In this edition of "Comparatively Speaking," Tony O’Lenick looks to Thomas O’Lenick, PhD, to describe the difference between critical molecular weight of entanglement (Mc) and critical concentration of overlap (C*). These concepts are important for formulators to grasp since they deal with altering the physical properties of a finished product.
Polymers differ from small molecules in many different ways. One of the most appealing is their drastic difference in physical properties. Many examples of this are seen in everyday life. For instance, ethylene, a small organic hydrocarbon, is a gas. Polyethylene is a polymer made up of repeat units of ethylene that varies from a weak stretchable solid, used in trash bags, to a hard brittle solid, used in laundry detergent containers. Polymer chemistry can provide an advantage to a formulating chemist by changing the physical properties of a product. A major component as to why polymers have such interesting properties is that the polymer chains entangle at certain concentrations and molecular weights and although this does not seem drastic, entanglement provides polymers with many unique properties.
When polymer chains entangle, the chains stop acting like individual chains and begin to act like a single unit. A good example of this is Crayola's Silly Putty product. Silly Putty is a lightly cross-linked polymer network but acts the same way as entangled polymer chains if one considers each cross-link as an entanglement. When Silly Putty is left in a ball on a table, the ball will slowly flatten out and become a flat layer of putty. On the other hand, if the ball is dropped on the table, the ball will bounce back. This is due to the rate of stress put on the system. There are two major ways to induce polymer entanglement; the first is increase the molecular weight of polymer chains and the second is to increase the concentration of the polymer in a formulation. The major difference between the two is that the molecular weight of the polymer is controlled by the polymerization and the concentration is controlled by the formula in which the polymer is used.
It is important to note that this is a brief overview of polymer chain-chain interactions. For a complete discussion, it is important to discuss the Flory-Huggins theory and the interaction parameter χ. To discuss the Flory-Huggins theory, this column would need to go into great detail about specific thermodynamic behaviors, which is a common topic in many polymer chemistry graduate school courses. Instead of going into complex physics, it would be better to go into a brief discussion of the basics; those interested in following the topic in more detail should consult a good polymer chemistry book.1
Molecular weight of entanglement: First it is important to understand the molecular weight of entanglement. Small molecules can mix together and easily pass by each other without entangling, resulting in minimal interaction with surrounding molecules. A great example of this is grains of rice. When rice is mixed together, a random mixture is formed but the rice gains are easily separated from one another. Low molecular weight polymers, much like grains of rice, do not have long enough chains to entangle. High molecular weight polymers, on the other hand, have long chains, thus they have much more difficulty passing by one another. As they try to squeeze past one another, they become entangled. This results in a higher viscosity and physical properties. A great example of this is cooked spaghetti noodles. When spaghetti noodles are mixed, a random mixture of entangled noodles results and this entanglement makes it more difficult to remove a single noodle from the entangled mass of noodles. To remove a single noodle, one needs to provide enough energy to overcome the entanglements it has with the surrounding polymer chains. High molecular weight polymers (i.e., the spaghetti noodles) do not possess the properties of a single chain, but instead possess properties of the whole bowl of noodles and one cannot change a single noodle without affecting the other noodles.
The critical point at which polymer chains become large enough to entangle is called the molecular weight of entanglement (Mc), which is the point where the physical properties of the polymer change. Most commonly, the molecular weight of entanglement is determined by plotting the log of the melt viscosity versus the log of the molecular weight. As the molecular weight increases, the plot will follow a linear path. Once the molecular weight of entanglement is reached, the slope of the line will increase drastically. It is important to note that the molecular weight of entanglement only applies when in bulk or neat conditions. Also, the Mc is specific to each polymer and controlled by the properties of that polymer including chain flexibility, pendant groups and intermolecular forces.
Critical concentration of overlap: Unlike the critical molecular weight of entanglement, the critical concentration of overlap (C*) can be adjusted by the formulation. As a polymer is added into a solvent, the polymer will take a random coil conformation and take up as much space as possible. de Gennes and Daoud et al.2 described a model that defined a polymer chain in solution as a “blob.” In this model, a polymer chain is divided into several different spheres, each having an average size, which are non-interpenetrating with other blobs. The best was to describe is is a string of pearls. The blobs are the pearl beads. As you move the string, the beads cannot pass through one other (i.e., they are non-interpenetrating).
As another polymer chain is added into the same volume of solvent, the first polymer chain does not have as much solvent as before, so it has to change its shape to avoid contact with the new polymer chain. As the concentration of polymer chains (in solution) increases, the original polymer has to collapse to minimize contact with other chains. Eventually, a concentration is reached where the polymer chains cannot collapse any further so they have to entangle. This value is the concentration of overlap (C*). Much like the molecular weight of entanglement, C* is commonly measured by the plot of viscosity versus concentration. At low concentrations, the plot will follow a linear path and once the C* is reached, the slope of the line drastically increases. It is important to note that the C* of the polymer is different for each solvent and can be easily adjusted by changing the solvent or its amount in a formulation.
These two concepts are important for formulators to grasp as both deal with altering the physical properties of a finished product. The critical molecular weight of entanglement is useful to a chemist when he or she is designing a new polymer for a certain application. If a low viscosity polymer for a given application is desired, the polymerization can be stopped below the critical molecular weight of entanglement. The opposite is also true; if a material is not rigid enough, the molecular weight can be increased to increase mechanical properties.
C* is a probably the more important parameter to understand for the formulation purposes. Take a hair spray, for example. To apply a hair spray, the consumer wants a low viscosity solution that is easily sprayed onto the hair. Once the product comes into contact with the hair, the product should reach the C* rapidly (through evaporation of the solvent) so the polymer can entangle. This entanglement locks the hair into place. This concept is very important for a formulating chemist.
However, both parameters must be considered when designing a product. If the concentration in a given formulation is above C*, the product might not be able to be sprayed, so the application will be limited. On the flip side, if the product is sprayed onto the hair but the polymer is not large enough to entangle, the product will be unsuccessful. For a more in-depth study including the physics of these phenomenona, consult Wool’s review.3
1. PC Painter and MM Coleman, Fundamentals of Polymer Science, 2nd ed, CRC Press, Boca Raton, FL (1997)
2. PD de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY (1979)
3. R Wool, Macromolecules, 26 (7) 1564–1569 (1993)