Collective dynamic length increases monotonically in pinned and unpinned glass forming systems

The Random First Order Transition Theory (RFOT) predicts that transport proceeds by cooperative movement of particles in domains whose sizes increase as a liquid is compressed above a characteristic volume fraction, \(\phi_d\). The rounded dynamical transition around \(\phi_d\), which signals a crossover to activated transport, is accompanied by a growing correlation length that is predicted to diverge at the thermodynamic glass transition density (\(> \phi_d\)). Simulations and imaging experiments probed the single particle dynamics of mobile particles in response to pinning all the particles in a semi-infinite space or randomly pinning (RP) a fraction of particles in a liquid at equilibrium. The extracted dynamic length increases non-monotonically with a peak around \(\phi_d\), which not only depends on the pinning method but is different from \(\phi_d\) of the actual liquid. This finding is at variance with the results obtained using the small wave length limit of a four-point structure factor for unpinned systems. To obtain a consistent picture of the growth of the dynamic length, one that is impervious to the use of RP, we introduce a multi particle structure factor, \(S^c_{mp}(q,t)\), that probes collective dynamics. The collective dynamic length, calculated from the small wave vector limit of \(S^c_{mp}(q,t)\), increases monotonically as a function of the volume fraction in glass forming binary mixture of charged colloidal particles in both unpinned and pinned systems. This prediction, which also holds in the presence of added monovalent salt, may be validated using imaging experiments.