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Register a new userSurface scaling analysis of hydrogels: From multiaffine to self-affine scaling
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We show that smoothing of multiaffine surfaces that are generated by simulating a crosslinked polymer gel by a frustrated, triangular network of springs of random equilibrium lengths [G.M. Buend{i}a, S.J. Mitchell, P.A. Rikvold, Phys. Rev. E 66 (2002) 046119] changes the scaling behavior of the surfaces such that they become self-affine. The self-affine behavior is consistent with recent atomic force microscopy (AFM) studies of the surface structure of crosslinked polymer gels into which voids are introduced through templating by surfactant micelles [M. Chakrapani, S.J. Mitchell, D.H. Van Winkle, P.A. Rikvold, J. Colloid Interface Sci., in press]. The smoothing process mimics the effect of the AFM tip that tends to flatten the soft gel surfaces. Both the experimental and the simulated surfaces have a non-trivial scaling behavior on small length scales, with a crossover to scale-independent behavior on large scales.
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We propose and study a simplified model for the surface and bulk structures of crosslinked polymer gels, into which voids are introduced through templating by surfactant micelles. Such systems were recently studied by Atomic Force Microscopy [M. Chakrapani et al., e-print cond-mat/0112255]. The gel is represented by a frustrated, triangular network of nodes connected by springs of random equilibrium lengths. The nodes represent crosslinkers, and the springs correspond to polymer chains. The boundaries are fixed at the bottom, free at the top, and periodic in the lateral direction. Voids are introduced by deleting a proportion of the nodes and their associated springs. The model is numerically relaxed to a representative local energy minimum, resulting in an inhomogeneous, ``clumpy bulk structure. The free top surface is defined at evenly spaced points in the lateral (x) direction by the height of the topmost spring, measured from the bottom layer, h(x). Its scaling properties are studied by calculating the root-mean-square surface width and the generalized increment correlation functions C_q(x)= <|h(x_0+x)-h(x_0)|^q>. The surface is found to have a nontrivial scaling behavior on small length scales, with a crossover to scale-independent behavior on large scales. As the vacancy concentration approaches the site-percolation limit, both the crossover length and the saturation value of the surface width diverge in a manner that appears to be proportional to the bulk connectivity length. This suggests that a percolation transition in the bulk also drives a similar divergence observed in surfactant templated polyacrylamide gels at high surfactant concentrations.
Diffusion-limited aggregation is consistent with simple scaling. However, strong subdominant terms are present, and these can account for various earlier claims of anomalous scaling. We show this in detail for the case of multiscaling.
We present a field-theoretic renormalization group (RG) analysis of a single flexible, screened polyelectrolyte chain (a Debye-Huckel chain) in a polar solvent. We point out that the Debye-Huckel chain may be mapped onto a local field theory which has the same fixed point as a generalised $n to 1$ Potts model. Systematic analysis of the field theory shows that the system is one with two interplaying length-scales requiring the calculation of scaling functions as well as exponents to fully describe its physical behaviour. To illustrate this, we solve the RG equation and explicitly calculate the exponents and the mean end-to-end length of the chain.
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1/r^{alpha}$ for $alphageq0$. Here, we review old nonextensive scaling. In addition, we propose a new Hamiltonian scaled by $2frac{(N/2)^{1-alpha}-1}{1-alpha}$ that explicitly includes symmetry of the lattice and dependence on the size, $N$, of the system. The new approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.
A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents $ u$ and $eta$, the crossover exponent $phi$, as well as the (related) wave-vector exponent $beta_q$, and the correction-to-scaling exponent $omega$ to second order in $epsilon_8=8-d$. These are compared with the authors recent $epsilon$-expansion results [{it Phys. Rev. B} {bf 62} (2000) 12338; {it Nucl. Phys. B} {bf 612} (2001) 340] for the general case of an $m$-axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ($m=d$) Lifshitz point all follow from the latter upon setting $m=8-epsilon_8$. This is so despite recent claims to the contrary by de Albuquerque and Leite [{it J. Phys. A} {bf 35} (2002) 1807].
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