Rapid Computation of Statistically Stable Particle - Computational ...

T. I. Zohdi Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740

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Rapid Computation of Statistically Stable Particle/ Feature Ratios for Consistent Substrate Stresses in Printed Flexible Electronics This paper develops a statistically based computational method to rapidly determine stresses in flexible substrates during particle printing processes. Specifically, substrate stresses due to multiple surface particle contact sites are statistically computed by superposing point load solutions for different random particle realizations (sets of random loading sites) within a fixed feature boundary. The approach allows an analyst to rapidly determine the number of particles in a surface feature needed to produce repeatable substrate stresses, thus minimizing the deviation from feature to feature and ensuring consistent production. Three-dimensional examples are provided to illustrate the technique. The utility of the approach is that an analyst can efficiently ascertain the number of particles needed within a feature, without resorting to computationally intensive numerical procedures, such as the finite element method. [DOI: 10.1115/1.4029327]

Introduction

1.1 Motivation: Printing Technologies on Flexible Substrates. Many additive manufacturing technologies employ deposition of particulate materials onto a surface, followed by pressure from a stamp or roller in order to press them into the substrate to ensure adequate bonding (Fig. 1). Due to the rise of printed flexible electronics involving sensitive substrates, which are oftentimes more compliant than the particles placed on their surface, subsurface stresses have now become a key concern. Common to many of flexible electronics applications is the fragility of the compliant substrates involved. For an early history of the printed electronics field, see Gamota et al. [1]. Applications include, for example, optical coatings and photonics [2], MEMS applications [3,4], and biomedical devices [5]. There have been many proposed processing techniques, and we refer the reader to Sirringhaus et al. [6], Wang et al. [7], Huang et al. [8], Choi et al. [9–12], Demko et al. [13, 14], and Fathi et al. [15] for details. 1.2 Specific Objectives. This work develops an efficient statistically based approach to compute stresses in flexible substrates, which can arise during particle printing technologies. This is achieved by utilizing classical point load elasticity solutions and superposing them at multiple sites within the desired feature boundary. This is a discretely loaded system, where the point loadings are located at the particle/substrate contact sites within the features to be constructed. Statistical analyses are undertaken to quantify the intensity of stress as a function of the number of particles within the surface feature pattern under the printing load. For a given number of particles on the surface, this is done repeatedly from different random realizations (sets of loading sites) in order to extract statistical moments such as the average, standard deviation, etc. The approach allows an analyst to rapidly determine the number of particles in a surface feature needed for repeatable average substrate stresses, thus minimizing the deviation from feature to feature and ensuring consistent production. The utility of the approach is that one can efficiently ascertain the Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received September 16, 2014; final manuscript received December 5, 2014; published online February 4, 2015. Assoc. Editor: Donggang Yao.

particle/feature size ratio without resorting to computationally intensive procedures such as the finite element method. Remark 1. We consider the particle (the loading sites in the upcoming analysis) placements as given. The dynamics of the particle deposition process will not be considered in this work. We refer the reader to Martin [16,17] for descriptions of the state of the art in industrial deposition processes, to Duran [18] for reviews of the physics of particulate media, and to Zohdi [19–25] for more computationally oriented techniques aligned with manufacturing processes involving particles. Remark 2. The interaction of the particles with the substrate is characterized through point loadings. As alluded to in the previous remarks, the radii of the particles and their mutual interaction (possible contact or overlap) do not enter into this simplified analysis. In the graphics throughout the paper, the particles are simply shown to illustrate from where the point loadings arise.

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Aggregate Multiparticle Stress Fields

2.1 Individual Particle Contributions. Our basic computational unit in this analysis will be a static normal point force on an infinite half space. The corresponding radially symmetric (h-independent) solution for a normal load at (x, y ,z) ¼ (0, 0, 0) in the z-direction is (in cylindrical coordinates, [26])

Fig. 1 Placing particles into a surface and applying pressure to properly bond them to the substrate

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Fig. 2 Left: a single particle; middle: a set of particles on the surface under loading; and right: the stress intensity

rrr rhh rzz rrz rrh rzh

    Fz 1 z 3zr 2 ¼ ð1  2 Þ 2  2  5 r cr 2p c   Fz 1 z z ¼  ð1  2 Þ 2  2  3 r cr c 2p 3 3Fz z ¼ 2p c5 3Fz rz2 ¼ 2p c5 ¼ 0 ðsymmetryÞ ¼ 0 ðsymmetryÞ

ffi def pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

where r ¼

(1)

def pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2

x þ y and c ¼

x þy þz .

2.2 Total Stress. The total stress at a point in the substrate is computed by summing of all point load (particle) contributions (I ¼ 1,2, ..., N, appropriately translated according to their position on the surface) rtotN ¼

N X

rI

(2)

I¼1

where the externally applied load is distributed equally over the set of surface particles. We assume that there is only normal loading. The presence of tangential loading is discussed in the conclusions. FromPEq. (1), one can determine the von Mises stress ðrtotN Þ0 ¼ NI¼1 r0I , where r0I ¼ rI  ðtrrI =3Þ1, which is usually important for failure assessment. 2.3 Computational Algorithm. The computational algorithm is as follows: • • •

• •

•

•

STEP 1: INITIALIZE THE STARTING NUMBER OF PARTICLES (N LOADING SITES) IN THE FEATURE. STEP 2: RANDOMLY PLACE THE N PARTICLES (LOADING SITES) IN THE FEATURE BOUNDARIES. STEP 3: COMPUTE THE STRESS FIELD CONTRIBUTION FROM EACH PARTICLE, I ¼ 1,2, ...N, IN THE REGION OF INTEREST UNDERNEATH THE FEATURE. STEP 4: SUM THE CONTRIBUTIONS OF EACH PARTICLE, I ¼ 1,2, ...N, TO COMPUTE THE TOTAL. STEP 5: REPEAT STEPS 2–4 M TIMES (REALIZATIONS (SETS OF LOADING SITES)) FOR THE SAME PARTICLE-IN-FEATURE NUMBER N, BUT WITH DIFFERENT RANDOM REALIZATIONS. STEP 6: COMPUTE THE STATISTICS: AVERAGE, STANDARD DEVIATION, ETC. FOR THE M REALIZATIONS. STEP 7: INCREASE THE NUMBER OF PARTICLES IN THE FEATURE AND REPEAT STEPS 2–6 UNTIL THE STATISTICAL MEASURES STABILIZE.

Figure 3 provides a corresponding flow chart for the process. The following statistical metrics are computed: 021019-2 / Vol. 137, APRIL 2015

Fig. 3 The algorithm for computation of statistically stable particle/feature resolution •

The volume average deviatoric stress metric in the volume with N surface load particles sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 kN ¼ ðrtotN Þ0 : ðrtotN Þ0 dV (3) jXj X

•

The ensemble average of the M realizations (sets of loading sites), each Phaving N surface load particles, is aN;M ¼ ð1=MÞ M K¼1 kN;K . The standard deviation of the realizations is dN;M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM ¼ ð1=MÞ I¼1 ðkN;K  aN;M Þ2 .

•

3

Examples and Statistical Metrics We considered three examples (Fig. 4):1 1

The Poisson ratio was set to  ¼ 0.3.

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Fig. 4 LEFT: Three feature examples tested: (a) a solid circular pattern, (b) a ring pattern, and (c) a figure 8 pattern. RIGHT: two random realizations (sets of loading sites) from the 100 tested for each N-set.

Fig. 5 Example 1—solid circle (with surface area of As 5 pð0:15LÞ2 ): overall quality metrics (the average, dN,M and the standard deviation, dN,M) as a function of the number of particles in the feature (for an applied unit load Fz 5 1). Results are shown for the deviatoric stresses (r0 ) and the total stresses (r). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary.

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Fig. 6 Example 2—ring (with surface area of As 5 pðð0:15LÞ2 2 ð0:1LÞ2 Þ): overall quality metrics (the average, dN,M and the standard deviation, dN,M) as a function of the number of loading sites in the feature (for an applied unit load Fz 5 1). Results are shown for the deviatoric stresses (r0 ) and the total stresses (r). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary.

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Fig. 7 Example 3—figure 8 (with surface area of As 5 2pðð0:15LÞ2 2 ð0:1LÞ2 Þ): overall quality metrics (the average, aN,M and the standard deviation, dN,M) as a function of the number of particles in the feature (for an applied unit load Fz 5 1). Results are shown for the deviatoric stresses (r0 ) and the total stresses (r). We note that the number of particles in the area should be interpreted as number of loading sites in the feature boundary. • • •

A solid circular pattern A circular ring pattern A figure 8 pattern.

were run in a matter of 1–2 min on a laptop. Figures 2–7 illustrate the statistical measures (for both the deviatoric and total stresses). While all of the feature examples had the same total imposed external loading, it is important to realize that the surface area of the features were different. The results indicate that, in a general, in a range of 50–100 loading sites within the tested features, the results roughly stabilized (according to the average stress metric). However, by studying the standard deviations of the three examples, it is clear that the solid feature takes longer to stabilize, because primarily of its larger surface area (and thus more possible deviations in patterns within the feature), relative to the other two examples. The figure 8 takes longer to stabilize (by observing again the standard deviation curves) than the ring pattern, again primarily because of the larger surface area, relative to that of the ring. We also note that the average deviatoric stress in the figure 8 is the highest (among all three examples), primarily because of the relatively more complex geometry leading to more intensity of stress deviation. Generally speaking, once one ascertains the number of loading sites, one can easily determine (“reverse engineer”) the size of the particles needed. Consistently, the average stresses per realization monotonically decreased for increasing particles per feature. As can be seen in the plots, after the respective curves stabilized, there was negligible gain for using finer (more) particles. For other types of features, there would be other stabilizing particle number per feature thresholds. The developed algorithm can easily handle a wide variety of complex features.

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These were selected to illustrate a solid feature (solid circle), a hollow feature (the “O”), and a nonsimply connected feature (a “figure 8”). A sampling domain of X ¼ Lð1  1  0:25Þ under the features in question was selected, where L was an arbitrary length scale.2 The sampling volume grid (to sum up the stresses) underneath the feature was comprised of a 30  30  15 mesh (13,500 sampling points). For the solid circular pattern, the radius was 0.15 L, thus leading to a surface area of As ¼ pð0:15LÞ2 , while for the circular ring and figure 8 (two rings placed side-by-side) examples, the inner radius was 0.1 L and the outer radius was 0.15 L, thus leading to As ¼ pðð0:15LÞ2  ð0:1LÞ2 Þ and As ¼ 2pðð0:15LÞ2  ð0:1LÞ2 Þ, respectively. For all examples, the intensity of the imposed external loading (Fz) was set to 1 N, but was irrelevant for these examples, since it scaled out of the analysis. Varying the number of particles within a feature (N), we computed M ¼ 100 realizations (random sets of loading sites) and ensemble averaged over the results, for each N-set. All computations

Conclusions

In summary, this paper developed a rapid, statistically based, computational technique to determine the substrate stresses which arise in printing technologies involving particles. Specifically, substrate stresses due to multiple particle contact sites were computed by superposing point load solutions for different particle realizations (sets of loading sites) within a feature boundary on the surface. Statistical analyses were undertaken to quantify the intensity of stress as a function of the number of particles within the surface feature. The approach allows an analyst to determine the density of particles per unit surface feature area needed, via the number of loading sites in a surface feature, for repeatable substrate stresses, thus minimizing the deviation from feature to feature and consequently ensuring consistent production. In the present analysis, only normal surface loading was included. Because some proposed manufacturing processes for flexible electronics involve rollers to press particles into substrates, such as roll-to-roll printing, tangential forces can be present. The effects of non-normal (tangential) loadings can be included by utilizing the solutions for a tangential point load in the x-direction (see Johnson [27] or Kachanov et al. [28] for reviews)

Fx x3 x 3x x3 2x3 þ þ 3 5 þ ð1  2 Þ 3  rxx ¼ 2 2 3 2 c 2p c cðc þ zÞ c ðc þ zÞ c ðc þ zÞ3

!!

Fx xy2 x x xy2 2xy2 þ þ 3 5 þ ð1  2 Þ 3  ryy ¼ 2 2 c 2p c cðc þ zÞ c3 ðc þ zÞ c2 ðc þ zÞ3 rzz ¼ 

3Fx xz2 2p c5

Fx x2 y y x2 y 2x2 y rxy ¼ þ þ 3 5 þ ð1  2 Þ  2 2 c 2p cðc þ zÞ c3 ðc þ zÞ c2 ðc þ zÞ3 3Fx xyz ryz ¼  2p c5 3Fx x2 z rzx ¼  2p c5

!!

(4)

!!

2

L scaled out of the results and was irrelevant.

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and performing a coordinate transformation (to a tilde-frame) x~ ¼ y, y~ ¼ x, ~z ¼ z to account for any Fy loading in the y–direction. The results from the loading in the x, y, and z directions can be superposed to produce the total loading. Often is convenient to move back and forth from Cartesian and cylindrical bases, which can be achieved by simply rotating the system with rcart ðhÞ ¼ RT ðhÞ  rcyl  RðhÞ

(5)

where RðhÞ is defined as 2

cosh

sinh

0

3

def 6 7 RðhÞ ¼ 4 sinh cosh 0 5

0

0

(6)

1

and where RT ðhÞ ¼ R1 ðhÞ, because it is an orthonormal matrix.3

References [1] Gamota, D., Brazis, P., Kalyanasundaram, K., and Zhang, J., 2004, Printed Organic and Molecular Electronics, Kluwer Academic Publishers, New York. [2] Nakanishi, H., Bishop, K. J. M., Kowalczyk, B., Nitzan, A., Weiss, E. A., Tretiakov, K. V., Apodaca, M. M., Klajn, R., Stoddart, J. F., and Grzybowski, B. A., 2009, “Photoconductance and Inverse Photoconductance in Thin Films of Functionalized Metal Nanoparticles,” Nature, 460, pp. 371–375. [3] Fuller, S. B., Wilhelm, E. J., and Jacobson, J. M., 2002, “Ink-Jet Printed Nanoparticle Microelectromechanical Systems,” J. Microelectromech. Syst., 11(1), pp. 54–60. [4] Samarasinghe, S. R., Pastoriza-Santos, I., Edirisinghe, M. J., Reece, M. J., and Liz-Marzan, L. M., 2006, “Printing Gold Nanoparticles With an Electrohydrodynamic Direct Write Device,” Gold Bull., 39(2), pp. 48–53. [5] Ahmad, Z., Rasekh, M., and Edirisinghe, M., 2010, “Electrohydrodynamic Direct Writing of Biomedical Polymers and Composites,” Macromol. Mater. Eng., 295(4), pp. 315–319. [6] Sirringhaus, H., Kawase, T., Friend, R. H., Shimoda, T., Inbasekaran, M., Wu, W., and Woo, E. P., 2000, “High-Resolution Inkjet Printing of All-Polymer Transistor Circuits,” Science, 290(5499), pp. 2123–2126. [7] Wang, J. Z., Zheng, Z. H., Li, H. W., Huck, W. T. S., and Sirringhaus, H., 2004, “Dewetting of Conducting Polymer Inkjet Droplets on Patterned Surfaces,” Nat. Mater., 3, pp. 171–176. [8] Huang, D., Liao, F., Molesa, S., Redinger, D., and Subramanian, V., 2003, “Plastic-Compatible Low-Resistance Printable Gold Nanoparticle Conductors for Flexible Electronics,” J. Electrochem. Soc., 150(7), pp. G412–G417.

[9] Choi, S., Park, I., Hao, Z., Holman, H. Y., Pisano, A. P., and Zohdi, T. I., 2010, “Ultra-Fast Self-Assembly of Micro-Scale Particles by Open Channel Flow,” Langmuir, 26(7), pp. 4661–4667. [10] Choi, S., Stassi, S., Pisano, A. P., and Zohdi, T. I., 2010, “Coffee-Ring EffectBased Three Dimensional Patterning of Micro, Nanoparticle Assembly With a Single Droplet,” Langmuir, 26(14), pp. 11690–11698. [11] Choi, S., Jamshidi, A., Seok, T. J., Zohdi, T. I., Wu, M. C., and Pisano, A. P., 2012, “Fast, High-Throughput Creation of Size-Tunable Micro, Nanoparticle Clusters Via Evaporative Self-Assembly in Picoliter-Scale Droplets of Particle Suspension,” Langmuir, 28(6), pp. 3102–3111. [12] Choi, S., Pisano, A. P., and Zohdi, T. I., 2013, “An Analysis of Evaporative Self-Assembly of Micro Particles in Printed Picoliter Suspension Droplets,” J. Thin Solid Films, 537(30), pp. 180–189. [13] Demko, M., Choi, S., Zohdi, T. I., and Pisano, A. P., 2012, “High Resolution Patterning of Nanoparticles by Evaporative Self-Assembly Enabled by In-Situ Creation and Mechanical Lift-Off of a Polymer Template,” Appl. Phys. Lett., 99(25), p. 253102. [14] Demko, M. T., Cheng, J. C., and Pisano, A. P., 2010, “High-Resolution Direct Patterning of Gold Nanoparticles by the Microfluidic Molding Process,” Langmuir, 26(22), pp. 16710–16714. [15] Fathi, S., Dickens, P., Khodabakhshi, K., and Gilbert, M., 2013, “Microcrystal Particles Behaviour in Inkjet Printing of Reactive Nylon Materials,” ASME J. Manuf. Sci. Eng., 135(1), p. 011009. [16] Martin, P., 2009, Handbook of Deposition Technologies for Films and Coatings, 3rd ed., Elsevier, Amsterdam, Netherlands. [17] Martin, P., 2011, “Introduction to Surface Engineering and Functionally Engineered Materials,” J. Vac. Sci. Technol., A2(2), p. 500. [18] Duran, J., 1997, Sands, Powders and Grains—An Introduction to the Physics of Granular Matter, Springer Verlag, Heidelberg, Germany. [19] Zohdi, T. I., 2003, “Genetic Design of Solids Possessing a Random-Particulate Microstructure,” Philos. Trans. R. Soc., A, 361(1806), pp. 1021–1043. [20] Zohdi, T. I., 2003, “On the compaction of Cohesive Hyperelastic Granules at Finite Strains,” Proc. R. Soc., 454(2034), pp. 1395–1401. [21] Zohdi, T. I., 2004, “A Computational Framework for Agglomeration in ThermoChemically Reacting Granular Flows,” Proc. R. Soc., 460(2052), pp. 3421–3445. [22] Zohdi, T. I., 2012, “Estimation of Electrical-Heating Load-Shares for Sintering of Powder Mixtures,” Proc. R. Soc., 468(2144), pp. 2174–2190. [23] Zohdi, T. I., 2012, Dynamics of Charged Particulate Systems: Modeling, Theory and Computation, Springer-Verlag, Heidelberg, Germany. [24] Zohdi, T. I., 2013, “Numerical Simulation of Charged Particulate ClusterDroplet Impact on Electrified Surfaces,” J. Comput. Phys., 233, pp. 509–526. [25] Zohdi, T. I., 2014, “A Direct Particle-Based Computational Framework for Electrically-Enhanced Thermo-Mechanical Sintering of Powdered Materials,” Math. Mech. Solids, 19(1), pp. 93–113. [26] Boussinesq, J., 1885, Application des Potentials a l’ etude de l’ equilibre et du mouvement des solides elastiques, Gauthier-Villars, Paris, France, Vol. 45, p. 108. [27] Johnson, K., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK. [28] Kachanov, M., Shafiro, B., and Tsukrov, I., 2003, Handbook of Elasticity Solutions, Kluwer, Springer, Verlag, Heidelberg, Germany.

3 We note that the quantity r0 : r0 is invariant under the rotational coordinate transformation, in other words, r0;car : r0;car ¼ ðRT ðhÞ  r0;cyl  RðhÞÞT : ðRT ðhÞ  r0;cyl  RðhÞÞ ¼ r0;cyl : r0;cyl , thus this metric remains perfectly acceptable to use in the presence of non-normal loading.

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