Thus, the formation of anisotropic NP phases must arise from higher-body interactions between NPs, that is, perturbative corrections to the free energy from the higher-order arrangement of particles beyond pairwise distances. Typically, the interaction free energy 𝐹 of an 𝑁 -NP system is formulated as a sum over all pairwise interactions between NPs, 𝐹 ≅ ∑ ∑ 𝑊 ! &𝛀 " , 𝛀 # ) #$" " , where 𝑊 ! &𝛀 " , 𝛀 # ) is the two-body PMF between particles 𝑖 and 𝑗 suitably averaged over all degrees of freedom associated with the grafted and matrix chains, leaving it dependent only on the positions and orientations of the two NPs, which we denote by 𝛀 " and 𝛀 # . Thus, the "effective" potential 𝑊 ! implicitly accounts for steric repulsion arising from the grafts and depletion-like attraction arising from the matrix, in addition to direct energetic interactions between the cores and grafts of the NPs. For spherical NPs grafted uniformly with polymer chains, 𝑊 ! is expected to be isotropic and hence a function of interparticle separation distance 𝑑 "# only. While such two-body description of interparticle interactions works well for many colloidal systems, higher-body interactions have been shown to become significant for NPs grafted with sufficiently long chains at sufficiently high density. When two NPs come into contact to form a dimer, their grafts are expelled from the region in between them due to the excluded volume of the NP cores. This leads to an increase in the graft segment density around the contact region of the dimer relative to its poles, causing a third approaching NP to experience additional steric repulsion in that region. This anisotropic repulsion is indeed what causes the NPs to sometimes form anisotropic structures like 1D strings and 2D sheets in a homogeneous polymer matrix. Other structural phases like gels have also been observed in NP-polymer systems at high particle loadings. The role of many-body interactions in promoting anisotropic assembly is best illustrated using the simple example of four NPs-the smallest set of particles capable of forming 1D, 2D, Figure . Role of three-body interactions in governing the morphology of clusters formed by polymergrafted NPs. The left panel shows the fully dispersed and idealized 1D, 2D, and 3D cluster morphologies (string, rhombus, and tetrahedron) formed by four such NPs. The middle and right panels show schematics of the two-and three-body interactions (free energies) between NPs that lead to these distinct morphologies. The free energies 𝐹 !" , 𝐹 #" , and 𝐹 $" of the three cluster morphologies relative to the dispersed state are also shown. These were calculated by enumerating the number of two-body contacts (blue arrows) and triangular three-body contacts (red triangles) in each cluster that respectively contribute attractive and repulsive energies, the maximum values of which are marked by Asterix. The stability conditions are then 𝐹 $" < 𝐹 #" (or