Design and Simulation of Novel Magnetic Tiles

Design and Simulation of Novel Magnetic Tiles Capable of Complex Self-Assemblies Urmi Majumder1 and John H Reif1 Department of Computer Science, Duke University [email protected], [email protected] Summary. As manufacturing and robotics reduces in size scale, there is an increasing need to make use of bottom-up self-assembly techniques rather than conventional top-down assembly techniques. This approach has been very successful at the molecular scale, resulting in complex patterning at that scale not previously achievable via conventional methods. However, the use of self-assembly techniques at a size range- between 100’s nm to 100’s of µm has so far been limited to the formation of patterns of quite limited complexity. For example, in a number of prior works self-assembling units or tiles had their sides fitted with magnets which we will call magnetic pads that allowed (or selectively disallowed) for the binding between distinct tiles to form assemblies of tiles. The complexity of the resulting assemblies was limited due to the very limited variety of magnetic pads that were currently used: namely just positive or negative polarity. This paper address the key challenge of increasing the variety of magnetic pads for tiles, which will allow the tiles to self-assemble into patterns of increased complexity governed by the selection of the pads of the tiles. In particular, we describe a family of barcode schemes that provide a much larger variety of magnetic pads. The introduction of these barcode schemes potentially allows for the generation of arbitrary complex structures using magnetic selfassembly at the meso-scale. The design and optimization of our barcode schemes for magnetic pads is the main concern and focus of this paper. We present a physical model based on Newtonian mechanics and Maxwellian magnetics and software simulation system that correctly models attachment, orientation and binding of macroscopic tiles using magnetic interaction as well as external forces (wind) which provide energy to the system. In particular, the simulation system allows us to determine the optimal parameters for the design and arrangement of the magnetic pads that selectively bind (or do not bind) with other tiles. We also demonstrate how we can use our simulation model to extract a thermodynamic model which can be used to predict yield of assembly on a larger scale and to provide better insight into the dynamics of our physical system.

1 Introduction

Self-assembly is the autonomous organization of components into structures without human intervention. It is a phenomenon that is prevalent on all scales, from molecules to galaxies. Though self-assembly is a bottom-up process not utilizing an overall central control, it is theoretically capable of constructing arbitrary complex objects, and nature uses it to assemble what may be the most complex thing in the universe: namely humans.

2

Urmi Majumder and John H Reif

One of most well-studied sub-fields of self-assembly is molecular self-assembly. However, many interesting applications of self-assembling processes can be found at a larger scale, such as meso-scale (submillimeter range- between 100’s nm to 100’s of µm). Massively parallel self-assembling systems present a promising alternative to conventional manufacturing (using by in large, sequential pick and place assembly). Examples of self-assembling systems at the meso-scale relevant to robotics and manufacturing include self-assembled monolayers, the patterned assembly of electronic components and MEMS devices, the assembly and arrangements of micro-robotics and/or sensors. In this paper we explore magnetic self assembly with the ultimate goal of discovering the practical limits for its use in the manufacturing and computing systems. Most of the related work described below in section 2 is focused on the demonstration of meso-scale self-assembly. However, here we emphasize more on the design issues relevant to the generation of more complex structures using our novel barcode scheme. Our testbed is an example of distributed system where a large number of relatively simple parts interact locally to produce interesting global behavior. Square “programmable tiles” float passively on a forced air-table, mixed randomly by oscillating fans. The tiles have magnetic encoding on each of their faces. So when they collide, if the facing poles are exactly complementary, the tile faces bind to each other and this process repeats to generate our desired final structure. We describe how a rigid body simulation environment can be used to model the testbed. We discuss techniques for minimization of undesired intra-tile and intertiles magnetic interaction using centered encoding, optimization of the ratio of tile size to dipole, long thin magnets, pairwise alternate arrangement of magnets and magnetic shielding. Combining the above techniques, we demonstrate the feasibility of using self-assembly of magnetic tiles (with barcoded pads) for the generation of modest size patterned lattices. We also present a kinetic/thermodynamic model based on our simulation of the physical system which provides us with the likelihood of a match and the rates of association and dissociation of tiles during the assembly while abiding by some pre-defined kinodynamic constraints.

2 Previous Work Meso-scale Self-Assemblies Recent relevant works in the field of meso-scale self-assembly include development of systems based on capillary interactions among millimeter-scale components either floating at a fluid-fluid interface or suspended in an approximately iso-dense fluid medium [1, 2, 3, 4, 5, 6, 7, 8, 9]. Of particular interest is the fabrication of 3D structures in the micron level [10, 11, 12]. In fact self-assembly based on capillary forces has been used for the fabrication of micro-electromechanical systems and microoptoelectromechanical systems [13, 9, 8, 14, 15, 16]. Some other micro-scale fluidic self-assemblies which deserves special mention are [17, 18, 19, 20, 21, 22, 23]. In fact, there has been commercial applications of fluidic assembly [24, 15]. Electrical conductivity has also been exploited to build self-assembling micro-electronic systems [25, 26, 27, 28, 24, 29, 30, 31, 32, 33, 34].Rothemund[35] demonstrated the use

Title Suppressed Due to Excessive Length

3

of hydrophobic and hydrophilic interactions to generate self-assemblies of moderate complexity and scale. His work was notable since it was only work at the meso-scale which involved some computational process during the self-assembly process.

Fig. 1. (Left) An example of Magnetic Assembly: Triangles, made of polydimethylsiloxane with magnetized strontium ferrite along the edges, (Right)Examples of Computational DNA Self-Assembly:Sierpinski Triangle, Binary Counter

Magnetic and Computational Assemblies Magnetic assembly[36, 37, 38][Figure 1(left)] is a form of meso-scale self-assembly that will be the central focus of this paper. Of particular interest is the spontaneous folding of at elastomeric sheets patterned with magnetic dipoles, into free standing, 3D spherical shells. The path of this self assembly is determined by a competition between mechanical and magnetic interactions[39]. The meso-scale self-assembling systems that have been assembled by magnetic assembly in this way (without control) have been limited to the generation of rather simple structures. This limited complexity was mostly due to the limited nature of the magnetic interactions. Here we address the key challenge of going beyond such limitations and aim at the design of more complex structures via magnetic assembly. To increase the complexity of magnetic assemblies, Erik Klavins group developed programmable units to ow passively on an air table and bind to each other upon random collisions. These programmable units have on-board processors that can change the magnetic properties of the units dynamically during the assembly of the units together. Once attached, they execute local rules that determine how their internal states change and whether they should remain bound using the theory of graph grammars [40]. However, we would like to execute complex magnetic assemblies without the use of on-board processors that can change the magnetic properties of the unit Computational Molecular Self-Assemblies Theoretical research in the 1960s proved that universal computation can be done via tiling assemblies[41]. This essentially proved that arbitrary complex structures can be generated by tiling assemblies. However, there was a very long gap in time before these theoretical ideas were put to practice. In 1982, Seeman[42] demonstrated that DNA nano-structures can be self-assembled by using Waston-Crick complementarity and thus it can form the basis of programmable nano-fabrication. A seminal paper by Adleman[43] in 1994 used onedimensional DNA self assembly to solve a computational problem (the Hamiltonian path problem), establishing the first experimental connection between DNA self assembly and computation. This work inspired a theoretical proposal by Winfree[44]

4


that two dimensional DNA self assembly is capable of performing Turing Universal computation, based on the prior work on Wang tiling, In collaboration with Seeman, we[45] demonstrated the first computation of Boolean vectors at the molecular scale. Our group gave the first experimental demonstration of patterned 2D barcode assemblies of DNA lattices in the paper[46]. Recent experimental demonstrations of 2D computational patterning via DNA tile self-assemblies include the Sierpinski triangle patterns by Winfree et al[47] in 2004 and counting tiling assemblies by our group (2006, to appear)[Figure 1(right)]. The goal of this paper is to develop techniques that will allow self-assembly of complex structures at the meso-scale. This task is quite challenging, since the available binding mechanisms (using magnetic and capillary) currently used at the meso-scale essentially provide only for binary binding (e.g., positive and negative in the case of magnetic binding and hydrophobic/hydrophilic interactions in the case of capillary binding). In contrast, DNA provides a large number of specific bindings by the use of complementary pairs of DNA sequence that can hybridize selectively to each other. Here, we will mimic the techniques and principles of molecular self-assembly (as described above) to build complex structures at the macroscopic level. The key challenge then for us to do self-assembly of complex structures at the meso-scale is to extend the binding mechanisms to a much larger number of specific bindings, rather than just two. We will achieve this by use of the magnetic barcode techniques described in this paper.

3 Design of a Magnetic Self Assembling System Self assembly at the meso-scale can happen through a wide range of forces viz. gravitational [48], electrostatic [29], centrifugal[49], osmotic[50], magnetic [38], capillary [2], entropic [51], light[52, 53], fluid shear [24], hydrodynamic[38], casimir [54], hydrophobic[55], Vander Waals[56], biospecific[57], etc. However, the choice of the type of interactions between components in a self-assembling system depends on several factors like scale and magnitude of the interactions, compatibility of the interactions with the environment, influence of the interactions on the function of the system. We have chosen magnetic force as the driving force for self assembly mainly because magnetic interactions are insensitive to the surrounding medium and are independent of surface chemistry. Also the distances over which the magnetic forces can act can be engineered so as to manipulate the long-ranged and short-ranged interactions among the components. This is important because a key issue in the design of programmable self-assembly is the recognition between components, governed by the design, surface chemistry and topology of the interacting surfaces. 3.1 The TestBed The main component of our self-assembling system is a set of square wooden tiles. Each edge of a tile is lined with a sequence of magnetic dipoles. The latter is perpendicular to each face and can either have their north pole or south pole facing out. A typical encoding on a tile face may be {NNNS} where N or S denotes whether the north pole or the south pole of the dipole is facing out of the tile. The tiles float on a custom made air-table. A set of fans are placed around the airtable and mix the tiles. Thus all the interactions between the tiles are due to chance collisions. The idea is


NSN S N S N S SNS N NSN S S S SNS NSN NSN N N S N N S N NSS N S SNS

N S NSN N N S N N S N SNS NSN N S S N S S N S SNS S

5

Tile A a

a

Break Contact

F

F Tile B

Slide Left

Slide Right

a

a

Stick

Fig. 2. A Typical Magnetic Assembly(left), Friction Model(right)

that if a tile face(say with encoding {NNNS}) collides with a tile face with matching encoding({SSSN}), they stick together, thus resulting in an assembly (figure 2(left)) Air Table The airtable provides a two dimensional fluid environment for the tiles. As tiles traverse the testbed, they lose kinetic energy due to friction and variations in their air cushion. Thus in our model, we assume that our airtable surface has a very low coefficient of friction. The Fan Model We have a Gaussian potential model for the fans. The potential energy E f is a function of the distance x from a tile to a fan It follows a Gaussian distribu2 tion and takes the form of Ef = e−x . Hence magnitude of the fan force Ff is 2 d Ef (x)| = | − 2xe−x | . A more realistic fan model based on Navier |Ff | = | dx Stokes equation is still under investigation. Tile Motion Model On the testbed, a tile’s motion is described in terms of its two dimensional linear 2 2 2 acceleration ( ddt2x , ddt2y ) and one dimensional angular acceleration ( ddt2θ ):  2     dx  d x − → −→ −µ 0 0 dt2 dt → − → − −→  d2 y    + Ff (x, y) + Fm (x, y) + τ ( r , Fm ) 0 −µ 0   dy  dt2  = dt m m I dθ d2 θ 0 0 −µ dt dt2

where m is the mass of the tile, I is the moment of inertia about the axis passing −→ through its centroid perpendicular to the bed, µ is the coefficient of friction, Fm is −→ − → the magnetic dipole force, τm is the torque exerted on the tile by the force Fm acting − at the magnetic point → r relative to the tile’s center of mass. Since for simplicity, we apply the fan force to the center of the tile so the torque due to the fan force is zero. Tiles also receive impulse forces and torques when they bump into each other or the sides of the airtable. The force and torque imparted during these events conserve linear and angular momentum but not kinetic energy, because the collisions are partially inelastic.

6


Handling Collision With an elastic collision, two colliding tiles, even with matching tile-faces will not be able to bind to each other. In case it is an inelastic collision, however, the short ranged magnetic forces can decide whether the tiles should stick together or not. So we have to guarantee that the coefficient of restitution should be small which can be altered by manipulating tile shape, mass and dimensions and by controlling impact velocity which depends on the arrangement of fans and their speeds. Friction Model Not only gliding on the air-table surface but also interactions between tile-faces incur frictional forces on a tile. An important observation here is that the coefficient of friction has to be low so that tiles do not bind to each other in case of a random collision irrespective of their magnetic encoding. We can reduce friction by lubricating the tile surface with oil or water. We can also use teflon instead of wood to make our tiles because teflon has a very low coefficient of friction and thus the tiles will be able to slide past each other easily. Our friction model is essentially an approximation of Coulomb’s friction model (Figure 2(right)). The parameter α = tan−1 µ captures the amount of friction by defining the friction cone. The tiles are stuck if they fall within the friction cone and slide if they are on the surface of the cone.

N

S

N S

S

N

S

N

S N

N S

S

S N

S

N

Long Thin Magnet

face 1 S

N

N

S

S

N

N

S

face 1

Pairwise Alternate Encoding

face 4

Tile B

face2 face 4

Tile A

face 2

S

S N

N

N

N S

S

N

N S

N

S

N

N

S

S

N

N

S

S

Complementary Shaped Markers

N

Centered Encoding

Use of Spacer Sequence

S

Tile(Large tile size to dipole size)

Barcode Scheme Iron Shield

face 3

face 3

Fig. 3. (Left)Techniques for minimizing magnetic crosstalk, (Right)Tile Face Convention

3.2 Magnetic Interaction Model Minimal Magnetic Crosstalk One of the main challenges in magnetic self-assembly is to arrange the magnets in such a way so that the likelihood of a correct match is maximized and that of an incorrect match is minimized. This can be achieved by insuring (i)Large Tile Size to Magnetic Dipole Size Ratio, (ii)Centered Encoding on each tile-face, (iii) Use of Spacer Sequences, (iv)Use of Long Thin Magnets, (v)Pairwise Alternate Encoding and (vi)Use of Magnetic Shields[Figure 3(left)]. Large Tile Size to Magnetic Dipole Size Ratio Magnetic interaction are quite local. Thus by increasing the tile size with respect to the bar magnet size, we can minimize the magnetic crosstalk among dipoles on different faces on the same tile. We also have an added benefit of decreased magnetic interaction between remote faces of interfacing tiles.


7

Centered Encoding on each Tile-Face Instead of arranging the dipoles along the entire length of a face, we concentrate the magnets around center of the face, thus obtaining similar intra-tile and inter-tile magnetic interaction minimization benefits as with the former technique. Centered encoding also leaves enough space along tile faces for carving out shape complementary markers, thus reducing partial matches to a large extent. Use of Spacer Sequences Spacer sequence is all about increasing the alphabet size from binary to a ternary one viz. {N, S, }. This increases the distance between the magnets on each tile face, thus decreasing the magnetic crosstalk on the same face and interfacing tiles. Spacer sequences also increase the code word library to O(3n ) where n is the encoding size. Pairwise Alternate Encoding For each tile face, this encoding technique chooses the outer polarity of first dipole from {N, S} at random. Thus the next dipole is the opposite of the first one and we repeat the process from n2 times(for simplicity assume length of the encoding n is even). Now we calculate the magnetic force between face 4 of a tile (say A) and face 2 of another tile (say B)(note that the faces are numbered in a clockwise direction starting from the top end[Figure 3](right)). Assume that the tiles directly face each other. Take the extreme left end dipole on face 1 of tile A and consider each successive pair of dipoles on the other tile. With monopole approximation(discussed later), 1 cλ2 an upper bound on the magnetic force between a pair is equal to c( d12 − d2 +λ 2 ) ∼ d4 consdiering λ d. So every successive pair of magnets cancel each other’s effect and hence for large d, the magnetic interaction is almost negligible. Even when d is not large, this technique is quite effective in reducing the overall impact on each dipole due to all the dipoles on the interfacing tile. Use of Long Thin Magnets The main benet of using long thin magnets is that geometry reduces the crosssectional area , so the force at the end of the magnets is quite concentrated, thus minimizing intra and inter-tile magnetic crosstalk. Also, if we approximate the bar magnet by a magnetic dipole, with long thin magnets, the effect of the distant pole is negligible on the pole that faces out of the tile. Use of Magnetic Shields Magnetic shielding, is a technique frequently used in electromagnetics to limit the coupling of electromagnetic fields between two locations. In our case, we cover most of the bar magnet with a ferromagnetic material like soft iron, which gets magnetized by the former and hence prevents coupling of flux lines of two neighboring bar magnets. Note that the pole of magnet which faces out of the tile is not covered, so that appropriate binding can take place at that end. Magnetic shielding is quite effective in minimizing undesired intra-tile and inter-tile interaction as will be demonstrated by our FEMM simulations shortly.

8


FEMM Simulations We used Finite Elements Method Magnetics[58] to verify the effect of all the techniques we described above for minimal magnetic interaction. This is because its capacity to represent a physical situation is much greater than the rigid body simulation. In Finite Element Method Magnetics from an input of a square shaped tile lined with n dipoles(orientation and the permeability of the dipoles are inputs too) on each face, the software can calculate and map the magnetic field intensity and the magnetic field lines. In the output of the simulation, the intensity of the magnetic field is depicted by color intensity. The strength of the interactions between the patterned tiles can be estimated from the strength of the magnetic field and the number of field lines surrounding the magnet.

Fig. 4. Effect of magnetic shielding(from left to right)(a)Comparison of a tile with shielded magnets with a tile with unshielded magnets, (b)Unshielded magnet(close view), (c)Shielded magnet(close view)

Fig. 5. Effect of magnetic shielding on magnetic binding(from left to right)(a)Correct binding of two tiles with polarity of the magnets facing out being the same), (b)Correct binding of two tiles with polarity of the magnets facing out being opposite), (c) Tile face interaction(close view), (d)Dipole-dipole interaction with shielding(close view)

Our FEMM simulations effectively demonstrate how by using the afore-mentioned techniques we can minimize undesired magnetic interaction. We conducted two-tile experiments with different magnetic encodings. The tiles were 80×50 cm 2 while the permanent magnets were 14 × 2cm2 and they were 13 cm apart. Thus the tile size to → − dipole size is 150 : 1. In an isolated environment, the flux density B is of the order → − of 10−1 T at the outer ends of the magnets while the value of B on a circle of radius 10 cm around the magnet is of the 10−2 T and 10−3 T beyond. In presence of another nearby tile(within 10 cm) and shielding(we used soft iron as shields), the intensity at the end of the magnets increases but is of the same order as before. Figure 4 and 5 effectively demonstrates the validity of our techniques in localizing magnetic field and minimizing magnetic crosstalk. Bar Magnet Model For the bar magnets, we present three models of increasing complexity and precision:(i)Monopole Approximation, (ii)Dipole Approximation and (iii)Magnetic model


9

based on Maxwell’s equations for macroscopic linear materials. In all the three models, since the magnets are glued to the tile surface so the magnetic force due to the dipoles on the same tile will have no effect on the magnetic dipole under consideration. For interfacing tiles, due to the optimization of the ratio of tile size to dipole size and shielding, only the magnets on the nearest face will have any effect on this dipole. Monopole Approximation In this model, we treat the magnets on the outer face of the tile as point magnets where forces are calculated between the points in the appropriate sense-like poles repel and unlike poles attract. The magnitude of the magnetic force between two poles as a function of distance d is given by |Fm (d)| = dc2 where c is the magnetic attraction constant. This approximation is valid if we use our techniques for minimizing magnetic interaction, in particular long thin bar magnets which makes the effect of the distant pole negligible. Magnetic Dipole Approximation Here we approximate our bar magnets as magnetic dipoles. We do not have any source of electric current in our system. So Maxwell’s equations for magnetostatics → − → → − → → − → − − − apply in this case viz. 5. B = 0 and 5 × B = 4π c J where B is the flux density of → − → − → − → − the magnet and J is its current density. If we define B = 5 × A , then for single → − geometries we can do Coulomb-like integrals for A and then a multipole expansion → − .ˆr)ˆr−→ − → − m at a distance → − r of it upto the dipole term , yielding flux density B = 3( m→ 3 − r → − due to a magnet with dipole moment m. From there we have the force on a dipole → − → − → − − in an external magnetic field as F = 5(→ m. B ). Thus we compute the total force F that the dipole experiences due to the magnetic flux density of another dipole at the center of mass of the first dipole. Precisely, suppose we are interested in computing → − the force experienced by a magnet M1 on tile T1 due to the magnetic field B 2 of a − magnet M2 on tile T2 located at a distance → r = xi + yj. Let the dipole moment of → − − a magnet M1 be m 1 = mx1 i + my1 j and that of M2 be → m 2 = mx2 i + my2 j. Then, − − → − 3(mx2 x2 +my2 xy)i+3(mx2 xy+my2 y 2 )j m i+m j 3(→ m r )ˆ r −→ m 2 .ˆ 2 B2 = = − x22 2 y23 and 5 → − r3 (x2 +y 2 ) 2 (x +y ) 2 → − → − − F = 5(→ m1 . B 2 ) = 5 3

(mx1 mx2 x2 + mx1 my2 xy) + (my1 mx2 xy + my1 my2 y 2 )

0

(x2 +

= @ − 15x +

5 y2 ) 2

+

(x2 +

7

(x2 + y 2 ) 2

5

+

(x2 + y 2 ) 2 0

(mx1 mx2 + my1 my2 )

(mx1 mx2 x2 + mx1 my2 xy) + (my1 mx2 xy + my1 my2 y 2 )

3(mx1 mx2 2x + mx1 my2 y) + 3my1 mx2 y

+@−

−

3x(mx1 mx2 + my1 my2 ) 5

(x2 + y 2 ) 2

15y(mx1 mx2 x2 + mx1 my2 xy) + (my1 mx2 xy + my1 my2 y 2 )

1

Ai

7

(x2 + y 2 ) 2

3mx1 my2 x + 3my1 mx2 x + 3my1 my2 2y 5

(x2 + y 2 ) 2

+

3y(mx1 mx2 + my1 my2 ) 5

(x2 + y 2 ) 2

1

Aj

3 y2 ) 2

!

10


We can compute the dipole moment of a bar magnet of length l and square crosssectional area with a = 4r 2 as follows. With long thin magnets, we can approximate the bar magnet with a cylindrical bar magnet which can be further approximated by a solenoid which is l units long, has N turns each of which is πr 2 sq units and i is the current in the coil. The magnetic field at the end of the coil is B0 = 2√µl02N+ri 2 and following the analogous calculation for an electric dipole, the magnetic dipole 2B0 al moment |M | = sqrtl 2 +r 2 and the direction is from the north pole to the south pole. In our case the |M | is same for all magnets and can be set to some pre-defined value. Model based on Macroscopic Maxwell’s equation A more correct model is the one based on the Maxwell’s equations with correction for macroscopic objects. Here too we consider only the equations for the magnetostatic → − → → − − → − → − → − case, viz. 5. B = 0 and 5 × H = 4π c J where H is the magnetic intensity of → − − → the bar magnet. For bar magnets J = 0 and magnetization density M is given 0 R 0 → − − 0 → − − → − → − → − − r −→ r0 ) + r )( → and fixed. Thus we can derive H as H (→ r ) = V d3 r (− 5.M )(→ − → − | r − r |3 0 R − − 0 − → → → − → − → − → − r −→ r da (m. ˆ M )( → ). Once we derive H , we obtain B as B = µ H where µ is 0 S − |− r −→ r |3 → − → − the permeability constant. We can derive force F as the gradient of B as discussed in the previous model. 3.3 Tile Programming By tile programming we essentially mean the code word design for each tile face. The goals for designing a ”good” library of words in the context of magnetic self assembly is that we need to maximize the energy difference between two completely matched tile faces and two partially matched or unmatched tile faces. So encoding scheme is a key factor determining the success of a correct assembly. One of the main features of this work is the barcode encoding scheme which will allow us to generate arbitrary complex structures. Figure 2(left) is an example of the application of our barcode scheme. With an encoding size of n we can generate a codeword library of size 2n−1 (Out of the 2n possibilities, half of them goes away for the complementary codes), which is significantly large for large n, thus providing us with considerable types of glues for generating arbitrary complex assemblies. Partial Matches Minimization A ”good” encoding scheme should minimize partial matches between two tile faces. For instance, if the length of the encoding is N, then the faces are mismatched at m positions where m < N . Mismatched location may not be consecutive. Ideally we would like to have such matches energetically more unstable than perfect matches and thus we generate a library of words abiding by these energy constraints. In general, the problem of partial matches can be handled by the use of prefix codes and Hamming distance maximization. The situation can be made much complicated by the presence of a slide match configuration. An example of a slide match configuration is NNNSSNSNSN matched with NNSNSNSNNS starting at the third location of the second sequence. One


11

technique for handling slide matches is to use shift-distinct codes like “NSSN” or “NNSS” Another technique would be to use complementary shapes for matching tiles along with our barcode to insure a higher yield of the assembled components. This brings us to the concluding part of our physical model based on Newtonian mechanics and Maxwellian magnetics. Evidently, the most determining factors for a successful assembly are: (i)The encoding scheme(number, orientation and arrangement of dipoles on each face), (ii)The total number of tiles, (iii)The arrangement of fans around the air table(number orientation and fan speed), (iv) Tile size to magnetic dipole size ratio, (v)Coefficient of friction, (vi)Coefficient of restitution in a collision.

4 Simulation Now that we have a physical model, we need to verify it either by simulation or with actual experiment. The latter calls for an elaborate patterning and suitable placement of fans with appropriate driving power along with fixing the values of several other parameters that we should take into account for a successful assembly. So it seems wiser to build a realistic simulation environment which would allow us to ”experiment” with all the parameters of the design. Our simulation uses the Open Dynamics Engine [59] library, which can routinely compute trajectories of all the tiles and determine the result of the collisions and contact situations.The simulation provides meaningful visual feedback, clearly indicating the distribution of tiles on the airtable and the current state of the assembly. 4.1 Simulation Model Since the actual assembly depends on so many factors, so for preliminary simulation we focused on a much simpler model viz a two-tile assembly model. Here, one tile has fixed position and the other tile has a random initial position. The goal is to find out the range of the magnetic force effective for tile binding in the absence of wind energy. Next we gave the second tile some initial velocity and computed the likelihood of a correct match, the rate of association and dissociation, given the kinodynamic(position, orientation and velocity) constraints on this tile. Note that by providing the initial velocity we are essentially simulating the Gaussian potential function of the wind energy source. It is important to note here that in our simulation, we call two tile faces connected if the corresponding matching dipoles are within some pre-defined distance. Also for estimating the likelihood of match in any simulation, we declare the tiles connected only when they remain connected until the end of the simulation. 4.2 Kinetic/Thermodynamic Model Besides serving as a novel tool for optimizing the parameters for an actual assembly experiments, our simulation model also provides us with a higher level kinetic/thermodynamic model. Kinetics in the context of physical chemistry is the study of reaction rates in a chemical reaction. Reaction rate, in general is dependent on concentration, physical state, temperature of the reactants and presence of catalysts. Molecular kinetics[60, 61], in particular kinetics of molecular self-assembly is quite well-understood[62]. The goal here is to extract a similar thermodynamic

12


Fig. 6. Two-tile assembly using magnetic interaction:(from left to right)(a)Random initial locations of the two tiles in the air-table (b)Due to the initial velocity the collide and move apart (c)+(d)+(e)Magnetic interaction and kinetic energy brings them closer gradually (e)Assembly takes place

model from our simulation results for the meso-scale self-assembly where the heat of reaction comes from the magnetic interaction of the tiles and the wind energy from the fans. Thus keeping the mass, dimensions, encoding scheme of the tiles constant we derive the probability of a successful assembly while following the kinodynamic constraints of relative inter-tile distance, velocity and orientation. A more important reason for a high level thermodynamic model is that the low level physical model based on Newtonian mechanics and Maxwellian magnetics, is difficult to scale and there are too many parameters to be optimized and thus the high level kinetic model extracted from the simple physical model will be very useful for analyzing the yields in larger assemblies and also understand the dynamics of the assembly better. For a two-tile assembly in two dimensions, we define the relative configuration space R3 as C where if r ∈ C then r is given as a triple (x, y, θ) where x, y are the relative Cartesian coordinates and θ is the relative orientation of the tiles. We define phase space as TC which is isomorphic to R2×3 . A point in TC is a (position, velocity) pair. Our goal is to obtain the likelihood of a correct match and also the rate at which tiles gets incorporated into a growing aggregate (on-rate) and the rate − − at which a bound tile gets knocked off from an aggregate(off-rate) for all ( → r ,→ v)∈ → − → − T C, where r and v denote the position and the velocity vector of center of mass of a tile respectively. Likelihood of the forward reaction The on-rate is dependent upon the number of tiles and the energy of the tile, If we model tile-assembly as a Poisson process then the “on-probability” or the likelihood of a tile getting correctly incorporated into the lattice can be computed based on its on-rate λon . For a Poisson model, the time in which a new tile comes and gets bound to the aggregate is exponentially distributed. Our goal is to extract the mean λon from the simulation results. So define λon ∼ f (o, v, d) where o, v and d are the initial relative orientation, velocity and distance between the two tiles respectively. ôn = 1 , where tci is the time taken by the tiles to bind to each Moreover, define λ i tci PN

ˆ λ

other in the ith simulation. Note that by Monte Carlo Integration, λon ∼ i=1N oni for large N where N is the total number of simulations. Likelihood of the backward reaction The rate(off-rate) at which a tile getting knocked off the lattice is dependent not only on its magnetic binding energy but also on the kinetic energy of a colliding tile. For


13

simplicity, we’ll assume that once attached the fan force is not enough to sweep off tiles from an aggregate and the coefficient of restitution of tile collision is small. Then, we can similarly define the mean rate λof f for computing “off-probability” 1 ˆ of f = based on a Poisson distribution as λ i tdci −tci where tci is defined as before and tdci is the time when the bound tile gets knocked off the aggregate and hence tdc i − tci is time interval between the moment when the tile gets attached and disconnected. Note that λof f = f (o, iv, µ, e), where o, v, µ and e are the initial relative orientation of the tile, impact velocity of the colliding tile, coefficient of friction and coefficient P ˆ of f λ i N

N i=1

of restitution respectively. Again by Monte Carlo Integration, λof f ∼ large N .

for

4.3 Results Our air-bed is 2 m wide and 2 m long. The air-table has a very small coefficient of friction viz 0.0005. The dimension of a tile is 43 × 43 × 1.3cm3 while that of each bar-magnet is 1 × 1 × 0.3cm3 . This ensures a large tile to dipole size ratio. Each tile weighs 100gm. Our first set of experiments were with the monopole approximation and hence bar-magnets dimensions are only relevant for their arrangement along the faces. The frictional coefficient on the tile surface is assumed to be 0.3 while the coefficient of restitution for inter-tile collision is 0.01. A snapshot of the two-tile assembly has been shown in figure 4.1. Figure 7(left) gives the probability distribution of a correct match in an assembly when the relative orientation of the two tiles is in (− π2 , π2 ), relative velocity is in (0, 1.3m/s)(based on tile mass and dimensions) and relative distance between (0, 2.82m)(based on the dimensions of the air-table). The overall probability for a correct match given the kinodynamic constraints is around 0.6. We also computed the mean on-rate and mean off-rates in our tile assembly. Figure 7(middle+right) gives the distribution of rates for tile connection and dissociation for the assembly. Note that, if the tiles fail to connect in the whole simulation period then its association rate is zero while if the tiles after being connected do not detach from each other until the end of the simulation then have a zero dissociation rate in that particular simulation. The data in figure 7(middle+right) only corresponds to the non-zero rates. We also computed the mean association rate over 104 simulations which is 5.334828 × 10−3s−1 while the mean dissociation rate is 2.717822 × 10−1 s−1 . Overall probability for a correct match

Rate of association "Probabilityx0y1.txt" "Probabilityx0y2.txt" "Probabilityx1y0.txt" "Probabilityx1y1.txt" "Probabilityx1y2.txt" "Probabilityx2y0.txt" "Probabilityx2y1.txt"

Overall Probability of a correct match

0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

Rate of dissociation "onProbability-x0-y1.txt" "onProbability-x0-y2.txt" "onProbability-x1-y0.txt" "onProbability-x1-y1.txt" "onProbability-x1-y2.txt" "onProbability-x2-y0.txt" "onProbability-x2-y1.txt"

rate of binding(per sec)

0.6

250

0.5

200

0.4

150

0.3 100

0.2

50

0.1 0

0

1.2

1.4 1.2

1 0.8 -2

"offProbability-x0-y1.txt" "offProbability-x0-y2.txt" "offProbability-x1-y0.txt" "offProbability-x1-y1.txt" "offProbability-x1-y2.txt" "offProbability-x2-y0.txt" "offProbability-x2-y1.txt"

rate of dissociation(per sec)

-1.5

-1

0.4

-0.5

0 Relative Orientation(radians)

0.5

1

0.2 1.50

0.6 Relative Velocity(m/s)

-2

-1.5

-1

-0.5

0 0.5 Relative Orientation(radians)

0.4 1

1.5

0.2 2 0

1 0.8 0.6 Relative Velocity(m/s)

-2

-1.5

-1

-0.5

0 0.5 Relative Orientation(radians)

0.4 1

1.5

1.4 1.2 1 0.8 0.6 Relative Velocity(m/s)

0.2 2 0

Fig. 7. (from left to right)Probability distributions for (a)Overall Probability of a match, (b)Rate of association (c)Rate of dissociation

14


5 Future Directions One of the immediate goals is to extend the simulation model to a multi-tile system with real fans. Again the significance of demonstration of an actual magnetic assembly cannot be undermined. So one possible future direction will be the actual demonstration of the assembly and then compare the experimental and the simulation results particularly the yield and the size of assembly. Another possible direction is to study the potential of a magnetic self-assembling system in three dimensions. The situation here is more complicated because we have six degrees of freedom as opposed to three degrees of freedom in the two-dimensional case. We would also like to study our encoding technique in a more general manner so that it can be applied to any meso-scale self-assembling system. For instance, study of complex self-assembled structures generation using the capillary interaction of MEMS tiles patterned with “wetting codes” would be quite interesting.

6 Conclusion In this work we have discussed a magnetic force driven meso-scale assembly technique with focus on the assembly and alignment process. We have presented the challenges involved in the design of such a magnetic force driven meso-scale selfassembling system, the most important contribution being the design of a barcode scheme which would allow us to generate arbitrary complex structures. We have presented a physical model based on Newtonian mechanics and Maxwellian magnetics that correctly describes self-assembly of macroscopic tiles using magnetic interaction. We have also discussed techniques like centered encoding, optimization of the ratio of tile size to dipole size, long thin magnets and pairwise alternate arrangement of magnets and magnetic shielding for minimizing magnetic crosstalk and maximizing correct binding or incorrect repulsion. We also presented a simulation software system that not only verifies our physical model but also provides us with the data to extract a higher level thermodynamic model of our self-assembly. Since the assembly depends on too many factors, their simultaneous optimization for a successful assembly even in a simulation environment becomes difficult. Hence the kinetic model based on simulation results on a small scale becomes useful for analyzing yields in a large self-assembling system. As an enabling technique, our hope is that this assembly approach will be applicable to generic tiles for the generation of arbitrary complex meso-scale systems.

7 Acknowledgements We thank Professor Erik Klavins and Sam Burden from the Computer Science Department, University of Washington for providing us with the basic simulation package. We also thank Thom LaBean, Chris Dwyer, Alex Hartemink, Sudheer Sahu and Erik Halvorson from Computer Science Department, Duke University for helpful suggestions and comments.

References 1. N B Bowden, A Terfort, J Carbeck, and GM Whitesides. Science, 276:233–235, 1997.


15

2. N B Bowden, S Oliver, and GM Whitesides. Journal of Phys. Chem, 104:2714–2724, 2000. 3. RJ Jackman, S T Brittain, A Adams, M Prentiss, and GM Whitesides. Science, 280:2089– 2091, 1998. 4. T D Clark, J Tien, D C Duffy, K Paul, and GM Whitesides. J. Am Chem. Soc., 123:7677– 7682, 2001. 5. Oliver S R J, Bowden N, and Whitesides G M. J. Colloid Interface Sci, 224:425–428, 2000. 6. Bowden N, Choi I S, Grzybowski B, and Whitesides GM. J Am. Chem, 121:5373–5391, 1999. 7. Grzybowski B, Bowden N, Arias F, Yang H, and Whitesides G M. J. Phys. Chem, 105:404–412, 2001. 8. Syms R R A and Yeatman E M. Electronics Lett., 29:662–664. 9. Harsh K F, Bright V M, and Lee Y C. Sens Actuators A, 77:237–244, 1999. 10. E W H Jager, E Smela, and O Inganas. Science, 290:1540–1545, 2000. 11. D H Gracias, V Kavthekar, J C Love, K E Paul, and G M Whitesides. Adv. Mater., 14:235–238, 2002. 12. T D Clark, M Boncheva, J M German, and G M Whitesides M Weck. J. Am. Chem. Soc., 124:18–19, 2002. 13. P W Green, R R A Syms, and E M Yeatman. J. Microelectromech. Syst, 4:170–176, 1995. 14. R R A Syms, C Gormley, and S Blackstone. Sens. Actuators A, 2839:1–11, 2000. 15. Srinivasan U, Liepmann D, and Howe R T. J. Microelectromech. Syst., 10:17–24, 2001. 16. R J Linderman, P Kladitis, and V M Bright. Sens. Actuators A, 95:135–142, 2002. 17. K F Böhringer. Surface modification and modulation in microstructures:controllinh protein adsorption,monolayer desoprtion, and micro-self assembly. IOP Journal of Micromechanics and Microengineering, 13(4):S1–S10, 2003. 18. X Xiong, Y Hanien, J Fang, Y Wang, W Wang, and K F Böhringer D T Schwartz. Controlled multibatch self assembly of micro devices. 19. J Lienarmann, A Greiner, JG Korvink, X Xiong, Y Hanein, and K F Böhringer. Modelling,simulation and experimentation of a promising newpackaging technology-parallel fluidic self assembly of micro devices. Sensor Update, 2003. 20. A Greiner, J Lienermann, J G Korvink, X Xiong, Y Hanein, and K F Böhringer. Capillary forces in micro fluidic self assembly. In Fifth Intl. Conf. on Modeling and Simulation of Microsystems, pages 198–201, 2002. 21. X Xiong, S Liang, and K F Böhringer. Geometric binding site design for surface tension driven self assembly. In IEEE International Conf. on Robotics and Automation, 2004. 22. S Liang, X Xiong, and K F Böhringer. Towards optimal designs of self-alignment in surface tension driven micro assembly. In IEEE Conf. on Micro Electro Mech. Systems, 2004. 23. K F Böhringer, U Srinivasan, and R T Howe. Modeling of fluidic forces amd binding sites for fluidic assembly. In IEEE Conf. on Micro Electro Mech. Systems, pages 369– 374, 2001. 24. Yeh H J J and Smith J S. IEEE Photon Technol Lett, 6:706–708, 1994. 25. A Terfort and G M whitesides. Adv. Mater., 10:470–473, 1998. 26. Gracias D H, Tien J, Breen T L, Hsu C, and Whitesides G M. Science, 289:1170–1172, 2000. 27. D H Gracias, M Boncheva, O Omoregie, and G M Whitesides. Appl. Phy. Lett., 80:2802– 2804, 2002. 28. M Boncheva, D H Gracias, H O Jacobs, and G M Whitesides. Proc. natl. Acad Sci. USA, 99:4937–4940, 2002.

16


29. K F Boehringer, K Goldberg, M B Cohn, R T Howe, and A Pisano. In Proc. IEEE International Conf. on robotics and Automation, pages 1204–1211. IEEE, New York, 1998. 30. M B Cohn. U.s patent no. 5,355,577, filed 23 june 1992, issued 18oct, 1994. 31. GTE Laboratories Inc. S Yando. U.s patent no.3439416, filed 3 feb 1966, issued 22 apr 1969. 32. H O Jacobs, A R Tao, A Schwartz, D H Gracias, and G M Whitesides. Science, 296:323– 325, 2002. 33. B A Grzybowski, A Winkelman, J A Wiles, Y Brumer, and G M Whitesides. Nat. Mat., 2003. 34. R F Ismagilov, D Rosmarin, D H Gacrias, A D Strook, and G M Whitesides. Appl. Phy. Lett., 79:439–441, 2001. 35. Rothemund P W K. Using lateral capillary forces to compute by self-assembly. PNAS, 2000. 36. B Gryzbowski and G M Whitesides. Nature, 405:1033–1036, 2000. 37. B Gryzbowski and G M Whitesides. Science, 296:718–721, 2002. 38. B Grzybowski, X Jiang, H A Stone, and G M Whitesides. Phy. rev. E, 64:111603/1– 111603/12, 2001. 39. M Boncheva, S A Andreev, L Mahadevan, A Winkleman, D R Reichman, M G Prentiss, S Whitesides, and GM Whitesides. Magnetic self assembly of 3d surfaces from planar sheets. PNAS, 102:3924–3929, 2005. 40. J Bishop, S Burden, E Klavins, R Kreisberg, W Malone, N Napp, and T Nguyen. Selforganizing programmable parts. In International Conference on Intelligent Robots and Systems. IEEE/RSJ Robotics and Automation Society, 2005. 41. Wang H. Bell System Tech. Journal, 40:1–42, 1961. 42. Seeman NC. Nucleic acid junctions and lattices. Journal of Theor. Biology, 99:237–247, 1982. 43. Adleman LM. Molecular computation of solutions to combinatorial problems. Science, 266:1021–1024, 1994. 44. Winfree E. On the computational power of dna annealing and ligation. DNA based computers, pages 199–221, 1996. 45. Mao C.and LaBean T.H, Reif J.H., and Seeman N.C. Logical Computation Using Algorithmic Self-Assembly of DNA Triple-Crossover Molecules. Nature, 407:493495, 2000. C. Erratum: Nature 408, 750-750(2000). 46. Hao Yan, Thomas H. LaBean, Liping Feng, and John H. Reif. Directed nucleation assembly of barcode patterned dna lattices. Proceedings of the National Academy of Science(PNAS), 100(14):8103–8108, 2003. 47. Rothemund Paul W. K, Papadakis Nick, and Winfree Erik. Algorithmic self assembly of dna sierpinski traingles. PLoS Biology, 2(12), December 2004. 48. M Cohn, C J Kim, and A Pisano. Digest of technical papers from transducers ’91. In Intl Conf.on Solid State Sensors and Actuators, page 493. IEEE, NY. 49. L B Kiss, R Vajtai, and R M Ajayan. Physica Status Solidi B: Basic Research, 214:R3– R4, 1999. 50. E S Kooji, E A M Brouwer, H Wormeester, and B Peolsema. Langmuir, 18:7677–7682, 2002. 51. M Adams, Z Dogic, S L Keller, and S Fraden. Nature, 393:349–352, 1998. 52. P Korda, G C Spalding, E R Dufrense, and D G Grier. Rev. Sci. Intr., 73:1956–1957, 2002. 53. M M Burns, J M Fournier, and J A Golovchenko. Science, 249:749–752, 1990.

Title Suppressed Due to Excessive Length 54. 55. 56. 57. 58. 59. 60. 61. 62.

17

S K Lamoreaux. Phys. Rev. Lett., 78:5–8, 1997. J N Israelachvili. Intermolecular and Surface Forces, Academic, San Diego, 1991. J T Feddema, P Xavier, and R Brown. J. Micromechatronics, 1:139–153, 2001. B Alberts, D Bray, J Lewis, M Raff, K Roberts, and J D Watson. Mol. Bio. of the Cell, Garland, NY, 1994. FEMM:Finite Element Method Magnetics. http://femm.foster-miller.net/index.html. ODE:The Open Dynamics Engine. http://www.ode.org. D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys, 22:403–434, 1976. D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem, 81:2340–2361, 1977. Winfree E. Simulations of Computing by Self-Assembly. published as Caltech CS Tech Report 1998.22.