# Building blocks of non-Euclidean ribbons: Size-controlled self-assembly via
discrete frustrated particles

Douglas M. Hall  Department of Polymer Science and Engineering, University of
Massachusetts, Amherst, MA 01003, US  Mark J. Stevens  Center for Integrated
Nanotechnologies, Sandia National Laboratories, Albuquerque, NM 87185, USA.
Gregory M. Grason  Department of Polymer Science and Engineering, University
of Massachusetts, Amherst, MA 01003, US

###### Abstract

Geometric frustration offers a pathway to soft matter self-assembly with
controllable finite sizes. While the understanding of frustration in soft
matter assembly derives almost exclusively from continuum elastic
descriptions, a current challenge is to understand the connection between
microscopic physical properties of misfitting “building blocks” and emergent
assembly behavior at mesoscale. We present and analyze a particle-based
description of what is arguably the best studied example for frustrated soft
matter assembly, negative-curvature ribbon assembly, observed in both
assemblies of chiral surfactants and shape-frustrated nanoparticles. Based on
our particle model, known as saddle wedge monomers, we numerically test the
connection between microscopic shape and interactions of the misfitting
subunits and the emergent behavior at the supra-particle scale, specifically
focussing on the propagation and relaxation of inter-particle strains, the
emergent role of extrinsic shape on frustrated ribbons and the equilibrium
regime of finite width selection. Beyond the intuitive role of shape misfit,
we show that self-limitation is critically dependent on the finite range of
cohesive interactions, with larger size finite assemblies requiring increasing
short-range interparticle forces. Additionally, we demonstrate that non-
linearities arising from discrete particle interactions alter self-limiting
behavior due to both strain-softening in shape-flattened assembly and partial
yielding of highly strained bonds, which in turn may give rise to states of
hierarchical, multidomain assembly. Tracing the regimes of frustration-limited
assembly to the specific microscopic features of misfitting particle shapes
and interactions provides necessary guidance for translating the theory of
size-programmable assembly into design of intentionally-frustrated colloidal
particles.

##  I Introduction

Geometric frustration (GF) occurs when the locally preferred ordering is
incompatible with geometric constraints of extending that order throughout the
assembly. [1, 2] Canonically, GF is associate in bulk systems, where it
requires the formation of extensive arrays of topological defects, as in
polytetrahedral sphere packings [3] or liquid crystal blue phases .[4] When a
self-assembling system has GF, the presence of free boundaries on potentially
forming finite-sized structures leads to distinct consequences and a range of
exotic, scale-dependent thermodynamic behavior. [5] Notably, finite and
sufficiently soft assemblies need not form defects as response to GF, which
may instead manifest in a superextensive accumulation of intra-assembly stress
that, in competition with the cohesive drive for assembly growth, may shape
the assembly’s equilbrium boundary and interior at length scales much larger
than the subunit (e.g. macromolular or colloidal) dimensions. [5, 6] Arguably,
the most notable emergent behavior is the ability of the GF to determine the
mesoscopic finite equilibrium size of assemblies. [7] This basic paradigm has
been explored in the context of a range of soft matter systems, from spherical
assemblies of colloids [8, 9] and protein shells, [10] to twisted bundles of
filamentous proteins or chiral fibers [11, 12, 13] and chiral ribbons. [14,
15, 16, 17, 18, 19] The specific dependence on long-range gradients in intra-
assembly stress and the resulting ability of thermodynamics to sense the
mesoscopic size of assemblies distinguishes geometrically frustrated assembly
(GFA) from other more familiar examples of size-selective assemblies, like
amphiphillic micelles or self-closing, curvature limited shells and tubules.

Models for size control in GFAs are generically predicated on continuum
elastic descriptions of the super-extensive growth in assembly energy. [7]
These models argue that elastic energy accumulates with size up to an upper
size limit, beyond which the assembly distorts away from the locally-preferred
packing (at finite energy cost) to maintain extensive energetic growth with
size. [6] At these large sizes, frustration is not able to restrain the
cohesive drive to larger size, and equilibrium assembly proceeds to unlimited
size, known as frustration escape. There are multiple possible structural
modes of assembly: elastic “shape flattening” of the preferred frustrated
packing into an unfrustrated one; [12, 20, 21] “filamentation” into structures
that remain finite in only a single direction of assembly but unlimited in
others; [8, 11, 22] and incorporation of topological defect arrays that screen
the far-field stresses responsible for cumulative frustration costs. [23, 24,
25, 26]

At a conceptual level, the possibility of thermodynamic self-limitation as
well as the existence of distinct modes of frustration escape that delimit the
range of self-limitation for any given GFA is well established. The potential
advantages posed by self-assembling systems that can “sense” their size at
ranges that exceed the subunits themselves raises the possibility of
intentionally engineering frustration into synthetically fabricated assemblies
as a means to “program” their assembly behavior. [27, 28, 29] In principle,
recent progress in the synthesis of colloidal-scale particles with programmed
shape can allow for tunable shape frustration that can more fully test the
continuum theory description of size control. [30, 31] Notably, advances in
DNA nanotechnology [32, 33] as well as synthetic protein engineering [34, 35,
36, 37] allow for both careful design and control of the shape frustration of
self-assembling nanoscale units as well as new opportunities for programming
the interactions to separately tune the strength of cohesion and costs
associated with distinct modes of assembly deformation. However, due to the
primary reliance on continuum descriptions of GFA, several basic challenges
remain to relate emergent thermodynamic behaviors in a particular system of
self-assembling frustrated subunits. In general, it remains to be understood
which specific structural mechanisms are responsible for frustration escape in
any particular system, and moreover, what are the size ranges, relative to the
subunit dimensions, at which frustration may limit the thermodynamic assembly
size. Finally, beyond the continuum descriptions whose predictions rely on
phenomenological constants of unknown value, how does the structure and
thermodynamic range of self-limiting GFA depend on physical properties of the
subunits themselves, their ill-fitting shapes, interactions and deformability?

In this study, we focus on a particular well-studied model of GFA: crystalline
membrane assemblies frustrated by preferred negative Gaussian curvature
shapes. Initial models of this type were motivated by observation of ribbon,
or tape-like, assemblies of chiral amphiphile exhibiting twisted, helicoidal
ribbon morphologies with well-defined ribbon width. [38, 39, 40] In these
assemblies preference for negative Gaussian curvature derives from the
chirality of the molecules. [41] Helicoidal ribbon morphologies have also been
observed in tetrahedral nanoparticles assembly. [42, 19]

The scale-dependent morphology of these structures has been described by a
continuum elastic theory that accounts for growth of intra-ribbon strains of
crystalline order in negatively curved ribbons, as well as the elastic
(bending) preference for negative curvature. The first model of this type was
developed by Ghafouri and Bruinsma for chiral membranes, but has been
subsequently elaborated on by several other studies. [16, 17, 43, 44] The key
predictions of the model can be divided into two-regimes: narrow- and wide-
ribbon regimes. Narrow-ribbons largely maintain their preferred negative
Gaussian curvature, and therefor incur elastic penalties (per unit area) for
crystal strains that grow with ribbon width $w$ as $\sim w^{4}\kappa_{0}^{4}$
where $\kappa_{0}$ is the preferred curvature radius. this super-extensive
elastic cost may, in balance with the cohesive drive for larger assembly due
to line tension, determine a thermodynamically optimal assembly width that
grows with decreasing curvature, $w_{0}\sim\kappa_{0}^{-4/5}$.

For wide ribbons, the in-plane elastic costs of negative Gaussian curvature
overwhelm the (bending) cost to deform ribbons to an unfrustrated shape,
leading to a shape transition from helicoids to spiral ribbons which expel
Gaussian curvature. This transition, which we refer to as shape-flattening
throughout this article, is predicted to occur at critical width,
$w_{*}\sim(B/Y)^{-1/4}\kappa_{0}^{-1/2}$ where $B$ and $Y$ are respective
bending and in-plane (2D Young’s) moduli for membrane. Notably this same
underlying mechanical transition has realized in a range of fabricated ribbon
architectures, [17, 45] is proposed as the basis of similar morphological
transitions in a range of elastic structures in biology, [46, 47] and has been
verified by a range of finite-element or finite-difference numerical
simulations. [39, 17] For the context of self-assembling ribbons, which can
adjust their widths via addition of free subunits, the helicoid-to-spiral
transition marks a shape-flattening transition, in which the shape
progressively expels Guassian curvature with increasing width. Since the
elastic energy becomes extensive in size for $w>w_{*}$, frustration cannot
limit the assembly size in wide ribbon regime, and $w_{0}\approx w_{*}$ marks
an upper limit to the possible range of frustration limited assembly. For
helicoidal assemblies, it is observed that optimal assemblies instead close
upon themselves into finite-diameter cylinders, a.k.a. tubules. [38, 48, 40]

Figure 1:  Assembly geomety and the role of $\phi_{0}$ are illustrated. In (a)
the local geometry is a saddle shape with principal curvature $\pm\kappa_{0}$.
Additionally, the principal curvature directions are oriented with respect to
the assembly’s close-packing directions by an angle $\phi_{0}$. (b) Particles
packed according to this geometry adopt angular differences with respect to
their neighbors, rotating parallel to the bonding directions when $\phi_{0}=0$
and twisting orthogonal to the bonding directions for $\phi_{0}=45^{\circ}$.
(c) When the assembly forms a ribbon with edges oriented along the close-
packed directions, the assemblies are described by the isometric family of
surfaces spanning between the catenoid and helicoid.

In this study, we aim to extend the understanding for frustration in
hyperbolic, crystalline membranes from the level of phenomenological continuum
descriptions to scale of shape-frustrated subunits from they form.
Specifically, we introduce a new class of “saddle-wedge monomers” (SWMs) whose
large scale interactions favor mesoscopic assembly geometries that map onto
the existing continuum models. As shown schematically in Fig. 1, the variable
shape of the SWM model encodes both tunable frustration (i.e. variable
$\kappa_{0}$) and also programmable relative directions of curvature with
respect to close-packed crystalline (bonding) directions. This broader class
of geometries has been recognized as equivalently frustrated, [46] and are
related to the isometric (Bonnet) family of minimal surfaces spanning the
catenoid and helicoid, [49, 50] parameterized by curvature angle $\phi_{0}$
between the crystalline rows and preferred curvature axis.

The aim of this discrete monomer study is several fold. First, we aim to
understand how features of the geometry and interactions of building blocks
govern the mesoscale shape and thermodynamics of optimal assemblies, and more
specifically, determine the mapping of particle-scale properties onto
parameters of the continuum description. Based on this, we analyze how the
range of accessible self-limiting widths compare to size of the building
blocks themselves. We show that the maximal size range of self-limitation is
critically delimited by the range of cohesive bonds between monomers. Second,
we analyze the responses to frustration that fall outside of linear elastic
descriptions, more specifically the distinct roles of strain softening and
yielding on size control, and the possibility of partial or incomplete
bonding. We show that the former slightly depresses the range of thermodynamic
self-limitation, relative to a purely Hookean elastic behavior, while the
latter may be associated with a range of heirarchical ground states possible
for sufficiently low temperatures.

The remainder of this manuscript is organized as follows. We first summarize a
continuum scale description that is expected to capture the mesoscale
structure and thermodynamics of SWM assembly, and then introduce the coarse-
grained, discrete particle model. Next, we present numerical results from
energy minimization calculations on the stress accumulation and flattening of
energetic ground states of varying width. We then analyze the limiting case of
the flattened state, the tube morphology, with numerics and continuum results
considering the effects of anisotropic bending stiffness and strain softening,
which are then used to construct the self-limitation phase diagrams in terms
of SWM geometry and interactions. Next, we analyze deviations from purely
linear-elastic behavior exhibited by the discrete subunit assembly, in
particular, show that finite-range interactions generically imply the
stability of internally-cracked or weakly-aggregated finite-domain
morphologies in regimes where self-limited structures are favored over
unlimited (bulk) structures. Finally, we conclude by discussing the relevance
of the results to assembly at finite temperatures, implications for
hierarchical assembly and we present preliminary evidence of assembly with
molecular dynamics (MD) results.

##  II Models of frustrated hyperbolic ribbons

We first summarize the key ingredients and predictions of a continuum elastic
model for assembly of hyperbolic, crystalline membranes followed by the
introduction of discrete particle model of SWM whose assembly forms these
frustrated morphologies.

###  II.1 Continuum Theory

Here we summarize a continuum elastic description for the frustrated ribbons
formed by 2D crystalline membranes with a preference for negative Gaussian
curvature shapes (see B for full details). The model, which we refer to as
“narrow ribbon” (NR) theory, is essentially an elaboration of the original
approach of Ref. [16], generalized to include arbitrary direction or curvature
axes relative to crystallographic axis, as in Ref. [17]. The approach assumes
slender assemblies with an assembly length $L$ and width $w$ such that $L\gg
w$: either ribbons of width $w$ much smaller than the (unlimited) assembly
length $L$ in the orthogonal direction, or instead closed rings with width $w$
much smaller than the assembly circumference $L\gg w$. The model includes
three ingredients,

$E_{\rm tot}=E_{\rm bend}+E_{\rm strain}+E_{\rm edge}$ (1)

corresponding, respectively, to elasticity of extrinsic (i.e. bending)
curvature, in-plane elastic strains of the 2D crystalline order, and the
cohesive cost of free edges of the ribbons, dominated by the two longer edges
(i.e. $E_{\rm edge}\sim 2\gamma L$). We consider the case of in-plane square-
lattice order, and due to the energetics of strong (nearest neighbor) bonding
along the lattice directions, assume that optimal ribbons form with their free
edges along the lattice directions (i.e. either the local $\hat{x}$ or
$\hat{y}$ direction of the ribbons, which are the low edge energy directions).
Here we take the $\hat{y}$ direction to be the long axis of the ribbon).

General considerations of the elasticity of anisotropic membranes, [41] imply
a coupling between free energy to the curvature tensor curvature tensor
$C_{ij}$ of the membrane. According to the narrow-ribbon approximation, for
which $|C_{ij}|w\ll 1$, we assume that curvatures are roughly constant across
the width of membrane, and described by the values at the mid-line: $C_{yy}$
along the ribbon’s length, $C_{xx}$ along the width, and $C_{xy}=C_{yx}$ the
off-diagonal element of the curvature tensor. Specifically our systems are
described by the following extrinsic curvature elasticity,

$E_{\rm
bend}\simeq\frac{wL}{2}\Big{(}C_{ij}-(C_{0})_{ij}\Big{)}B_{ijkl}\Big{(}C_{kl}-(C_{0})_{kl}\Big{)}.$
(2)

where $B_{ijkl}$ is the tensor of elastic bending constants and $(C_{0})_{ij}$
is the locally preferred curvature. The nearest neighbor binding square-
lattice model leads to two non-zero elasticity constants,

$B_{\parallel}=B_{xxxx}=B_{yyyy};\ B_{\perp}=B_{xyxy}=B_{yxyx}$ (3)

for deformations that alter bending and twisting (of the tangent plane) along
lattice rows; the remaining elastic constants are zero. The preferred (or
“target”) shape can be written in matrix form,

${\bf
C}_{0}=\kappa_{0}\left({\begin{array}[]{rr}\cos(2\phi_{0})&\sin(2\phi_{0})\\\
\sin(2\phi_{0})&-\cos(2\phi_{0})\\\ \end{array}}\right),$ (4)

where $\kappa_{0}$ sets the magnitude of the preferred principle curvatures
and $\phi_{0}$ parameterizes the angle between the lattice directions and the
principle curvature directions (see Fig. 1). Notably, this preferred curvature
targets minimal surfaces with a mean curvature $H_{0}\equiv{\rm Tr}[{\bf
C}_{0}]/2=0$ and a preferred negative Gaussian curvature $K_{G,0}\equiv{\rm
det}[{\bf C}_{0}]=-\kappa_{0}^{2}$. While the original NR apporach of Ghafouri
and Bruinsma [16] for chiral membranes corresponds to the case of
$\phi_{0}=\pi/4$, it was pointed out by Armon and coworkers that a larger
family of target minimal surfaces (corresponding to the Bonnet family of
minimal ribbons) are generated simply by rotation of the preferred curvature
axis relative to its pitch axis. [17] We explore the implications of this
broader control over frustrated shape for the design of the SWM and its
ultimate assembly below.

The strain elastic energy takes the from

$E_{\rm strain}=\frac{1}{2}\int dA~{}\sigma_{ij}u_{ij}$ (5)

where $u_{ij}$ and $\sigma_{ij}=\lambda u_{kk}\delta_{ij}+2\mu
u_{ij}+\lambda_{\perp}\big{(}\delta_{ix}\delta_{jx}u_{yy}+\delta_{iy}\delta_{jy}u_{xx}\big{)}$
are the in-plane 2D strain and stress tensors for a square crystal. In-plane
strains are coupled to the out-of-plane deflection the membrane through its
intrinsic curvature, i.e. non-zero Gaussian curvature generates in-plane
stress gradients. [51] As described in Ref. [16] and in the Appendix, these
may be solved for long-ribbons assuming uniform stress along $y$ and constant
Gaussian curvature $K_{G}$ resulting in an elastic cost that grows
superextensively with width yielding,

$E_{\rm strain}/A\simeq\frac{Y}{1440}K_{G}^{2}w^{4},$ (6)

where $Y=(2\mu-\lambda_{\perp})(2\lambda+2\mu+\lambda_{\perp})/(\lambda+2\mu)$
is the 2D Young’s modulus of the membrane.

The thermodynamics of the NR approximation follow from minimization of the
total free energy density with respect to curvature and ribbon width and are
summarized schematically in Fig. 2. For narrow ribbons (corresponding to small
$\gamma$), the ribbon adopts a shape close it is target hyperbolic shape,
${\bf C}(w\to 0)\simeq{\bf C}_{0}$, so that the dominant elastic costs derive
from in-plane strains. As a result, the stretching energy is super-extensive,
growing faster than the assembly size $A$, according to $E_{\rm strain}/A\sim
Y\kappa_{0}^{4}w^{4}$. In this regime, the optimal width $w_{0}$ is set
(approximately) by the balance between in-plane stretching of the target shape
and the edge energy (per unit area), $E_{\rm edge}/A\sim\gamma/w$, leading to
an optimal (self-limiting) width that grows with edge energy and decreases
with increasing target curvature, $w_{0}\sim(\gamma/Y\kappa_{0}^{4})^{1/5}$.
When ribbons grow sufficiently large, the strain energy cost to maintain the
preferred negative Gaussian curvature overwhelms the cost to unbend that
assembly into an isometric (i.e. $K_{G}\to 0$) shape. Roughly speaking this
occurs at a characteristic width scale, $w_{*}\sim(B/(Y\kappa_{0}^{2}))^{1/4}$

Figure 2:  (a) Ribbons of increasing width $w$, for the case
$\phi_{0}=45^{\circ}$, maintain approximately helicoidal shape up until a
width $w_{*}$ at which point they flatten to a cylindrical shape. The
flattened ribbons may continue widening until the edges meet and a closed
tubule is formed. (b) Schematic plots of $E_{\rm{bend}}$ (yellow),
$E_{\rm{strain}}$ (brown) and $E_{\rm{elastic}}=E_{\rm{bend}}+E_{\rm{strain}}$
(solid blue curve), as a function of the assembly width $w$. Assymptotic
limits are shown for narrow ribbons $E_{\rm{elastic}}(w\to 0)=E_{\rm
narrow}\propto Y\kappa_{0}^{4}w^{4}$ (dotted blue) and wide ribbons
$E_{\rm{elastic}}(w\to\infty)=E_{\infty}$ (dashed blue). (c) A self-limiting
state is a minimum in the model free energy $E_{\rm tot}=E_{\rm
elastic}+E_{\rm edge}$. With increasing line tension $\gamma$, larger self-
limiting sizes may be achieved up until a point when the minimum’s energy is
equal to the flattened state’s energy $E_{\infty}$ at which point the self-
limiting state is metastable. Further increase of the line tension results in
larger metastable finite minima until the point at which the minimum
disappears entirely near $w=w_{*}$.

The model energy functional gives the (approximate) optimal shape $C_{ij}(w)$
and resulting elastic energy $E(w)$ as a function of ribbon width (shown
schematically in Fig. 2), when optimized over values of the curvatures. For
small curvatures, $E(w)/A\simeq Y\kappa_{0}^{4}w^{4}/1440$ as the curvature
remains close to the preferred value. When ribbons reach a critical width
$w_{*}$, the strain energy cost to maintain the preferred negative Gaussian
curvature overwhelms the cost to unbend that assembly to reduce $K_{G}$, and
the elastic ground states undergo a symmetry breaking bifurcation. For chiral
ribbons ($\phi_{0}=\pi/4)$, this shape transition corresponds to a
transformation from helicoids to spirals. In B we show that the supercritical
shape transition occurs in GB theory for any value $\phi_{0}$ and for
$0<\phi_{0}<\pi/4$, leading to two different spiral equilibria (degenerate
within GB theory). See for example the two stable branches for $w>w_{*}$ for
$\phi_{0}=22.5^{\circ}$ in Fig. 11, which different in terms of helical pitch
and radius.

In the limit $w\to\infty$, the the Gaussian curvature vanishes at the expense
of bending spiral membranes into cylindrical shape with energy,

$\displaystyle\frac{E_{\infty}}{A}=\frac{1}{2}B_{\parallel}\kappa_{0}^{2}\left[1+\frac{B_{\perp}-B_{\parallel}}{B_{\perp}+B_{\parallel}}\sin^{2}(2\phi_{0})\right].$
(7)

The flattened cylindrical shape is identified with frustration escape, as the
assembly can grow without increasing elastic energy density. At large enough
widths, the actual assembly will close up so that the flattened state is a
closed tubule. Whereas the initial stretching cost is independent of
$\phi_{0}$, the expression for the flattening cost $E_{\infty}$ may depend on
$\phi_{0}$ when the associated curvature moduli differ $B_{\perp}\neq
B_{\parallel}$. One might naively expect that the larger flattening cost can
extend the range of super-extensive elastic energy with growing size, and thus
increase the range of size control; that is, the mechanical equilibrium would
shift from self-limiting ribbon shapes to flattened, tube morphology roughly
speaking when the ribbon elastic energy $E/A\simeq Y\kappa_{0}^{4}w^{4}/1440$
was equal to $E_{\infty}/A$, so that
$w_{\mathrm{max}}\sim(E_{\infty}/Y\kappa_{0}^{4})^{1/4}\approx w_{*}$ would
increase with increasing flattening cost. That is, based on this model, the
mechanics of unbending the membrane away from its curved shape sets an upper
limit to size scales where frustration can provide a thermodynamic limitation
to the ribbon width. Analysis of equation 1 predicts a moderate reduction in
the range of self-limitation with increasing curvature angle, $\phi_{0}$, as a
consequence of the mechanical flattening transition occurring at a smaller
value of $w_{*}$ with increasing flattening cost $E_{\infty}$. The central
goal of this study is to directly assess variation of the range of
frustration-limiting widths with $\phi_{0}$, as the target shape is varied
from catenoidal to helicoidal, for a discrete subunit model of hyperbolic, 2D
crystalline membrane assemblies.

We note that the assumptions of the NR theory, namely that curvatures are
sufficiently uniform across the width of ribbons, do not strictly hold across
the full range of ribbons widths. This is because torque-free boundary
conditions require a boundary layer of characteristic size proportional to
$w_{*}\sim(B/Y)^{1/4}\kappa_{0}^{-1/2}$, [52, 17, 53] so that through-width
curvature variation becomes non-negligible for $w>w_{*}$. We show this for an
explicit solution for exact (boundary layer) solution below (for $\phi_{0}=0$
in Appendix B). This boundary layer correction modifies predictions of elastic
energy, particularly in the large $w\gtrsim w_{*}$ regime. Notably, finite-
element calculations for ribbons $0<\phi_{0}<\pi/4$ suggest that boundary-
layer corrections break the degeneracy between the two large-$w$ equilibria.
These detailed corrections for intermediate-$w$ notwithstanding, we argue that
the NR approximation works reasonably well for both the self-limiting (i.e.
$w$ small) and the asymptotically flattened ($w\to\infty$) regimes. The simple
and analytically tractable solutions of the NR theory therefore provide a
useful means to survey how thermodynamics of self-limitation varied with
geometric and mechanical properties of the membrane in the continuum elastic
description.

Figure 3:  Monomer design for programming assembly size control. (a) The
attractive site pair potential $u(r)$ is plotted as a function or separation
$r$ for varying potential range $r_{a}$ with respect to monomer size $d$ and
bond energy $u_{0}$. The yield point for each case is indicated with an open
circle and the harmonic approximation plotted as a dashed curve. (b) A sphere
representation of the monomer design viewed along the ${\bf c}_{1}$ and ${\bf
c}_{2}$ monomer bonding axes. Gray spheres have radius representing the range
of their excluded volume interaction. Attractive sites are spheres of
different colors, with four attractive sites on each of the four bonding faces
of the monomer. The four attractive sites on a given side define a frame with
attractive site spacing $t$ between opposite pairs. The frame between opposite
sides has distance $d$ defining the monomer size, and rotation
$\theta_{\parallel}$ projected along the frame displacement and
$\theta_{\perp}$ orthogonal to the displacement. (c) Monomer geometries for
$\phi_{0}=0,22.5,45^{\circ}$ are shown viewed along the ${\bf c}_{1}$ bonding
axis and along the $c_{3}$ axis (red) orthogonal to the assembly surface.
Sphere representations are shown alongside respresentations with the excluded
volume of each monomer represented as a polygon.

####  II.1.1 Discrete Model and Methods

To connect discrete-monomer properties and design to assembly behavior, we
developed a coarse-grained simulation model, building from a model previously
developed to study microtubule assembly. [54, 55, 56, 57, 58] The saddle-wedge
monomer (SWM) is designed for energy minimization and dynamical assembly
simulation using the LAMMPS software. [59, 60, 61] The basic shape of the SWM
is a “double-wedge” geometry: four binding faces that promote curvature of
opposite signs in the orthogonal directions of the assembly, as illustrated in
Fig. 3. The rigid monomer consists of 27 sites of a single type and purely
repulsive interactions, surrounded by 16 attractive sites of 8 types, with the
attractive sites arranged in a planar square on each of four bonding sides of
the monomer. The square diagonal of the attractive sites on each face defines
the thickness parameter $t$ that in principle may be used to tune the relative
costs of changes in assembly curvature, bending, with respect to assembly
stretching. For the results presented in this study, we consider the case
$t=0.568d$. The monomer width $d$ is defined by the distance between
respective centers of mass of attractive sites on opposite sides. For the
attractive sites, pairwise binding only acts between sites of the same type on
different monomers (as denoted by distinct colors of binding sites in Fig. 3).
The repulsive sites interact according to a Weeks-Chandler-Anderson (WCA) pair
potential, [62] and define the monomer excluded volume and shape in the low-
energy minimized structures. Their arrangement, with coordinates, are
described in more detail in Appendix A. The attractive sites on a given face
are arranged in a plane parallel to the adjacent plane of repulsive sites on
the monomer, so that pairwise attractions of all four sites are possible
without overlaps from purely repulsive sites. The attractive site interactions
each have the form of

$\displaystyle
u(r)=\left\\{\begin{array}[]{ll}\displaystyle-\frac{1}{8}u_{0}\left[1+\cos\left(\frac{\pi
r}{r_{a}}\right)\right],&r\leq r_{a}\\\ 0,&r>r_{a}\\\ \end{array}\right.,$
(10)

where $r$ is the distance between interacting sites, $r_{a}$ is the
interaction range and $u_{0}$ defines the potential well depth, such that the
minimum energy for two monomers binding with all four attractive sites ideally
placed is $-u_{0}$. Figure 3(a) shows the shape of the attractive interaction
with varying interaction range. Full details of the monomer geometry and
interactions are given in Appendix A. Importantly, $r_{a}/d$, the range of
interaction with respect to the monomer width, controls both the relative
stiffness of the assembly via the elastic moduli defined below and also the
strain necessary for a single bond to reach the point of yielding.

The orientational geometry relating pairs of SWMs bound together is defined in
terms of the orthonormal frame $\\{{\bf c}_{1},{\bf c}_{2},{\bf c}_{3}\\}$
associated with each monomer as shown in Fig. 3(b) with the first two
directions pointing along neighbor bonding axes and the third direction point
along the vertical (non-bonding) direction. The preferred binding geometry is
determined by angles defined in a single rigid monomer. Attractive sites are
arranged so that $\theta_{0}$ is the preferred angle between the ${\bf c}_{3}$
axis of bonded neighboring monomers, when all four interacting sites on their
respective faces coincide. The $\phi_{0}$ angle can be understood as a twist
of binding directions (i.e. orientation of the square of attractive sites)
around the axes connecting the SWM centers to their binding faces, i.e. by +
or - around ${\bf c}_{1}$ and ${\bf c}_{2}$, respectively (the twist sense in
one bonding direction is chosen to be opposite that in the other direction to
be compatible with membrane geometry of zero mean curvature). Taking the ${\bf
c}_{3}$ direction to be normal to the mid-surface of multi-particle membrane
assemblies formed by SWMs, we can relate wedge angle $\theta_{0}$ and
curvature angle $\phi_{0}$ to the target curvature tensor of the membrane as
follows. Projecting the rotation sense ${\bf c}_{3}$ between ${\bf c}_{i}$
faces into the ${\bf c}_{j}$ gives the preferred surface curvature $C_{ij}$
times the particle width $d$, or the angles $\theta_{\parallel}=\theta_{0}\cos
2\phi_{0}$ and $\theta_{\perp}=\theta_{0}\sin 2\phi_{0}$ as illustrated in
Fig. 3(b). Hence, orientational geometry of SWMs map onto preferred curvature
of the form eq. (4) with target principle curvature

$\kappa_{0}=\theta_{0}/d.$ (11)

The attractive site arrangement defines both a monomer width $d$ that is
approximately the preferred distance between neighboring monomers and a
monomer thickness $t=0.568d$ that controls the cost of bending deformations.
As described in Appendix B, the effective elastic constants of membrane
assemblies of SWM are determined by consideration of the local deformations on
ideally bounded neighbors, imposed by distortions of a crystalline membrane.
As shown schematically in Fig. 10, $Y$ corresponds to stretching/compressing
of inter-face spacing, while $B_{\parallel}$ and $B_{\perp}$ correspond to
dihedral and twist angle distortions between bound SWM. Modeling bound
attractive sites as effective springs of stiffness $\pi^{2}u_{0}/r_{a}^{2}$
leads to

$\displaystyle Y=\frac{\pi^{2}u_{0}}{2r_{a}^{2}};\
B_{\parallel}=\frac{\pi^{2}t^{2}u_{0}}{16r_{a}^{2}};\
B_{\perp}=\frac{\pi^{2}t^{2}u_{0}}{8r_{a}^{2}}.$ (12)

Notably, as SWM are modeled as rigid bodies, eq. (12) highlights the role of
the range of attraction in controlling the deformability of the assembly.
Additionally, we note that the characteristic ratio of bend to stretch moduli
$B/Y\propto t^{2}$ is independent of interaction parameters, controlled only
by the geometric thickness of the SWM particles. Last, it is important to note
that, distinct from previously studied models of anisotropic bend-elasticity,
off-diagonal bending is stiffer that bending along the lattice directions
(i.e. $B_{\perp}=2B_{\parallel}$. The greater twist stiffness relative to row
bending, is generic consequence of attraction only binding, and has
consequences in the thermodynamics of frustration escape for distinct
$\phi_{0}$ values of SWM. As discussed further below, one consequence is the
dependence of the flattening transition on $\phi_{0}$,

$w_{*}=(360)^{1/4}\frac{\sqrt{t/\kappa_{0}}}{\sqrt{1+\frac{7}{9}\sin^{2}(2\phi_{0})}}=3.28~{}\frac{d/\sqrt{\theta_{0}}}{\sqrt{1+\frac{7}{9}\sin^{2}(2\phi_{0})}},$
(13)

which is derived in Appendix B, eqs. (22)-(34). As we consider assemblies to
form with open edges only along low energy directions in the bond lattice, it
is straightforward to compute the edge energy per unit length

$\gamma=\frac{u_{0}}{2d},$ (14)

as (half) the ideal bond energy needed to separate membranes along their
nearest neighbor direction.

To explore the the groundstate thermodynamics of size control of this model,
the LAMMPS simulation software was used to minimize the energy of preassembled
initial configurations. The LAMMPS minimize command was used with default,
conjugate gradient method. Structures were successively minimized from a soft,
relatively long-range interaction $r_{a}/d=0.199$ down to the target range of
interaction, decrementing $r_{a}/d$ by $0.007$ and re-minimizing at each step.
Additionally, the monomers were first minimized at a softer state where intra-
monomer geometry was maintained with springs: (see appendix) minimizations
with incrementing $r_{a}$ were run at intra-monomer bond stiffness
$k_{\mathrm{bond}}=885~{}u_{0}/d^{2}$ and subsequently the minimization at the
final value of $r_{a}$ was re-minimized at successive values
$k_{\mathrm{bond}}/(u_{0}/d^{2})=890,4400,8900,44000$ and $89000$. The intra-
monomer bonds are found to contribute negligible total energy compared to the
total inter-monomer interaction potential energy (less than 1 part in
$10^{5}$) after minimization at the larger bond stiffness. The final
minimization was run until reaching a force tolerance of $0.3\times
10^{-4}u_{0}/d$. Intermediate steps at higher $r_{a}$ and lower
$k_{\mathrm{bond}}$ were run until either the same force tolerance was
reached, or $10^{8}$ steps of minimization.

Multiple pre-assembled initial configurations were sampled with varying
lateral dimensions $w/d$ corresponding to the ring or ribbon width, which is
the number of monomer rows as measured in the shorter assembly direction. The
initial configurations for varying width were cylindrical geometry
$\phi_{0}=0$ and flat rectangular geometry for $\phi_{0}>0$. The flattened
tubule state’s energy was found from initially cylindrical geometry for longer
tubes. For all starting configurations, monomers were arranged at a slightly
dilated spacing of $1.05d$. For $\phi_{0}=0$, the cylinder circumference was
chosen to be $360^{\circ}/\theta_{0}$, the cylinder length to be the target
assembly size of $w/d$ monomers, with monomer bond directions aligned along
the circumferential and longitudinal directions. For large-size energetics of
all $\phi_{0}$, cylindrical configurations were prepared with successively
varying lengths of 100, 110, 120, 130, 140, and 150 monomers and bond
directions along the cylindrical surface making an angle $\phi_{0}$ with the
tube circumferential direction and axis. For $\phi_{0}>0$, a rectangular
geometry was used to sample smaller assembly size. The bonding directions were
chosen in-plane and parallel to the boundaries of the rectangle, one side
length was kept to be $100$ monomers while the other chosen to be the target
size of $w/d$ monomers.

To determine optimal zero-temperature size of small-width assemblies, varying
cylindrical ring or rectangular ribbon assembly widths $w$ were sampled up to
$2w_{*}$ where $w_{*}$ is the theoretical transition width from Eq. 13.
Structures rendered in figures are shown with effective strain energy
calculated from the average soft interaction energy, subtracting off the
reference value of $-u_{0}$ for each bond that was present in the initial
configuration. Structures were analyzed to determine if any bond initially
present in the starting configuration exceeds the yield point in the final
relaxed state.

The total assembly energy $U_{\mathrm{a}}$ is evaluated in terms of all
pairwise interactions between sites on different monomers,

$\displaystyle
U_{\mathrm{a}}=\frac{1}{2}\sum_{i,j}\Big{(}u(r_{ij})+u_{\mathrm{WCA}}(r_{ij})\Big{)}.$
(15)

Following similar analyses of geometrically frustrated assemblies, [7] we
define the excess energy as energy of the assembly relative to the cohesive
bulk and edge energetics, and compute it by subtracting the ideal energy
$u_{0}$ (of an unstrained bond) for every bond in the assembly

$\displaystyle E_{\mathrm{ex}}=U_{\mathrm{a}}+2u_{0}wL/d^{2}-u_{0}(w+L)/d.$
(16)

In the following sections, we consider the comparison of the excess energy of
the discrete model $E_{\mathrm{ex}}$ to the continuum model predictions
derived from equation 1.

##  III Results

###  III.1 Stress accumulation and flattening, beyond harmonic and isotropic
elasticity

In this section, we compare simulated ground states of discrete-SWM assembly
to the predictions of the continuum theory, illustrating how strain
accumulation and elastic shape-flattening depend on on arrangement of
attractive sites and the tapered shapes SWM binding.

Figure 4:  Excess energy accumulation and flattening for assemblies of
increasing width, for $\phi_{0}=0,22.5,45^{\circ}$,
$\theta_{0}=2.5,5,10^{\circ}$ and $r_{a}/d=0.14,0.17,0.20$. (a-c) Average
excess energy density $E_{\rm ex}/Nu_{0}$ after numerical minimization
(points), for varying assembly width $w/d$ and monomer parameters
$\theta_{0},r_{a}$, is compared to the continuum theory $E/Nu_{0}$ prediction
(dashed curves). (d-e) The progression of structures with increasing width
generically shows a transition to flattened shape, close to a critical width
$w_{*}$ defined in Eq. 13. Structures have widths $w/d=4,8,10,12,14$,
$r_{a}/d=0.14$ and $\theta_{0}=5^{\circ}$. Monomers are colored by relative
excess energy $E^{({\rm wedge})}_{\rm ex}$.

We focus on three values of curvature direction, with $\phi_{0}=0$ the closed
ring that approximates a catenoid surface in the narrow limit,
$\phi_{0}=45^{\circ}$ approximating a helicoidal ribbon in the narrow limit,
and $\phi_{0}=22.5^{\circ}$ an intermediate case. The excess elastic energy
$E_{\rm{ex}}$ computed from energy minimizations are plotted along with
(narrow-ribbon) continuum model predictions in Fig. 4(a-c), for varying taper
angle ($\theta_{0}$), curvature direction ($\phi_{0}$), and interaction range
($r_{a}/d$). In each case, the continuum model accurately captures the excess
energy in of discrete assemblies in the $w\to 0$ regime as well as the
transition from super-extensive growth at small $w$ to extensive growth (i.e.
saturated $E_{\rm{ex}}/N\sim w^{0}$) at large $w$. Typical structures for the
progression through the mechanical transition are rendered in Fig. 4(d-f). The
cases for $\theta_{0}\neq 0$ both show an apparently sharp shape transition
between low- to high-$w$ values, as highlighted by the dashed lines in Fig.
4(e-f). In Fig. 11, we compare the predicted curvatures of from NR theory to
shapes from the SWM ribbon minimizations in Fig. 11 for
$\phi_{0}=22.5^{\circ}$ and $45^{\circ}$. The general $w$-dependence of
simulated ribbon shapes is well-captured by the NR model, with abrupt changes
in shape occuring near to the predicted values of $w_{*}$. We note that our
energy minimizations seemed to resolve only the larger pitch solution for
large-$w$ cases of the $\phi_{0}=22.5^{\circ}$, and attempts to seed and
sample the lower-pitch branch were unsuccessful in finding these equilibria.
This, combined with the observation from finite-element simulations [17] that
the larger pitch branch has higher elastic energy, would seem to account for
an apparent jump in the computed $E_{\rm ex}$ value for
$\phi_{0}=22.5^{\circ}$ as the ribbon shape transitions from low- to high-$w$
equilibrium shapes (i.e. visible in between $w/d=8$ and 9 in Fig. 4b). In Fig.
2a-b, we show, nevertheless, that the basic dependence of $w_{*}$ on curvature
angle $\theta_{0}$ is well captured by NR theory for both
$\phi_{0}=22.5^{\circ}$ and $\phi_{0}=45^{\circ}$ ribbons. As there is a no
symmetry breaking transition for catnoidal ($\phi_{0}=0^{\circ}$) membranes,
the exact (boundary layer) formalism summarized in eqs. (36)-(47) shows that
the shape evolution with increasing $w$ is fully-continuous for this case. No
attempt to extract a shape-flattening size from simulated $\phi_{0}=0^{\circ}$
membranes was made.

Figure 5:  Flattening energetics with nonlinear and anisotropic flattening
cost. (a) Rescaling the same data (points) in Fig. 2 for $r_{a}$ and
$\theta_{0}$, shows the collapse and agreement with continuum theory for
assembly widths up to the transition, where stress accumulation grows
$E_{ex}/N\sim Y\kappa_{0}^{4}w^{4}$. Dashed curves show the continuum theory
for varying $\phi_{0}$, while the single dotted curve shows a correction in
the shape solution for $\phi_{0}=0$ that agrees with the simpler solution
(dashed curve) for $w\rightarrow\infty$. Dotted flat lines indicate values of
$E_{\infty}$ from the model prediction for varying $\phi_{0}$. The vertical
line indicates the predicted value for the flattening transition $w_{*}$. (b)
Extrapolations of numerical minimizations of tubes to infinite length, shown
as discrete points, compared to flattened-state solutions of the nonlinear
continuum bending energy shown as dashed curves. (c) Minimized tubule
assemblies of length $w/d=100$, optimal radius and bond orientations,
illustrating the fully flattened geometry of wedge assembly for differing
$\phi_{0}$, away from the assembly boundaries. For $\phi_{0}=0$, bonds are
aligned parallel to the tube axis upon flattening and parallel to the
principal directions in the ideal ‘target’ geometry. For
$\phi_{0}=45^{\circ}$, bond directions are at $45^{\circ}$ with respect to the
tube axis (target principal directions) for the flattened (target) geometries.

The comparison of discrete and continuum results for $E_{\rm ex}(w)$ is made
clearer when results are rescaled by the parameter combination
$Nu_{0}d^{2}\theta_{0}^{2}/r_{a}^{2}$ (proportional to shape flattening
energy) and rescaling widths by the characteristic elastic scale
$d/\sqrt{\theta_{0}}\sim w_{*}$ in Fig. 5(a). Here, the results show good
agreement with the approximate continuum theory, plotted as dashed lines, for
the limit of small $w$. In this limit, the curves coincide for varying
$\phi_{0}$, showing the monomers are equivalently frustrated with $E_{\rm
a}/A\sim Y\kappa_{0}^{4}w^{4}/1440$ independent of $\phi_{0}$. Beyond the
flattening transition, a noticable discrepancy is due to the continuum model
approximation of uniform curvature, whereas a boundary layer forms for wider
structures, lowering the elastic energy accumulation below the NR theory
approximation. The exact boundary layer solution for the catenoidal case of
$\phi_{0}=0$, eq. (43), is plotted in 5(a), showing better agreement with the
discrete SWM numerics. [63] Notwithstanding the discrepancy at intermediate
scale, the flattening cost, $E_{\infty}=E_{\rm ex}(w\to\infty)$, is the same
in the narrow-ribbon approximation and boundary layer solutions as indicated
on the right of 5(a). The difference in energies with $\phi_{0}$ for large $w$
is interpreted to be largely due to the differing values of $E_{\infty}$. Some
numerical results at small $\theta_{0}$, in the case of $\phi_{0}=45^{\circ}$
were at unexpectedly large energy after minimization. This is attributed to
the limited resolution of the minimization for the smallest values of
$\theta_{0}$, which reach low force tolerance to meet the stopping criterion
without fully resolving the small residual strains in the structure.

The elastic energy due to shape flattening $E_{\infty}$ in the limit
$w\to\infty$ was further analyzed by minimization of tubule assemblies of
SWMs. Closed tubules were prepared with monomer bonds aligned to minimize the
bending energy, e.g. monomers with $\phi_{0}=0$ had bonds aligned parallel and
perpendicular to the tube axis whereas for $\phi_{0}=45^{\circ}$ the bonding
directions are at $45^{\circ}$ with respect to the tube axis. For intermediate
$\phi_{0}$ values we analyzed the higher pitch helical geometry, but confirmed
that both branches are degenerate in the $w\to\infty$ limit (i.e. in the limit
of vanishing boundary layer contributions). To account for the boundary layer
relaxation of finite-length tubes, tubes of varying length were minimized,
from $100d$ to $150d$ in increments of $10d$. The flattening energy was then
found by extrapolation to infinite length. The results for five values of
$\phi_{0}$ and varying $\theta_{0}/r_{a}$ are shown in Fig. 5(b). The
dependence of flattening cost on $\phi_{0}$ via the anisotropic bending costs
is captured by the harmonic approximation, eq. (7). However, results with
significant strains associated with flattening, $\delta r\approx
t\theta_{0}/2$, relative to the range of interaction $r_{a}$, show a reduction
in the flattening cost due to strain softening as the interaction potential
drops significantly below its harmonic approximation (see Fig. 3(a)). The
modified predictions for flattening cost, plotted for varying
$\theta_{0}d/r_{a}$ as dashed curves in Fig. 5(b), are computed by minimizing
attractive interactions over monomer orientations while enforcing uniform
flattening with monomers maintaining spacing $d$ (i.e. numerical minimization
of eq. (28) over $C_{ij}$ subject to $K_{G}=0$ using the fully non-linear form
of soft attractive potential). The flattening geometry for $\phi_{0}=0$ and
$45^{\circ}$ is illustrated in 5(c), where the monomers in the tube are
colored by excess energy to show the significant strain relaxation near the
boundary and uniform strain in the interior. The comparison between
(cylindrically) flattened and (hyperbolic) target geometry are illustrated for
a $9\times 9$ cross array of SWMs blocks. Notably, this highlights that shape
flattening for catenoidal SWMs ($\phi_{0}=0$) membranes generates row
unbending, whereas for helicoidal SWMs ($\phi_{0}=45^{\circ}$) rows are
untwisted from their target binding. The combined effects of shape flattening
transitioning from unbending to untwisting as $\phi_{0}$ increases with a
greater twist stiffness than row-bending stiffness
($B_{\perp}=2B_{\parallel}$), leads to the ($\sim 30\%$) increase in elastic
shape-flattening energy from catenoidal to helicoidal assembly observed in
Fig. 5(b).

To summarize, ring and ribbon morphologies of SMWs exhibit ground-state
energetics that are well-described by the NR theory as summarized in Sec.
II.1. Additionally, we find that the wide-ribbon regime, where shape-
flattening leads to saturation of the frustration cost, is also well described
to a first approximation by the continuum model, eq. (7), although strain
softening affects reduce this energy by up to $\sim 10\%$ for large wedge
angles. Building from these results, we consider the zero-temperature
thermodynamics of width limitation in the next section.

###  III.2 Self-limitation and range of size control

By including the effect of edge energy due to missing bonds at the assembly
boundaries (i.e. the effects of $E_{\rm edge}$), we develop predictions for
the possible equilibrium self limitation in the SWM model in the limit of zero
temperature. The competition between surface energy and super-extensive
elastic energy may result in minima in the energy-density landscape $U(w)/A$
at finite $w$. This minima is the self-limiting state, when its energy is less
than the bulk flattened state $U_{\infty}/A=E_{\infty}/A$, which in this case
is a self-closing tubule. [7] In Fig. 6, typical data from minimizations for
varying $\phi_{0},\theta_{0},r_{a}$ are shown. The corresponding linear-
elastic, narrow-ribbon continuum model predictions are shown as dashed curves,
along with flat dotted lines for the prediction of $E_{\infty}$ according to
the strain-softened flattening (bend) energy shown in Fig. 5(b) . Typical SWM
ground state structures are rendered in Fig. 6(d-f), for
$\phi_{0}=0^{\circ},22.5^{\circ}$ and $45^{\circ}$.

Figure 6:  Self-limiting assembly and yielded states. (a-b) Average energy
density $U_{\rm a}/N$ of minimized structures shown as points, for
$\theta_{0}=2.5^{\circ}$ and $r_{a}/d=0.014$, 0.036, and 0.057. Continuum
theory predictions with harmonic elasticity are shown as dashed curves and
nonlinear flattening energy predictions are shown as dotted curves only for
the cases where flattened state is stable to yielding. Both size selection
behavior and the transition to escape show agreement with theory, but yielded
structures denoted with open markers in (a-b) exist that have similar or even
lower energy. (c-d) Typical structures are shown, with monomers colored by
relative potential energy $E^{\rm(wedge)}_{\rm ex}$. Structures (ii) and (iii)
are yielded for both (c) and (d).

We start by noting the existence of a well-defined minimum at
$w_{0}=(5\mbox{-}6)d$ for the smallest attraction range ($r_{a}=0.014d)$, or
equivalently higher stiffness to cohesion ratio $Y/\gamma\propto r_{a}^{-2}$,
for both catenoidal and helicoidal assemblies, with quantitative agreement
between discrete SWM model and the continuum model. These minimal energy
states fall well below the expected flattening energy from continuum theory,
suggesting the thermodynamic ground state has frustrated-limited finite width
111$E_{\infty}$ is not shown in Fig. 6 for the cases with smallest $r_{a}$,
when the corresponding strains exceed the inflection points of the attractive
potential, $\epsilon\approx\theta_{0}t/2=r_{a}/2$.. Notably, and as discussed
in detail in Section III.3 below, ground state structures for larger widths
than the minimum fall off the curve predicted by the continuum theory, an
effect which can be attributed to non-linear yielding of highly strained SWM
bonds (locally yielded bonds appear as high-energy density bands in Fig.
6(c-d)).

For larger values of $r_{a}$, the SWM ground states show general agreement
with the predictions of the continuum model, including an optimal $w_{0}$ and
energy substantially increasing for successively larger $w$ notwithstanding
the instability associated with yielding for large enough structures. With
increasing $r_{a}$, which leads to effectively softer assemblies, the minimum
becomes more shallow and eventually metastable to the defrustrated (i.e.
$K_{G}\to 0$) state. Metastable minima can be resolved for increasingly soft
assembly parameters, but cannot be resolved beyond $w=w_{*}$ despite
minimizations extending up to $w=2w_{*}$. Self-limiting minima beyond $w_{*}$
are not predicted for the NR continuum model(see B).

The predictions for equilibrium self-limiting size are presented in Fig. 7,
for the full range of parameters investigated. Specifically, we denote SWM
assembly self-limiting for parameters where we resolve a local minimum in the
energy density and that minimum falls below the predicted shape flattening
energy density. For this range of SWM taper angles ($\theta_{0}\geq 2.5$), we
find the range of size control agrees with the continuum model (including
strain-softening corrections to $E_{\infty}$), with equilibrium sizes up to
$9$ monomer widths in length, consistent with $w_{0}\leq w_{*}$ 222The
boundary layer corrected prediction for moderately larger sizes achievable for
$\phi_{0}=0$ is not apparent, and may be too small of an effect to appear in
the discrete model. Taken together, these show that anharmonic (i.e. strain-
softening) effects of bonds in the discrete SWM model, which having little
effect on the small, finite-width assembly energetics, lead to measurable
reductions in the range of size control, relative to purely linear-elastic
model descriptions.

Figure 7:  Phase diagrams and range of self-limiting assembly. (a-c) Left
column shows self limitation phase diagrams in the design space of interaction
range relative to monomer size $r_{a}/d$ and amount of monomer shape misfit,
monomer wedge angle $\theta_{0}$, for three values of $\phi_{0}$, where points
are colored according to the optimal assembly size $w_{0}/d$ or gray if there
was no minimum up to $2w_{*}$. Open markers indicate that the flattening
energy prediction $E_{\infty}$ was lower than energies of minimized finite-
size assemblies. The continuum prediction is plotted with contours. On the
right column, the same data is plotted to show predicted size $w_{0}/d$ as a
function of interaction range $r_{a}/d$ for curves of constant misfit
$\theta_{0}$. Curves represent prediction from continuum theory with solid
curves where minima are stable and dashed for metastable states.

###  III.3 Role of bond yielding in self-limiting assembly

The structures with large internal strains shown in Fig. 6 are a consequence
of bond yielding, which occurs at the inflection point in Fig. 3. In general,
models of self-assembly with geometric frustration and also with finite-range
interactions can exhibit regimes where the internal strains associated with
stress accumulation are greater than the interactions can support. This can
result in a distinct mode of thermodynamic escape from self limitation, such
as the nucleation of low-symmetry, cracked assemblies in curvature frustrated
tubules.[21] In this regime, complex branched morphologies are expected, which
are composed of stronger bound and elastically coupled regions, weakly bound
together by partially yielded, yet at least slightly cohesive zones, as we
observe in the partially yielded SWM ground states in Fig. 6(c-d).

To further rationalize the energetics of yielded structures, we consider the
simpler case of the low-energy yielded structure (ii) from Fig. 6(d), a
catenoidal ring. Here, the yielded bonds are approximately in the middle of
the structure, and the structure of total width $w/d=10$ is nearly twice the
width of the self-limiting structure at width $w/d=6$. We consider this
structure as a composite of two non-yielded structures with yielded bonds
acting as weak, partial bonding between the two structures. For this
particular monomer geometry and parameters, we further explore the energetics
of partial-bonding between self-limiting rings, with results presented in Fig.
8. Additional minimizations were conducted, starting with the minimized and
non-yielded structures of width $w/d=4$, 5, and 6. For each starting width new
structures were prepared by arranging stacked copies of that structure, such
that the copies did not interpenetrate but partial bonds were made, where one
of the four attractors on a binding face coincided with its neighbor. After
further minimization to the same force tolerance $0.3\times 10^{-4}u_{0}/d$,
the results are presented, with colors corresponding to the number of copies.
The energetics of this composite ring morpholgies are well-described by an
augmented model that uses equation 1 to describe the energies of the
constituent non-yielded rings and fits a constant energy of $0.15u_{0}$ to
each of the partial bonds. These partially bonded structures are of lower net
cohesive energy than the isolated, non-yielded structures.

More general considerations (i.e. at least weakly cohesive binding between
elastically coherent self-limiting membranes) imply that such hierarchical
morphologies are possible for any SWM assembly, at least at sizes sufficiently
larger than $w_{0}$, and generically such structures should have at least
slightly lower total energy than the elastically self-limiting states (i.e.
associated with the minimum in the energy density). However, due to the
relatively weaker cohesive energy binding the structures at these yield bonds,
it is expected these hierarchical structures may be broken up due to entropic
considerations at sufficient high temperature, leading to an equilibrium state
dominated by the elastically self-limiting morphologies identified in the
phase diagram shown in Fig. 7.

Figure 8:  Role of highly deformed bonds in hierarchical assembly. (a) The
self-limiting structures of optimal width, studied here, are expected to
coexist with stacked structures, which gain some additional cohesion. (b) Pre-
built structures of width $w$ = 16, 20, and 24 $d$ are generated from the
previously relaxed structures of width $w$ = 4, 5, and $6d$ at
$r_{a}/d=0.014$, $\phi_{0}=0$, and $\theta_{0}=2.5^{\circ}$. (c) The energies
of pre-yielded structures are plotted versus the width with curves generated
from the non-yielded harmonic theoryincluding an additional contribution from
yielded bonds fit to $u_{\mathrm{yield}}=0.15u_{0}$.

##  IV Discussion and conclusions

In summary we have developed and studied a discrete SWM model of hyperbolic
membrane assemblies with crystalline order. Detailed analysis of the minimal
energy density morphologies is compared to predictions of linear-elastic,
continuum theory illuminating the connection between microscopic features of
the frustrated particles and their mesoscopic structure and thermodynamics. To
conclude we discuss the implications of these results for the understanding
and engineering of self-limiting assembly of geometrically frustrated building
blocks.

###  IV.1 Controlling self-limiting dimensions through discrete building block
shape and interactions

In this work, we identify the role of extrinsic geometry to shape equilibrium
self-limitation via, $\phi_{0}$, the angle relating bonding directions to the
direction of rotation between bonded monomers. While the SWM model realizes
equivalent stress accumulation, $E\sim Y\kappa_{0}^{4}w^{4}$ independent of
$\phi_{0}$, the elastic cost of escaping frustration through shape flattening
can vary with $\phi_{0}$ when there is anisotropic bending cost (i.e.
$B_{\perp}\geq B_{\parallel}$) 333Notably, while the priori continuum analyses
[17] are based on isotropic bending elasticity ($B_{\perp}=B_{\parallel}$) the
anisotropic case $B_{\perp}\geq B_{\parallel}$ is fairly generic for purely
attractive binding geometries, as twist deformations load all cohesive bonds
in proportion to their distance form the rotation axis, whereas bend
distortions only load bonds in proportion to projected distance to the neutral
axis of bending.. For this reason, the extrinsic geometry has the potential to
influence assembly energetics because $\phi_{0}$ relates the directions of
preferred curvature to the bonding directions, determining the extent to which
different modes of bending are activated with associated moduli
$B_{\perp},B_{\parallel}$. This is most clear in dependence of flattening
energy $E_{\infty}$ on $\phi_{0}$ shown in Fig. 5b, which derives from the
transition from unbending ($\phi_{0}=0$) to untwisting ($\phi_{0}=45^{\circ}$)
in the flatten tubule geometry. A consequence of the increasing escape energy
with $\phi_{0}$ is that the flattening size $w_{*}$, and to a smaller extent
$w_{\rm max}$, are predicted by NR theory to decrease with $\phi_{0}$, as
shown in Fig. 12c. A countervaling trend might be anticipated, based on the
previously reported effects of the boundary-layer corrections on the shape-
flattening tranisition [17] for equal bend and twist constants
($B_{\perp}=B_{\parallel}$), as shown Fig. 12d, which shows that $w_{*}$
instead increases by $\sim 20\%$ with $\phi_{0}$. To assess whether the shape-
flattening width should decrease with increasing $\phi_{0}$ as suggested by NR
theory or whether boundary layer corrections dominate and lead to the opposite
dependence, we extracted, we extracted the apparent $w_{*}$ (for $\phi_{0}\neq
0$) and $w_{\rm max}$ from SWM ground state simulations. Plotting these in
Fig. 12c, we note that there may be a slight tendency for $w_{*}$ to decrease
with $\phi_{0}$, but this is obscured by the resolution limits imposed by
discreteness of the changes in $w$ possible for the SWM model. We observe no
measurable changes in the maximum self-limiting size with curvature angle.
Hence, notwithstanding these two possible mechanisms for $\phi_{0}$
dependence, the size-range of self-limitation and stress-accumulation appear
to relatively insenstive to intrinsic geometry of the SWM ribbon morphology.

Beyond the role of extrinsic geometry (i.e. the direction of curvature axes),
the SWM model highlights the special dependence of self-limitation on the
range of interactions via the stiffness associated with accumulating
frustration costs to the assembly. Consistent with the generic predictions of
continuum theory, results in Fig. 7 confirm that the maximum self-limiting
width, $w_{\rm max}$, for a given block geometry is dictated by the flattening
size scale, $w_{\rm max}\lesssim w_{*}\propto(B/Y)^{1/4}\kappa_{0}^{-1/2}$,
simply because the energetics approach extensive scaling in this limit. In the
simplest case, where interactions are purely cohesive, the ratio of bend to
stretch ratio is controlled by the thickness (i.e. $B/Y\propto t^{2}$) which
itself is of order of the particle size $d$. Hence, this suggests the maximum
self-limiting size $w_{\rm max}\approx d/\sqrt{\theta_{0}}$, which implies
that the self-limiting dimensions that far exceed the size of the building
block require small taper angles, $\theta_{0}\ll 1$. Additionally, the
relations determining equilibrium size
$Y\kappa_{0}^{4}w_{0}^{5}\approx\gamma/w_{0}$ imply that that decreasing the
degree of frustration through the wedge angle also decreases the range of edge
energies for equilibrium, self-limited structures, which are characterized by
the maximum edge energy $\gamma_{\rm max}/Y\approx\kappa_{0}^{4}w_{\rm
max}^{5}\approx d\theta_{0}^{3/2}$. Notably, the ratio $\gamma/Y\propto
r_{a}^{2}/d$ is a cohesive elastic length scale, which is most strongly
dependent on the range (more strictly, the stiffness) of the cohesive
interactions between subunits. Taken together, these two relations show that
decreasing the degree of frustration through reduced wedge angle increases the
size range of frustrated limited assembly, but does so at the expense of
requiring narrowed range of interaction stiffness. In particular,
thermodynamic self-limitation by frustration is only possible for
$r_{a}/d\lesssim(w_{\rm max}/d)^{-3/2}\propto\theta_{0}^{-3/4}$, implying that
self-selection on larger, multi-subunit dimensions requires increasingly
shorter range (i.e. stiffer) cohesive interactions, which is consistent with
the shift to smaller $r_{a}$ range with decreasing $\theta_{0}$ shown in Fig.
7.

Notably, this basic result is predicated on two assumptions: (i) that
deformations in assembly primarily strain interactions while subunits are
considered rigid and (ii) binding interactions are purely attractive. Relative
to the case considered here, including additional deformability in the
subunits themselves as in models of Refs. 22, 21, 6, the expectation may be to
reduce the cohesive-elastic ratio $\gamma/Y$ below the (upper bound) limited
by interaction stiffness, such that for any finite interaction range, inter-
subunit deformability should only further depress the feasible size-range. On
the other hand, it can be shown that more complex binding geometries,
incorporating distinct spatial patterns of local attraction and repulsion can
give rise to effective elasticity where $B/Y\gtrsim t^{2}$, [20] thereby
extending the elastic scale where shape flattening takes place.

Applying these elementary considerations to the experimental
$C_{12}$-$\beta_{12}$ gemini amphiphile system studied in Ref. 18 where width-
dependent helicoidal ribbons morphologies were carefully characterized, we
note, of course, that macromolecular subunits are both highly deformable and
realize highly complex interactions. Nevertheless, we may assess the apparent
effects of the likely ranges of interactions on the overall morphologies and
likely equilibrium states. In these experiments, at early times, helicoidal
ribbons are observed with mesoscopic pitches of order $\sim 100$ nm. Based on
a considerations of local packing in the “twisted crystal” of amphiphiles,
[18] It was estimated that $d\approx 0.6$ nm, $\kappa_{0}\approx 0.03~{}{\rm
nm}^{-1}$. Assuming the naive estimate $\sqrt{B/Y}\approx d$, this suggests a
shape-flattening size scale $w_{*}\approx 40$ nm, and at this shape-flattening
size, ribbons are of order $w_{*}/d\approx 60-70$ sub-units across. Notably
the estimate of $w_{*}$ is consistent with the fact that at longer times, as
ribbons grow larger than this size range, they exhibit a shape-transition to
spiral ribbons, consistent with the basic predictions of the continuum elastic
theory, but ultimately suggesting that thermodynamic equilibrium does not
correspond to a regime where frustration stabilizes the open-boundary
helicoidal ribbon, presumably because the assembly is too ductile. This raises
a basic question: presuming cohesive interactions govern the elasticity of
assembly, how close might such a molecular system be to an equilibrium state
of frustration limitation? Applying the estimate of the upper limit on
interaction range for the gemini amphiphile membrane, we find
$r_{a}/d\lesssim(w_{*}/d)^{-3/2}~{}10^{-2}$. Notably, as subunits are
molecular in dimension, this corresponds to a limiting interaction range that
is sub-Angstrom, clearly much shorter range than what might reasonably be
expected from van der Waals or hydrophobic interactions which bond the
amphipillic subunits together. That interactions likely far exceed this range
is consistent with the observed long-time growth of chiral amphiphiles
assemlbies into shape-flattened tubules at long times, and more generally
suggests that the possibility of frustration-limited assembly in molecular
crystalline membranes may be difficult to achieve, if at all possible.

The restrictions placed on the elasticity and range of frustration
accumulation by the interaction range suggest the feasible avenue for
engineering self-limiting systems requires the combination of (larger)
colloidal-scale particles bound by shorter range interactions. We point out a
recent example of DNA origami particles 67, 32, 33 designed to be triangular
subunits, $\sim 50$ nm in size with controllable inter-particle geometry, and
large $w\gg d$, assemblies driven by short-range base-stacking interactions,
whose interactions range may be less than $\sim 1$ nm. [68] Beyond the
requirement of shorter-range interactions relative to subunit size,
frustration-limitation will also require engineered colloidal particles with
the combination of precise geometry binding and also stiffness comparable to
or exceeding that from the short-range stacking interactions.

###  IV.2 Role of finite interaction ranges: hierarchical aggregation

The results presented here show features of discrete systems that are not
captured by the linear-elastic continuum description. Two possible features
have been proposed to augment the continuum description. Firstly, we
identified the relevance of nonlinear elasticity, which becomes relevant as
the scale of deformations in the equilibrium structures $\sim d\kappa_{0}t$
becomes comparable with the range of interactions $\sim r_{a}$. Strain
softening was shown to reduce the cost of tubule formation, with results
presented in Fig. 5 that were well-described by the nonlinear elastic
description of bending costs. Secondly, as the scale of deformations
approaches the range of interactions, structures become mechanically unstable
when individual interactions reach the yield point of the interactions. For
our soft-binding model this corresponds to $r_{y}=r_{a}/2\approx
d\kappa_{0}t$. The zero-temperature optimal size $w_{0}$ presented in Fig. 7
do not show a deviation from theory due to the breakup of bonds at high
strains. However, the weakly-cohesive aggregation of otherwise elastically
deformed, cohesive and finite assembly domains suggested here (as well as the
low-symmetry internal cracking exhibited in the model of Ref. 21) constitute
an alternative manner of escaping the self-limiting thermodynamic consequences
of frustration, that will occur in any realistic, particle-based description
of frustrated assembly.

In summary, if there is a minimal energy density at width $w_{0}$, then two of
these structures can at least weakly bind together (i.e. through partial
yielding of cohesive bonds, or some set of bonds that do not transmit effects
of frustration between domains) without introducing additional elastic costs
into those two structures. Hence, it is straightforward to argue that at
$T=0$, the energy density of multiple weakly-aggregated domains of size
$w_{0}$ will fall at least slightly below the single domain minimum. This
simple argument suggests that generically even in states where a self-limiting
domain minimum falls below the energy of the (smooth) shaped-flattened states,
at sufficiently low temperature (and high concentration), self-limiting
aggregates would be unstable to some condensed, multi-aggregate morphology
that is effective unlimited in size, and points to the importance of
finite-$T$ entropic effects in stabilize any putative regime of self-
limitation.

###  IV.3 Modeling frustrated assemblies at finite temperature

While the core analysis of this study focuses on purely energetic ground
states at $T=0$, the conclusions detailed above imply some important open
challenges for understanding frustrated self-assembly at finite temperature.

For one, while the frustrated assembly seemingly offers an attractive paradigm
for controlling equilibrium self-limiting assembly through engineering misfit
of building blocks, [27] we find that realizing such self-limitation at non-
trivial size scales places important restrictions on the range of
interactions. In particular, thermodynamic conditions where frustration-
limited morphologies out compete shape-flattened (i.e. defrustrate)
morphologies rely on attractive interactions between subunits that are short-
ranged, ideally much shorter-range than the size scale of the SWM subunits.
Using molecular dynamics to sample equilibration of discrete particle models,
like SWM, in the self-limiting regimes faces additional challenges. Not only
are the time scales to equilibrate such systems long, but stiff potentials
require shorter time steps increasing the total computational cost. In
addition, simulations of assembly of free particles will have a short capture
radius, which will increase the simulation time. [69, 70]

Figure 9:  (a) Monomers with angles $\phi_{0}=0$ and $\theta_{0}=10^{\circ}$
assemble into clusters at moderate interaction range $r_{a}/d=0.14$ and
temperature $k_{\rm B}T/u_{0}=0.054$. The clusters have edges predominantly
oriented in the closed-packed directions. (b) Averaging the interaction
energies over many timesteps reveals the gradients in strain energy that
develop in clusters.

As a preliminary picture into the finite temperature self-assembly behavior of
the discrete particle model, we carried out MD simulations of SWM model
starting from disassociated configurations. In these simulations, 500 free
subunits are randomly placed in the simulation box and the temperature is
$0.8\epsilon$. The simulation system has 500 subunits at a density of
$6.35\cdot 10^{-4}\,{\rm particles}/\sigma^{3}$, and the subunit parameters
are $\phi_{0}=0,r_{a}=0.5\sigma$ (i.e. relatively large range) and
$\theta_{0}=10^{\circ}$. Fig. 9 shows that intial assembly does occur for this
system in this time window, with wultiple rectangular assemblies have formed
after 1 billion time steps. While further work is needed to determine the
equilibrium structures and more fully explore the kinetics assembly, it clear
that SWM parameters can be identified where MD simulation is viable.

Notably, while the free SWM simulations are free to sample much more irregular
morphologies, the assembled clusters shown in Fig. 9(a) are mostly rectangular
with boundaries aligned along the close-packed directions, which is favorable
for cohesive assemblies to optimize the number of bonds formed for a given
number of particles. This feature suggests the possibility that larger
assemblies will maintain boundaries along the close-packed directions, which
was implicit in the choice of pre-assembled structures of varying-$w$ that
were generated for the $T=0$ results presented here. Coloring monomers by
their respective interaction energy in Fig. 9(a) reveals that in addition to
bonds forming and breaking, there are varying degrees of strain in the
interior of each cluster, evidenced by patches of lighter color corresponding
to more strained bonds. Further analysis revealed gradients in strain within
the clusters assembled at finite $T$, shown in Fig. 9(b). These more regular
patterns of coherent strain gradients were extracted by averaging local
binding in clusters over multiple time steps. Time sampling was chosen to be
sufficiently short with respect to the lifetime of bonds but longer than the
apparent correlation of elastic fluctutations or phonons within the structure.
The appearance of these coherent strain gradients, which grow in magnitude
with size, are consistent with the fact accumulation of frustration costs in
the SWM model shape finite temperature pathways.

What remains to be determine are how these effects of strain accumulation also
influence the kinetics of reaching equilibrium states, and further, whether
and how finite temperature effects shift the expects phase boundaries between
equilibrium self-limiting states and states of bulk assembly. Not only do we
expect that especially soft and weakly frustrated systems to escape frustrated
to unlimited, shape-flatted structures, but we also expect, based on the
analysis described in Sec. III.3 that a low temperatures, otherwise
frustration limited structures may condense or aggregate into (presumably low-
symmetry) clusters held together by partially yeilding bonds. Hence, a further
open challenge is to identify how non-linear features of the inter-particle
binding that control yield, shape the critical temperature and concentration
conditions at which translation entropy favors break-up (or melting) of
heirarchical clusters into free, size-controlled aggregates.

## Acknowledgements

The authors are grateful to M. Wang, B. Tyukodi and N. Hackney for valuable
discussions on this work. DH and GG acknowledge support for this work through
US National Science Foundation through award NSF DMR-2028885, as well as
through the Brandeis MRSEC on Bioinspired Materials NSF DMR-2011846. This work
was performed, in part, at the Center for Integrated Nanotechnologies, an
Office of Science User Facility operated for the U.S. Department of Energy
(DOE) Office of Science. Sandia National Laboratories is a multimission
laboratory managed and operated by National Technology & Engineering Solutions
of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc.,
for the U.S. DOE’s National Nuclear Security Administration under contract DE-
NA-0003525. The views expressed in the article do not necessarily represent
the views of the U.S. DOE or the United States Government. Simulation studies
of SWM model were performed on the UMass Cluster at the Massachusetts Green
High Performance Computing Center and computers at Sandia.

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##  Appendix A SWM design details and pairwise elasticity

This section details the SWM geometry, the interactions defined by repulsive
and attractive potentials, and the resulting elastic response of a bound pair
of monomers. The SWM geometry is designed so that pairs of bonded monomers
prefer to adopt a target configuration with opposite rotation sense of
neighbors in the two bonding directions, a discrete analogue to local
curvatures of a minimal surface and the rotation of that surface’s normal
vector. Here, the monomer’s coordinate frame is denoted by unit vectors ${\bf
c_{1}}$ and ${\bf c_{2}}$ that are orthogonal and point in directions of
bonding faces (see the coordinate frame illustrated in Fig. 3 in the main
text) and ${\bf c_{3}}$ is assumed to point along the local normal to the mid-
surface spanned by the 2D assembly.

Figure 10:  Pairwise elasticity due to attractor interactions. The energy of a
pair of monomers for different deformations away from the preferred
configuration is shown for $\theta_{0}=10^{\circ},r_{a}=0.11d$, with the
harmonic approximation shown as a dashed curve. The repulsive interactions
become nonzero at deformations much larger than seen for assemblies in this
study, e.g. for compression $\delta r<-0.1d$ or bending deformation angle
$|\Delta\theta|>20^{\circ}$.

The rotation axes of the target configuration, relating a monomer’s coordinate
frame to those of neighbors bonded in the ${\bf c_{1}}$ and ${\bf c_{2}}$
directions are, respectively,

$\displaystyle{\bf r_{1}}\equiv-\sin 2\phi_{0}~{}{\bf c_{1}}+\cos
2\phi_{0}~{}{\bf c_{2}},$ $\displaystyle{\bf r_{2}}\equiv\cos 2\phi_{0}~{}{\bf
c_{1}}+\sin 2\phi_{0}~{}{\bf c_{2}}.$ (17)

In the target configuration, the frame of a monomer is related to its four
neighbors’ frames by a displacement along the bond direction $\frac{1}{2}d{\bf
c_{\alpha}}$ ($\alpha=1$ or $2$), rotation of angle $\theta_{0}$ about ${\bf
r_{\alpha}}$, and additional displacement of $\frac{1}{2}d{\bf
c_{\alpha}}^{\prime}$ where ${\bf c_{\alpha}}^{\prime}$ is the neighbor’s
frame after the rotation about ${\bf r_{\alpha}}$. Here, $d$ is the nominal
monomer width related to the actual monomer geometry and the arrangement its
of attractive sites, detailed below. To leading order in $\theta_{0}$, this
transformation is a bending angle between neighboring monomers of
$\theta_{\parallel}\equiv\theta_{0}\cos 2\phi_{0}$ projected along the bond
direction ${\bf c_{\alpha}}$ and a twist angle
$\theta_{\perp}\equiv\theta_{0}\sin 2\phi_{0}$ of rotation about ${\bf
c_{\alpha}}$. The definition of ${\bf r_{\alpha}}$ enforces opposite sense of
bending angle in the two bond directions, analogous to a minimal surface
having curvature tensor with zero trace, and opposite sense of twist, which is
true for a smooth surface that necessarily has symmetric curvature tensor.

The monomer design consists of an array of repulsive sites that define an
excluded volume to monomer overlap, and attractive sites on four sides that
determine the geometry of bound neighbors. This design is inspired by previous
work on tubule-forming monomers in Ref. 54, 55, 56. Repulsive sites are of a
single type and interact with the repulsive sites on other monomers according
to the Weeks-Chandler-Anderson potential,

$u_{\rm
WCA}(r)=\left\\{\begin{array}[]{ll}4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}-\left(\frac{\sigma}{r_{c}}\right)^{12}+\left(\frac{\sigma}{r_{c}}\right)^{6}\right],&r\leq
r_{c}\\\ 0,&r>r_{c}\\\ \end{array}\right.$ (18)

where $r$ is the distance between interacting sites, $\sigma$ and $\epsilon$
are the reference distance and energy units chosen for simulations. For this
study, the soft attractive interaction was chosen so that $u_{0}=14\epsilon$,
consistent with attractive interaction strengths which favored finite
temperature assembly in the tube-forming system. [54] The cutoff length
$r_{c}=2^{1/6}\sigma$ yields a purely repulsive interaction that reaches zero
energy and zero force at $r=r_{c}$. The nominal monomer width is related to
$\sigma$ by $d=(2.4+2^{1/6})\sigma$, which follows from the attractive site
arrangement specified below.

The arrangement of a monomer’s 27 repulsive sites can be specified in terms of
the orthonormal coordinate frame for a given monomer with ${\bf c}_{1},{\bf
c}_{2}$ the bonding directions and ${\bf c}_{3}={\bf c}_{1}\times{\bf c}_{2}$.
The arrangement varies with curvature angles $\theta_{0},\phi_{0}$ to maintain
separation between repulsive sites of bound neighbors that are close to the
target configuration. Adopting indices $i_{1},i_{2},i_{3}$ that each take
values $-1,0,1$, the repulsive site coordinates are

${\bf R}_{i_{1},i_{2},i_{3}}^{\rm
rep}=\sigma\Big{[}i_{1}+i_{1}i_{3}\cos(2\phi_{0})\tan\left(\frac{\theta_{0}}{2}\right)+i_{2}i_{3}\sin(2\phi_{0})\tan\left(\frac{\theta_{0}}{2}\right)\Big{]}{\bf
c}_{1}\\\
+\sigma\Big{[}i_{2}-i_{2}i_{3}\cos(2\phi_{0})\tan\left(\frac{\theta_{0}}{2}\right)+i_{1}i_{3}\sin(2\phi_{0})\tan\left(\frac{\theta_{0}}{2}\right)\Big{]}{\bf
c}_{2}+\sigma\,i_{3}\,{\bf c}_{3},$ (19)

Reference: https://www.physicsforums.com/threads/wide-equation-in-
revtex-4-2-is-overlapping-with-other-content.992578/

Repulsive sites in the midplane $z=0$ are arranged on the corners and edge
centers of a square of side length $2\sigma$, the repulsive interaction has
range $2^{1/6}\sigma$, so the attractive sites are chosen to sit approximately
$0.2\sigma$ from the effective excluded volume of the monomer. Note that for
$\phi_{0}\neq 0$ the location of the repulsive sites do not lie along planar
faces (as shown schematically in Fig. 3c), but more accurately instead skew
quadralateral surfaces which deviate slightly from planarity. In Fig. 10,
plots of pairwise energy under small deformations illustrate that repulsive
sites do not play a role for the deformations up to the point of yielding
(which can be roughly identified with the inflection point for energy vs.
displacement), and for the parameters used in this work.

The interaction between attractive sites of the same type is given by the pair
potential defined in Eq. 10 and there is no interaction between sites of
different type. The 8 types at different locations (see Fig. 3) are used to
control monomer face-face binding orientation. Based on the form of Eq. 10,
the depth of each attractive potential pair is $-u_{0}/4$, so that an
unfrustrated ($\theta_{0}=0$) bulk assembly has potential energy $-2u_{0}$ per
monomer (there is energy $-u_{0}$ per bond in the assembly, consisting of four
attractive site pairs). There are four attractive sites on each side of the
monomer (arrayed on a common plane), with corresponding types on opposite
sides (see attractive sites colored by type in Fig. 3b-c).

The attractive sites on each side are arrayed in a strictly planar
configuration, which is not generally the case for the repulsive sites
arranged according to Eq. 19. For the side binding with respect to the ${\bf
c_{\alpha}}$ direction, the four attractive sites have center of mass at
$\frac{1}{2}d{\bf c_{\alpha}}$ with respect to the monomer center. From the
center of mass, each attractive site is displaced a distance of $\approx
t=\sigma$ in either the $\pm{\bf c}_{3}^{\prime}$ (i.e. vertically on the
face) or $\pm{\bf c_{\alpha}}^{\prime}\times{\bf c}_{3}^{\prime}$ (i.e.
horizontally on the face), where ${\bf c}_{3}^{\prime}$ and ${\bf
c_{\alpha}}^{\prime}$ are the monomer’s coordinate frame after rotation about
${\bf r}_{\alpha}$ by $\theta_{0}/2$.

The two attractive sites per side that are activated by both bending and twist
deformations discussed below, displaced “vertically” from the center of each
binding face, are defined using the monomer coordinate frame ${\bf c}_{1},{\bf
c}_{2},{\bf c}_{3}$ by

${\bf R}_{i,j}^{\rm
vert}=\Big{[}\frac{d}{2}\cos\left(\frac{i\pi}{2}\right)+\sigma(-1)^{i}(-1)^{j}\tan\left(\frac{\theta_{0}}{2}\right)\cos\left(\frac{\pi
i}{2}+2\phi_{0}\right)\Big{]}{\bf c}_{1}\\\
+\Big{[}\frac{d}{2}\sin\left(\frac{i\pi}{2}\right)+\sigma(-1)^{i}(-1)^{j}\tan\left(\frac{\theta_{0}}{2}\right)\sin\left(\frac{\pi
i}{2}+2\phi_{0}\right)\Big{]}{\bf c}_{2}+\sigma(-1)^{j}{\bf c}_{3},$ (20)

for $i=1..4$ indexing the four bonding directions and $j=1,2$ referencing
sites above/below the monomer midplane. From the coordinates above, the
magnitude of the displacement from the midplane is $t_{\rm vert}/2=\sigma
2\sqrt{1+\tan^{2}(\theta_{0}/2)}$.

The attractive sites activated with twisting rotation about the bond axis but
not bending rotations about the orthogonal direction, displaced “horizontally”
from the center of the binding face, are defined by

$\displaystyle{\bf R}_{i,j}^{\rm horiz}$
$\displaystyle=\Big{[}\frac{d}{2}\cos\left(\frac{i\pi}{2}\right)-\sigma(-1)^{i}(-1)^{j}\sin\Big{(}\frac{\pi
i}{2}\Big{)}\Big{]}{\bf c}_{1}$ (21)
$\displaystyle+\Big{[}\frac{d}{2}\sin\left(\frac{i\pi}{2}\right)+\sigma(-1)^{i}(-1)^{j}\cos\left(\frac{\pi
i}{2}\right)\Big{]}{\bf c}_{2}$
$\displaystyle+\sigma(-1)^{1+i}(-1)^{j}\sin(2\phi_{0})\tan\Big{(}\frac{\theta_{0}}{2}\Big{)}{\bf
c}_{3},$

for $i=1..4$ indexing bond directions again and $j=1,2$ indexing sites
displaced in opposite directions on the same monomer face. From these
coordinates, the displacement of these attractors from the binding face center
is $t_{\rm horiz}/2=\sigma\sqrt{1+\sin^{2}(2\phi_{0})\tan^{2}(\theta_{0}/2)}$.

While our elastic theory treats each monomer as perfectly rigid, SWM is
enforced via harmonic potentials of the form
$\frac{1}{2}k_{\mathrm{spring}}(r-r_{0})^{2}$ where $k_{\mathrm{spring}}$ is
the spring constant and $r_{0}$ is the rest length to maintain site
separations according to the geometry above. Springs are applied between
nearby repulsive sites, for all pairs with indices $i_{1},i_{2},i_{3}$ such
that each index differ by at most one (i.e. springs between nearest, next-
nearest, and next-next nearest neighbors). Each attractive site has springs
enforcing its distance to the six nearest repulsive sites. All springs are
kept at the same spring constant, which is incremented during energy
minimization as detailed in Sec. II.1.1.

The attractive site positioning described above determines the relative cost
of different elastic deformations of the assembly. In Fig. A10, the potential
energy of a bound pair of monomers is plotted for deformations away from the
optimal (target) pair configuration with energy $-u_{0}$, corresponding to all
four attractive sites in contact with respective sites on the other monomer.
Compression or stretching deformations, due to center-of-mass displacement by
a distance $\Delta r$ that equally affects all four attractive sites, change
the potential energy to leading order by
$\frac{1}{2}k_{\mathrm{stretch}}\Delta r^{2}$ with
$k_{\mathrm{stretch}}=4\partial_{r}^{2}u(r)={\pi^{2}u_{0}}/{2r_{a}^{2}}$.
Bending rotation, of an angle $\Delta\theta$ away from the preferred bend
angle $\theta_{\parallel}$ and about an axis through the horizontal attractive
sites, displaces vertical attractive sites by a distance $\Delta\theta t_{\rm
vert}/2$ and changes the potential energy of the configuration to leading
order by $\frac{1}{2}k_{\parallel}\Delta\theta^{2}$ with
$k_{\parallel}={\pi^{2}t_{\rm vert}^{2}u_{0}}/{16r_{a}^{2}}$. Twisting
rotation of an angle $\Delta\theta$ away from the preferred twist changes the
potential energy by $\frac{1}{2}k_{\perp}\Delta\theta^{2}$ with
$k_{\perp}={\pi^{2}(t_{\rm vert}^{2}+t_{\rm horiz}^{2})u_{0}}/{16r_{a}^{2}}$
due to displacement of vertical and horizontal attractive sites. These three
pairwise elastic constants determine the continuum model elastic constants
relevant to stress accumulation in the assembly, discussed in the next section
on the continuum theory. These constants depend weakly with $\theta_{0}$ and
$\phi_{0}$ via $t_{\rm vert}$ and $t_{\rm horiz}$, which range from $t_{\rm
vert},t_{\rm horiz}=2\sigma$ up to $t_{\rm vert},t_{\rm horiz}\approx
2.008\sigma$ for $\theta_{0}=10^{\circ}$, so that it approximately holds that
$t_{\rm vert}\approx t_{\rm horiz}\approx 2\sigma$. We therefore use a single
elastic thickness parameter $t\equiv 2\sigma$ independent of monomer shape
parameters.

##  Appendix B Continuum elastic model for anisotropic membranes

In this section, continuum theory predictions of wedge assembly energetics are
derived, namely the harmonic elastic flattening energy $E_{\infty}(\phi_{0})$,
the harmonic elastic energy of ribbons and rings $E(w,\phi_{0})$,
characteristic shape-flattening size $w_{*}(\phi_{0})$ and numerical solutions
for zero-temperature escape size $w_{\mathrm{max}}(\phi_{0})$. These show the
role of anisotropy of the assembly, as presented along with the discrete-SWM
numerical results in the main text: the in-plane stretching cost depends only
on a single elastic modulus $Y$ whereas the out-of-plane bending and the range
of size control are dependent on the anisotropic bending costs
$B_{\parallel},B_{\perp}$ and preferred direction of curvature $\phi_{0}$.

The continuum model presented here follows previous theory that captured
flattening of frustrated membranes [16, 46, 43] and the equivalent description
of shape selection in frustrated elastic sheets. [45] The model predictions
relevant to this study, including stress accumulation in the self-limiting
regime (narrow limit), flattening transition and flattening cost are captured
in an approximate description of the assembly geometry described by a single
curvature tensor with components $C_{xx},C_{yy},C_{xy}$. This is an
approximation to the shape of either slender ribbon or ring assembly, both
with translational symmetry along the midline, and it is exact for the tubule-
shaped membrane which is achieved in the limit of flattening $w\to\infty$
without yielding. A correction to the theory that accounts for varying
curvature throughout the assembly, in the closed-ring case $\phi_{0}=0$, is
detailed at the end of this appendix. To map the discrete wedge model to the
continuum model, we additionally assume wedge monomer orientations stay
aligned with the surface, so the monomer frame direction ${\bf c}_{3}$ is
aligned everywhere with the local surface normal of the assembly.

The elastic energy of the assembly is partitioned into two terms, applicable
for the wedge assembly in the case of small deformations when the harmonic
approximation to the attractive potential results in separate terms for
stretching, bending and twist deformations of wedge bonds derived in the
previous section. The continuum elastic energy is

$\displaystyle E_{\mathrm{elastic}}=E_{\rm strain}+E_{\mathrm{bend}}$ (22)

describing respective elastic costs due to monomer spacing deviating from $d$
and gradients in monomer orientations deviating from the preferred angles
$\theta_{\parallel}$ and $\theta_{\perp}$.

The stretching cost is derived from the in-plane response to given curvatures
$C_{xx},C_{yy},C_{xy}$ and the monomer preferred configuration, i.e. a square
lattice with preferred spacing $\approx d$. For a 2D material with square
symmetry, the full elastic energy has the form

$E_{\mathrm{strain}}=\int
dA\,\Big{\\{}\frac{\lambda}{2}(u_{xx}+u_{yy})^{2}+\mu(u_{xx}^{2}+2u_{xy}^{2}+u_{yy}^{2})\\\
+\lambda_{\perp}u_{xx}u_{yy}\Big{\\}},$ (23)

where $u_{ij}$ is the strain tensor and $\lambda,\mu,\lambda_{\perp}$ elastic
constants. [71] The corresponding stress tensor $\sigma_{ij}$ satisfying that
$E_{\mathrm{strain}}=\int dA\sigma_{ij}u_{ij}$, is related to the strain by
$u_{xx}=({1}/{Y})\Big{(}\sigma_{xx}-\sigma_{yy}({\lambda+\lambda_{\perp}})/({\lambda+2\mu})\Big{)},u_{yy}=({1}/{Y})\Big{(}\sigma_{yy}-\sigma_{xx}(\lambda+\lambda_{\perp})/(\lambda+2\mu)\Big{)},$
and $\sigma_{xy}=u_{xy}/(2\mu)$ where

$\displaystyle
Y\equiv\frac{(2\mu-\lambda_{\perp})(2\lambda+2\mu+\lambda_{\perp})}{\lambda+2\mu}$
(24)

is the Young’s modulus measured upon loading parallel to either of the square
lattice close-packed directions. The strain depends on out-of-plane
deflections, via
$u_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i}+\partial_{i}f\partial_{j}f)$
where $u_{i}$ is the in-plane displacement field and $f(x,y)$ is the
deflection out-of-plane. [51] A condition for a single-valued displacement
field, the compatibility condition, is derived from applying
$\epsilon_{ik}\epsilon_{jl}\partial_{k}\partial_{l}u_{ij}$ to the relations of
strain to both the stress and the displacements. The special case of
translational symmetry in the $y$ direction, so that gradients in $y$ vanish,
implies $\sigma_{xx},\sigma_{xy}$ are constants by mechanical equilibrium
$\partial_{i}\sigma_{ij}=0$, and compatibility becomes

$\displaystyle\epsilon_{ik}\epsilon_{jl}\partial_{k}\partial_{l}u_{ij}=\frac{1}{Y}\partial_{x}^{2}\sigma_{yy}=-K_{G}$
(25)

where $K_{G}=C_{xx}C_{yy}-C_{xy}^{2}\simeq\det\partial_{i}\partial_{j}f$ is
the Gaussian curvature, with the relation to $f(x,y)$ under the small-slope
approximation. One finds the constants of integration that fully determine
$\sigma_{ij}$ by minimizing over $E_{\mathrm{strain}}$. Thus, the continuum
model expression for stretching cost of slender ribbons or rings of length $L$
and narrow width $w$ is

$\displaystyle\frac{E_{\mathrm{strain}}}{wL}=\frac{Y}{1440}(C_{xx}C_{yy}-C_{xy}^{2})^{2}Lw^{5}.$
(26)

We note that different prefactors appear in the same “narrow-ribbon” constant
curvature calculations in other Refs. [43, 9, 18, 72], with specific Poisson
ratio dependence, while our result in eq. (26) is inagreement with the
(Poisson ratio-indendent) results of Refs. [16, 8]. We believe this
discrepancy to derive from the neglect of vanishing longitudinal net stress
along the ribbon (i.e. $\int_{-W/2}^{+W/2}dx~{}\sigma_{yy}(x)=0$) in the
former group of references. The fit-free agreement of numerical results
presented in Fig. 5 with the eq. (26) is consistent with this conclusion.

The Young’s modulus $Y$ is the response to uniaxial stress applied in $x$ or
$y$ coordinate directions, which are the bonding directions for the wedge
model. The stretching response of the assembly then derives from the pairwise
stretching of bonds described in the previous section, so that taking the
continuum limit of a square lattice of springs with spacing $d$, and extension
along the bond direction, we have form of the the Young’s modulus given in eq.
(12), $Y=k_{\mathrm{stretch}}$.

The harmonic bending energy is also anisotropic. Tme most general form for the
harmonic bending energy of membranes with anisotropy arising from distinct in-
plane directions is given

$E_{\mathrm{bend}}=\int
dA\Big{\\{}\frac{1}{2}B_{xx}(C_{xx}-C_{xx}^{0})^{2}+\frac{1}{2}B_{yy}(C_{yy}-C_{yy}^{0})^{2}\\\
+B_{xy}(C_{xy}-C_{xy}^{0})^{2}+\frac{1}{2}{\bar{B}}(C_{xx}C_{yy}-C_{xy}^{2})\Big{\\}},$
(27)

equivalent to the most general form given in Helfrich and Prost [41], where it
was emphasized that $C_{xy}^{0}\neq 0$ arises from molecular chiral asymmetry
despite the achiral symmetry of the curvature quadratic form. The case of
$C_{xx}^{0}=C_{yy}^{0}$, with zero Gaussian curvature modulus ${\bar{K}}=0$
was considered in Ref. 16 to explain chiral amphiphile assemblies. The more
general case with varying $C_{xx}^{0},C_{yy}^{0}$ was connected to internally
stressed elastic solids in Ref. 46, where $C_{xy}^{0}\neq 0$ could be realized
from stretched bilayers without material chirality, instead the assignment of
$x,y$ via the boundary breaks symmetry and results in chiral shapes. The
interaction specificity in the wedge model distinguishes bonding in the two
lattice row directions, so wedge monomers are notably chiral for the case
$\phi_{0}>0$. If the attractive sites were arranged differently on the $x-$
and $y-$ oriented faces, the monomers could in principle have $B_{xx}\neq
B_{yy}$. The saddle-wedge design presented in this study has
$B_{xx}=B_{yy}=B_{\parallel}\neq B_{xy}=B_{\perp}$, with consequences for size
control derived below. The role of Poisson’s ratio in the elastic sheet
description of Refs. 43, 46 can be mapped to the Helfrich-Prost bending energy
functional of the assembly above, effectively modulating both
$B_{\parallel}/B_{\perp}$ and $C_{xx}^{0},C_{yy}^{0}$. Here, we further
develop the analysis with varying $\phi_{0}$ but constant preferred principal
curvatures set by $\kappa_{0}$.

The bending elastic cost derives from the monomer attractive site interactions
and the target pairwise configuration derived in the previous appendix.
Assuming monomers have their frame direction ${\bf c}_{3}$ aligned with the
assembly surface normal, the deviation from preferred bend angle in the $x$
direction is taken to be $d\times C_{xx}-\theta_{0}^{\parallel}$, in the $y$
direction $d\times C_{yy}+\theta_{0}^{\parallel}$ and the deviation of twist
angle in either direction is $d\times C_{xy}-\theta_{0}^{\perp}$. The cost of
bend deformations for a single SWM with these local curvature values, is then
the sum of potential energy from displacements of the two vertical attractive
sites in either direction and of the four horizontal attractive sites

$E^{\rm(wedge)}_{\mathrm{bend}}=2u\Big{(}\frac{1}{2}t_{\rm
vert}\sqrt{(d~{}C_{xx}-\theta_{0}^{\parallel})^{2}+(d~{}C_{xy}-\theta_{0}^{\perp})^{2}}\Big{)}\\\
+2u\Big{(}\frac{1}{2}t_{\rm
vert}\sqrt{(d~{}C_{yy}+\theta_{0}^{\parallel})^{2}+(d~{}C_{xy}-\theta_{0}^{\perp})^{2}}\Big{)}\\\
+4u\Big{(}\frac{1}{2}t_{\rm
horiz}(d~{}C_{xy}-\theta_{0}^{\perp})\Big{)}+2u_{0}.$ (28)

where the attractive potential is, again,
$u(r)=-\frac{1}{8}u_{0}\Big{[}1+\cos\Big{(}{\pi r}/{r_{a}}\Big{)}\Big{]}$ when
$r<r_{a}$. Summing over all wedges (per unit area $d^{2}$) and assuming small
deformations, $u(r)\approx-\frac{1}{4}u_{0}+\pi^{2}u_{0}r^{2}/(16r_{a}^{2})$
and the bending energy takes the (harmonic) form of Eq. (2) with the moduli
given in Eq. (12). Thus, the SWM design has anisotropic bending constants,
with $B_{\perp}=2B_{\parallel}$ arising from the configuration with two
attractive sites displaced under bending and all four attractive sites
displaced under twisting deformation.

The NR theory elastic cost of the flattened state $E^{\infty}$ is found by
optimizing $E$ under the constraint that the curvatures describe a cylindrical
surface with zero Gaussian curvature, $C_{xx}C_{yy}-C_{xy}^{2}=0$. Thus,
monomers arrange with preferred spacing $d$ and there is zero stretching
energy so $E=E_{\mathrm{bend}}$ (minimized over subject $K_{G}=0$).
Minimization $E_{\mathrm{bend}}$ subject to $K_{G}=0$ leads to two degenerate
solutions, shown Fig. 11 for $\phi_{0}=22.5^{\circ}$ and $45^{\circ}$. In the
harmonic approximation described by equation 2, one finds that both the
optimal shape and energy depend on both $\phi_{0}$ and the twist-bend
anisotropy. When $B_{\parallel}=B_{\perp}$ (as studied in the case of Ref.
[17]), one of two equal-energy shape solutions is
$C_{xy}/\kappa_{0}=\frac{1}{2}\sin(2\phi_{0})$,
$C_{xx}/\kappa_{0}=\frac{1}{2}(\cos(2\phi_{0})-1)$, and
$C_{yy}/\kappa_{0}=-\frac{1}{2}(\cos(2\phi_{0})+1)$, giving energy
$E_{\infty}/A=\frac{1}{2}B_{\parallel}\kappa_{0}^{2}$ that is independent of
$\phi_{0}$.

For the present case of SWM assembly, were $2B_{\parallel}=B_{\perp}$, one of
the two shape solutions is

$\displaystyle C_{xy}/\kappa_{0}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sin(2\phi_{0})$ $\displaystyle C_{xx}/\kappa_{0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\cos(2\phi_{0})-\frac{1}{2}{\sqrt{\cos^{2}(2\phi_{0})+4C_{xy}^{2}}}$
$\displaystyle C_{yy}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\cos(2\phi_{0})-\frac{1}{2}{\sqrt{\cos^{2}(2\phi_{0})+4C_{xy}^{2}}}$
(29)

corresponding to (Hookean elastic) continuum result for flattening energy,

$\displaystyle\frac{E_{\infty}}{A}=\frac{1}{2}B_{\parallel}\kappa_{0}^{2}(1+\frac{1}{3}\sin^{2}(2\phi_{0})),(\textrm{when
}B_{\perp}=2B_{\parallel}),$ (30)

and more generally,

$\displaystyle\frac{E_{\infty}}{A}=\frac{1}{2}B_{\parallel}\kappa_{0}^{2}\Big{[}1+\frac{B_{\perp}-B_{\parallel}}{B_{\perp}+B_{\parallel}}\sin^{2}(2\phi_{0})\Big{]}.$
(31)

This is the expression plotted in Fig. 5b for the harmonic limit
$\theta_{0}/r_{a}\rightarrow 0$. To get the corrected result accounting for
non-linear strain softening of the potential, also plotted in Fig. 5b, eq.
(28) was numerically optimized over $C_{xy},C_{xx}$ with
$C_{yy}=C_{xy}^{2}/C_{xx}$ to produce the curves with nonlinear bending energy
and nonzero $\theta_{0}/r_{a}$.

The full NR elastic energy for the SWM model is the sum of the narrow-ribbon
limit $E_{\mathrm{strain}}$ and the harmonic approximation of
$E_{\mathrm{bend}}$ evaluated with constant curvatures everywhere:

$\frac{E}{wL}=\frac{Y}{1440}(C_{xx}C_{yy}-C_{xy}^{2})^{2}w^{4}\\\
+\frac{1}{2}B_{\parallel}\Big{[}(C_{xx}-\kappa_{0}\cos(2\phi_{0}))^{2}+(C_{yy}+\kappa_{0}\cos(2\phi_{0}))^{2}\Big{]}\\\
+B_{\perp}(C_{xy}-\kappa_{0}\sin(2\phi_{0}))^{2},$ (32)

which is then optimized over the curvatures $C_{ij}$ for varying $w$.

When $B_{\parallel}=B_{\perp}$, the branches of the equilbriumsolution are
characterized by a length
$w_{*}=\Big{[}2880{B_{\parallel}}/({Y\kappa_{0}^{2}})\Big{]}^{1/4}$ associated
with the transition from unflattened shape at small $w$ with zero mean
curvature to one of two flattened branches of degenerate energy with
analytical forms given in Ref. 43 up to a constant arising from different
approximations for stretching energy. One can verify that the energy in both
narrow and wide branches is independent of $\phi_{0}$. In general, we find

$\displaystyle
w_{*}=\Big{[}\frac{2880B_{\parallel}(B_{\perp}+B_{\parallel})^{2}}{Y\kappa_{0}^{2}((B_{\perp}+B_{\parallel})^{2}+(B_{\perp}-B_{\parallel})(3B_{\perp}+B_{\parallel})\sin^{2}(2\phi_{0}))}\Big{]}^{1/4},$
(33)

and the specific presnet case of the SWM model with anisotropic curvature
moduli,

$\displaystyle
w_{*}=\Big{[}\frac{2880B_{\parallel}}{Y\kappa_{0}^{2}(1+\frac{7}{9}\sin^{2}(2\phi_{0}))}\Big{]}^{1/4},\
\ \ \ \textrm{for }B_{\perp}=2B_{\parallel}$ (34)

so that increasing $\phi_{0}$, i.e. increasing preferred twist curvature,
results in the flattening at a smaller characteristic width. The energy of the
wide branch has the form,

$\displaystyle\frac{E_{\rm elastic}(w\geq
w_{*})}{A}=\frac{E_{\infty}}{A}-\frac{360B_{\parallel}^{2}}{Yw^{4}},\ \ \ \
\textrm{for }B_{\perp}=2B_{\parallel}.$ (35)

Figure 11:  Two flattened shape minima are predicted in the wide branch of
ribbon minima. Theory prediction and numerical results are shown for the case
$\theta_{0}=5^{\circ}$ for centerline helical pitch $P$ and radius $R$ in (a)
and (d), and centerline curvatures $C_{xx},C_{yy}$ in (b) and (e) both pre-
and post-flattening extracted from simulated SWM ground states. Dashed curves
denote secondary branches that are of equal energy in the approximate theory.
For the case $\phi_{0}=22.5^{\circ}$ shown in (a-c), corrections to the theory
predict in Ref. 52 predict that the dashed, smaller-pitch branch will be of
lower energy. Numerical results (points) consistenly land in the higher-pitch
shape solution. For the case $\phi_{0}=45^{\circ}$, shown in (d-f), the two
flattened shapes are of equal energy, pitch and handedness but related by
rotation about the helical axis $R\to-R$.

The wide branch approaches the harmonic approximation to the flattening energy
$E_{\infty}$ that itself depends on $\phi_{0}$. The decay of the residual
energy $\propto w^{-4}$ has the consequence that there is no surface energy
$\gamma$ such that the model energy density $U/A=E(w)/A+2\gamma/w$ has a
minimum in the wide branch. Because of this, $w_{*}$ is strict upper bound for
the optimal size $w_{0}$ that can be obtained from this theory, and the
analytical expression for $w_{*}$ decreases with $\phi_{0}$ despite the
increasing flattening cost with $\phi_{0}$ for this model.

Local minima exist for any $w<w_{*}$ in the narrow branch, with a
discontinuity in the second derivative of $E(w)$ present at $w=w_{*}$. In the
narrow branch, the maximum self-limiting size $w_{\mathrm{max}}$ is taken to
be the minimum to $U/A=E(w)/A+2\gamma/w$ at which the local minimum has equal
energy to the bulk tube state. That is, following the framework in Ref. 7,
self-limiting size is achievable when the accumulant is increasing,
$\partial_{w}{\mathcal{A}}=\partial_{w}(w(E_{\infty}-E(w))/A)>0$. This is
found to always occur at a size smaller than $w_{*}$ in the model. The
solution $w_{\mathrm{max}}$ satisfying $\partial_{w}{\mathcal{A}}=0$ from
numerically solving for the narrow branch is plotted in Fig. 12.

In general, a more accurate description of shape solutions with varying
curvatures is expected to include a boundary layer that develops as
$w\to\infty$, so the residual energy of wide ribbons
$E_{\mathrm{boundary}}\sim 1/w$ allows for locally optimal sizes at any width.
The calculation at the end of this section, however, shows that the correction
in $w_{\mathrm{max}}$ is small for $\phi_{0}=0$. The boundary layer correction
may have an effect on the dependence of $w_{\rm max}$ with $\phi_{0}$ : even
for the isotropic case $B_{\perp}=B_{\parallel}$, the boundary layer length
was shown to scale depending on the tangential curvature of the flattened
ribbon $l_{\rm boundary}\sim\sqrt{\sqrt{B/Y}/C_{yy}}$ in Ref. 52, and as a
consequence $w_{*}$ increases with increasing $\phi_{0}$ supported by finite-
element numerical results in Ref. 46. The weak dependence of
$w_{\mathrm{max}}$ on $\phi_{0}$ in the numerics presented in this study may
be affected by the effect captured in this boundary layer scaling : increasing
$\phi_{0}$ decreases tangential curvature $C_{yy}$ along the boundary of the
flattened ribbon. Whereas the initial elastic energy growth $E\sim
Y\kappa_{0}^{4}w^{4}$ depends only on the intrinsic geometry, the extrinsic
geometry, $\phi_{0}$, can affect the energetics, and range of equilibrium size
control, via the mechanics at the boundary.

Figure 12:  (a) Theory predictions (curves) are compared to numerical results
(points) on the apparent critical width of the shape transition $w_{*}$. (b)
Sequences of numerically minimized structures are shown, where numerical
estimates of $w_{*}$ (vertical dashed line) are determined from the apparent
change in structure based on these visualizations. (c) The solid black curve
is the maximum size predicted by the continuum theory with shapes restricted
to uniform curvature, rescaled by material parameters,
$w_{\rm{max}}(B_{\parallel}/Y\kappa_{0}^{2})^{1/4}$. The dashed line is the
correction for $\phi_{0}=0$ from allowing curvature to vary along the ring
width. The characteristic flattening size $w_{*}$ is an upper bound to the
maximum size, plotted as a dashed red curve. Open circles are $w_{*}$ results
from finite element numerics taken from Ref. 17. (d) The corresponding curves
for the theory when $B_{\parallel}=B_{\perp}$ have no dependence on $\phi_{0}$
as a consequence of the isotropic flattening cost. Numerical results from Ref.
17 are plotted as open circles to show the magnitude of the affect that the
authors attribute to the role of the boundary layer, supported by scaling
arguments in Ref. 52. (e) Curves of constant $\phi$ are plotted for the
equilibrium size $w_{0}$ as a function of line tension $\gamma$. The curves
are plotted as solid up until $w_{\rm{max}}$, when the minimum at finite $w$
becomes unstable to the bulk tube energy $E_{\infty}$. The dashed curves track
metastable $w_{0}$ up until $w_{*}$.

For the case of the $\phi_{0}=0$ ring assembly we derive the boundary layer
corrected solution starting from harmonic elastic energy described by
equations 23 and 2, limited to axisymmetric shapes, i.e. translation along the
$y$ coordinate. Here, the optimal shape is allowed to adopt in-plane
displacement $u_{x}(x)$ transverse to the ring and out-of-plane displacement
$h(x)$ in the radial direction, expanding around a cylindrical shape with
principal curvature $C_{yy}=-\kappa_{0}$ with linearized solution for
$h(x)\kappa_{0}\ll 1$. One finds that

$\displaystyle C_{yy}$ $\displaystyle\simeq$
$\displaystyle-\kappa_{0}+\kappa_{0}^{2}h$ (36) $\displaystyle C_{xx}$
$\displaystyle\simeq$ $\displaystyle\partial_{x}^{2}h$ (37) $\displaystyle
C_{xy}$ $\displaystyle=$ $\displaystyle 0$ (38) $\displaystyle u_{xx}$
$\displaystyle=$
$\displaystyle\partial_{x}u_{x}+\frac{1}{2}\Big{(}(\partial_{x}h)^{2}+(\partial_{x}u_{x})^{2}\Big{)}$
(39) $\displaystyle u_{xy}$ $\displaystyle=$ $\displaystyle 0$ (40)
$\displaystyle u_{yy}$ $\displaystyle=$
$\displaystyle\kappa_{0}h+\frac{1}{2}\kappa_{0}^{2}h^{2}$ (41)

and defining characteristic length $\lambda$:

$\displaystyle\lambda$ $\displaystyle\equiv$
$\displaystyle\Big{(}\frac{4B_{\parallel}}{Y\kappa_{0}^{2}}\Big{)}^{1/4}$ (43)
$\displaystyle E_{\infty}$ $\displaystyle=$
$\displaystyle\frac{1}{2}B_{\parallel}\kappa_{0}^{2}wL$ (44) $\displaystyle
E_{\rm elastic}$ $\displaystyle=$ $\displaystyle
E_{\infty}\Big{(}1+2(\lambda/w)\frac{\cos(w/\lambda)-\cosh(w/\lambda)}{\sin(w/\lambda)+\sinh(w/\lambda)}\Big{)}$
(45) $\displaystyle\frac{E}{wL}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{1440}Y\kappa_{0}^{4}w^{4},(\textrm{when }w\ll\lambda).$
(46)

Where the $w\to 0$ limit on the last line is in agreement with the result of
the uniform curvature approximation. Following the accumulant analysis of Ref.
7, the maximum possible self-limited size becomes

$\displaystyle w_{\mathrm{max}}=\pi\lambda.$ (47)

For $\phi_{0}=0$, $w_{*}=(720)^{1/4}\lambda\approx 5.18\lambda$, and
approximately $1.3\%$ greater than $w_{\mathrm{max}}$ from the uniform
curvature result, as shown by the comparison in Fig. 12. In this way,
corrections to the model results with the uniform curvature approximation are
expected to be small for the energetics up to $w_{\mathrm{max}}$. A similar
result for the case of $\phi_{0}=45^{\circ}$ was included in Ref. 53.