Nematic polymer networks (NPNs) are nematic elastomers within which the nematic director is enslaved to the elastic deformation. The elastic free energy of a NPN sheet of thickness has both stretching and bending components (the former scaling like , the latter scaling like ). NPN sheets bear a director field imprinted in them (usually, uniformly throughout their thickness); they can be activated by changing the nematic order (e.g., by illumination or heating). This paper illustrates an attempt to compute the bending energy of a NPN sheet and to show which role it can play in determining the activated shape. Our approach is approximate: the activated surface consists of flat sectors connected by ridges, where the unit normal jumps and the bending energy is concentrated. By increasing the number of ridges, we should get closer to the real situation, where the activated surface is smooth and the bending energy is distributed on it. The method is applied to a disk with imprinted a spiraling planar hedgehog. It is shown that upon activation the disk, like a tiny hand, is able to grab a rigid lamina.
REFERENCES
1.
T.-S.
Nguyen
and J.
Selinger
, “Theory of liquid crystal elastomers and polymer networks
,” Eur. Phys. J. E
40
, 76
(2017
). 2.
O.
Ozenda
, A. M.
Sonnet
, and E. G.
Virga
, “A blend of stretching and bending in nematic polymer networks
,” Soft Matter
16
, 8877
(2020
). 3.
Y.
Guo
, M.
Jiang
, C.
Peng
, K.
Sun
, O.
Yaroshchuk
, O.
Lavrentovich
, and Q.-H.
Wei
, “High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals
,” Adv. Mater.
28
, 2353
(2016
). 4.
G.
Babakhanova
, T.
Turiv
, Y.
Guo
, M.
Hendrikx
, Q.-H.
Wei
, A. P. H. J.
Schenning
, D. J.
Broer
, and O. D.
Lavrentovich
, “Liquid crystal elastomer coatings with programmed response of surface profile
,” Nat. Commun.
9
, 456
(2018
). 5.
C. D.
Modes
, K.
Bhattacharya
, and M.
Warner
, “Disclination-mediated thermo-optical response in nematic glass sheets
,” Phys. Rev. E
81
, 060701
(2010
). 6.
C. D.
Modes
, K.
Bhattacharya
, and M.
Warner
, “Gaussian curvature from flat elastica sheets
,” Proc. R. Soc. A
467
, 1121
(2011
). 7.
H.
Aharoni
, E.
Sharon
, and R.
Kupferman
, “Geometry of thin nematic elastomer sheets
,” Phys. Rev. Lett.
113
, 257801
(2014
). 8.
C.
Mostajeran
, “Curvature generation in nematic surfaces
,” Phys. Rev. E
91
, 062405
(2015
). 9.
C.
Mostajeran
, M.
Warner
, T. H.
Ware
, and T. J.
White
, “Encoding Gaussian curvature in glassy and elastomeric liquid crystal solids
,” Proc. R. Soc. A
472
, 20160112
(2016
). 10.
T. J.
White
and D. J.
Broer
, “Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers
,” Nat. Mater.
14
, 1087
(2015
). 11.
O.
Ozenda
and E. G.
Virga
, “On the Kirchhoff-Love hypothesis (revised and vindicated)
,” J. Elast.
143
, 359
(2021
). 12.
M.
Warner
and E. M.
Terentjev
, Liquid Crystal Elastomers, International Series of Monographs on Physics Vol. 120 (Oxford University Press, New York, 2003).13.
A.
Pedrini
and E. G.
Virga
, “Ridge energy for thin nematic polymer networks
,” Eur. Phys. J. E
44
, 7
(2021
). 14.
C. D.
Modes
and M.
Warner
, “Blueprinting nematic glass: Systematically constructing and combining active points of curvature for emergent morphology
,” Phys. Rev. E
84
, 021711
(2011
). 15.
C. D.
Modes
and M.
Warner
, “Negative Gaussian curvature from induced metric changes
,” Phys. Rev. E
92
, 010401
(2015
). 16.
C.
Mostajeran
, M.
Warner
, and C. D.
Modes
, “Frame, metric and geodesic evolution in shape-changing nematic shells
,” Soft Matter
13
, 8858
(2017
). 17.
C. L.
van Oosten
, K. D.
Harris
, C. W. M.
Bastiaansen
, and D. J.
Broer
, “Glassy photomechanical liquid-crystal network actuators for microscale devices
,” Eur. Phys. J. E
23
, 329
(2007
). 18.
C. D.
Modes
and M.
Warner
, “Responsive nematic solid shells: Topology, compatibility, and shape
,” Europhys. Lett.
97
, 36007
(2012
). 19.
J. J.
Stoker
, Differential Geometry, Pure and Applied Mathematics (Wiley-Interscience, New York, 1969).20.
P.
Plucinsky
, M.
Lemm
, and K.
Bhattacharya
, “Programming complex shapes in thin nematic elastomer and glass sheets
,” Phys. Rev. E
94
, 010701
(2016
). 21.
B. A.
Kowalski
, C.
Mostajeran
, N. P.
Godman
, M.
Warner
, and T. J.
White
, “Curvature by design and on demand in liquid crystal elastomers
,” Phys. Rev. E
97
, 012504
(2018
). 22.
M.
Warner
and C.
Mostajeran
, “Nematic director fields and topographies of solid shells of revolution
,” Proc. R. Soc. A
474
, 20170566
(2018
). 23.
H.
Aharoni
, Y.
Xia
, X.
Zhang
, R. D.
Kamien
, and S.
Yang
, “Universal inverse design of surfaces with thin nematic elastomer sheets
,” Proc. Natl. Acad. Sci. U.S.A.
115
, 7206
(2018
). 24.
P.
Plucinsky
, M.
Lemm
, and K.
Bhattacharya
, “Actuation of thin nematic elastomer sheets with controlled heterogeneity
,” Arch. Ration. Mech. Anal.
227
, 149
(2018
). 25.
L. T.
de Haan
, C.
Sánchez-Somolinos
, C. M. W.
Bastiaansen
, A. P. H.
J. Schenning
, and D. J.
Broer
, “Engineering of complex order and the macroscopic deformation of liquid crystal polymer networks
,” Angew. Chem. Int. Ed.
51
, 12469
(2012
). 26.
T. H.
Ware
, Z. P.
Perry
, C. M.
Middleton
, S. T.
Iacono
, and T. J.
White
, “Programmable liquid crystal elastomers prepared by thiolene photopolymerization
,” ACS Macro. Lett.
4
, 942
(2015
). 27.
K.
Momma
and F.
Izumi
, “VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data
,” J. Appl. Cryst.
44
, 1272
(2011
). 28.
M. E.
Gurtin
, E.
Fried
, and L.
Anand
, The Mechanics and Thermodynamics of Continua
(Cambridge University Press
, Cambridge
, 2010
).29.
D. P.
Kingma
and J.
Ba
, “Adam: A method for stochastic optimization,” in 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7–9, 2015, Conference Track Proceedings, edited by Y. Bengio and Y. LeCun (ICLR, 2015).30.
M.
Abadi
, P.
Barham
, J.
Chen
, Z.
Chen
, A.
Davis
, J.
Dean
, M.
Devin
, S.
Ghemawat
, G.
Irving
, M.
Isard
, M.
Kudlur
, J.
Levenberg
, R.
Monga
, S.
Moore
, D. G.
Murray
, B.
Steiner
, P. A.
Tucker
, V.
Vasudevan
, P.
Warden
, M.
Wicke
, Y.
Yu
, and X.
Zhang
, “Tensorflow: A system for large-scale machine learning,” CoRR abs/1605.08695 (2016
).31.
J.
Langer
and D. A.
Singer
, “The total squared curvature of closed curves
,” J. Differ. Geom.
20
, 1
(1984
). 32.
J.
Langer
and D. A.
Singer
, “Curve-straightening in Riemannian manifolds
,” Ann. Global Anal. Geom.
5
, 133
(1987
). 33.
D. A.
Singer
, see http://www.cwru.edu/artsci/math/singer/publish/elaslecs.pdf for “Lectures on Elastic Curves and Rods” (2007).34.
A.
Pedrini
, M.
Piastra
, and E. G.
Virga
, “Non-monotonic, lily-like twist distribution in toroidal nematics
,” Soft Matter
15
, 633
(2019
). 35.
Thus, as also pointed out in Ref. 15, the problem of minimizing the bending energy of a conical NPN sheet ultimately reduces to the problem of minimizing the elastic energy of the elastica on a sphere. This problem has been extensively studied in great generality;31–33 an analytic solution can be given for the optimal geodesic curvature in terms of Jacobi elliptic functions. Here, we prefer to phrase it directly in terms of a parameterization of the optimal curve and to find for it a numerical solution, to be compared with that found in Sec. IV.
36.
For example, director patterns are imposed onto a NPN sheet by the plasmonic photoalignment technique. After photopolymerization, the imprinted pattern remains parallel to the plane of the bounding surfaces with no escape out of that plane.
37.
The abuse is pardoned by thinking of the flat, undeformed configuration as being endowed with the prescribed metric, which is kept by the immersion in a three-dimensional space.
38.
Unfortunately, we do not yet possess a formal convergence result; we shall be contented with considering a sufficiently large number of ridges.
39.
Hedgehog is the name usually given to a point defect in three-dimensional space; in our two-dimensional setting, a planar hedgehog is what is also called a disclination.
40.
Here, fe is scaled to an elastic modulus having physical dimensions of an energy per unit volume, see Ref. 13.
41.
The reader may observe that for λ1 = λ2 all solutions to (14) are straight line segments.
42.
This heuristic reasoning will be made formal shortly below.
43.
It might be noted that, since c1 = 0 for α = 0, the former case corresponds to a rare occasion when (15) represents indeed a straight line segment.
44.
A tiny gap must be present between the rigid lamina and the elastic disk to allow the latter to glide freely in the (x1, x2) plane.
45.
By symmetry, indeed all inner circles of S are mapped into deflated images of the rim of .
46.
This formally retraces the heuristic argument given above.
47.
Here, we take both λ1 and λ2 in (8) as constants but not necessarily such that λ1λ2 = 1. The latter constraint can easily be enforced whenever needed.
49.
Angles γj's play here a role similar to that played in Ref. 13 by the single angle β, which designated the inclination on the (x1, x2) plane of all j's, in the very special case where the number N of ridges is also the number of macroscopic folds of .
50.
In the actual process, the initialization was slightly changed to avoid any initial angle to be zero.
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