ ν b cos φ ) − c ( s + b ) λ − − ν sin φ + λ ν cos φ , (77)where φ , b and s are functions of y defined in (76). Figure 10(b) illustrates the non-dimensionalizedconcentrated GC dΩ / d y against the position y on the reference domain. One can observe that theconcentrated GC has positive and negative regions, where the negative GC is concentrated near y = 0. To compare with the numerical simulation more illustratively, we provide another plotshowing the concentrated GC against the arclength of the actuated crease in Fig. 10(c). The plothas features similar to the previous one. The simulation in Fig. 10 shows that the GC is ‘smearedout’ over a neighborhood around the crease, indicating localised non-isometry and correspondingto the ‘blunting’ effect discussed in Section 3. The trend of the value of GC (small positive → largepositive → large negative → small negative) shown in the simulation agrees with the analyticalsolution in Fig. 10(c). The localisation of non-isometry suggest that isometric deformations can beused to analyse the evolution of the surfaces outside the non-isometric neighborhood of the crease.27igure 10: (a) The straight-line interface passing through the bisector between two spiral patternswith α = − α = 1 .
25, a special value where the actuation is to a flat state, but with rotations.There are no tips and hence no point sources of GC to obscure the crease contributions. (b) Thenondimensionalized concentrated GC c dΩ / d y has positive and negative regions along the nondi-mensionalized position y/c of the reference interface. The positive (negative) GC is concentratednear the position y (cid:47) y (cid:39) c dΩ / d l A against the nondimensionalized actuated arclength l A /c . (d)-(e) The simulated GC rescaled by hc .The simulation shows the smearing out of the GC, which leads to the blunting of the theoreticalinterface indicated by the thin black line. Establishing the coordinates in Fig. 11(a), the centers of the two equal spiral patterns with α = α = α are at ( − c,
0) and ( c, − d to d . To be consistent with the parameter y of the interface in the previous28igure 11: (a) The horizontal continuous-director interface between two spiral patterns with α = α = 1 . y ∈ [ − d, d ] , d = 8, along the horizontal axis to be consistentwith the previous example. The spiral centers are at ( − c,
0) and ( c,
0) with c = 2. (b) The non-dimensionalized, concentrated GC c dΩ / d y has the positive region ( y ∈ [ − d, − c ) ∪ ( c, d ]) and negativeregion ( y ∈ ( − c, c )) along the non-dimensionalized position y/c of the reference interface. The GCapproaches positive infinity as y → − c − or y → c + , and negative infinity as y → − c + or y → c − .(c) The plot of non-dimensionalized, concentrated GC c dΩ / d l A against the non-dimensionalized,actuated arc length l A /c has similar features. (d)–(e) The simulated GC rescaled by hc . Thesimulation shows the smearing out of the GC, leading to the blunted interface indicated by the thinblack line.example, we define the horizontal axis as y here. The parameters we choose for this example are α = 1 . d = 8 and c = 2.Recalling the GC distribution (68) for the horizontal interface and replacing u with y , we havedΩd y = ( λ ν − λ − − ν ) c sin(2 α ) c − y . (78)The plots of the non-dimensionalized, concentrated GC per unit length against the reference position y and the actuated arc length l A are provided in Figs. 11(b) and (c) respectively. One can observethat the GC is positive for y ∈ [ − d, − c ) ∪ ( c, d ] and negative for y ∈ ( − c, c ), and approaches29ositive infinity as | y | → c + , and negative infinity as | y | → c − . These features are consistent withthe analytical form (78) with the chosen parameters. Note that the reference state calculation ofgeodesic curvature, eqn (57), leading to (62), (64), and (78), does not contain the singular defect-likecontributions from the activated tip regions.Another interesting observation from (78) is that dΩ / d y = 0 when α = ± π/ y (cid:54) = ± c , whichmeans the horizontal continuous-director interface between two circular patterns bears no GC, apartfrom that of the circle centres themselves. This discrepancy between the (lack of) concentrated GCin circular patterns and that for spiral patterns relies on the fact that the splay and bend arediscontinuous across the interface only for spirals, not circles.The simulation in Fig. 11 qualitatively agrees with the theoretical result and also shows thelocalisation of the smeared GC in a neighborhood of the crease. Integrating (78) from − y to y (taking principal parts etc.), for y > c , gives ( λ ν − λ − − ν ) sin(2 α ) ln [( y + c ) / ( y − c )] whichvanishes for large y . However, as mentioned earlier, this calculation does not account for the defect-like semi-tip contributions, which can be calculated as follows: In the far field of a split-spiral conepattern such as in Fig. 11, one perceives just a single spiral cone whose integrated GC is 2 π (1 − sin ϕ ).This quantity must therefore equal the total integrated GC arising from the split-spiral pattern,consistent with the vanishing of the integral of (78). One single spiral pattern evolves into a cone or an anticone with GC localized at its tip uponactuation. In this work, we have calculated all possible metric-compatible interfaces between twocenters of GC, i.e., spiral patterns. By employing the metric compatibility equation and ellipticcoordinates, we have found there are four, and only four, types of interfaces between spiral patterns:hyperbolic-type, elliptical-type, continuous-director, and concentric spiral interfaces. Since theseinterfaces are generically inconsistent under cone deformations, the deformed interfaces for eachpattern are further isometrically or non-isometrically deformed in order to join one with the othercontinuously. The simulations suggest that the non-isometric deformation only occurs in a smallneighborhood of the actuated interface, so as to trade the bending energy of the fold with thestretching energy, leading to blunted interface. Except for the non-isometric neighborhood, the restof the surface exhibits nearly isometric evolution which further reduces the bending energy of thesurfaces. This phenomenon has been observed numerically in a previous work [16].Inspired by seminal prior work in isometric curved-fold origami [20, 13, 11], the geometrical evo-lution of the fold between cones can be further understood, assuming the non-isometric deformationis localized in the neighborhood of the fold. Figures 12(a) and (b) sketch the evolution of LCE foldsbetween two cones. The original fold in Fig. 12(a) is constructed by cutting off a narrow strip thatcontains the fold between two cones and isolated from the point defects of the concentrated GC,i.e., the tips. The relaxed fold in Fig. 12 is deformed isometrically from the original one. Followingthe results of differential geometry [20], the opening angle of the fold is determined by the totalcurvature and the geodesic curvature of the curved crease. For the isometric origami case, theopening angle is symmetric with respect to the crease by the fact that the geodesic curvatures fromthe two sides are identical. But for the non-isometric, the opening angle is generically asymmetric,and is potentially programmable by programming the geodesic curvature.On the mechanics front, determining the final state of a non-isometric curved-fold strip is es-30igure 12: (a) The original fold is cut from the cone intersection in Fig. 1(b). (b) The relaxed foldis deformed isometrically from the original fold in (a). Two views are provided for each case toillustrate the curved creases.sentially an energy minimization problem. Without stretching, the total elastic energy E total of thefolded strip with developable facets is given by a simplified model [13, 26] with contributions fromthe bending energy E b and and the energy of the crease E c : E total = E b + E c = 12 D (cid:90) (2 H ) d A A + 12 K (cid:90) ( θ ( l A ) − θ ( l A )) d l A , (79)where D is the bending modulus, K is the crease stiffness, H is the mean curvature distributed overthe union of the two developable surfaces with area A A , l A is the arclength of the actuated crease, θ ( l A ) is the opening angle of the crease which is assumed to be close to the rest angle θ ( l A ). Thebending energy, i.e. the first term of (79), is usually treated as the Wunderlich functional [48, 12] byintegrating along the generator of the surface first and then lowering the integral to one dimensionalong the curved crease. The bending energy is divergent when the surface has point defects (e.g.cone tips) [46, 55]. The rest angle θ ( l A ) is usually treated as a constant angle in isometric origami[13, 26], but is expected to vary with the arclength in the non-isometric origami case because of theconcentrated GC.Unlike isometric folds, non-isometric folds are not Gaussian flat, so their evolution is constrainedby the non-zero GC. For example, one can not unfold it onto the 2D plane without stretching. Thestretching energy will emerge when deforming the folded strip beyond the threshold of isometricdeformation. In other words, the concentrated GC will lead to a stronger folded strip by preventing itfrom entering the stretching regime. Moreover, as mentioned above, a general director field does notresult in a developable surface necessarily. Though the metric-compatible interface between generaldirector patterns still exists, determining the actuated state consisting of curved interface and non-developable surfaces is challenging because of the coupling of the stretch, bend, and geometry.According to these discussions, we propose several interesting questions for the future study: • What is the physically relevant model for the energy of a crease bearing non-vanishing GC? • Can we program the geodesic curvature of the crease and the distribution of point defects toinverse design a non-isometric folded strip that can approximate a target curve in space? • Can we have a theory for the mechanics of a non-isometric folded strip with two non-developable side surfaces and a curved crease deformed from a general director pattern?31nderstanding all these questions is useful for LCE engineering applications such as designing newLCE actuators, lifters, and robotics.
Acknowledgement
F.F. and M.W. were supported by the EPSRC [grant number EP/P034616/1]. D.D. was supportedby the EPSRC Centre for Doctoral Training in Computational Methods for Materials Science [grantno. EP/L015552/1]. J.S.B. was supported by a UKRI “future leaders fellowship” [grant numberMR/S017186/1].
Authors’ Contributions
F.F., J.S.B. and M.W. performed the theoretical calculations. D.D. performed all the simulations.D.D and J.S.B. developed the simulation platform.
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