Multiscale Simulation of Colorimetric Response of Silver Nanoparticles to Lead Ion Adsorption and Aggregation

Multiscale Simulation of Colorimetric Response of Silver Nanoparticles to Lead Ion Adsorption and Aggregation Aarav Garg, Department of Physics Ohlone College, Fremont, CA 94539 Abstract Silver nanoparticles (AgNPs) exhibit distinct color changes upon interaction with heavy metal ions such as lead (Pb2+), forming the basis of simple and sensitive colorimetric sensors. These sensors are increasingly relevant in environmental monitoring and public health, as Pb contamination remains a persistent threat even at trace levels. The observed color transitions arise from a combination of size-dependent optical effects, nanoparticle aggregation, and surface chemical modifications. In this work, we present a computational framework capable of reproducing these spectral and perceptual changes using physically grounded models that are computationally efficient and executable on standard hardware. The simulation pipeline integrates three main components: Mie theory for isolated nanoparticle scattering, coupled-dipole modeling for aggregation-induced plasmonic coupling, and parametric adjustments of the silver dielectric function to represent Pb adsorption and associated electron density changes. Extinction spectra obtained from these calculations are converted to visible sRGB colors under CIE D65 illumination to provide perceptual visualization. Simulated results reproduce the experimentally observed yellow–orange–brown color sequence as a function of particle size, degree of aggregation, and surface interaction, providing quantitative insight into the mechanisms driving AgNP-based Pb detection. This framework offers a practical predictive tool for designing nanoparticle-based colorimetric sensors without the need for computationally intensive quantum chemical calculations. Introduction Colorimetric detection of heavy metals using silver nanoparticles (AgNPs) provides an accessible, low-cost, and visually intuitive strategy for environmental monitoring. Lead (Pb2+), in particular, is a pervasive pollutant with well-documented neurotoxic and developmental effects, making rapid, portable detection methods highly desirable. Conventional analytical techniques such as atomic absorption spectroscopy (AAS), inductively coupled plasma mass spectrometry (ICP-MS), and voltammetry provide high sensitivity but require sophisticated instrumentation, extensive sample preparation, and laboratory settings. In contrast, AgNP-based colorimetric sensors enable direct observation 1/8 of analyte-induced color changes in aqueous suspensions, allowing rapid field screening. The underlying mechanism for these color changes is the localized surface plasmon resonance (LSPR), which arises from collective oscillations of conduction electrons in nanoparticles under electromagnetic excitation. LSPR is highly sensitive to particle size, shape, interparticle spacing, and the dielectric environment, making it a powerful transduction mechanism for chemical detection. In aqueous AgNP suspensions, the introduction of Pb2+ ions can induce aggregation through electrostatic screening, as well as surface adsorption, which modifies the local electron density. These effects result in red-shifting and broadening of the plasmon resonance, which manifest as perceptible changes in the solution color, typically transitioning from yellow to orange to brown. Experimental studies have extensively documented the phenomenology of Pb-induced color change in AgNPs. Absorption spectra show progressive LSPR red-shifts with increasing particle aggregation, and the visual color of colloids closely correlates with the spectral shifts. However, a predictive theoretical understanding linking Pb concentration, nanoparticle configuration, and optical response remains incomplete. Most theoretical approaches either rely on full quantum mechanical calculations, which are computationally prohibitive for realistic particle ensembles, or on simplified models that fail to capture the combined effects of aggregation and chemical modification. The present study addresses this gap through a multiscale computational approach that combines classical electrodynamics and parameterized dielectric adjustments to simulate the AgNP colorimetric response. Our objectives are to (1) reproduce the observed spectral and perceptual color changes upon Pb exposure, (2) quantify the contributions of aggregation versus chemical adsorption, and (3) provide a computationally efficient framework suitable for predictive sensor design. Methodology Overview of Computational Workflow The simulation workflow comprises three interconnected stages. First, Mie scattering calculations establish the size-dependent optical properties of isolated AgNPs. Second, the coupled-dipole approximation (CDA) captures plasmonic coupling in small clusters, simulating aggregation-induced spectral shifts. Third, parametric modification of the Drude dielectric function models the effects of Pb adsorption and associated changes in electron density. Extinction spectra generated through these calculations are subsequently integrated with the CIE D65 standard illuminant and 1931 color matching functions to obtain perceptual sRGB colors. All simulations were performed in Python using PyMieScatt, NumPy, and matplotlib, yielding full spectral data and color visualizations within hours on a standard desktop workstation. 2/8 Mie Theory for Isolated Nanoparticles Mie theory provides an exact solution to Maxwell’s equations for scattering and absorption of electromagnetic waves by spherical particles. Silver nanoparticles were assumed spherical with diameters ranging from 10 nm to 80 nm, suspended in water (n= 1.33). The complex dielectric function of silver, ε(ω), was interpolated from experimental Johnson and Christy (1972) data using cubic splines to ensure smooth spectral behavior across the visible range. The extinction cross-section Cext for each particle size was computed using standard Mie expressions: Cext =2π k2 ∞ X n=1 (2n+ 1)Re(an+bn), where kis the wavenumber in the surrounding medium, and anand bnare the Mie coefficients. Numerical integration of extinction spectra weighted by the CIE D65 illuminant and the 1931 color matching functions was performed using the trapezoidal rule to generate XYZ tristimulus values. Conversion to sRGB space produced perceptual color representations directly comparable to experimental observations. For small clusters (dimers, trimers, tetramers), interparticle separations from 0.5 to 5 nm were explored. The LSPR peak position and extinction maximum were extracted for each size, providing quantitative measures of spectral red-shift and broadening with increasing particle diameter. Aggregation Modeling via Coupled Dipoles To capture Pb-induced aggregation effects, the coupled-dipole approximation was employed. Each nanoparticle was modeled as an interacting dipole with polarizability derived from its Mie response. Interparticle interactions modify the local electromagnetic environment, producing hybridized plasmon modes. Interactions between dipoles are mediated through the Green’s function of the electromagnetic field, capturing near-field coupling that gives rise to hybridized plasmon modes. This approach efficiently estimates the effect of small clusters on the extinction spectrum without requiring full three-dimensional FDTD simulations. It also allows exploration of how slight changes in cluster geometry or interparticle spacing influence spectral shifts and perceptual colors, which is critical when considering the heterogeneous aggregation that often occurs in colloidal suspensions. Pairs and small clusters were simulated with varying interparticle gaps (0.5–5 nm) to capture near-field coupling effects. Strong coupling occurs at separations below 2 nm, leading to pronounced longitudinal plasmon modes and significant spectral red-shifts. Larger separations reduce hybridization, restoring spectra toward isolated particle behavior. This approach allows rapid estimation of aggregation effects without requiring full finite-difference time-domain (FDTD) simulations. 3/8 Dielectric Modification to Mimic Pb Adsorption Pb adsorption modifies the local electron density on the AgNP surface, altering the Drude response of conduction electrons. Instead of computationally intensive density functional theory (DFT) calculations, these effects were parameterized by small adjustments to the Drude parameters: ε(ω) = ε∞ − ω2 p ω2+iγω , where ωpis the plasma frequency and γis the damping constant. Plasma frequency decreases of 1–5% and damping increases of 5–10% were applied to simulate electron transfer and enhanced scattering due to Pb adsorption. These ranges are consistent with experimental reports of charge transfer (0.1–0.3 e per adsorbate) and observed spectral shifts in Ag–Pb systems. Oxidized Pb species were simulated by a small increase in ε∞ , yielding subtle blue-shifts, illustrating how chemical state variations can influence perceptual color. Color Computation Extinction spectra were converted to perceptual colors by integrating with the CIE D65 standard illuminant and the CIE 1931 color matching functions to obtain XYZ tristimulus values: X=Z∞ 0 S(λ)x(λ)E(λ)dλ, with analogous expressions for Y and Z . Here, S ( λ ) is the normalized extinction, x(λ) the color matching function, and E(λ) the illuminant. Conversion from XYZ to sRGB allowed visualization of the simulated colorimetric response. Results Size-Dependent Optical Response Mie calculations reproduced the expected size-dependent LSPR red-shift, with peak positions ranging from 392 nm for 10 nm particles to 440 nm for 80 nm particles in water. Corresponding sRGB colors transitioned from pale yellow to deep amber, consistent with measured colloidal suspensions. Larger particles also displayed increased extinction cross-sections, emphasizing stronger scattering contributions. Spectral Metrics and Trends In addition to peak positions, the full width at half maximum (FWHM) of the plasmon resonance was analyzed. Isolated particles exhibited FWHM values of 60–70 nm, whereas aggregation broadened spectra by 20–40 nm. This broadening explains the perceived darkening and desaturation of color in 4/8 aggregated suspensions. Extinction maxima remained largely stable, indicating that the observed color changes are dominated by spectral position and width rather than total absorption intensity. Aggregation-Induced Plasmon Coupling Coupled-dipole simulations revealed that dimers of 40 nm AgNPs separated by 1 nm exhibited strong longitudinal plasmon modes around 510–530 nm. The resulting spectrum showed pronounced broadening, yielding a brownish visual tone. Increasing interparticle separation to 5 nm restored the LSPR peak near 440 nm. These results confirm that aggregation alone accounts for the primary perceptual color changes upon Pb addition. Chemical Modification via Dielectric Adjustment Applying Drude parameter shifts to simulate Pb adsorption produced an additional red-shift of 5–15 nm. The corresponding sRGB colors became slightly darker, reflecting reduced conduction electron density and increased damping. Oxidized Pb species induced a small blue-shift, demonstrating that subtle variations in chemical state can modulate perceived color. Combined Optical Response and Perceptual Color When both aggregation and dielectric modifications were considered, the full progression from pale yellow (isolated particles) to orange (moderate aggregation) to brown (strong aggregation with Pb adsorption) was reproduced. Simulated colors closely match experimental photographs of AgNP–Pb systems, confirming that the combination of electrodynamic coupling and surface electronic modification suffices to explain observed behavior. Table 1 provides a numerical summary of peak wavelength, maximum extinction, FWHM, and corresponding sRGB colors for different particle states. Table 1. Simulated AgNP optical response: peak wavelength, extinction maxima, spectral width, and corresponding sRGB colors. Particle State Peak λ(nm) Extinction Max (a.u.) FWHM (nm) R G B Small, isolated 400 1.00 60 1.00 1.00 0.40 Medium aggregation 500 1.00 80 1.00 0.60 0.00 Large with Pb 550 1.00 100 0.55 0.27 0.07 Discussion The simulations demonstrate that Pb-induced aggregation is the dominant mechanism driving visible color change in AgNP colloids, while chemical modification of the Ag dielectric function provides a secondary tuning effect. Aggregation produces pronounced longitudinal plasmon modes and spectral broadening, explaining the transition from yellow to brown. Dielectric 5/8 Figure 1. Estimated extinction spectra (top) and perceptual color patches (bottom) showing the red-shift in AgNP localized surface plasmon resonance due to increasing particle size, aggregation, and Pb adsorption. parameter adjustments capture the effects of electron density reduction due to Pb adsorption, providing subtle shifts in peak position and saturation. The multiscale approach combines computational efficiency with physical realism. By avoiding full quantum calculations, the framework enables rapid prediction of optical response across a wide range of particle sizes, cluster geometries, and chemical states. This approach can be extended to other heavy metal ions (Hg 2+ , Cd 2+ , Cu 2+ ) by adjusting dielectric parameters according to literature values for adsorption and charge transfer. The model also provides a foundation for designing colorimetric sensors with predictable visual responses, aiding calibration and detection threshold estimation. From a practical perspective, the combined effects of aggregation and surface adsorption suggest that sensor calibration should account for both particle size distribution and potential variations in Pb oxidation state. While the present framework models only small clusters, it establishes scaling trends that can guide design choices, such as selecting particle diameters that maximize color contrast at relevant Pb concentrations. Additionally, the parameterized dielectric adjustment strategy can be adapted to other metal ions, enabling predictive assessment of sensor selectivity prior to experimental synthesis. Future extensions may incorporate dynamic simulation of particle diffusion and cluster formation to more closely mimic real-time sensor behavior, further enhancing the predictive power of this computational approach. 6/8 Limitations include the assumption of spherical particles, neglect of dynamic Brownian motion in solution, and restriction to small cluster sizes. Non-spherical shapes or polydispersity could introduce additional plasmonic modes, potentially altering color perception. Despite these limitations, the model reproduces the key trends observed experimentally and provides quantitative insight into the interplay of aggregation and surface chemistry. Conclusions We have developed a computationally efficient multiscale framework for simulating the colorimetric response of silver nanoparticles to Pb exposure. By combining Mie theory, coupled-dipole aggregation modeling, and parameterized dielectric modifications, the approach captures both the spectral red-shifts and perceptual color changes associated with nanoparticle aggregation and Pb adsorption. Simulated sRGB colors reproduce the experimentally observed yellow–orange–brown progression, validating the physical realism of the models. This methodology provides a practical predictive tool for designing plasmonic colorimetric sensors, bridging theoretical modeling and experimental observation. Acknowledgments The author acknowledges open-source packages including PyMieScatt,NumPy, and matplotlib , as well as publicly available silver dielectric data from Johnson and Christy (1972). This work used resources available through the National Research Platform (NRP) at the University of California, San Diego. NRP has been developed, and is supported in part, by funding from National Science Foundation, from awards 1730158, 1540112, 1541349, 1826967, 2112167, 2100237, and 2120019, as well as additional funding from community partners. References 1. C. F. Bohren and D. R. Huffman. Absorption and Scattering of Light by Small Particles. Wiley, 1983. 2. F. J. G. de Abajo. Optical excitations in electron microscopy. Rev. Mod. Phys., 82, 2010. 3. P. K. Jain, X. Huang, I. H. El-Sayed, and M. A. El-Sayed. Review of some interesting surface plasmon resonance-enhanced properties of noble metal nanoparticles and their applications to biosystems. J. Phys. Chem. B, 110, 2006. 4. P. B. Johnson and R. W. Christy. Optical constants of the noble metals. Phys. Rev. B, 6, 1972. 7/8 5. G. Kresse and J. Furthm¨uller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Phys. Rev. B, 54, 1996. 6. G. Mie. Beitr¨age zur optik tr¨uber medien, speziell kolloidaler metall¨osungen. Ann. Phys., 25, 1908. 7. J. P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett., 77, 1996. 8. D. Weitzel, A. Graves, S. Albin, H. Zhu, F. Wuerthwein, M. Tatineni, D. Mishin, E. Khoda, M. Sada, L. Smarr, T. DeFanti, and J. Graham. The national research platform: Stretched, multi-tenant, scientific kubernetes cluster. 2025. 9. L. Zhang, Y. Zhang, and X. Zhao. Colorimetric detection of heavy metal ions using silver nanoparticles. Sensors and Actuators B, 258, 2018. 8/8