We explore the impact of excluded volume interactions on the local assembly of linear physical networks, where nodes are spheres and links are rigid cylinders with varying length. To focus on the effect of elongated links, we introduce a minimal 3D model that helps us zoom into confined regions of these networks whose distant parts are sequentially connected by the random deposition of physical links with a very large aspect ratio. We show that the nonequilibrium kinetics at which these elongated links, or spaghetti, adhere to the available volume without mutual crossings is logarithmic in time, as opposed to the algebraic growth in lower dimensions for needle-like packings. We attribute this qualitatively different behavior to a delay in the activation of depletion forces caused by the 3D nature of the problem. Equally important, we find that this slow kinetics is metastable, allowing us to analytically predict the kinetic scaling characterizing an algebraic growth due to the nucleation of local bundles. Our findings offer a theoretical benchmark to study the local assembly of physical networks, with implications for the modeling of nest-like packings far from equilibrium.
