doi:10.1016/j.mejo.2005.05.009 
Copyright © 2005 Elsevier Ltd All rights reserved.
Surface scaling analysis of hydrogels: From multiaffine to self-affine scaling
G.M. Buendíaa, 
, 
, S.J. Mitchellb and P.A. Rikvoldc
aDepartamento de Física, Universidad Simón Bolívar, Caracas 1080, Venezuela
bDepartment of Chemistry and Chemical Engineering, Shuit Institute for Catalysis, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
cDepartment of Physics, Center for Materials Research and Technology, School of Computational Science and Information Technology, Department of Physics, Center for Materials Research and Technology and School of Computational Science. Florida State University, Tallahassee, Florida 32306-4350, USA
Available online 27 June 2005.
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Abstract
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. Buendía, 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. 258 (2003) 186]. 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.
Keywords: Self-affine scaling; Multiaffine scaling; Hydrogels; Atomic force microscopy
PACS: 61.43.Hv; 89.75.Da; 82.70.Gg; 68.37.Ps
Fig. 1. A typical portion of the spring-network surface (20% vacancies) before and after smoothing by convolution with a Gaussian. The smoothing removes the vertical discontinuities without significantly altering the overall surface features. Such smoothing should mimic the measurement process of using a weakly interacting AFM tip to probe the structure of a soft surface.
Fig. 2. The qth root of Cq(r) from the spring-network surfaces for q=0.5 to q=4.0 in steps of 0.5. For graphical simplicity, the q labels have been omitted, but Ci(r)>Cj(r), when i>j. The surfaces are averaged over 8, 8, and 10 independent realizations of the equilibrium spring lengths for the networks with 0, 40, and 49% vacancies, respectively. The linear regions of the plot indicate power-law scaling, and for the original simulated surfaces, the different power-law exponents (slopes in the log–log plots) indicate q-dependent multiaffine scaling. After smoothing, the power-law scaling at small length scales is self-affine (parallel lines in the log–log plots) with Hq=1 for all q.
Fig. 3. Hq vs. 1/q for the spring-network surfaces. Open symbols indicate the original, multiaffine surfaces, while corresponding filled symbols indicate the smoothed, self-affine surfaces. As expected, Hq=1.0 for all of the smoothed surfaces. The observed multiaffine scaling of the original surfaces is consistent with the behavior caused by vertical discontinuities as expected from Ref. [5]. The data are averaged over 8 independent realizations of the equilibrium spring lengths for 0–47% vacancies, 10 realizations for 49% vacancies, and 14 realizations for 49.5% vacancies.
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