This lab will begin with simple explanations of viscosity (internal friction) and the basics of its application in polymers. As a simple example, we provided an animation with two funnels, each containing a different fluid, for example, orange juice and maple syrup.

You can press the "GO" button to start the fluids in motion. This simple demonstration should serve as a solid display of the fundamental properties of two different types of fluids. After watching the animation, it is obvious that maple syrup flows at a much slower rate than orange juice. Polymer solutions can be compared in a similar way. Flow properties in polymer solutions depend almost entirely upon their molecular weight.

To demonstrate flow properties in general, the user will use
what could be called the ‘dropping-ball method.’ In
this method, a vessel is filled with a solvent, like water, in
which a metal ball of known volume and mass is dropped
into—the distance the ball travels over a time period *t*
is measured. How long the ball takes to travel the distance for
different solvents is a simple way to compare the levels of flow
between materials. This little experiment will help to further
illustrate flow properties, and is a little bit less scientific
in its appearance. In this animation, simply pull the lever to
drop the metal balls into the two beakers and watch the
difference in time between the two solvents.

Now that you have some basic background information on the
flow properties of polymers and liquids in general—you can
perform a polymer solution virtual experiment. We have provided a
capillary-type, drop-time viscometer lab. You are probably
wondering just what is a capillary-type, drop-time viscometer.
First of all, viscosity determines the rate at which a liquid
flows - in our case the liquid is a
diluted, polymer solution. The funnel animation above showed that
maple syrup is more *viscous *than orange juice. A *viscometer*
is simply a device used to measure viscosity. In this virtual
lab, you will run time trials for polymer solutions, use this
information to find the viscosity, and finally, use the viscosity
to determine the molecular weight. You are probably wondering why
the end result of a viscosity experiment is a molecular weight?
Molecular weight and viscosity have more in common than you might
think. Think of the polymer molecules as if they were tiny balls.

If you fill a funnel with large balls, it takes a long time for them to leave through the funnel’s opening. On the other hand, tiny balls in a funnel ‘flow’ out much faster. You can see how this may prove useful in the study of polymers - the viscosity can be a dead giveaway of the molecular weight.

In the previous paragraphs, we used animations and some general information to get you up to speed with flow, viscosity, and molecular weight - now its time to learn the formulas that make it all work. Flow is the irreversible, viscous property of a material as opposed to the reversible, elastic behavior of some materials. Viscous fluids support shear stress and exhibit shear strain and the viscosity is defined by this ratio:

Stress(s ) = [Viscosity(h )] * [Strain Rate]

If this equation is true for a material, the stress versus shear rate flow curve is linear and the material is known as a Newtonian fluid. Only a small number of specific examples exist, as only a few simple fluids have the molecular make-up to behave in this manner.

We can illustrate the terms in this definition for the case of laminar flow in which the fluid layers move smoothly over one another.

Here, the portion of the fluid abcd is steadily deformed as the upper plate is pulled along by force F. The instantaneous deformation illustrated here is described by a strain bb'/h. The strain rate, then, is the rate of change of bb'/h = v/h. The applied shear stress is just s = F/A where A is the area of each plate.

Previously, we used the example of funnels filled with orange juice and maple syrup. When released, the orange juice will obviously pour out of the funnel at a much faster rate. It is said that maple syrup is a more viscous fluid than orange juice. This of course is elementary when compared to polymer viscosity, but it is a good example to demonstrate flow properties and how it could be tied in to the study of polymers. In the field of polymers, a solution of high molecular weight carries a viscosity that is noticeably higher than that of a solution of lower molecular weight. We can understand this using a simplified model of a ball-like configuration for the polymer molecules in a dilute solution. Because of the velocity gradient in a direction perpendicular to the flow, a larger ball will experience a wider range of shear rates and, as it deforms, provide greater drag than does a smaller (lower molecular weight) ball.

This leads us to the task of measuring viscosity of dilute
solutions. In the past, a drop-time capillary viscometer was the
instrument of choice for the measurement of characteristics of
polymer solutions. In this sort of viscometer, the solution is
released from the capillary in which the time *t* is
recorded for the meniscus of the solution to travel some distance
*d.* This time is called an *efflux time,* and it is
measured for several concentrations of the polymer solution, as
well as for the solvent with no polymer in it. In recent times,
there is more of a push towards dual-capillary viscometers that
are arguably more efficient. At any rate, the goal when using a
viscometer is to, of course, obtain a viscosity for the solution,
and ultimately characterize the solute molecules. The efflux time
measured when using the viscometer is used to find the *relative
viscosity *(RV or h _{rel}),
which is the ratio of solution viscosity divided by solvent
viscosity; so that, in terms of efflux times:

h _{rel }= t
/ t_{o}

where *t* is the efflux time of the solution and *t*_{o}*
*is the efflux time of the solvent. From the relative
viscosity all other solution viscosity quantities can be
obtained. However, before we use the relative viscosity, there is
another quantity, the *specific viscosity* (h _{sp}), that can be calculated
using the efflux times of solution and solvent. The specific
viscosity can be defined as the fractional increase in viscosity
caused by the presence of the dissolved polymer in the solvent.

h _{sp }= [t
- t_{o}] / t_{o}

From the specific viscosity and the concentration ( c ) of the
solution, the *reduced viscosity* (h
_{red}) can be obtained.

h _{sp} / c
= h _{red}

The reduced viscosity changes as a function of concentration. A plot of reduced viscosity versus concentration would look something like this:

h _{red} =
k’[h ]^{2}c + [h ]

The y-intercept, [h ], is the *intrinsic
viscosity* which is related to the slope which is k’[h ]^{2}*. *The intrinsic
viscosity is the quantity that relates viscosity to the molecular
weight and the intrinsic structural differences of the solute
molecules—it is obviously important in determining the
molecular weight. There is, however, a second way to calculate
the intrinsic viscosity. This method involves calculating the *inherent
viscosity* (h _{inh}) from
the relative viscosity, and finally, using it to calculate the
intrinsic viscosity.

ln h _{rel}
/ c = h _{inh}

Once again, the inherent viscosity changes as a function of concentration. So, the graph of inherent viscosity plotted versus concentration looks like this:

And again, the intrinsic viscosity, [h
], is the y-intercept, but in this case the slope is
different—it is k"[h ]^{2}.
In the following equation, we can relate the intrinsic viscosity
to the inherent viscosity:

h _{inh} =
k"[h ]^{2}c + [n]

Now, we have obtained the intrinsic viscosity using two different methods. It is normal procedure to calculate it using both methods; if the results from each calculation match, then the measurements taken were most likely correct. Displaying both plots of intrinsic viscosity on the same graph should also back up your results—with the two lines meeting at the same y-intercept.

Now, you have the information necessary to calculate the molecular weight. The equation:

[h ] = K’ M^{a}

is all you need to complete the calculations. This is called
the Mark-Houwink equation. *M *represents the viscosity
average molecular weight, and *K’* and *a* are
Mark-Houwink constants. These constants depend upon the
polymer-solvent combination you are dealing with. One last thing
to remember—these equations and this experiment in general
deals with *diluted* solutions. If you try to use this data
for a polymer solution that is too concentrated, the calculations
will not be as accurate. For example, if the solution is too
concentrated, the molecules may be close enough to each other to
start to interact. This could cause the viscosity to increase in
ways that these equations do not support.

Viscosity can be used to determine flow properties other than
molecular weight. For example, the *melt flow index* (MFI)
is another useful piece of information when studying polymers
that also ties in to molecular weight. The melt flow index is a
measure of how much polymer flows through an orifice over a given
time period under a constant pressure. This index also gives you
an indication of what the molecular weight is.

For a more complete discussion of solution viscosity, see
Ferry, J. D. (1980)* Viscoelastic Properties of Polymers, 3rd
ed.* (JohnWiley & Sons, New York) or Sperling,
L. H. (1986) *Physical Polymer Science.* (John Wiley &
Sons, New York)