Fluctuations of the local magnetic field in frustrated mean-field Ising models

We consider fluctuations of the local magnetic field in frustrated mean-field Ising models. Frustration can come about due to randomness of the interaction as in the Sherrington - Kirkpatrick model, or through fixed interaction parameters but with varying signs. We consider central limit theorems for the fluctuation of the local magnetic field values w.r.t. the a priori spin distribution for both types of models. We show that, in the case of the Sherrington - Kirkpatrick model there is a central limit theorem for the local magnetic field, a.s. with respect to the randomness. On the other hand, we show that, in the case of non-random frustrated models, there is no central limit theorem for the distribution of the values of the local field, but that the probability distribution of this distribution does converge. We compute the moments of this probability distribution on the space of measures and show in particular that it is not Gaussian

Grafting Bimodal Polymer Brushes on Nanoparticles Using Controlled Radical Polymerization

RAFT (reversible addition–fragmentation chain transfer) polymerization has been widely used to synthesize different polymer architectures such as polymer brushes on nanoparticles for incorporation into polymer nanocomposites. It is believed that these polymer brushes, with the same chemistry as the matrix polymer, can be employed to improve filler dispersion by compatibilizing unfavorable enthalpic interactions between the inorganic nanoparticles and their organic host matrices. However, monomodal brush graft nanoparticles are found to aggregate into a range of isotropic and anisotropic morphologies, formed due to a delicate balance between enthalpic and entropic interfacial interactions. This coupling of enthalpy and entropy leaves only a small window of graft densities and molecular weights to obtain randomly dispersed filler morphologies. These issues can be countered by using a bimodal polymer brush that contains a small number of long homopolymer chains that can entangle, and a high density of short brushes that screens the particle/particle attraction, thereby aiding in decoupling the interfacial enthalpic and entropic interactions. In the present work, we demonstrate a robust step-by-step technique using RAFT polymerization to synthesize these bidisperse/bimodal polymer brush-anchored nanoparticles. A layer of dense brush of the first population was initially prepared using surface-initiated RAFT polymerization from colloidal silica nanoparticles. After cleavage of the chain transfer agent from the first population of chain ends, a second RAFT agent was attached onto the silica nanoparticles and then a monomer, which may be the same or different from the first brush, was polymerized. This versatile and widely applicable route enables us to independently control the molecular variables of the attached chains, such as composition, molecular weights and graft densities of the individual populations. The bimodal brush-grafted colloidal silica nanoparticles show superior dispersion and interaction with a homopolymer matrix when compared to monomodal brush-grafted particles