Classical Liquids in Fractal Dimension

Abstract

We introduce fractal liquids by generalizing classical liquids of integer dimensions d ¼ 1; 2; 3 to a noninteger dimension d l . The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here, we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semianalytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results. DOI: 10.1103/PhysRevLett.115.097801 PACS numbers: 61.20.Gy, 61.20.Ja, 61.43.Hv The liquid state, an intermediate between gas and solid, exhibits short-ranged particle pair correlations in isotropic shells around a tagged particle [1]. While the shell structure is lost for gases, solids exhibit long-ranged correlations and anisotropy. Particle correlations are accessible by experiments [2,3] and contain valuable information about the particle interactions. A fundamental task in theory and computer simulation of the liquid state is to predict particle correlations for given interactions. In this respect, one particularly successful approach is liquid integral equation theory [1,4].Molecular and colloidal liquids can be restricted to one or two spatial dimensions [5,6] by confining them on substrates [7], at interfaces [8], between plates [9,10], or in channeled matrices [11] or optical landscapes [12]. The thermodynamic properties change accordingly [13] and the systems are called one-dimensional or two-dimensional (2D) liquids. While these classical examples of confinement involve integer-dimensional configuration spaces, there are also cases where liquids are confined in porous media [14][15][16][17], or along quenched polymer coils [18], and where the configuration space exhibits noninteger (fractal) dimension at suitable length scales. Much work has been devoted to understanding the limiting cases of low and high density-namely, the motion of a single particle on a fractal [19][20][21][22][23] and the structure of a fractal aggregate itselfcorresponding to an arrested high-density particulate system [24][25][26][27][28]. In all of these situations, particles interact in the embedding integer-dimensional space and interaction energies depend on Euclidean particle distance.In this Letter, we break new ground by considering fractal particles in a fractal configuration space, both of the same noninteger dimension. An alternative model in which the particle dimension differs from the configuration-space dimension is briefly mentioned near the end of the Letter. Figure 1 is a snapshot from one of our Monte Carlo (MC) simulations. The interaction between two particles is not described by a pair potential in Euclidean spa...

Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation

“…As we will see, despite the simplicity of the theory, the results agree fairly well with both MC simulations and PY numerical results obtained in Ref. [5] at d = 1.67659.…”

“…IV with those of the generalized PY theory and the MC data presented in Ref. 5. The paper concludes in Sec.…”

“…Motivated by the recent work of Heinen et al [5], in this paper we have constructed a heuristic approximation for the RDF g (d) (r, φ) and its associated structure factor S (d) (k, φ) of hard-core fluids in fractal dimension 3,exact are below ±1%. Further, the comparison between our results for g (d) (r, φ) and the available MC and PY results [5] for hard-core fluids in fractal dimension (d = 1.67659), indicates that our proposal is much simpler and as accurate as the PY theory, and in some instances it shows a slightly better agreement with the MC data than the PY theory.…”

“…Such a generic model of fractal liquids can describe, for instance, microphase separated binary liquids in porous media and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Heinen et al [5] performed Monte Carlo (MC) simulations of fractal hard * andres@unex.es; http://www.unex.es/eweb/fisteor/andres/Cvitae/ † malopez@unam.mx; http://xml.ier.unam.mx/xml/tc/ft/mlh/ "spheres" lying on a near-critical percolating lattice cluster. In this context, a fractal sphere of "diameter" σ is defined by those lattice sites at a chemical distance (taxicab metric) from the sphere center site smaller than In that solution, the corresponding radial distribution function (RDF) and thermodynamic properties represent the analytic continuations of the standard PY theory with respect to the dimension.…”

Flexibly imposing periodicity in kernel independent FMM: A multipole-to-local operator approach

“…If the requirement of accuracy can be relaxed, M 2L can be used to approximate a ring-channel geometry to study the spontaneous coherent flows created by active suspensions [33; 34]. The flow of liquids in fractal geometries [35] is another example of interesting problems that could be studied.…”