Lattice animal

The chirality quantification technique proposed by Harary and Mezey [54,55] is motivated by the Resolution Based Similarity Measure (RBSM) approach used in more general molecular similarity analysis [243]. This method does not rely on a single reference object. Instead, it characterizes shape on any desired finite level of resolution by considering various A(J,n) parts of square lattices, called lattice animals or P(G,n) parts of cubic lattices called polycubes which can be inscribed within the two- or three-dimensional objects J or G, respectively. In the above... 

Figure 6.2 Lattice animals of less than six cells. The mirror images of chiral animals are given in pairs, with respect to horizontal reflection planes. The animals of four cells are "Skinny", "Fatty , "Knobby", "Elly , and "Tippy", in the order listed in the figure.

For both two- and three-dimensional chirality, a formal degree of chirality has been introduced [54,55], based on a discretization of shape features using lattice animals and polycubes [240,243]. These definitions are based on chiral animals and chiral polycubes, for which chirality can be detected by simple algebraic means. 

Since the smallest chiral lattice animals have four cells [54], the minimum possible value for chirality index is n (J)=4. The degree of chirality X(J) of a Jordan curve J is defined as... 

The actual determination of a set M for some chiral set T and the calculation of the volume v(T) are usually rather difficult problems (see some relevant comments in references [51-53,58,240,242]), and the same applies for superset N. However, within a RBSM framework, the analogous chirality measures given in terms of a discretization procedure using polycubes (or lattice animals in 2D) [240] do not require the explicit determination of a maximal volume (area) achiral subset M and the calculation of its exact volume (or area) v(M). 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

Assuming that = 3.37 ( lattice animals [31]), we obtain d, = 0.93-1.98. This interval agrees with the range of d found experimentally [66]. 

Equation (11.18) has two peculiar features firstly, D depends appreciably on d and, secondly, there exists a critical dimension of Euclidean space d = % for which D = 4 in accordance with the ideal statistical model, i.e., the model withont correlations. At d > 8, the correlations cansed by the effect of excluded volume are no longer significant and Df does not change. The value d = % was found in studies of branched polymers and lattice animals [79]. Calcnlations using formula (11.18) are in good agreement with the known results for lattice animals. ... 

As it is known [52], for the lattice animals case in ( dimensional Euclidean spaces the value D is given by the equation ... 

Euclidean dimension Flory formula lattice animals Monte-Carlo method phantom fractal true self-avoiding walk... 

There is quite a funny muddle of terms here a tree is the generally accepted word for what we have described, yet its lattice model is known as a lattice animal. Does this indicate how well-informed physicists are about biology ... 

Figure 1. Fractal dimension of clusters on square lattice. The uppermost arrow shows maximal dimension d = D = 2. The percolation threshold point is shown by the pair of arrows below the upper right corner of the diagram. The lattice animal limit is marked by the o symbol, the dashed line marks the (uncertain) course of the 0 dependent dimensionality at /3 < 0. The two arrows along the left vertical axis mark the vedues of self-avoiding random walk at d = 4/3 and the limiting value of the dimensionality (d = 1) of straight clusters in the limit /3 — —oo.

Figure 2. Fractal dimension of clusters on cubic lattice. See also the caption of Fig. 1. Note that in three dimensional case there are three values of fractal dimensions in close vicinity lattice animal limit d = 2.08, marked by the o symbol), self-avoiding random walk (d = 1.7) and ordinary random walk (d = 2) represented by the arrows at the left vertical axis.

Figure 4- Snapshot of a cluster with 5000 lattice points at the lattice animal limit.

The value V = 0.5 is consistent with a random-walk structure for the gel, as well as any other structure of fractal dimension 2. This result was frequently found in colloidal silica aggregates and was confirmed by X-rays experiments performed on dry silica. Structures of fractal dimension 2 include random-trees (lattice animals) with excluded volume. The random-tree would seem a likely suggestion for the short-scale structure of a silica gel. 

Consider a single crosslinked cluster, where the crosslinks can be everywhere along the chains. The mean field theory of Flory and Stockmayer calculates for the size of the cluster the typical N law, where N is the total amount of monomers in the cluster. This law is true in ideal noninteracting systems for any kind of branched molecule or lattice animal. " The cluster of size R has then... 

In the percolation problem one has for the spectral dimension approximately, d = 4/3, independent of the space dimensions according to the Alexander and Orbach conjecture. It is also the mean field value for lattice animals (branched structures defined on lattices) or Cayley tree-like structures. Hence the Cayley tree corresponds to the mean field solution to percolation. ... 

The consideration of microgels as a new class of polymers has been put forward mainly by Funke who has developed a great deal of expertise on these systems. Microgels will be defined as crosslinked polymers roughly of spherical structure with dimensions of the order of the size of ordinary polymer molecules, branched polymers or lattice animals. Thus microgels are microscopic networks. Their physical properties depend on crosslink density, their connectivity and presence of solvent, etc. The formation of these objects is therefore a microgelation problem , and we may use the usual theories of gelation or vulcanization. 

---