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The mystery of permeation theory: Why can liquids flow freely in porous materials?
In the research of materials science and applied physics, penetration theory plays an indispensable role. When liquids are poured into porous materials, a critical question often arises: Can the liquid penetrate the material smoothly and reach the bottom? This problem involves not only physics but also mathematical modeling and has broad applications in various fields of science and engineering.
Percolation theory studies the behavior of a network when additional nodes or links are added, especially when a critical point is reached where originally separate small pieces merge into large connected sets.
The root of all this lies in the understanding of random networks. Suppose we pour a liquid onto a porous material. Our goal is to determine whether the liquid can find a path between the porous pores. Mathematically, this process is modeled as a three-dimensional network of n × n × n vertices, where each edge (or "link") between two adjacent vertices (called "sites") ”) can be open (i.e., liquid can pass) or closed (i.e., liquid cannot pass) with a certain probability.
The fundamental problem in this situation, called edge penetration, was first proposed in the mathematical literature by Broadbent and Hammersley in 1957.
This model provides a mathematical framework for thinking about liquid flow in porous materials. By varying the value of p, the model captures the probability of liquid flow available from the upper part of the material to the lower part. Research has shown that when p approaches a certain critical value, the prediction of flow rapidly increases from almost zero to a high probability close to one. This is not only applicable to mathematical models, but also reflects the physical reality of liquids flowing in porous structures. characteristics.
Historical background of penetration theory
The development of permeability theory can be traced back to the needs of the coal industry. Since the Industrial Revolution, research into the properties of coal has fueled many scientific quests to understand its composition and optimize its use. In 1942, when Rosalind Franklin began studying the density and porosity of coal at the Coal Utilization Research Association (BCURA), she delved into the porosity of coal and came up with various test results, which showed that the microstructure of coal and its The size of the pores varies depending on the carbonization process.
Franklin's research showed that the pores in coal can be used as tiny screens to filter gases based on their molecular size.
The theory was further developed in the early 1950s by the statistical work of Simon Broadbent, whose work at BCURA led him to ask the question of how liquids diffuse in pores in coal. This question further led him to discussions with John Hammersley, which ultimately led to the formulation of a mathematical model of the osmotic phenomenon.
Calculation of critical parameters
Although the critical probability pc cannot be calculated accurately for most infinite grids, some specific grids have clear critical values. For example, in a two-dimensional planar grid, the critical probability of edge penetration is known to be 1/2. This result was determined by Harry Kersten in the early 1980s and has been verified in numerous simulations and theoretical models.
These research results not only deepen the understanding of permeability theory, but also provide a valuable mathematical basis for the behavior of liquids in porous structures.
The behavior of critical points has a long and complex history across different network types and their structural properties. The clustering degree, degree distribution and other characteristics of the network will correspondingly affect the threshold and characteristics of the penetration process. These further understandings allow scientists to apply the theory in fields as diverse as biology, ecology and virology to shed light on mobility issues in different systems.
Application scope of penetration theory
The application of penetration theory in various fields continues to expand. In biology and biochemistry, osmosis theory is used to predict the fracture behavior of biological virus shells, as in the study of the hepatitis B virus shell, with the random removal of key subunits, which can lead to rupture of the shell.
Such results are similar to the common jigsaw puzzle Jenga, helping to reveal a complete picture of the virus's decomposition process.
In ecology, the study of the impact of environmental fragmentation on animal habitats, as well as the application of plague bacteria diffusion models, have shown the practicality of osmosis theory. These examples not only demonstrate the importance of percolation theory in theoretical physics but also highlight its potential for practical applications.
As research continues, permeability theory continues to provide insights into the behavior of material flow, challenging our understanding of porous materials and fluid dynamics. If liquids can flow freely through these materials, does it mean we can explore more deeply how fluid dynamics behave in different environments?
