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50mer polymer

Coordination Number



The coordination number, z, is the number of neighboring points adjacent to a lattice point that make contact with the lattice point. It is used in the lattice model.

In the model below, the coordination number is 6. For this example, the six neighbors are located:



It could be asked what about the points in a model that don't have six contacts, like on the edges, or the corners?

Simple answer: their contribution to the overall average is so small that it can be neglected. This statement is probably fairly easy to accept without proof. However, the math exercise might be used later in a review of algebra operations.

We will build several cubes for this z=6 model, and compare the number of adjacent neighbors to the coordination number, 6.

There are four types of points in the model, identified by color as used in the graphic above:

  1. corner points, yellow
  2. edge points, orange (anything on the edge that is not a corner)
  3. face points, blue (anything on a face that is not on the edge and is not on a corner)
  4. interior points, green

2 x 2 x 2- There are 8 corner points, and each corner has 3 neighbors. Average number of neighbors- 3.

3 x 3 x 3- There are 8 corner points, 12 edge points, 6 face points, 1 interior point. An edge point has 4 neighbors, a face point has 5 neighbors, and an interior point has 6 neighbors. Average number of neighbors- 4.

4 x 4 x 4- There are 8 corner points, 24 edge points, 24 face points and 8 interior points. Average number of neighbors- 4.5.

5 x 5 x 5- There are 8 corner points, 36 edge points, 54 face points and 27 interior points. Average number of neighbors- 4.8.

6 x 6 x 6- There are 8 corner points, 48 edge points, 96 face points and 64 interior points. Average number of neighbors- 5.0.


Putting this into a table:

n
corners
edge
face
interior
2
8
0
0
0
3
8
12
6
1
4
8
24
24
8
5
8
36
54
27
6
8
48
96
64


corners: 8
edges: 12(n-2)
faces: 6(n-2)2
interior: (n-2)3




How many lattice points are in a real application? For an example of polystyrene dissolved in toluene, the volume surrounding a lattice point is defined by the volume of a molecule of toluene.

A 250 ml reactor filled with toluene, contains 1.42 x 1024 molecules, so there are over 1 x 1024 lattice points.

This would require a 108 x 10 8 x 108 lattice, and the average number of neighbors is 5.99999994.


Basic information: toluene- density 0.867 g ml-1, molecular weight 92.1402 g mol-1; Avogadro's number 6.0221415 x 1023