Polymer Viscosity

Here's a simple problem. You have a polymer and a solvent and you want some idea of how the viscosity of that solution depends on the polymer's MWt and how "happy" it is in the solvent.

It turns out that the answer is far from simple and I am not aware of any really good ways to get an answer. So I have created this plausible app based on a formula¹ that makes plenty of sense but which does not seem widely known or used.

The viscosity we want is the Relative Viscosity, η_r which is that of the polymer solution η_p divided by that of the pure solvent η_s. We want to know how it depends on concentration, C, (in units of g/dl, i.e. g/100ml which is ~ weight %) up to a maximum value, C_max. The first of many equations tells us that:

`η_r=η_p/η_s`

First start with the default inputs and vary MWt and then the χ parameter (0=perfect solvent up to 0.5=theta solvent) get a feel for what happens, then read more details below.

Now we can take things step by step. The key facts of polymer life is that at a concentration C* the polymer chains start touching each other, significantly increasing viscosity, transitioning from the "dilute" to "semi-dilute" regime, then at Ce they enter the entangled domain and viscosity really starts to increase. It is this concentration zone where most formulators find themselves. Incidentally, most polymer scientists are much happier discussing things in the entirely uninteresting dilute regime below C*.

We can calculate C* by knowing the MWt, Avogadro's number Av and the Radius of Gyration, R_g.

`C^** = (MWt)/(Av.R_g^3 )`

So now we need to know R_g. There is a well-known formula based on the number N of pseudo-monomers of length b, and in the bottom row you can try out this formula. What is b? Typically it's a few real monomer units, but for your specific polymer, who knows. Use your instincts. If you set things so that the MWt of this pseudo-polymer is 40K then you can use the estimated R_g in the real calculation. Alternatively R_g can be measured via scattering experiments for those who are keen.

But R_g depends on MWt and on how "happy" the polymer is in the solvent. To simplify the app as much as possible, you input R_g @0.5+40K which means the value in a theta solvent (χ=0.5) for a 40K MWt polymer, a value I chose for no special reason. As you change the MWt and χ, a new R_g@χ+MWt is calculated and shown in the row below.

To calculate the viscosity in the domain above C* we use the Rouseian formula:

`η_r = (|η|C)^1.3`

Which means that we need to calculate the Intrinsic Viscosity |η| which is a limiting value of (η_p-η_s)/(C.η_s) in units of 1/concentration. We get it from:

`|η| = (5AvRg^3)/(MWt)`

Finally, we can calculate our relative viscosity by multiplying the Rouseian viscosity with a term dependent on C/Ce:

`η_r = (|η|C)^1.345(C/"Ce")^3`

Ah, but for this we need Ce which comes from:

`Ce = Ne^(3v-1)C^**/6.6`

And for this we need v (which varies from 0.5 at theta to 0.585 for χ=0) and Ne, which is the number of monomer units between entanglements which depends on the two inputs we've not mentioned so far, M_c, the critical entanglement MWt above which the polymer tangles, and M_mono which is the monomer MWt.

`Ne = M_c/(3M_(mono))`

We know rather few M_c values and the table (data kindly provided by my colleague Dr YAMAMOTO, Hiroshi) are only for some relatively pure polymers. They cannot (in practice) be calculated so they have to be measured. Hopefully this app will help people to measure M_c values for some real-world polymers, though I admit that the approach is rather indirect.

Should we bother?

Thank goodness for apps (and for some VB code I wrote 10 years ago when I first tried to understand all this). At least the tedious calculations are taken care of in a plausible manner. But how does it apply to your polymer in your solvent system? If you don't know R_g and M_c (and most of us don't) then it might all seem a waste of time. But hold on, we're not completely lost. If you happen to have just a single η_r measurement and you know your MWt and can guess whether the solvent is really good (χ=0), neutral (χ=0.5) or in between, then you can make some plausible guesses for M_c and R_g, so you can at least get a feel of what might happen if you changed concentration or MWt. With two datapoints then you will have a quite reasonable idea of the whole system. And, finally: