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Chapter 5: Viscosity

Last updated

Jan 28, 2025
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- Chapter 4: Flory Huggins
- Chapter 6: Osmometry

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Measuring Properties of Polymers:

We have run into several key properties of polymers like molecular weight, end-to-end distance, etc., and there are a number of other key polymer behaviors that depend on some of these key properties like

Non-cross linked rubber elasticity
Shear thickening
Elastic modulus of cross linked networks
Electrical conductivity

The question then becomes if these polymer properties are so critical to behavior how to do we measure them? This question is quite difficult for example it will be important in terms of rubber elasticity to know the distance between cross-links or how do we measure χ? Well we will be talking about several techniques to

Determine the relationship between the viscosity of dilute solutions and the molecular weight of the polymers
Understand the difference between the free draining and non-draining regimes of viscosity.

Measuring Viscosity:

Solution and Melt Viscosity η:

Most of you are likely familiar with the concept of viscosity (peanut butter is more viscous than water) and have likely encountered this in a fluid mechanics class however we will give a very brief review here. We will use η to describe viscosity and from mechanics we know that

$\tau = \eta \frac{d\gamma}{dt} \nonumber$

where τ is shear force, η is viscosity, and $\frac{d\gamma}{dt}$ is strain rate. You can think of this equation in the context of a channel filled with some fluid. When you apply a shear force to one plate the plate will apply a shear force to the liquid. You’ll notice that this equation has a very similar form as Hooke’s law but we replace strain with strain rate and Young’s modulus with viscosity. We see that the faster the strain rate the larger the resistive shear force and the larger the viscosity the larger the shear force. This makes sense as pulling a plate through water is easier than pulling it through peanut butter. At very slow strain rates the shear force goes to 0 which is a property of liquids whereas solids can be approximated as having infinite viscosity so they will not flow. You can see this scenario schematically here

Figure $\PageIndex{1}$ : Shearing a Fluid.

When the visocity is independent of strain rate or total strain as is the case above that material is Newtonian. Polymers however, are typically nonNewtonian, i.e. viscosity is not a constant quantity but is dependent on strain rate or total strain we can see an example in one of my favorite Youtube Videos in the lecture slides

Stokes Law and Viscosity Scaling

So to start with our viscosity discussion let’s start with a typical scenario. We have our polymer (dry, solid, or lypholized) and add some liquid solvent. How will this effect viscosity? We will determine this using Stokes law and see how the viscosity of a solution will scale with molecular weight and identify two different scaling regimes

Freely Draining Regime
Non-Draining Regime and these two regimes lead to the non-Newtonian behavior we saw in the video.

So to start let’s think of an idealized scenario that physicists’ love, a sphere with radius, R_sfalling through a fluid with a viscosity η₀at a velocity v. The viscous force acing on the sphere is simply

$F_{viscous} = fv \nonumber$

where f is the friction factor of the solution. You can see this equation is very similar to our shear stress and shear strain relationship thus we will expect f ∝ η and the scaling laws should also scale similarly. Now you might be asking yourself well what is the expression for the friction factor in terms of the variable in our system. Well luckily Stokes already solved this in 1851 with some fairly complex math and found that

$f = 6\pi \eta_0 R_s \nonumber$

Figure $\PageIndex{2}$ : Stokes Drag Force.

You might see in some texts this frictional force denoted as F_dand referred to as Stokes Drag Force. Now we have one issue and that is while we can maybe get away with the fact that we can approximate the shape of a coiled polymer as a sphere we can’t always approximate the polymer as a solid sphere. There are two size extremes or regimes we must consider which we mentioned previously. The non-draining regime where the polymer is coiled into one impenetrable sphere where fluid cannot pass, hence non-draining. The freely-draining regime assumes the polymer is permeable to solvent and in this case the polymer is composed of a larger number of small spheres (monomers) and each has some friction factor associated with it. Note: Stokes law is derived from solving the Navier Stokes equation and beyond the scope of the class but again I can provide resources if you are interested.

Non-Draining Regime:

Here each monomer is a solid sphere drags some solvent and zooming out the polymer is a highly coiled bunch of monomers that is effectively a single large sphere with a radius that approximates the radius of gyration. Now each monomer will affect the fluid flow and frictional force around the other monomers, thus the overall friction force applied to the entire coil is lower than for a single similar sized impenetrable sphere. The monomers can only affect or effectively shielding neighboring monomers from the fluid flow effects (i.e. hydrodynamic interactions) if the neighboring monomer is very close.

Thus this model is accurate/appropriate for highly coiled polymers. So in the non-draining case the fluid flow is highly perturbed by the monomers inside the coiled up polymer so that it effectively acts as a solid sphere. Let’ go ahead and calculate the friction factor for this regimes which is Stoke’s law directly

$f = 6\pi \eta_0 R_H \nonumber$
$R_H \approx \gamma \langle R_G^2\rangle ^\frac{1}{2} \nonumber$
$f \approx x_2^\frac{1}{2} \nonumber$

You will notice that we simply replaced the radius of the sphere with the hydrodynamic radius, R_H, which we can approximate as radius of gyration (or the RMSD) with some constant prefactor γ (typically around 0.85-1). But again we are focused primarily on scaling in this course so we see that the friction will scale as n^1/2in theta solvent and as n^3/5in a good solvent.

Freely Draining Case

In the freely-draining case the polymer is idealized as an elongated rod where each monomer is a single sphere and each monomer/sphere will not effect the fluid flow or effective friction on neighboring monomers. The total effective friction will be the sum of the friction force felt by each monomer and will thus be larger than the impenetrable sphere or non-draining case as there is no shielding. This model is appropriate for highly elongated polymers, polymers in good solvents, or polymers at high strain rates.

Figure $\PageIndex{4}$ : Freely Draining Regime.

To calculate the friction factor for the free draining case we have to account we have to account for the friction of all the beads so the friction will be the friction per monomer multiplied by the number of monomers

$f \approx \zeta x_2 \nonumber$

where ζ is the constant monomeric friction factor (assumption). We can clearly see here that in the non-draining regime the friction scales with the square root of the number of monomers while in the freely draining regime scales linear with the number of monomers. This leads to an extremely larger friction factor for the freely draining case and thus larger viscosity!

So if the friction factor differs greatly in these two regimes we need to understand when each regime is applicable/occurring. When we have very high shear forces and shear rates the polymer will elongate and thus it will be described very well by the freely-draining model. The friction factor will increase as will viscosity. This explains the non-Newtonian behavior of polymers because when we change shear rates the viscosity will change and increase as well. Alternatively when we have a high molecular weight flexible chain, that can be highly coiled, the non-draining model is applicable. So we can start compiling scaling factors below:

f ≈ x₂- low MW / rod-like polymers - freely-draining case
$f \approx x_2^\frac{1}{2}$ - theta conditions - Gaussian coil - non-draining
$f \approx x_2^\frac{3}{5}$ - good solvent - also non-draining

Intrinsic Viscosity Measurements

Clearly viscosity and molecular weight are critical parameters and we can measure intrinsic viscosity which will then reveal other important polymer properties like solvent conditions and viscosity-average molecular weight. To start with this technique we need to introduce the Einstein equation for viscosity of a solution containing impenetrable (hard) spheres of volume fraction Φ_hs

$\eta = \eta_0(1 + 2.5\Phi _{hs}+...) \nonumber$

Here η is still the polymer viscosity, η₀is the solvent viscosity and Φ_hsis the hard sphere volume fraction which can be calculated by (number of spheres)(vol. of sphere)

$\Phi_{hs} = \frac{\text{(number of spheres)(vol. of sphere)}}{\text{Total volume of Polymer}} = \left ( \frac{c_2}{M/N_{av}} \right ) \bigg (\frac{4}{3}\pi\gamma^3\langle R_G^2 \rangle^\frac{3}{2}\bigg ) \nonumber$

where the total number of spheres is the ratio of the concentration of polymer c₂by the molecular weight of the polymer M (scaled by Avogadro). Note that Φ_hsis not equivalent to Φ₂due to different assumptions in Flory-Huggins and this model. The volume of the sphere is your typical equation for sphere where we substitute the radius of our polymer from the discussions above.

We can now rearrange the Einstein equation to define the specific viscosity

$\eta_{sp} = \frac{\eta}{\eta_0} - 1 = 2.5\Phi_{hs} \nonumber$

Again the physical meaning in this equation describes that the specific viscosity is the increase in viscosity beyond that of the solvent due to polymer additives. Now the intrinsic viscosity is the limit of the specific viscosity as the polymer concentration goes to 0

$[\eta] = \lim_{c_2 \to 0} \frac{\eta_{sp}}{c_2} = \lim_{c_2 \to 0} \frac{2.5\Phi_{hs}}{c_2} \nonumber$

The intrinsic viscosity is the infinitesimal increase in solution viscosity as you first add the polymer or alternatively the viscosity increase in the limit of an infinitely dilute solution. So this a property of a polymer, not a property of the solution and we know that the Chemist’s Chain model gives us

$\langle R_G^2 \rangle^\frac{1}{2} \approx \langle r^2 \rangle^\frac{1}{2} \approx l C_\infty^{1/2} M^{1/2} \alpha \nonumber$

Note that I’ve used M instead of n because I want to represent the molecular weight of the chain, which is related to the length of the chain. We can now substitute this expression back into our intrinsic viscosity equation and get

$[\eta]_\theta = K_\theta M^\frac{1}{2} \nonumber$

here all the constants (not dependent on molecular weight) have been encapsulated into K_θfor theta conditions and we see that the intrinsic viscosity scales with.

We can make this more general if we include the α parameter in the above substitution and get

$[\eta] = \alpha^3 [\eta]_\theta = KM^a \nonumber$

where you can see if were are in θ conditions we get our original θ expression back but note that a is how the molecular weight will scale depending on α. This equation is known as Mark-Houwink equation. Here we can see that if a = 0.5 we are in θ conditions. This illustrates how important of a parameter intrinsic viscosity is as this parameter relates directly to solvent conditions and molecular weight.

We know some values of a in previous lectures

a < 0.5 - poor solvent
a = 0.5 - theta conditions
a > 0.5 - good solvent

We thus see that intrinsic viscosity relates directly to both molecular weight and solvent quality, two important properties we would like to measure.

Experimentally Determining Intrinsic Viscosity

Now comes the key question you might be asking, well how do we actually perform these measurements in order to obtain these key values. Well first you will typically measure the specific viscosity for a series of different concentrations. Typically the way these measurements are taken is by extruding molten polymer and measuring the flow rate and back calculating the specific viscosity. Once you have these measurements you will make a plot of specific viscosity normalized by concentration as a function of concentration and the functional form of the graph is

$\frac{\eta_{sp}}{c_2} = [\eta] + \text{slope}c_2 \nonumber$

So once you have this plot the key thing to do will be to extract the y-intercept which will give you the intrinsic viscosity because of our previous relationship that

$[\eta] = \lim_{c_2 \to 0} \frac{\eta_{sp}}{c_2} \nonumber$

Nice! But we are not done yet. We can also get information about the solvent quality as well by running this same set of experiments for a series of molecular weights. So you would run the same set of experiments previously for different molecular weights

Figure $\PageIndex{5}$ : Intrinsic Viscosity Measurement.

and then you would create a Log-Log plot of intrinsic viscosity as a function of molecular weight remembering we have the following relationship

$[\eta] = KM^{a} \nonumber$

So on this Log-Log plot the slope will give us our solvent quality parameter a and we can also find our prefactor term via the y-intercept if that was a parameter of interest.

Summary: Viscosity

We first studied the characterization of polymer solutions by determining the intrinsic viscosity of a solution - using the Einstein relation, we could then derive the Mark-Houwink equation, which gives an idea of the viscosity-average molecular weight as well as solvent quality. We also talked about two different regimes of polymer solution viscosity (not to be confused with intrinsic viscosity): the freely draining regime, where the polymer is treated as an elongated rod with separate, distinct monomers, and the non-draining regime, where the polymer is treated as a single, spherical, coiled up molecule. We identified the different conditions under which these models are applicable, and derived how polymer viscosity scaled with molecular weight in the two regimes.

Figure $\PageIndex{6}$ : Solvent Quality Determination.

Intrinsic viscosity

Key Ideas: Einstein relation allows us to determine viscosity of solution based on concentration of additives; relate intrinsic viscosity to size of added polymer; relate size of added polymer to solvent quality, viscosity-average molecular weight.
Key Equations: Mark-Houwink equation - $[\eta] = K\overline{M}_V^a$ . K is constant, M_Vis viscosity average molecular weight, a is scaling exponent that gives solvent quality.
Key Experimental Method: Take two measurements - first plot viscosity of solution vs. polymer concentration, obtain intrinsic viscosity from extrapolation to origin. Next, obtain intrinsic viscosity (again from plot of viscosity vs. concentration) for variety of molecular weights, plot log[η] vs. logM_V. Obtain K from intercept, a from slope, and with this can determine M_Vfrom any given value of [η].
Key Insights: obtaining solvent quality from exponent a - can compare scaling to theta solvent (a = 0.5). Can obtain M_V, which falls between number-average and weight-average molecular weights.

Viscosity Averaged Molecular Weight

We have previously discussed both number averaged and weight averaged molecular weight but in measuring intrinsic viscosity we can derive a viscosity averaged molecular weight which is given below

$\overline{M}_V = \left ( \frac{\sum_i n_i M_i^{1+a} } {\sum_i n_i M_i } \right )^{1/a} \nonumber$

You’ll notice that the viscosity averaged molecular weight is M_N≤ M_V≤ M_W. Experimentally Measuring Viscosity

To determine the intrinsic viscosity experimentally will require two steps:

Measure the viscosity of a polymer solution for a series of polymer concentrations.
Measure viscosity for a series of molecular weights

So we have the previously relationship for intrinsic viscosity

$\frac{\eta_{sp}}{c_2} = [\eta] + \text{slope}c_2 \nonumber$

We can find [η] by measuring a number of polymer concentrations and fitting the slope and extrapolating the line back to the origin (finding the intercept). Then, we measure viscosity for a series of molecular weights and plot as a log-log plot

$\log [\eta] = a \log M + \log K \nonumber$

where we can find a from the slope of the log-log plot and K from the intercept. We can find these critical parameters which will then allow us to determine the viscosity-averaged molecular weight and our solvent quality a.