Second Virial Coefficient Control Enables Quantitative Spatial Matching in Polymer-Supported Cooperative Catalysis

Cooperative catalysis has emerged as a powerful strategy for accessing reactivity beyond the reach of single-catalyst systems through synergistic effects between multiple catalytic centers. However, achieving precise and quantitative spatial control over inter-catalyst distances, critical for cooperative enhancement, remains a fundamental challenge, as existing approaches lack tunable molecular-level design parameters. Here, we demonstrate a quantitative spatial matching strategy on dual nanobrush-grafted hollow nanospheres, where bifunctional initiating sites enable grafting of two different brushes that selectively anchor carbene and phosphoramidite catalytic centers. Using the second virial coefficient ( A 2 ) as a quantitative descriptor of nanobrush interactions, we established the relationship between brush architecture and catalytic center proximity. The optimally matched system ( A 2 = 7.29 × 10 −5 mol·cm 3 ·g −2 ) delivers 94% yield, >99% ee, and 20:1 dr in [3 + 2] annulation, whereas mismatched architectures exhibit rate constants up to 3.4-fold lower. This work establishes A 2 as a quantitatively predictive descriptor for cooperative catalysis within the studied dual-brush system, where two ligands directly coordinate to a shared metal center.