Author: Woods, R C
Date published: April 1, 2010
Journal code: SNTD
(ProQuest: ... denotes formulae omitted.)
Self-assembly techniques are becoming increasingly important as multiple-component integration of microsystems and microcircuits is developed. A basic problem is how to ensure that a silicon (or other material) chip (subsequently referred to herein as the die, plural dice) may be automatically placed in accurate registration or alignment on top of a substrate without fine mechanical handling, either manual or automated. The objective is to ensure that the die will self-assemble into the correct position.
Yeh and Smith  reported self-assembly of GaAs LED dice onto a substrate using an inert fluid carrier (otherwise unidentified), where the final position of the LEDs was defined by physically etching the substrate to match the LED profile in only one orientation. A similar method was used by Zheng and Jacobs  using molten solder as the fluidic component. Tong and Gösele  have indicated mechanisms for enhancing die-to-substrate bonding following fluidic treatment.
Yeh and Smith  explicitly regarded surface tension in their carrier fluid as a hindrance, as it drew some dies out of correct alignment. However, a promising new area of investigation is a contrasting approach that makes a virtue out of this necessity (at readily accessible temperatures and pressures), by using capillarity and surface tension to induce accurate alignment of individual dice on a planar substrate. The basic technique is illustrated in Fig. 1 . The correct position of the die is defined by a processed hydrophilic region on the substrate, exactly the same size and shape as the die to be placed over it, and also in the exact position required of the die when self-assembled. The area surrounding the hydrophilic region is treated to be hydrophobic. Then, a small volume of fluid (e.g., liquid water) placed between the die and the substrate will aggregate on the hydrophilic region and will tend to draw the die to the correct position as the liquid evaporates. The substrate-water contact angle is larger than usually expected because the droplet surface must be fixed at the boundary between the hydrophilic and hydrophobic regions. When no liquid is left, the die remains held by van der Waals forces. Clearly, this technique cannot align a die to much better accuracy than its dimensional tolerance, which may hamper the utility of the method in practice. Nevertheless, it has sufficient promise that a number of workers have been investigating the method.
In a recent paper, Berthier, Grossi and di Cioccio  presented some simulations of the behaviour of a die positioned (using fluidic capillary forces from a hydrophilic area) on a planar substrate during selfassembly. Their work used the software package "Surface Evolver" to give qualitative pictures of the surface formed by water under various misalignments, and these authors also published quantitative graphs of the self-alignment force and restoring torque under translational and rotational displacements of the die respectively.
In the present paper, a simple analytical model is used to calculate the die's restoring force and torque under similar conditions, and the results are compared with the numerical simulations of Berthier, Grossi and di Cioccio .
2. Floating Height of Die
The basic property of free surface energy due to surface tension is used here. In this model, the free energy of a liquid-air interface is written as yS, where S is the liquid-air surface area and ? is the surface tension of the liquid (= 0.073J m^sup -2^ for water at 18 0C ). For a rectangular die, the effects at corners are ignored as they will be dwarfed by the effects associated with the straight sides. For the die to be supported as drawn in Fig. 1 it is required that the excess pressure of the liquid droplet can exactly support the weight of the die. Ignoring the weight of the liquid, the radius of curvature of the liquid droplet surface (r in Fig. 1) along the straight sides can be calculated. As did Berthier, Grossi and di Cioccio , it is assumed here that the liquid surface is pinned at the die edges and also at the edges of the hydrophilic region, so that the positions of the extremes of the liquid surface are known. Along the straight sides of the rectangular die, the liquid surface will take a cylindrical shape with a uniform radius of curvature unchanging along the length of each side. For a complete cylinder of liquid of length L, the free surface energy is given by 2πyLr, neglecting end effects. So, if the radius r changes by an increment dr, then a change of free surface energy will occur given by
The free energy of the cylindrical droplet E = pV, where Fis the volume of liquid and/? is the excess hydrostatic pressure of the liquid inside the droplet caused by the surface tension, analogous to the excess pressure inside a soap bubble, also caused by surface tension. (Colloquially, it can be said that the excess pressure is caused by the compression force of the liquid surface around the cylindrical surface.) Therefore, equating the change in free energy gives
Gathering terms, the resultant equation is
which is an exact differential equation whose solution is
where C is a constant of integration which must be equal to zero as the excess pressure must be zero if γ = 0. This basic result is also obtainable immediately as a special case of the Young-Laplace equation . The weight of the liquid will introduce an error into this calculation since the liquid-air surface will bulge towards the bottom and it will no longer be cylindrical, but the weight of liquid will be negligible in cases where the liquid volume is small.
For a die supported by liquid, the excess pressure caused by the weight of the die must be equal to mg/(XY), where m is the mass of the die, g is the acceleration due to gravity, and X and Y are the dimensions of the (rectangular) die in the ? and y directions respectively. The liquid/air interfaces at all of the four edges of a rectangular die must have the same radius of curvature (apart from corner effects that will be small for a reasonably sized die), as the excess hydrostatic pressure must be uniform throughout the droplet (neglecting gravitational pressure). This gives the radius of curvature
and for low die mass Eqn. (5) can always be satisfied for any liquid volume, since the floating height of the droplet will adjust until the radius of curvature is correct.
Inter alia, Berthier, Grossi and di Cioccio  concluded that the most stable final position in practice for a die floating on a liquid droplet is a "dihedral" configuration where one edge of the die rests on the substrate and the opposite edge is raised by the liquid occupying a wedge-shaped volume. This is easy to deduce from Eqn. (5) as the curvature of the liquid/air interfaces must all be the same. For the same curvature, the downwards resolved force component on the die is lower for the higher end than for the lower end, so any infinitesimal initial tilt is unstable and results in one end touching the substrate. This may represent a serious drawback of the technique, since if an edge of the die touches at high angle then there may be a possibility of damage, and also this will create drag that will tend to prevent accurate self-alignment as the liquid evaporates. However, subsequently in the present work the die will be assumed to be parallel to the substrate surface.
For large die mass, if too much liquid is present, it is no longer possible to satisfy Eqn. (5) since the required radius of curvature is too small and some liquid must be expelled to reduce the floating height. Roughly, for a floating height of h, produced by liquid volume
V = XYh (6)
neglecting the cylindrical contributions, the minimum possible radius of curvature will be A/2 so that the maximum possible volume of liquid usable will be
for a die thickness Z and density p. For the case investigated by Berthier, Grossi and di Cioccio , the die area XY was 25×10^sup -6^m^sup 2^ and silicon density is 2330 kg m^sup -3^ , so for a typical die thickness Z= 0.5 mm the maximum water volume is around 640 µ?. This means that their results for a simulated maximum water volume of 2.5 µl use considerably less than the maximum possible water volume, so that the die will float in equilibrium at a height determined to a good approximation assuming that the radius r is large and the sidewalls of the supporting water droplet are almost planar (and therefore vertical). This is borne out by their published pictures of the simulated behaviour . It also simplifies further calculation, since the liquid volume in the cylindrical portions can usually be neglected.
3. Translational Restoring Force
Berthier, Grossi and di Cioccio  presented quantitative results for the translational restoring force in the case of a die displaced from its correct position in a horizontal direction assumed to be parallel to two of the edges of the die. These results may be calculated as follows. Firstly, there is assumed no extra downwards pressure as a result of the translation, so the droplet excess pressure is unchanged. Therefore, the radius of the cylindrical edges of the droplet is unchanged and the walls will continue to be almost planar. The edges parallel to the displacement of the die, x, do not contribute to the restoring force. Therefore, the only contributions are from the two edges perpendicular to the translation.
Using the geometry shown in Fig. 2, the slant height of the liquid droplet is given by
and for one die edge length G perpendicular to the displacement x, the liquid/air interface area is Ys and the free surface energy is γYs. Adding an identical contribution from the opposite edge, the restoring force may be found immediately as
since the floating height is unchanged and is still given by Eqn. (6).
The final expression of Eqn. (9) represents an analytic formula for the force calculated numerically by Berthier, Grossi and di Cioccio  in their Fig. 6. Using the parameter 7=5 mm allows a direct comparison, as shown in Fig. 3.
The variable parameter on the graphs is the liquid volume, V. The two sets of results have very similar general shapes. The close agreement of the general shapes of the curves validates the method used and shows that the approximations made have not affected the functional form of the results significantly. The maximum value of the restoring force is larger in the analytical model than in the numerical model, for some reason unexplained. From first principles the asymptotic restoring force at high displacements should be accurately predictable, since it will simply be equal to -2γY, so Berthier, Grossi and di Cioccio  may have reported their numerical results for a fluid having ?= 0.0 Um-2 rather than for water.
4. Restoring Torque
To calculate the restoring torque when the die is rotated about a vertical axis by an angle #, it is necessary to calculate the angular dependence of the liquid surface area. The situation is illustrated in Fig. 4. The die has width X in the x-direction and, as before, the liquid surface is assumed pinned to the edges of the die and the hydrophilic region.
In Fig. 4, the non-trivial coordinates of the important points B, D and E are:
Since the radius of curvature of the liquid/air interface is very large, the liquid/air interface takes up the surface in Fig. 4 defined by AECDBFGA and three analogous additional surfaces around the die. Each of these four surfaces is a doubly-ruled hyperbolic paraboloid , and its doubly-ruled property may be exploited for calculation of the area without error by flattening it onto a plane to produce a "net" or "development", as shown in Fig. 5. The length c of CD is given by ... and is equal to FG, and BEC and FBE are right angles. First, the quadrilateral area Q of ECDBE is found and doubled to give the area of the complete shape AECDBFGA, then the analogous area 90° rotated can be found by extension, and finally by symmetry the total liquid surface area is twice the sum of these components.
Unfortunately, quadrilateral ECDBE does not permit simple calculation of its area, and a general expression must be used. Of several available, the one apparently due to Bretschneider  is easiest to apply here:
and α and β are the solutions to
The total area of the net in Fig. 5 is then 2Q including the equal area on the left side. To this must be added the corresponding area 90° rotated, which is found using the same expressions but with X and Y interchanged (but not x' x'', or y''). The final total area is double this sum to account for the opposite sides of the die. Multiplying by γ gives the surface free energy. This result may then be differentiated with respect to θ to obtain the restoring torque on the die.
Apart from corner effects, this result is exact although the expressions are rather unwieldy and awkward to handle. In the general case some computational assistance will probably be needed at this point. Fortunately, Berthier, Grossi and di Cioccio  presented quantitative numerical results for the restoring torque in a very specific special case, i.e., a square die, and under circumstances in which the torque was proportional to angle. For this special case, write X= Y, use the small angle approximations tan ..., and similar approximations for functions of a and ß, and expand the results as far as quadratic terms in θ, to give immediately:
These results give finally from Eqn. (1 1) the area of quadrilateral ECDBE in Fig. 5:
to second order in θ. The total area of the net in Fig. 5 is therefore .... This expression has no linear term in θ, agreeing with the expectation on symmetry grounds. Multiplying by the four sides of the square and by the surface tension to give the free surface energy, then differentiating with respect to T, finally gives the torque on a square die:
The torque calculated from this final simple expression is shown in Fig. 6 and compared with the numerical results of Berthier, Grossi and di Cioccio . Unfortunately, no exact comparison is possible in this case since Berthier, Grossi and di Cioccio  did not record the liquid volumes (or die thickness) in their simulations, but the linear torque-angle relation and the similar trend with fluid volume and floating height is clear. Of course, an angle of θ = 40° is not small enough for the approximations in Eqn. (14) to be accurate, but is adopted as the maximum plotting range so that Fig. 6 is directly comparable with the results of Berthier, Grossi and di Cioccio .
The discrepancy in torque values may be examined in the context of the published numerical results  for free surface energy as a function of angle. Manual graphical estimation gives the grathent -24.7 nN m = 0.43 nJ deg^sup -1^ for hlZ = 0.04 at θ = 10° (compared with torque -0.6 nN m  as shown in Fig. 6 (top)). At θ = 0 the free surface energy should be equal to AXZyQiIZ). The numerical results  show three values of free surface energy, estimated from the graph  as 65.3 nJ, 59.5 nJ, and 54.0 nJ, corresponding to hlZ = 0.16, 0.1, and 0.04 respectively. This is a linear relationship but with an offset free surface energy of 50.2 nJ extrapolated to hi Z= 0. From its slope, ignoring the zero offset, assuming fixed Z, and using the surface tension value γ=0.01 J m^sup -2^ (as apparently for the translational restoring force) gives a reasonable value for the die thickness Z= 0.47 mm. Using this value of Z gives fluid volumes 1.9 µl, 1.2 µl, and 0.47 µl respectively. Finally, the free surface energy is quadratic in ? at low values but the numerical results become linear at high angles , implying a torque plateau not apparent  in Fig. 6 (top).
5. Restoring Force for Vertical Displacement
Although not calculated explicitly by Berthier, Grossi and di Cioccio , the analytical model permits finding the restoring force caused by a vertical displacement d about the equilibrium floating height h. A restoring force is produced because downwards displacement makes the droplet surface bulge outwards, decreasing the radius of curvature, so that by Eqn. (4) the excess pressure in the droplet must increase, opposing the original displacement. To analyze this it is no longer possible to ignore the cylindrical contributions to the droplet volume. In Fig. 1, and once again neglecting corner effects, the total liquid volume is
This is a non-linear equation and further analytical progress requires an approximation. In the limit of large r so that r^sup 2^ » h^sup 2^, the power series approximation arcsin ... may be used to third order, so that after some manipulation Eqn. (19) becomes
Solving for r and finding the force from the excess pressure given by Eqn. (4) gives the static restoring force to first order as
In practice, to investigate a dynamic system, the mass and viscosity of the liquid will also need to be taken into account.
The previous report by Berthier, Grossi and di Cioccio  used a numerical simulation to calculate the horizontal restoring force and torque on a die self-located by a liquid droplet constrained by hydrophilic and hydrophobic regions. Their quantitative results have been re-calculated analytically and good agreement with the functional forms has been found in both cases. Therefore, the analytic results may be used in future investigations of the performance of fluidic self-assembly methods. In addition, a simple analytic form for the vertical restoring force has been found.
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R. C. WOODS
Department of Electrical and Computer Engineering, Louisiana State University,
Baton Rouge, LA 70803-590 1 , USA
Tel.: (225) 578 8961, fax: (225) 578 5200
Received: 4 December 2009 /Accepted: 20 April 2010 /Published: 2 7 April 2010