We present the results of Monte Carlo lattice simulations of a model symmetric diblock copolymer wherein a fraction of segments of one block, p, corresponds to ionic species, and the other block does not contain ions. We use experimentally determined Flory−Huggins interaction parameters, χ, to quantify the interactions between ionic and nonionic monomers. Analysis of the experimental data indicate that χ between poly(styrenesulfonate) and polystyrene is about 5, a value that is orders of magnitude larger than that obtained in mixtures of nonionic polymers. Our model predicts that clustering of ionic monomers in the disordered state results in stabilization of the disordered phase and the product p 2 χN is well above the mean-field value of 10.5 at the order−disorder transition (N is the total number of monomers per chain). Network morphologies and hexagonally packed cylinders are observed in the ordered state at large p values while more traditional lamellar phases are found at small values of p.
This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.
The dynamics of spatial hierarchical processes displays fascinating images of the evolution of attraction basins of towns. Assuming in the present paper a random character of such complex processes, gravitational modeling is employed to depict changes in the nature of fuzzy boundaries between attraction basins in a hexagonal world, i.e. the geographical pattern central in the renowned and influential Christaller–Lösch central place theory in geography. The paper presents various interesting stylized maps of such stochastic processes. Our findings demonstrate that the collective behavior of actors in geographic space shows a white noise pattern that mirrors a fuzzy gravitation towards large population concentrations. These results support the validity of the conventional central place theory.
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