Diffusion and Flexibility in Networks

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This award is co-funded by the Divisions of Materials Research and Mathematical Sciences under the umbrella of the NSF-wide Mathematical Sciences Priority Area.The properties of networks of atoms are determined to a large degree by the constraints present, which include fixed bond lengths and angles. This theoretical research extends previous work on statics to study the effect of constraints on dynamical behavior of networks. The emphasis will be on the diffusion of sticky spheres that are packed together. The theory of constraints will also be extended to finite temperature as it relates to the thermal properties of network glasses. Constraint theory replaces the strong forces in a network by hard constraints. This allows the rigid regions, both unstressed and stressed, to be determined, as well as the location of the hinges, especially near the rigidity phase transition, where the whole network transforms from flexible to rigid.The present theory of constrained networks in three-dimensions is restricted by the necessity to include bond bending forces everywhere along with central forces. While this is appropriate for some networks, it has been found to be restrictive to wider applications. We therefore will develop a general theory for the rigidity of generic networks in three-dimensions, with the aim of finding a very integer algorithm. This will be added to the software on flexibility that is available free to academic users at the website that has been set up at flexweb.asu.edu.Diffusion in sphere packs can be studied experimentally using a confocal microscope, and it is found that certain regions are locked so that no diffusion occurs over the time of the experiment, while diffusive motion is possible in other regions. We will predict these regions, using a single initial configuration to determine constraints, and the theory as generalized above. Such an approach will provide a bridge between statics and dynamics.A new model will be introduced that will allow the effect of constraints to be tracked as the temperature increases up through the glass transition temperature. This will lift the current restriction to zero temperature, and allow entropic effects to be included. The potential is constructed with the sticky spheres locked together at low temperature, so that constraint theory is applicable, but as the temperature is increased, entropic effects become important and can be tracked thermodynamically. This work will help to explain the differences in the fragility of glasses as seen through the viscosity as a bulk glass is formed from the melt.The new theory that is developed has broader implications, particularly to bio-molecules, where function is often associated with flexibility, and diffusive motion. Proteins in the native state unfold as the temperature is increased above room temperature to become denatured, and so entropic effects are important.%%%This award is co-funded by the Divisions of Materials Research and Mathematical Sciences under the umbrella of the NSF-wide Mathematical Sciences Priority Area.The properties of networks of atoms are determined to a large degree by the constraints present, which include fixed bond lengths and angles. This theoretical research extends previous work on statics to study the effect of constraints on dynamical behavior of networks. The emphasis will be on the diffusion of sticky spheres that are packed together. The theory of constraints will also be extended to finite temperature as it relates to the thermal properties of network glasses. Constraint theory replaces the strong forces in a network by hard constraints. This allows the rigid regions, both unstressed and stressed, to be determined, as well as the location of the hinges, especially near the rigidity phase transition, where the whole network transforms from flexible to rigid.The new theory that is developed has broader implications, particularly to bio-molecules, where function is often associated with flexibility, and diffusive motion. Proteins in the native state unfold as the temperature is increased above room temperature to become denatured, and so entropic effects are important.***