# Abstract

Random jammed dipole scatterers are natural composite and common byproducts of various chemical synthesis techniques. They often form complex aggregates with nontrivial correlations that influence the effective dielectric description of the medium. In this work, we investigate the packing dynamic of rectangular nanostructure under a close packing protocol and study its influence on the optical response of the medium. We show that the maximum packing densities, maximum scattering densities, and percolation threshold densities are all interconnected concepts that can be understood through the lens of Onsager’s exclusion area principle. The emerging positional and orientational correlations between the rectangular dipoles are studied, and various geometrical connections are drawn. The effective dielectric constants of the generated ensembles are then computed through the strong contrast expansion method, leading to several unintuitive results such as scattering suppression at maximum packing densities, as well as densities below the percolation threshold, and maximum scattering in between.

## 1 Introduction

Random packing persists to be an alluring topic, pertinent to fundamental questions in physics, chemistry, and biology [1], [2], [3]. Within the field of optics and photonics, in particular, understanding light–matter interactions in random packed media is crucial and urged by the growing usage of optical sensors and imaging systems in probing complex living cells, liquids, and granular media. In addition, the thriving genre of disordered photonics domesticates randomness toward various applications in light trapping [4], radiative cooling [5], and random lasing [6].

In this work, we investigate the optical response of packed rectangular nanostructures on a surface, as they are commonly employed as dipole scatterers in optical devices [7], [8], [9] for various applications including light harvesting [10] and biosensing [11]. However, a detailed electromagnetic simulation of such an ensemble is a computationally expensive task to perform and one rather seeks the effective medium description as an approximation. Many homogenization theories with varying degrees of applicability and complexity have been developed toward this aim [12], [13]. Bruggeman’s theory models aggregate structure with constituents that are treated on an equal footing and therefore cannot be applied in this case [14]. Maxwell–Garnett approximation, on the other hand, models inclusions dispersed in a continuous host medium. Its analytic simplicity arises from the consideration of only the one-point probability function (density) where convergence is assured under the dilute and long wavelength limit. However, at large packing density, the positional and orientational correlations between the dipoles are not negligible anymore and can drastically alter the effective dielectric constant of the ensemble. The *strong contrast expansion method* presented in the study by Rechtsmanand and Torquato [13] is rather a generic and exact approach that includes the contribution of high-order point probability functions and thus captures the correction due to the emerged correlations.

In this article, we investigate the influence of the packing dynamic on the optical response of jammed rectangular nanostructures. A comprehensive workflow chart can be found in the supplementary material (section S1). In section 2, we define the packing protocol used in the study and compute the maximum achieved packing densities at various aspect ratios. We proceed in section 3 with point process statistical analysis to unravel the short-range ordering and spontaneous alignment between the packed rectangles. In section 4, we model the ensemble as a two-phase isotropic medium and estimate the effective dielectric constant

## 2 Random close packing

We consider the packing of hard rectangles of length *l* and width *w* on a square substrate, where the interaction potential *N* identical rectangles of aspect ratio *A*_{s} with an initial packing fraction *l*] and *n* attempts has been reached after which the process is terminated, and the rectangle is removed from the packing process. To calculate the maximum packing density *n* = 100,000, and particle number to *N* = 10,000. Periodic boundaries were used to avoid finite size effects, and the results were averaged over multiple realizations to ensure convergence. The maximum packing density achieved for different aspect ratios is shown in Figure 1B. We note that *lw* to what it excludes on average

where *c*_{1} and *c*_{2} coefficients, as shown in Figure 1B. However, the model fails to fit the cusp and *M&M’s* experiment [19], [20]. The conjecture states that the mean contact number between packed elements is on average twice the number of degrees of freedom in a jammed configuration. Consequently, deviating from squares to rectangles introduces an additional degree of freedom (orientation) to the packing process, which increases the average contact numbers and consequently

### Figure 1:

## 3 Point process statistical analysis

The collective rearrangement packing protocol produces a statistically homogeneous medium that we assume ergodic (any single realization of the ensemble is representative of the ensemble in the infinite area-limit). We start our investigation by performing a stochastic point process analysis. Each rectangle is represented by its two midpoint coordinates (*x*, *y*) and the angle *pair correlation* function, which is defined as follows:

where *r* and *r* + *dr* from a reference rectangle. Thus, a deviation of *g*_{2}(*r*) from the unity provides a measure of positional correlation or anticorrelation between the rectangles. The second statistical descriptor is the orientational correlation function

which is an average measure of the degree of alignment between two rectangles within a distance of *r* and *r* + *dr*. Thus, *w* = 1) without loss of generality, and the descriptors are plotted as shown in Figure 2A and B. We observe the emergence of three distinct features on both surfaces that can be fitted by three lines (*p*_{1}, *p*_{2}, *p*_{3}) that intersect at *x*, *y*) and three DOFs for rectangles *g*_{2} correlation in the triangular region between *p*_{1} and *p*_{2} is a direct consequence of the excluding area principle. In other words, as the aspect ratio increases, it becomes statistically difficult to pack rectangles within close proximity. Figure 2C shows a cross-sectional plot of both *g*_{2} and *g*_{2} at *g*_{2}(*r*) that is damped with distance. We also note that *p*_{2} and *p*_{3}, whereas *g*_{r} is negatively correlated. The statistical interpretation indicates that it is highly constrained to place two rectangles in proximity, yet if it is deemed necessary, they must be well aligned. However, such constraint is lifted at *p*_{2} and further relaxed at *p*_{3}. The three constraints can be traced geometrically, as illustrated in Figure 2D. The *p*_{2} line equals to the minimum distance when two perpendicular rectangles are not overlapping, that is, when *p*_{3} line equals to the minimum distance of two stacked rectangles on their longer axes, that is, when *p*_{1}, *p*_{2}, *p*_{3}), a transition in the sign of

### Figure 2:

We conclude from this analysis the lack of long-range translational or nematic order in the ensemble. The effective permittivity in the 2D plane is thus macroscopically isotropic and polarization independent at all aspect ratios. In addition, high aspect ratios have a destructive behavior on short-range positional order, and therefore, their scattering features will be weaker. Furthermore, the average rectangle orientation after lifting the *p*_{2} constrain is approximately *h*), as illustrated in the middle configuration of Figure 2D.

## 4 Strong contrast expansion of the effective dielectric constant

The statistical properties of phase *i* in two-phase heterogeneous media can be specified by an infinite set of *n*-point probability functions *i**r* in phase *i*. In the following discussion, we will drop the superscript (*i*) and implicitly refer to the rectangle phase. The scattering behavior of the ensemble is captured by the spectral density function *k* wave vectors. This is attributed to the suppression of long-range density fluctuations due to the positional ordering of the packed elements. The attenuation becomes weaker as

### Figure 3:

From the calculated autocovariance function, we can proceed in calculating the effective-dielectric constant by the strong contrast expansion method. The expressions presented in the study by Rechtsman and Torquato [13] were formulated for 3D random structures. We rederive the method for two-phase medium in 2D and truncate the expansions up to the second order to include the 2-point probability function

where *p*) with respect to the environment (phase *q*), and

where ^{−6} for *r* > 5. Therefore,

### Figure 4:

## 5 Conclusion

Random packed media are a ubiquitous and natural outcome of various chemical synthesis techniques. In the subwavelength limit, the complex inhomogeneous medium can be described by an effective homogeneous one with great accuracy. In this work, we statistically analyze jammed rectangular dipoles under the random close packing protocol for various densities and aspect ratios. The arising microscopic correlations were traced and shown to have direct and indirect consequences on the effective dielectric constant of the medium. Statistical tools and concepts such as Onsager’s excluded area principle, the positional correlation function, and the orientational correlation function, are of great utilities in describing the state of the ensemble and deduce some of the optical characteristics such as polarization dependence and spectral shifts. To study the influence of structural correlations on the macroscopic optical response, we accommodate the strong contrast expansion method to two-dimensional structure and use it to estimate the effective dielectric constant for the generated ensembles. This allows us to capture various effects beyond what Maxwell–Garnett approximation can, such as scattering enhancement and suppression as well as correlation-induced spectral shift. This work paves a systematic path toward engineering random medium with tailored optical properties.

**Funding source: **National Science Foundation Career Award

**Award Identifier / Grant number: **ECCS-1554021

**Funding source: **Office of Naval Research Young Investigator Award

**Award Identifier / Grant number: **N00014-17-1-2671

**Funding source: **ONR JTO MRI Award

**Award Identifier / Grant number: **N00014-17-1-2442

**Funding source: **DARPA DSO-NLM

**Award Identifier / Grant number: **HR00111820038

**Author contribution**: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding**: This work was supported by the National Science Foundation Career Award (ECCS-1554021), the Office of Naval Research Young Investigator Award (N00014-17-1-2671), the ONR JTO MRI Award (N00014-17-1-2442), and the DARPA DSO-NLM Program no. HR00111820038.**Conflict of interest statement**: The authors declare no conflicts of interest regarding this article.

### References

[1] J. D. Bernal, “A geometrical approach to the structure of liquids,” *Nature*, vol. 183, pp. 141–147, 1959. https://doi.org/10.1038/183141a0.Search in Google Scholar

[2] P. Schaaf and J. Talbot, “Kinetics of random sequential adsorption,” *Phys. Rev. Lett.*, vol. 62, pp. 175–178, 1989. https://doi.org/10.1103/physrevlett.62.175.Search in Google Scholar

[3] J. Feder, “Random sequential adsorption,” *J. Theor. Biol.*, vol. 87, pp. 237–254, 1980. https://doi.org/10.1016/0022-5193(80)90358-6.Search in Google Scholar

[4] H. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” *Nat. Mater.*, vol. 9, pp. 205–213, 2010. https://doi.org/10.1038/nmat2629.Search in Google Scholar

[5] Y. Zhail, Y. Ma, S. David, et al., “Scalable-manufactured randomized glasspolymer hybrid metamaterial for daytime radiative cooling,” *Science*, vol. 355, pp. 1062–1066, 2017. https://doi.org/10.1126/science.aai7899.Search in Google Scholar

[6] N. Lawandy, R. Balachandran, A. Gomes, et al., “Laser action in strongly scattering media,” *Nature*, vol. 368, pp. 436–438, 1994. https://doi.org/10.1038/368436a0.Search in Google Scholar

[7] M. Dupre, L. Hsu, and B. Kante, “On the design of random metasurface based devices,” *Sci. Rep.*, vol. 8, p. 7162, 2018. https://doi.org/10.1038/s41598-018-25488-4.Search in Google Scholar

[8] H. Nasari, M. Dupré, and B. Kanté, “Efficient design of random metasurfaces,” *Opt. Lett.*, vol. 43, pp. 5829–5832, 2018. https://doi.org/10.1364/ol.43.005829.Search in Google Scholar

[9] J. Park, A. Ndao, W. Cai, et al., “Symmetry-breaking-induced plasmonic exceptional points and nanoscale sensing,” *Nat. Phys.*, vol. 16, pp. 462–468, 2020. https://doi.org/10.1038/s41567-020-0796-x.Search in Google Scholar

[10] Y.-Z. Zheng, X. Tao, J.-W. Zhang, et al., “Plasmonic enhancement of light-harvesting efficiency in tandem dye-sensitized solar cells using multiplexed gold core/silica shell nanorods,” *J. Power Sources*, vol. 376, pp. 26–32, 2018. https://doi.org/10.1016/j.jpowsour.2017.11.072.Search in Google Scholar

[11] A. Abbas, L. Tian, J. J. Morrissey, et al., “Hot spot‐localized artificial antibodies for label‐free plasmonic biosensing,” *Adv. Funct. Mater.*, vol. 23, pp. 1789–1797, 2013. https://doi.org/10.1002/adfm.201202370.Search in Google Scholar

[12] M. Safdari, M. Baniassadi, H. Garmestani, et al., “A modified strong-contrast expansion for estimating the effective thermal conductivity of multiphase heterogeneous materials,” *J. Appl. Phys.*, vol. 112, p. 114318, 2012. https://doi.org/10.1063/1.4768467.Search in Google Scholar

[13] M. Rechtsman and S. Torquato, “Effective dielectric tensor for electromagnetic wave propagation in random media,” *J. Appl. Phys.*, vol. 103, pp. 1–15, 2008. https://doi.org/10.1063/1.2906135.Search in Google Scholar

[14] G. A. Niklasson, C. G. Granqvist, and O. Hunderi, “Effective medium models for the optical properties of inhomogeneous materials,” *Appl. Opt.*, vol. 20, pp. 26–30, 1981. https://doi.org/10.1364/ao.20.000026.Search in Google Scholar

[15] A. Bertei, C. C. Chueh, J. G. Pharoah, et al., “Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: towards engineering applications,” *Powder Technol.*, vol. 253, pp. 311–324, 2014. https://doi.org/10.1016/j.powtec.2013.11.034.Search in Google Scholar

[16] J. Perez-Justea, I. Pastoriza-Santosa, L. Liz-Marzana, et al., “Gold nanorods: synthesis, characterization and applications,” *Coord. Chem. Rev.*, vol. 249, pp. 1870–1901, 2005. https://doi.org/10.1016/j.ccr.2005.01.030.Search in Google Scholar

[17] L. Onsager, “The effects of shape on the interaction of colloidal particles,” *Ann. N. Y. Acad. Sci.*, vol. 51, pp. 627–659, 1949. https://doi.org/10.1111/j.1749-6632.1949.tb27296.x.Search in Google Scholar

[18] I. Balberg, C. H. Anderson, S. Alexander, et al., “Excluded volume and its relation to the onset of percolation,” *Phys. Rev. B*, vol. 30, pp. 3933–3943, 1984. https://doi.org/10.1103/physrevb.30.3933.Search in Google Scholar

[19] A. Donev, I. Cisse, D. Sachs, et al., “Improving the density of jammed disordered packings using ellipsoids,” *Science*, vol. 303, pp. 990–993, 2004. https://doi.org/10.1126/science.1093010.Search in Google Scholar

[20] P. Chaikin, A. Donev, W. Man, et al., “Some observations on the random packing of hard ellipsoids,” *Ind. Eng. Chem. Res.*, vol. 45, pp. 6960–6965, 2006. https://doi.org/10.1021/ie060032g.Search in Google Scholar

[21] Z. Ma and S. Torquato, “Hyperuniformity of generalized random organization models,” *Phys. Rev. E*, vol. 99, p. 022115, 2019. https://doi.org/10.1103/physreve.99.022115.Search in Google Scholar

[22] S. Torquato, *Random Heterogeneous Materials*, Berlin, Springer, 2002.10.1007/978-1-4757-6355-3Search in Google Scholar

[23] S. Torquato, “Hyperuniform states of matter,” *Phys. Rep.*, vol. 745, pp. 1–95, 2018. https://doi.org/10.1016/j.physrep.2018.03.001.Search in Google Scholar

[24] W. Man, M. Florescu, K. Matsuyama, et al., “Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast,” *Opt. Express*, vol. 21, pp. 19972–19981, 2013. https://doi.org/10.1364/oe.21.019972.Search in Google Scholar

[25] W. Man, M. Florescu, E. P. Williamson, et al., “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” *Proc. Natl. Acad. Sci. U.S.A.*, vol. 40, pp. 15886–15891, 2013. https://doi.org/10.1073/pnas.1307879110.Search in Google Scholar

[26] O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” *Optica*, vol. 3, pp. 763–767, 2016. https://doi.org/10.1364/optica.3.000763.Search in Google Scholar

[27] F. Bigourdan, R. Pierrat, and R. Carminati, “Enhanced absorption of waves in stealth hyperuniform disordered media,” *Opt. Express*, vol. 27, pp. 8666–8682, 2019. https://doi.org/10.1364/oe.27.008666.Search in Google Scholar

[28] S. Torquato and J. Kim, “*Nonlocal effective electromagnetic wave characteristics of composite media: beyond the quasistatic regime*,” arXiv preprint arXiv:2007.00701.Search in Google Scholar

## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0431).

**Received:**2020-07-28

**Accepted:**2020-10-12

**Published Online:**2020-11-24

© 2020 Mutasem Odeh et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.