I.
Introduction
Researchers
studying the use of ceramic/graphene composites in diverse industrial
applications include material scientists. Very few theoretical studies on the
rGO in nano-ceramic matrix are published to date, despite substantial research
on rGO nano-ceramics in diverse applications and in describing the rGO ceramics
nano-structure. Because they are influenced by the proportion and orientation
of rGO in the ceramic matrix, mechanical and electrical characteristics of rGO
nano-ceramics are crucial.
A
recent theoretical examination of the connection between the proportion of rGO
in a ceramic matrix and the structure's electrical and mechanical
characteristics is described in [1]. This theoretical work increases the understanding
of electrical conductivity and fracture toughness using sophisticated
mathematical methodologies. It is possible to undertake a thorough theoretical
investigation of the structure's structural entropy and electrical conductivity
by expressing the rGO-included ceramic structure as a polyhedral complex. The
paper also shows a theoretical method for figuring out the best rGO percentage
to use in order to maximise structural and electrical attributes. The
combinatorial structure of various orientations of rGO inclusion in ceramics
matrix can lead to variable structural entropy and electrical conductivity
values. The suggested method can produce electrical conductivity and structural
entropy values that are ideal. The simulation study's findings support the
theoretical framework of the rGO nano-ceramics analysis. Future studies can use
this data to identify the proper proportion of rGO in the nano-ceramic matrix.
II. Preliminaries
Over
the past 10 years, a large body of research has been conducted on reduced
graphene oxide (rGO) ceramics, including multiple papers that have been
published in a variety of publications. The development of applications for
certain industries depends heavily on the utilization of rGO ceramics. The
applicability and exploration of choosing the right rGO inclusion for certain
applications are crucial for commercially viable products.
There
are scientific and commercial uses for studying rGO/ceramic nano-composites.
There have been very few theoretical investigations to better understand the
behaviour of rGO nano-composites [1, 2], despite the fact that disconnected
rGO's characteristics are well understood [2,]. The existing theoretical
investigations in numerous research publications do not provide a strategy for
logically building rGO composites with the necessary mix of mechanical and
physical characteristics. The areas of materials science and design might
greatly benefit from modern mathematical techniques. There are several
techniques for modelling and characterising material design topologies in
advanced discrete mathematics [3]–[7]. Recently, computational materials
structure has made use of machine learning [8], neural networks [9], fractal
theory [10], and biological pattern production [11]. Because of the intriguing
discrete topologies of connected 1D, 2D, and 3D substructures in bulk
nanocrystalline composites, such as rGO plates, mathematical topology is very
helpful in enhancing computer analysis. In material science and material
design, contemporary mathematical methods are an invaluable tool. Bulk
nano-crystalline composites with 2D inclusion of rGO plates may be described as
complex discrete topologies of linked substructures.
A.
Polyhedron
The
collection of solutions to a system of linear inequalities with non-negative
variables that makes up a polyhedron may be written as,
……….
Hence,
we can state that, a polyhedron P can
be defined as a set such that
Numerous
mathematical categories, including 0-dimensional faces (vertices),
1-dimensional faces (edges), 2-dimensional faces (planar polygons), 3-dimensional
faces, and others, can be used to categorise the polyhedron's substructures.
This is so that the polyhedron's topology of linked substructures may be
articulated. We are now restricting the topological structure to a small convex
3-dimensional polyhedron in order to simplify the computational analysis.
III. Computational Model
A. Polyhedral complex
A
collection of polyhedra in a real vector space that fit together in a
particular way are known as polyhedral complexes in mathematic [1]. It is helpful
to discretize complicated forms and see them as the union of basic building
pieces that have been adhered together in a "clean way" in order to
examine and handle them. Simple geometric shapes such as points, lines,
segments of lines, triangles, tehrahedra, and more generally simplices, or even
convex polytopes, should be used as the building blocks. Polyhedral complexes
generalise simplicial complexes and appear in a number of polyhedral geometric
situations, such as tropical geometry, splines, and hyperplane configurations.
Regular quasi-convex discrete topological complexes are separated into
polyhedral complexes [6, 12, 13], where 0 and 1 cells are represented by points
or vertices, 2 and 3 cells by planar polygons, and so on. We are particularly
interested in polyhedral 3-complexes with precisely three convex polyhedrons
acting as 0-cells and three convex polyhedrons acting as 3-cells (Fig.1). Using
the Voronoi tessellating approach, these complexes are created by surrounding
each 1-cell with precisely three 2-cells and three 3-cells, and each 0-cell
with precisely four 1-cells, six 2-cells, and four 3-cells [1]. This kind of
mathematical description is widely utilised in molecular dynamics simulations
[14]–[16] because it precisely mimics the microstructures of actual materials.
Fig 1: Polyhedral complex and its elements [1]
B. Structural entropy
The
2-cells of the 3-complex being occupied or not by rGO plates serves as a
representation of the rGO/ceramic nanocomposite microstructure. It is
reasonable to compare rGO plates to 2-cells since their breadth is just a few
atomic layers, which is insignificant when compared to the size of the 3-cells.
Calculating the rGO-occupied 2-cells requires knowing the proportion (p) of
feasible 2-cells in the structure. pN2 is the combinatorial realisation of rGO
inclusion in an N-sized vector, where N2 is the total number of 2-cell
configurations that are feasible. rGO 2-cells' pN2 number may be the
consequence of several geometric realisations. Based on the quantity of rGO
inclusions, the 1-cell may be divided into four different categories, just as
the 2-cell.For the sake of simplicity, consider the four types of 1-cells as J_{ω} for ω = 0,1,2,3.
Fig 2: Triple junction of rGO plate[1]
As per [1], the structural entropy can be described
as,
This definition is closely related to the Shannon
informational entropy [17] and, on the other hand, to the alloy configurational
entropy [18].
C. Conductivity
Electrical conductivity is the main physical
characteristic of interest in nanostructured ceramics. The conductivity and
fraction p are inversely related. The 2-cell random distribution provided in
rGO is a rather straightforward and well-developed issue. The proportion of
peculiar 2-cells defines the percolation threshold in this example to be p =
0.08. Above the threshold, the graph's conductivity exhibits a quasilinear
dependency on p as,
Here
for considered 3-complex [1].
is the conductivity for p = 1 and H represents
the Heaviside step function. The equation shows a linear growth of conductivity
at large enough p.
IV. Simulation results
For the polyhedral complex, we have simulated the
structure using 500X500 odes in which 2-cell nodes are of fraction and
p fraction of f represents the number of rGO inclusions in the matrix. We
can iterate the the structure with different random configuration and can
obtain the conductivity value for various p and f values at each iteration of
the process. The theoretical curve of rGO portion vs Structural Entropy is as
follows (fig 3):
Figure 3: rGO portion vs Structural Entropy
Whereas rGO portion vs
Conductivity is linear except of those values where the Heaviside function is
considered zero (fig 4).
Figure 4: RGO portion vs Conductivity
Using the given equation, the structural entropy and conductivity can be found as fig 5 (a). The simulation runs for various random configurations
with different p and f values. The analysis requires the maximum conductivity
in each fixed p and f. The structural entropy vs conductivity values for a
Gaussian distribution are shown in fig 5.(b):
(a)
(b)
Figure 5:
Structural entropy vs Conductivity for rGO inclusion using (a) theoretical
equation and (b) using computation for Gaussian distribution with mean zero and
variance 0.010 and 0.001