CALIFORNIA STATE UNIVERSITY, SACRAMENTO

Department of Mechanical Engineering

Paskowitz Spring 2000

Course Web Page http://webpages.csus.edu/~passki/

Spreadsheet Exercise 1

Calculating Ionic Bonding Forces

ENG-45
 
Introduction
 
Spreadsheets are valuable tools for any student or practicing engineer who does calculations. They can aid in performing complex calculations and graphing the results, for database applications and for importing data from experiments for analysis and graphing. Modern spreadsheets provide a number of advanced numerical tools which allows one to solve complex equations. Oddly, many students and even practicing engineers don't know how to use a spreadsheet. Therefore we offer this exercise to put these valuable tools in your hands to help you solve many of the problems presented during this course and in future courses you take.

This exercise involves generating a spreadsheet which will produce a graph that illustrates a fundamental concept in materials science. It utilizes three equations and their associated parameters plus several physical constants. We approach this calculation in a simple, progressive manner which will allow one to examine each aspect of the program as it is added and to build upon its basic foundation. The approach taken here is a good basic approach to building any reusable spreadsheet program, especially those which will continue to grow.
 

The Problem: Ionic Bonding Potentials
 

The forces involved in ionic bonding are described in section 2.5 of Van Valck's text. Briefly, the force holding two oppositely charged ions at a particular distance from each other consists of two parts. The first is the force of the coulombic attraction between the oppositely charged ions

where ko is a constant, a is the interatomic spacing, Z is the valence and q the charge (±1.602xl0^-19 C) on each ion. The second part is the repulsion by the overlapping of negatively charged electric fields and the positively charged nuclei
 

where, l and r are both constants. The total bonding force is equal to the sum of the repulsive and attractive forces.
 
 

The General Layout of the Spreadsheet
 
 

Define the Constants and Parameters Used by the Equations

All parameters used by the equations above should be defined in the spreadsheet independently of the equations. While one could enter them as constants in the cells containing the equations they should occupy cells in another part of the spreadsheet. The reason for this is that we'd like to be able to examine and experiment with the values of these parameters.

The parameters used by these equations are actually functions of temperature. Instead of entering the value for these constants, we could enter the temperature dependent function instead and define the temperature in yet another cell. That way we could experiment with the effect of temperature on ionic bonding. While this would be a much more interesting calculation we would not have to alter any other part of the spreadsheet. The equations for the interatomic forces would still reference the same cells to obtain the values of the parameters.

In general, it is better to define the parameters used by an equation and other constants such as physical constants in cells rather than in the equations themselves. This makes it easier to debug the program, to change the units and to experiment with the equations.
 
 

Enter the Equations for the Interatomic Forces in Columns

The main part of the spreadsheet will be a series of columns of equations for the independent and dependent variables. In this exercise the independent variable is the interatomic spacing while the dependent variables are the forces of attraction, repulsion and their sum which is the total bonding force. The dependent variables could be combined into one large equation, and one column, but long equations are difficult to debug so it is best to keep them simple. Plus we'd like to plot each force to see how they relate to the total bonding force.
 
 

Construct a Graph

The graph will be a standard x-y graph with interatomic spacing as the x axis and the interatomic forces plotted along the y axis. The line colors and styles plus the legend will be used to identify the different forces.
 
 

Construct the Spreadsheet

Header

Start by entering the basic details about this spreadsheet program such as the title, owner and date it was created. Next, enter a brief description of this calculation and any literature citations that might be important. This header can be very useful when referring to this work at a later date.
 
 

Parameters

Enter the values for each parameter for each equation in cells just below the spreadsheet's header. Include the names of the parameters and the units you are using. If the units are not those you will need to use in the equations then include a cell with the appropriate units. This cell should be a simple calculation using a units conversion factor. Your equations will reference these cells, saving you the trouble of including the conversion factors in the equations.
 

The Independent Variable - Interatomic Spacing

Put the interatomic spacing in the first column. Start with a value of 0 at the top of the column and in the next cell add to it the increment da. Repeat this for the next 300 or so cells in the column, referencing the cell above for the value to add da to. For maximum flexibility the increment da should be defined in a separate cell in the parameters section of this spreadsheet. This will let you easily change the range and resolution of your calculations.
 
 

The Dependent Variables - Attractive and Repulsive Bonding Forces

In adjacent columns enter the equations for the attractive and repulsive forces. Enter it at the top of the column and replicate it so that it uses each of the cells in first column (interatomic spacing).
 
 

Total Bonding Force

Add columns 2 and 3 together in column 4.
 

Annotate

Label each column. Don't forget to include the units.
 

Graph

Set up an xy graph where column 1 contains the values for the x axis and columns 2 through 4 contain the values for the first 3 series of the y axis. Label the axes appropriately and include a legend. Insert the graph into the spreadsheet, above the columns and to the right of the parameters.
 

Print

Print the graph and also the top portion of the spreadsheet which includes the parameters, results and the inserted graph.
 
 

Further Analysis

This basic spreadsheet can be used to calculate the equilibrium spacing between the ions, Young's modulus and the bulk modulus. One can even calculate the energy of interatomic bonding and then estimate the melting point of the material.

1 . Examine the column containing the total bonding force and find the value of the interatomic spacing where the force is equal to zero. This is the equilibrium interatomic spacing of the ions.

where n is the number of atoms in the unit cell. Compare this to the bulk modulus calculated using

Calculate these bonding energies and plot the results. 

Special Thanks to Dr. Mike Meier from UC Davis for providing this material for the lab section.

© Dr. Michael L. Meier 1999