Collective Mechanical Behavior of Multilayer Colloidal Arrays of Hollow Nanoparticles

The collective mechanical behavior of multilayer colloidal arrays of hollow silica nanoparticles (HSNP) is explored under spherical nanoindentation through a combination of experimental, numerical, and theoretical approaches. The effective indentation modulus <i>E</i>_ind is found to decrease with an increasing number of layers in a nonlinear manner. The indentation force versus penetration depth behavior for multilayer hollow particle arrays is predicted by an approximate analytical model based on the spring stiffness of the individual particles and the multipoint, multiparticle interactions as well as force transmission between the layers. The model is in good agreement with experiments and with detailed finite element simulations. The ability to tune the effective indentation modulus, <i>E</i>_ind, of the multilayer arrays by manipulating particle geometry and layering is revealed through the model, where <i>E</i>_ind = (0.725<i>m</i>^–3/2 + 0.275)<i>E</i>_mon and <i>E</i>_mon is the monolayer modulus and <i>m</i> is number of layers. <i>E</i>_ind is seen to plateau with increasing <i>m</i> to <i>E</i>_{ind_plateau} = 0.275<i>E</i>_mon and <i>E</i>_mon scales with (<i>t</i>/<i>R</i>)², <i>t</i> being the particle shell thickness and <i>R</i> being the particle radius. The scaling law governing the nonlinear decrease in indentation modulus with an increase in layer number (<i>E</i>_ind scaling with <i>m</i>^–3/2) is found to be similar to that governing the indentation modulus of thin solid films <i>E</i>_{ind_solid} on a stiff substrate (where <i>E</i>_{ind_solid} scales with <i>h</i>^{<i>‑</i>1.4} and also decreases until reaching a plateau value) which also decreases with an increase in film thickness <i>h</i>. However, the mechanisms underlying this trend for the colloidal array are clearly different, where discrete particle-to-particle interactions govern the colloidal array behavior in contrast to the substrate constraint on deformation, which governs the thickness dependence of the continuous thin film indentation modulus