10.1 Fundamentals of polymer rheology

Introduction to Polymer Rheology

Polymer rheology is the study of how polymers flow and deform when forces are applied to them. This matters because nearly every polymer manufacturing process (extrusion, injection molding, fiber spinning) involves forcing material to flow, and the rheological properties of the polymer dictate how well that process works. Rheology also affects the final product: its mechanical strength, surface finish, and dimensional stability all trace back to how the material behaved during processing.

Concepts of Stress, Strain, and Viscosity

These three quantities form the foundation of everything else in rheology. You need to be comfortable with what each one means and how they relate to each other.

Shear stress ($\tau$) is the force applied per unit area parallel to a surface. Think of it as a sliding force rather than a pushing force. It's measured in pascals (Pa).

Shear strain ($\gamma$) is the relative deformation a material undergoes when shear stress is applied. Specifically, it's the change in angle between two lines that were originally perpendicular. Because it's a ratio of lengths, shear strain is dimensionless (no units).

Shear rate ($\dot{\gamma}$) is how quickly that deformation happens:

$\dot{\gamma} = \frac{d\gamma}{dt}$

It's measured in reciprocal seconds ($s^{-1}$).

Viscosity ($\eta$) ties stress and rate together. It measures a material's resistance to flow:

$\eta = \frac{\tau}{\dot{\gamma}}$

Viscosity is measured in pascal-seconds (Pa·s). A high viscosity means the material strongly resists flowing. Honey has a much higher viscosity than water, and polymer melts can have viscosities many orders of magnitude higher still.

Newtonian vs. Non-Newtonian Flow

A Newtonian fluid has a constant viscosity no matter how fast you shear it. The relationship between shear stress and shear rate is perfectly linear. Water, simple oils, and very low molecular weight polymers behave this way.

Most polymers are non-Newtonian, meaning their viscosity changes depending on the shear rate. This is one of the things that makes polymer processing tricky. There are three main types of non-Newtonian behavior you should know:

1. Shear-thinning (pseudoplastic): Viscosity decreases as shear rate increases. This is by far the most common behavior in polymer melts and solutions. It happens because polymer chains progressively align in the flow direction at higher shear rates, reducing their resistance to flow. This is why polymer melts flow more easily when pushed faster through a die.

2. Shear-thickening (dilatant): Viscosity increases as shear rate increases. This is rare in pure polymers but can show up in highly concentrated polymer suspensions.

3. Yield stress behavior (Bingham plastic): The material won't flow at all until the applied stress exceeds a minimum threshold called the yield stress. Once that threshold is crossed, flow begins. This is observed in filled polymers, gels, and concentrated suspensions.

Factors Affecting Polymer Rheology

Two factors dominate how a polymer's viscosity behaves: temperature and molecular weight.

Temperature controls how much thermal energy the polymer chains have. More thermal energy means chains can move past each other more easily, so viscosity drops as temperature rises. The most dramatic change happens around the glass transition temperature ($T_g$):

• Below $T_g$, the polymer is glassy and brittle. Chain mobility is extremely limited, and the material behaves more like an elastic solid with very high viscosity.
• Above $T_g$, chains gain enough mobility to slide past one another. The polymer becomes rubbery or melt-like, with much lower viscosity and increasingly viscous (flow-like) behavior.

Molecular weight determines chain length and, critically, how much chains entangle with each other. The relationship between zero-shear viscosity and molecular weight follows a power law:

$\eta \propto M^a$

where $M$ is molecular weight and $a$ is a constant. The value of $a$ changes at a threshold called the critical molecular weight ($M_c$):

• Below $M_c$, chains are short enough that entanglements are minimal. Viscosity increases roughly linearly with molecular weight ($a \approx 1$).
• Above $M_c$, chains are long enough to form a dense network of entanglements. Viscosity shoots up much more steeply, with $a \approx 3.4$ for most polymers.

This is why even small increases in molecular weight above $M_c$ can dramatically change how a polymer processes. It's also why manufacturers carefully control molecular weight distribution to balance processability (lower viscosity) against final mechanical properties (which often improve with higher molecular weight).