A density functional approach to ferrogels

Abstract

Ferrogels consist of magnetic colloidal particles embedded in an elastic polymer matrix. As a consequence, their structural and rheological properties are governed by a competition between magnetic particle-particle interactions and mechanical matrix elasticity. Typically, the particles are permanently fixed within the matrix, which makes them distinguishable by their positions. Over time, particle neighbors do not change due to the fixation by the matrix. Here we present a classical density functional approach for such ferrogels. We map the elastic matrix-induced interactions between neighboring colloidal particles distinguishable by their positions onto effective pairwise interactions between indistinguishable particles similar to a "pairwise pseudopotential". Using Monte-Carlo computer simulations, we demonstrate for one-dimensional dipole-spring models of ferrogels that this mapping is justified. We then use the pseudopotential as an input into classical density functional theory of inhomogeneous fluids and predict the bulk elastic modulus of the ferrogel under various conditions. In addition, we propose the use of an "external pseudopotential" when one switches from the viewpoint of a one-dimensional dipole-spring object to a one-dimensional chain embedded in an infinitely extended bulk matrix. Our mapping approach paves the way to describe various inhomogeneous situations of ferrogels using classical density functional concepts of inhomogeneous fluids.

Classical density functional theory for a two-dimensional isotropic ferrogel model with labeled particles

“…In one spatial dimension, the particles described by a pseudo-spring model are automatically "labeled" through the steric hard-core repulsions [54,55], which fixes their mutual ordering. Hence, within a proper range of the spring length at sufficiently high density [12], where the particles usually interact with their two nearest neighbors, the mapping works perfectly. Note that there is no phase transition due to the strong thermal fluctuations in one dimension.…”

“…Following Ref. [12], we consider a mapping of H el onto a pseudo-spring potential between unlabeled particles of the form…”

“…Finally, the red solid line presents the elastic energy of pseudo-springs u pel (r) defined in Eq. (12). p = k B T /a 2 and determine the volume V and the responses to elastic deformations ∇ u, where u is the displacement field.…”

“…Such a scenario can emerge if the particles are permanently localized with * segun.goh@hhu.de † hlowen@thphy.uni-duesseldorf.de respect to their neighbors, thereby rendering the system non-ergodic on the relevant energy scale. It is thus a major challenge for statistical mechanical theories to describe a model with labeled particles [12] or to keep track of a single localized particle [13].…”

“…However, due to the particle labeling, a direct application of the machinery of DFT to ferrogels is not possible. To this end, we will map the elastic interaction onto an appropriate pairwise pseudopotential [12] between unlabeled particles, which allows us to formulate a DFT. Ultimately, we aim at investigating the elastic properties of the dipole-spring systems within the DFT framework.…”