Zimm Plot Equation



Hiementz (p. 710) details how to take data and construct a Zimm plot. The data workup can be done on a spreadsheet. It takes hours, and is a tedious operation; you will use the 'copy' function of your spreadsheet many times. However, once completed, you can recycle the same spreadsheet for other light scattering molecular weight, , determinations.

The Zimm Plot data comes from Light Scattering data.



The above equation with all its variables has an obscene appearance. Notice that the term in brackets, shown below,



goes to unity (one) if the second half of the addition (the term with 16, , , 3, , and the sine function) goes to zero. We can make the squared sine function goe to zero by chosing to be zero. Experimentally, we need to plot points for a variety of scattering angles, , at a constant concentration, , and then we can extrapolate to a zero scattering angle. You will see this further on.

Having forced the expression in brackets to go to unity, we are left with


It makes sense that we would now want to plot a line, picking variables from the above such that the slope of our line will have a meaning, and the y-intercept of our line will have a meaning. We look at the graph below, and may be surprised to see two lines, one of which is an extrapolation for =0 (sky blue), and the other an extrapolation for =0 (sea green.) To construct a Zimm plot you will do math involving limits to obtain each line.
Some background information is needed now: We can vary , and hold constant, which will produce points as shown below:
We can hold constant, and vary , which will produce points as shown below:
We are going to do both. What you see below is a Zimm plot. You have a 2x2 grid of data points, and a scattering value corresponding to each point. For our example, the are five scattering angles and five concentrations, so it makes sense that we have twenty five data points.

I make no guarantees at this time that the above procedure is correct. I do not have the time to double check it.

We have all these points, and now we start extrapolating as shown in the graphic below with the orange lines.

For each scattering angle you will do an extrapolation. The extrapolation involves building data points, and then calculating a least squares line such that the y-intercept provides the needed information. It should make sense that the extrapolation line for =0 should select concentration to be the independent variable x, so that when x=0, the y-intercept value will be a value of interest.

Calculating the =0 line: (construction alert! this is a first draft--mistakes possible)
  1. Let the y values of your data be .
  2. Select a scattering angle, and that each (R(theta), , ) datum and calculate vs. , and the y-intercept of this least squares line is value extrapolated to =0 for each scattering angle that you used in the experiment. On your graph, you will plot the =0 line on the Zimm Plot. Take note that the Zimm plot is the final graph, but that these least square line determinations that you need are separate graphs that provide information you need for the Zimm plot.
  3. You then plug =0 into sin (theta over 2) + kc2, which of course is just Sin (theta over 2), and that is your x-coordinate value. You already know I make no guarantees at this time that the above procedure is correct. I do not have the time to double check it.

    Each orange line connects points of with the same scattering angle, but changing concentration. At the end of the line, the sky blue point is the extrapolation to zero concentration.

    We now turn our attention to the yellow extrapolation lines used to generate the theta=0 line. This time we take the data for a given concentration and make an extrapolation, and we will have an extrapolation for each concentration. It makes sense that this time, we should build our least squares line using x = the scattering angle theta, and y = Kc2 over R(theta.) The y-intercept gives the value of Kc2 over R(theta) corresponding to theta=0 and you can calculate the x-value for the extrapolation, sin (theta over 2) + kc2, which is just kc2. This builds the theta=0 line.
    Each yellow line connects points of the same concentration, but changing scattering angle. At the end of the line, the sea green point is the extrapolation to zero scattering angle. All this builds the bilinear plot we saw at the beginning of this web page.


    The slope of the = 0 line is used to calculate , the second virial coefficient.

    The slope of the = 0 line is used to calculate the radius of gyration, . We note that the green and blue lines intersect at the value of where is zero and is zero. We recall



    which was made possible by = 0, and note that if is zero, then



    from which we can say that is the inverse of molecular weight.

    Earlier we've said that the Zimm plot is used to determine weight average molecular weight. But it's not that simple. We assume the blue line, which was the extrapolation for =0, is a straight line, but what if it deviates? As you can see below, if you are plotting a straight line near =0 you will get , but if you are some distance from zero concentration, you could get a straight line that would give you 2 times .     Hmmm... this makes me wonder, has anyone been clever and run a lot of points and ended up using the Zimm Plot to get both and , or is this just too risky for theoretical reasons...

    There should be more here, but I've run out of time. I hope you found this useful.





    Last Update- September 9, 1995- wld