Mass Transport Phenomenon Modeling

Introduction

This project delves into the comprehensive study of mass transport phenomena, taking a distinctive approach from traditional models that often employ the diffusion equation. At the microscopic level, these phenomena involve intricate processes such as fragmentation, adsorption, diffusion, and aggregation, which are prevalent in nature.

The study of mass transport phenomena has garnered widespread interest among researchers. These non-equilibrium phenomena pose challenges in determining steady states using Gibb’s distribution. The significance of these studies extends beyond simulation and mathematical analyses, finding applications in various physical phenomena like growing interfaces, colloidal suspensions, polymer gels, river networks, granular materials, traffic flows, and more.

Project Overview

The fundamental movement of mass in this project is termed "chipping," where a unit of mass nn chips off from a site having mass mi(≥n)mi​(≥n), aggregating to a randomly chosen adjacent site. While a simplified "conserved mass model" considers only fragmentation and aggregation, our project adopts a more general approach. It includes not only fragmentation and diffusion but also the adsorption of a unit of mass at a randomly chosen site, adding complexity to the model.

The study employs a 1-dimensional lattice where mass is randomly distributed at different lattice sites. For simplicity, we assume that the mass at one site is not affected by the mass present at adjacent lattice sites, enabling the treatment of each site individually. The movement of mass from one site to a neighboring site is considered under mean field (MF) approximation, which tracks the distribution of masses while ignoring correlations in the occupancy of adjacent sites. Despite the shortcomings of the MF theory, our Monte Carlo simulations demonstrate its accuracy in describing the model, particularly in a 1-dimensional case.

Model Dynamics

Our model incorporates two crucial processes: fragmentation and adsorption, governing the system's evolution over time. The dynamic equation that describes transport and steady-state behavior depicts the evolution of $P(m,t)$ with time. Here, $P(m,t)$ represents the probability of having mass mm at time tt. The study of mass-transport phenomena is thus focused on understanding the evolution of $P(m,t)$. Rate equations, encompassing all elementary moves in the system, have been solved for different chipping kernels, either analytically or numerically using Monte Carlo methods.

(Note: Only the simulation part is presented here!)

Rate Equations

The rate equation. Let $P(m, t)$ denote the probability that a site has mass $m$ at time $t$. In Mean Field (MF)* approximation, which keeps track of the distribution of masses, ignoring the correlations in the occupancy of adjacent sites, $ P(m, t) $ evolves as follows:

$$ \frac{dP(m, t)}{dt} = -P(m, t) \sum_{m_1=1}^{\infty} g_m(m_1) - P(m, t) \sum_{m_2=1}^{\infty} P(m, t) \sum_{m_1=1}^{\infty} g_{m_2}(m_1) - q $$

$$(m, t) + \sum_{m_1=1}^{\infty} P(m + m_1, t) g_{m + m_1}(m_1) + \sum_{m_2=1}^{\infty} P(m - m_2, t) \sum_{m_1=1}^{\infty} P(m_2, t) g_{m_2}(m_1) + q P(m - k, t) \quad \text{for } m \geq 1 $$

$$ \frac{dP(0, t)}{dt} = -P(0, t) \sum_{m_2=1}^{\infty} P(m_2, t) \sum_{m_1=1}^{\infty} g_{m_2}(m_1) + \sum_{m_1=1}^{\infty} P(m_1, t) g_{m_1}(m_1) - q P(0, t) \quad \text{(2)} $$

These equations enumerate all possible ways in which a site with mass ( m ) may change its mass. Various terms on the R.H.S of equation (1) can be interpreted as follows:

1st term: loss term due to chipping of a mass ( n ) from a site having mass $( m (\geq n) )$

2nd term: loss term due to coalescence chipped mass after chipping from neighbour

3rd term: Loss term due to adsorption of a unit mass with rate ( q ) at a randomly chosen site, where ( q ) is the adsorption of 'k' units of mass, and $( k = 1, 2, 3, \ldots )$ in various ((k) + 1) and ((k) + k).

4th term: Gain term due to fragmentation of excess mass from a site having mass greater than ( m ).

5th term: Gain term due to coalescence to gain the deficit mass at a site having mass less than ( m ).

6th term: Gain term due to adsorption of a unit mass with rate ( q ) at a randomly chosen site having mass less than ( m ).

*MF approximation allows treating each site independently; otherwise, rate equations will contain complicated terms which may make the equation hard to solve.

computational

Simple Jupyter File:

There

The repository includes a simple Jupyter file for running and experimenting with the provided code snippets.

Getting Started:

Clone the repository: git clone [repository-url]
Navigate to the desired code block directory.
Run the Python script or Jupyter notebook: python filename.py or jupyter notebook filename.ipynb

Dependencies:

Python 3.x
Required Python packages (NumPy, Matplotlib, Jupyter)

Acknowledgements

I would like to express my sincere gratitude for the guidance and support I received during this project, which was an integral part of my master's course at the Indian Institute of Technology Delhi, India (2011-2013). I extend my deepest appreciation to Professor Versha Banerjee, under whose supervision I conducted my thesis. The invaluable insights and mentorship provided by Professor Banerjee played a pivotal role in the successful completion of this project