Lattice model for self-folding at the microscale
mmanuscript No. (will be inserted by the editor)
Lattice model for self-folding at the microscale
T. S. A. N. Simões a,1,2 , H. P. M. Melo b,1 , N. A. M. Araújo c,1,3 Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal Centro de Física das Universidades do Minho e do Porto, Campus de Gualtar, 4710-057 Braga, Portugal Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, PortugalReceived: date / Accepted: date
Abstract
Three-dimensional shell-like structures can be obtained spontaneously at the microscale from the self-folding of2D templates of rigid panels. At least for simple structures, the motion of each panel is consistent with a Brownian processand folding occurs through a sequence of binding events, where pairs of panels meet at a specific closing angle. Here, wepropose a lattice model to describe the dynamics of self-folding. As an example, we study the folding of a pyramid of N lateral faces. We combine analytical and numerical Monte Carlo simulations to find how the folding time depends on thenumber of faces, closing angle, and initial configuration. Implications for the study of more complex structures are discussed. Folding is considered a promising strategy to design shape-changing materials with fine tuned mechanical, electrical,and optical properties [1–4]. The idea is to act over a singledegree of freedom to promote a reversible change in shape,as in the case of the Miura-Ori fold [5–7]. The folding trajec-tories are usually deterministic and the final folded configu-ration unique. However, with technology putting pressure togo to smaller and smaller scales as, for example, in encapsu-lation and soft robotics [8–11], new challenges are posed. Atthe microscale, thermal fluctuations are no longer negligibleand so the folding process is stochastic. Experimental andnumerical results show that the final folded configuration isno longer unique, but rather dependent on the experimentalconditions [12–15].Thermal fluctuations have also advantages. They can drivethe folding process and, if the templates are properly de-signed, there is no need to control any degree of freedom.Recently, the spontaneous folding of a pyramid from a pla-nar template consisting of rigid micron-size panels connectedthrough flexible hinges has been studied using molecular dy-namics simulations [16]. The numerical results suggest that,when thermal fluctuations dominate, the motion of individ- a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] ual panels is well described by a Brownian process and fold-ing evolves through a sequence of binding events betweenpairs of faces, which correspond to first-passage processes.The first binding event occurs when the first pair of lat-eral faces meet at the closing angle φ . The characteristictime is the first binding time T F and can be estimated bymapping the problem into a two-dimensional first-passageprocess [16]. The remaining N − T L . The total fold-ing time T is then T = T F + T L .Due to the stochastic nature of the folding process, thetimescale of binding is typically much larger than the one ofthermal jiggling of the individual panels. Thus, to observethe complete folding, one needs very long molecular dynam-ics simulations. In order to access the folding timescales ef-ficiently, here, we propose a lattice model. We also considerthe folding of a pyramid with N lateral faces. We show thatthe dynamics of folding can be mapped into a set of ran-dom walks on a lattice, each one representing a lateral face,and the binding events occur when two random walks visit apredefined set of lattice sites simultaneously. We show thatour model is consistent with existing results and reproducesthe relation between folding time and number of faces N ,reported previously [16]. The proposed lattice model allows a r X i v : . [ c ond - m a t . s o f t ] F e b Fig. 1
Schematic representation of the folding of a pyramid of threelateral faces and closing angle φ , starting from a flat template, andthe corresponding dynamics on the lattice model. The horizontal axisrepresents the angle θ of each face at a certain time t , in the domain θ ∈ [ , π ] with reflective boundaries, and the vertical axis is the time t . The motion of each face is mapped into the movement of a ran-dom walker in a 1D. The first binding time T F is the time at whichthe first pair of walkers meet at the closing angle φ and bind together.The folding time T is when all walkers meet at the closing angle, thuscompleting the folding process. The last binding time T L is definedto be the time interval between first binding and the folding time, so T L = T − T F . The excluded volume interaction between walkers andwith boundaries is described by elastic collisions as depicted. The cor-responding configuration for four times ( t =
0, 0 < t < T F , t = T F and t = T ) is also shown. us to explore the folding dynamics for different structuresand initial conditions. The model is generic and can be gen-eralized to simulate the folding of more complex structures,by including new interaction zones and/or changing the in-teraction rules.The paper is organized as follows. In sect. 2, we intro-duce the lattice model and we define the first and last bind-ing times. In sect. 3 we present numerical and analytical re-sults for the total folding time. We draw some conclusionsin sect. 4. We map the spontaneous folding of a regular pyramid with N lateral faces into a set of random walks on a one-dimensionallattice. Each face i is a random walk and the angle θ i relativeto the base is given by the position on a 1D lattice of l sites.To study the folding of a regular pyramid without misfold-ing, we consider that the domain of the lattice ranges from θ ∈ [ , π ] , such that the site j corresponds to an angle in therange (cid:2) j π l , ( j + ) π l (cid:3) , see model scheme in Fig. 1.We consider a rejection-free kinetic Monte Carlo scheme[17–19], where, at each time step, a randomly selected walkertries to hop to one of its neighbors. Time is incremented by ∆ t = ∆θ D N F , where ∆ θ = π l is the lattice spacing, D is theangular diffusion coefficient of the lateral faces, and N F isthe number of walkers that are still free. For simplicity, weconsider an unbiased folding, and thus the hopping proba-bility is the same in both directions.The binding between any two lateral faces occurs whentwo random walkers first meet at a specific site correspond-ing to the closing angle φ . Additionally, when two faces arein the region beyond the closing angle ( θ > φ ) there is a ge-ometrical constraint that they can not overpass each other.To include this interaction, we assume that in the region θ ∈ ] φ , π ] , random walkers interact through excluded vol-ume: If a walker tries to hop into an already occupied site, itgoes back to the original site. For each number of random walkers N , we performed 10 independent samples, starting with all walkers at the ori-gin of the lattice, which correspond to the open templateshown in Fig. 1. Figure 2 shows the average folding time (cid:104) T (cid:105) , where we recover the non-monotonic dependence onthe number of lateral faces N , reported previously [16]. Thisbehavior stems from a different functional dependence on N of the mean first binding time (cid:104) T F (cid:105) and the mean last bindingtime (cid:104) T L (cid:105) , as discussed below.3.1 First binding timeWe first estimate the dependence of (cid:104) T F (cid:105) on N . For a pyra-mid with N lateral faces, the time that the first two faces bindcan be estimated in the following way. Let us define g ( t ) as the probability density function that a pair of Brownianparticles meet at an angle φ at time t for the first time. As-suming there is no correlation between the movement of the N p = (cid:0) N (cid:1) = N ( N − ) pairs, for each sample, T F = min ( t , t , t ... t N p ) ,where t i is a random number drawn from the distribution g ( t ) , for each pair i . From the theory of order statistics [20] Fig. 2 Dependence of the folding time on the number of lateralfaces N . (cid:104) T F (cid:105) , (cid:104) T L (cid:105) and (cid:104) T (cid:105) are the average first binding, last bindingand folding time, respectively. Time is in units of Brownian time, i.e. π D , where D is the diffusion coefficient of each face. We started froma planar condition, where θ i = i , and a fixed closingangle to φ = π . Results are averages over 10 independent sampleson a lattice of l =
181 sites and the error bars are given by the standarderror. the expected time for the first binding is, (cid:10) T F ( N p ) (cid:11) = N p (cid:90) ∞ tg ( t ) G ( t ) N p − dt , (1)where G ( t ) = (cid:82) ∞ t g ( t (cid:48) ) dt (cid:48) is the survival function. Using par-tial integration we obtain, (cid:10) T F ( N p ) (cid:11) = (cid:90) ∞ e N p ln G ( t ) dt . (2)Equation 2 shows that, since ln G ( t ) ≤
1, the first bindingtime T F is a monotonic decreasing function of N , as also ob-served with the lattice model (see Fig. 2). The value of (cid:104) T F (cid:105) for N → ∞ can be estimated for different geometries andinitial conditions [21–25]. For all faces starting at θ i ( ) = (cid:104) T F (cid:105) ∼ / ln ( N p ) [22,26], which is in agreement with the simulation results shownin Fig. 3(a).For the random initial condition, all walkers start at alattice site selected uniformly at random. Since there is achance of two faces to start close to φ , we can assume that G ( t ) decays rapidly with t , and for large values of N themost significant contribution for the integral in Eq. 2 is from t near zero. Therefore, using the approximation ln G ( t ) ≈ ln ( G )+ G (cid:48) G t , where G = G ( t = ) = G (cid:48) = dGdt ( t = ) ,the first binding time can be calculated as (cid:104) T F ( N ) (cid:105) ≈ − N ( N − ) G (cid:48) . (3)Since G (cid:48) <
0, Eq. 3 implies that the first binding timedecays with 1 / N ( N − ) in perfect agreement with the sim-ulation results shown in Fig. 3(a). Fig. 3 Dependence on the initial conditions.
We consider two ini-tial conditions: all faces start at the origin (planar) or uniformlydistributed (random). (a) First binding time as a function of N forthe two initial conditions. The solid and dashed lines are given by T F = ω P + ω P / ln ( N ( N − ) / ) where ω P = − . ± . ω P = . ± .
002 for the planar, and T F = ω R / N ( N − ) with ω R = . ± .
007 for the random, all obtained by least mean squarefitting. (b) Last binding time as a function of N . Asymptotically forboth initial conditions time scales logarithmically with N , as predictedby Eq. 5. The solid and dashed lines given by T L = τ P + τ P ln ( N − ) where τ P = . ± .
002 and τ P = . ± . T L = τ R + τ R ln ( N − ) where τ R = . ± .
002 and τ R = . ± . φ = π . Results are averages over 10 indepen-dent samples on a lattice of l =
181 sites and the error bars are givenby the standard error. φ , while the remaining N − (cid:104) T L (cid:105) , we calculate the average time for the binding ofthe last face. From a first-passage time probability f ( t ) , wehave T L = max ( t , t , t ... t N − ) , where the average time is given by (cid:104) T L ( N ) (cid:105) = ( N − ) (cid:90) ∞ t f ( t ) [ − F ( t )] ( N − ) − dt , (4)where F ( t ) = (cid:82) ∞ t f ( t (cid:48) ) dt (cid:48) . We assume an uniform distribu-tion of θ i < φ for all the N − t , the distribution offirst-passage times is well described by f ( t ) ≈ e − t / τ L , with τ L = φ / D π [27], and the average time for the last bindingis (cid:104) T L ( N ) (cid:105) = τ L N − ∑ i = i . (5)For large N , (cid:104) T L ( N ) (cid:105) ≈ τ L ln ( N − ) + γτ L , where γ is theEuler-Mascheroni constant. In Fig. 3(b) we show that thelast binding time grows with the logarithm of N as predictedby our analytical calculations for both initial conditions.For low values of N , the first binding time is large and,therefore, after the first binding the N − T L is the same for the planar and randominitial condition (see Fig. 3(b)). However, for large values of N , we have a fast first binding and the remaining N − θ i = T L is larger for theplanar than the random initial condition (see Fig. 3(b)).Since the average folding time is the sum between (cid:104) T F (cid:105) and (cid:104) T L (cid:105) , the different functional dependencies for these twoquantities explains the non-monotonic behavior of (cid:104) T (cid:105) shownin Fig. 3(c) for both initial conditions, in line with previousresults obtained using molecular dynamic simulations [16].3.3 Closing angleSince the overall dynamics of all faces is diffusive, we wouldexpect that the time scale that controls the binding process isthe Brownian time φ / D . Figure 4 shows simulation resultsfor the average first (a) and last (b) binding times as a func-tion of φ , for different values of the number of faces N . Thesolid and dashed lines are given by fitting the simulation datato a power law (cid:104) T F (cid:105) = A F φ m F and (cid:104) T L (cid:105) = A L φ m L , using leastmean square method, with fitting parameters ( A F , m F ) and ( A L , m L ) . However, the results show that the binding timetends to be proportional to φ only for large values of N . Inparticular, for N = N =
3, we see that there is a slightdeviation from the quadratic scaling. This deviation is dueto the fact that the first binding is a two-dimensional first-passage process where the binding site is located inside thedomain, and the time scale is controlled by the size of theboundary. However, for large values of N the path for thefirst binding will be close to a straight line [28], recoveringthe timescale φ / D as shown in Fig. 4(c). Fig. 4 Time dependence with the closing angle φ . (a) First bindingtime (cid:104) T F (cid:105) as a function of φ for N = { , , } . (b) Last binding time (cid:104) T L (cid:105) as a function of φ for N = { , } . In (a) and (b) the data was fit-ted by a power law (cid:104) T L / F (cid:105) ∼ φ m , represented by the dashed lines. Fromthe theory of first-passage processes, the time scale should be propor-tional to the square of the domain size, however since the adsorbingsite is not on the boundary of the lattice we can have deviations fromthat scaling, corresponding to m (cid:54) = .
0. We show in (c) that for both thefirst and last binding times, m converges to 2 . N , sincefor large N the first face to reach φ closely follows a straight path [28].All times are in units of Brownian time and φ ∈ (cid:2) π , π (cid:3) . Results areaverages over 10 independent samples with planar initial conditionson a lattice of l =
181 sites and the error bars are given by the standarderror.
We proposed a lattice model to simulate the spontaneousfolding of a pyramid at the microscale, driven by thermalfluctuations. We map the angular motion of each face into arandom walk on a lattice. The pairwise binding of faces cor-responds to having two walkers at a specific lattice site. Werecover a recent result obtained from Molecular Dynamicssimulations, namely, that the average folding time is a non-monotonic function of the number of lateral faces N . The folding dynamics involves two types of first-passage pro-cesses: The first binding, where the first two faces bind to-gether; and the last binding, where the last face binds. Weshow that the first binding corresponds to a two-dimensionalfirst-passage process, where two random walkers have tobind for the first time at a specific lattice site. After that,the remaining free walkers bind through a sequence of one-dimensional first-passage processes. The total folding timeis the sum of the first and last binding times.Describing folding as a sequence of binding events, wedemonstrated that the characteristic time of the first bind-ing has to decrease with N , while for the last binding it in-creases. It is the balance between these two processes thatleads to a non-monotonic dependence on the number of faces.We show that this non-monotonic dependence is robust tochanges on the initial conditions or value of the closing an-gle. Although the initial condition could affect both first andlast binding times, it only affects significantly the first bind-ing time and only for large N . For a small number of lateralfaces the binding times slightly deviates from a quadraticdependence on φ , however for large N the quadratic depen-dence is recovered, as expected for a diffusive process.The lattice model allows to evaluate not only the fold-ing time, but also the yield of more complex structures byextending the domain and including new closing angles. Forexample, an additional closing angle at − φ corresponds tothe possibility of folding in two different sides of the baseof the pyramid, with possible misfolding for a pyramid with N >
3, where the final folded configuration combines facesat φ and − φ . Analytical techniques developed for first-passageprocess with multiple adsorbing boundaries can be used tocalculate the yield as a function of N . Acknowledgements
We acknowledge financial support from the Por-tuguese Foundation for Science and Technology (FCT) under Con-tracts No. PTDC/FIS-MAC/28146/2017 (LISBOA–01–0145–FEDER–028146), UIDB/00618/2020, and UIDP/00618/2020.
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