Chapter 11: Polymer Dynamics, Diffusion, and Kinetics

Polymer Dynamics, Diffusion, Kinetics:

Today we will tackle the concept of polymer dynamics in solution and in the melt state. The general idea to remember is that polymer chains tend to be highly intertwined so that a single polymer chain will interact/overlap/interpenetrate with the volume occupied by other polymer chains. We can then imagine chains becoming stuck on one another. Essentially, a single chain can act as a topological obstacle preventing the free motion of a different chain. We callthis obstacle an entanglement. Again, you have seen this at the macroscale with spaghetti noodles becoming stuck, etc. Since these obstacles are inherently kinetic in nature we will be focusing on polymer properties that reflect system kinetics specifically, polymer viscosities.

Viscosity and Diffusivity of Polymer Melts:

We will start our discussion on the viscosity and self-diffusion properties of polymers in the melt state (no solvent). A perplexing problem that bothered polymer scientists for a long time was the change in slope of polymer viscosity in the melt as a function of molecular weight. At some particular molecular weight the viscosity of the melt changes its scaling exponent with respect to polymer molecular weight. Since this occurs for polymers in the melt (only polymer), solvent effects cannot matter.

Below this critical molecular weight (M_c) the viscosity tends to scale as

however above the M_c the viscosity scales as

here the increase in viscosity is dramatically larger as molecular weight increases! The low MW regime is called the Rouse regime and the high MW regime is called the Reptation regime. As we will see, the critical molecular weight for the crossover between these two regimes will depend on the presence of entanglements in the system, which we described above as topological constraints due to the tendency of polymers to interpenetrate when concentrated. The critical molecular weight is thus also referred to as the entanglement molecular weight.

Typically the M_c is on the order of 10⁴, which does not correspond to a large number of repeat units per chain - for example, for polystyrene, only about 200 repeat units are necessary to reach the critical molecular weight. We can look at the structure of the repeat unit to gain some insight into the critical molecular weight - typically, more flexible chains will lead to entanglements more easily or at lower molecular weight. Hence, polyethylene will probably have a much smaller critical molecular weight than polystyrene, given that polystyrene has a large phenyl group which inhibits chain flexibility.

The diffusivity of polymer samples also shows similar molecular weight behavior.

In small molecule liquids, diffusive motion is accomplished via random, stochastic jumps of molecules between adjacent free volume, similar to monomer motions through free volume as described in our discussion of the glass transition temperature. In polymers, however, we observe much much lower diffusivities than small molecules - the diffusion coefficients of small molecules tend to be on the order of 10⁻⁵cm²/s, while the diffusion coefficients of polymers are usually between 10⁻¹⁴ and 10⁻¹⁸cm²/s, a 9-13 order of magnitude difference! Let’s think about what the physical implications of these much smaller diffusion coefficients. Remembering back to ENGR045 the mean-squared displacement from diffusion can be related to diffusivity by:

This relation is the fundamental equation of Brownian motion, and tells us that in a time t the mean-squared displacement of a molecule in 1 dimension is linear with its diffusion coefficient. So if we let t = 1sec we guess that a typical small molecule would move

Thus we can say that in one second, small molecules (with a size of about 1 nm) move a distance about 10⁴ times their size! This is incredibly fast, and gives some insight into the fast rearrangement of small molecules when perturbed. Now we can repeat this same exercise for polymers, with their much lower diffusivity

So the polymer tends to move about 1 nm in one second - if we assume that your average polymer is about 10 nm in size, then the polymer only displaces 1/10 times its size. Hence, the takeaway is that on average, polymers move much more slowly than small molecules due to diffusive motion. If we repeat this calculation for a polymer in the regime where diffusivity is on the order of 10⁻¹⁸, then this relative motion decreases even more - so the polymer experiences very little diffusive motion at all. You can also think instead of how long it would take for a polymer to diffuse a distance equal to its own size, which would be on the order of hours. This fact clearly has processing implications, and we also see that we can tune the diffusion time based on the scaling regime. So we can get a very rough estimate of the relaxation time by looking at the time necessary for a polymer to diffuse a length equal to its size/radius of gyration.

Now we need to understand how to calculate the diffusion coefficient for these two different regimes.

Rouse Regime:

Before we get into talking about the Rouse Regime we need to remember back to to the two regimes that we described when talking about the viscosity of a polymer solution, additionally let’s consider some key equations that we have mentioned throughout this course.

1. $\langle x^2 \rangle = Dt$ − Brownian motion fundamental equation
2. $D = \frac{kT}{\xi}$ Brownian particle diffusion with a friction factor in host medium
3. F = ξv−Frictional force for particle moving with velocity v in host medium The first regime being the non-draining model which was appropriate for long, highly coiled polymers, where the polymer coil was thought of as a sphere impenetrable to solvent. The individual monomers on the interior of the coiled polymer experience strong shielding from solvent interactions due to hydrodynamic interactions. Thus, in the non-draining regime these hydrodynamic interactions make the spherical coil appear as a single, impenetrable sphere, and we found that the viscosity scaled with the size of the sphere.

The other model, freely-draining model, was appropriate for short, highly elongated polymers, where each chain was thought of as a collection of connected monomers that independently interact with solvent. In this regime, no hydrodynamic interactions are present as monomers do not see each other due to the elongation of the chain. We thus found that the viscosity scaled with the number of monomers.

Both of these models were originally discussed in the context of a polymer in solution with a small molecule solvent. In a polymer melt, however, the polymer is its own solvent. Because a given monomer cannot distinguish between monomers from the same chain or other chains, in the melt state there will be significant penetration of the solvent into any given polymer coil. Hence, in the melt state the polymer coils can be described using the freely-draining model, where we can think of a single polymer chain as a flexible connected string of particles moving in the presence of a featureless viscous background. The key here is featureless, as the polymer effectively does not see individual solvent molecules (other chains) other than their influence on the viscosity; at higher molecular weights (in the reptation regime) we will see that solvent background can no longer be considered featureless due to the presence of entanglements. So in this regime the molecular weight is low enough so the chains are effectively able to move past one another. They can unentangle.

In this freely-draining model / Rouse regime the viscosity is given as

We can relate the diffusivity to the friction factor of the entire polymer chain via the Einstein relation recognizing that the friction factor of the entire chain scales as the number of monomers

This simple model thus correctly obtains the scaling of the diffusion coefficient! Finally, note that we can now obtain the time scale over which a polymer chain diffuses a distance equivalent to its own size, which is called the Rouse time in the Rouse regime

This is the time scale over which the polymer coil is able to effectively rearrange, it is qualitatively the same as the characteristic relaxation time. Note that the Rouse time increases with increasing molecular weight, implying that higher molecular weight polymers take longer to respond to perturbations. We can thus correctly explain scaling behavior for the Rouse regime (M < M_c) using the freely-draining model, and recover the scaling exponents of the viscosity and diffusivity observed experimentally.

However, we still need to understand the higher molecular weight regime called the reptation regime.

Reptation Regime:

In the reptation regime, we imagine a physical picture of a single polymer chain surrounded by fixed entanglements which act as impenetrable obstacles through which the chains cannot move. In the entanglement regime, the distance between these obstacles is much smaller than the length of the chain, such that the chain’s path of motion is heavily influenced by these obstacles. Unlike the Rouse regime, where the solvent background was featureless, in the reptation regime the motion of a single polymer is thus highly constrained by other chains due to the presence of entanglements, and we might qualitatively expect a higher viscosity and lower diffusivity due to the constraints.

We can begin by imagining the polymer coil as constrained in a tube of free volume that surrounds the polymer chain. You can imagine the tube drawn through a sea of topological constraints, representing the entanglements, such that the tube is as thick as possible without overlapping any entanglements. Within this tube, the polymer is capable of freely diffusing. We know that we can approximate the size of our polymer coil by

We also know that the contour length of the chain is

Since our tube is all around our coil, we can thus say that the tube contour length is the same as the chain contour length, and the tube end-to-end distance is the same as the chain end to end distance. If we let Z equal the number of tube segments and a equal the length of each tube segment, then we can say that:

As the polymer moves, the front of the tube moves forward into available free volume, while the back of the tube is destroyed, reflecting conservation of volume. We call this movement reptation - the name reflects the snake-like motion that we are trying to convey. What we want to find is the time it takes for a tube at some given instant to be completely destroyed, and replaced by a new tube - this would correspond to the time it takes for a polymer coil to diffuse its own length. We call the time necessary for the formation of a completely new tube the disengagement time. Inside the tube, we assume that the chain is free to diffuse within the constraints of the tube walls since the interior of the tube is free of entanglements/obstacles. Hence, inside the tube the motion is akin to that of the Rouse regime, since the polymer will tend to be elongated as a result of the tube constraints. We can thus zoom-in on one section of tube and calculate the time necessary for the polymer to diffuse completely out of a tube section - this will give us the time for tube creation/destruction that we described above. Since this is like the Rouse regime we can say

which is the same as Rouse-like polymers with number of monomers N. Note that the physical motion we are picturing is the random 1-dimensional diffusion of the polymer chain back and forth inside the tube (with some radial movement as well within the tube volume). Next, we want to know the time the coil takes to move its entire contour length, where motions are given by the diffusivity of the Rouse regime

τ is also called the longest relaxation time in addition to the disengagement time, and is the time at which the polymer starts to flow after feeling a stress - that is, τ is how long it takes for rearrangement to occur, qualitatively equivalent to the characteristic relaxation time in the reptation regime or the reptation time. We can see from the equations above that

Additionally, we know that viscosity scales linearly with flow time (we will show this in the mechanics section)

In the reptation regime, then, our simple derivation gives approximately correct scaling, experimentally it is M^3.4.

We can derive the diffusion coefficient for entangled chains in the reptation regime as well. We can think that in time τ, the entire polymer coil has moved a distance approximately given by its own size, which we can approximate using the radius of gyration, so

Hence, we get the proper scaling of the diffusivity for entangled melts. Note that it is proportional to 1/M² versus the 1/M scaling in the Rouse regime

Our qualitative intuition that in the reptation regime viscosity will be higher (∝ M³ vs ∝ M) and the diffusivity will be lower (∝ 1/M² vs ∝ 1/M) than in the Rouse regime is correct - the topological constraints imposed by entanglements slow down polymer dynamics.

Note carefully the distinction between how we derived D_rept and how we derived the longest relaxation time τ. τ was found based on the assumption of Rouselike motion in each tube segment, and corresponded to the time necessary for the polymer coil to slide through a tube length equal to the contour length. We then essentially zoomed out and realized that if the coil moved through a tube equal to the contour length, then the entire center of mass must have moved a distance approximately equal to the radius of gyration - that is, the polymer moved a distance approximately equal to its own size. We used this to find the diffusivity. The motion of the polymer IN the tube is effectively 1-dimensional along the tube length, while the motion of the tube itself is a 3D random walk. The major contrast between the Rouse regime and the reptation regime is not in the motion of the center of mass of the polymer but rather in the process by which this center of mass moves; in the Rouse regime we can imagine any given segment of the polymer as effectively random walking without constraints, while in the reptation case the fixed entanglements force only limited 1D motion. These two different views/length scales of the polymer coil’s motion - i.e. the motion of individual parts of the polymer vs. motion of the entire polymer coil - are what make reptation a unique form of motion!

Polymer-Polymer Diffusion

So far we have been talking about only the diffusion of a single polymer species, i.e. self-diffusion. What happens if we want to think about the diffusion of two different polymer species into each other? What about if the two species have different molecular weights? Hold your horses with all the questions, we can deal with this by thinking about two different scenarios

1. Fixed obstacles - you can think of a background matrix of large, entangled chains corresponding to the larger of the two polymers. The shorter of the two polymers then diffuses among these topological constraints. Since the larger chains have a lower diffusivity, they are apparently fixed compared to the faster moving short chains.
2. Constraint release - the small chains act as a featureless background, and we can envision the long polymer constraints gradually diffusing against the viscous background without encountering topological obstacles. The main idea is that because the short polymers diffuse much more quickly than the large chains, any topological constraints associated with the short chain would disentangle more quickly than the large chains could interact with them anyway and hence would not significant constrain the motion of the large chains.

Note that these same arguments/descriptions may be made for a polydisperse sample of a single species - in either case, the large polymers, with their much slower diffusion times, dominate the overall diffusion process.

However, it should be apparent that the diffusion of either species is going to be different in a blended case than in the self-diffusion case. Specifically, we can recognize that enthalpic effects can play a large role - if two polymer species enthalpicly prefer to be mixed (as would be the case if there are some strong intermolecular interactions between the two species), then diffusion will tend to speed up. Recall that the χ parameter essentially reflects the enthalpic contribution to polymer mixing. If χ > 0, then mixing is not preferred enthalpicly. As a result of this enthalpic barrier, interdiffusion is slowed down. In the case of large χ, interdiffusion is stopped completely, as would be expected for immiscible polymer blends.

In the case of χ < 0, where mixing is preferred enthalpicly, we can refer back to

Flory-Huggins theory to find the change in free energy of mixing, given by

When discussing Flory-Huggins theory we only really discussed the χ > 0 case where mixing entropy dominated mixing. If χ < 0, enthalpy dominates and we can assume that:

Hence, we can see that the free energy change scales directly with N. It was found that this factor of N also then increases the rate of diffusion by a factor of N as well, increasing the molecular weight scaling above what would be expected from the simple derivation before. You can think of this in terms of a driven motion of the polymer coils into each other due to the free energy difference, which then acts on top of the normal diffusive motion, leading to a net increase in diffusive speed.

Supplemental Qualitative Description of Rouse and Reptation:

When we discussed viscosity way back we found that that the scaling behavior of a polymer’s viscosity was dependent on whether the solvent was able to effectively interact with every monomer in a polymer coil (freely-draining model) or only interact with the coil as a whole (non-draining model). In the non-draining model, we said that hydrodynamic interactions related to some monomers ”shielding” others from the influence of the solvent led a coiled-up polymer to look like a single sphere to the solvent; in the freely-draining model, we said that in relatively elongated polymers there were no such hydrodynamic interactions and every monomer interacted with the solvent independently.

In a polymer melt, there is no additional solvent, so we effectively can think of a melt as a single chain dissolved in other chains. In this case, a given monomer has no ability to distinguish monomers along the same chain from monomers in other chains (i.e. ”solvent” monomers). As a result, we have described polymers as highly interpenetrating, and thus the ”solvent” interacts with all of the monomers in a given chain - i.e. the freely-draining model is appropriate to describe the viscosity of a melt. We called the application of the freely-draining model to the polymer melt the Rouse regime and effectively repeated the derivation from Lecture 5 to find the scaling of the diffusion coefficient and viscosity.

While the Rouse regime makes sense in the context of a single polymer chain moving in a viscous solvent consisting of other chains, an implicit assumption of the model is that the diffusion of this chain is completely unconstrained - that is, other chains only act as viscous solvent but do not otherwise obstruct motion. However, as polymer chains get longer and longer, they will increasingly intertwine and form what are called entanglements, or topological constraints where polymer chains are stuck due to steric barriers and effectively cannot move. Since entanglements can be thought of as fixed, any given chain that is diffusing in the presence of entanglements must move to avoid steric hindrance from these fixed points; in other words, the entanglements constrain the motion of other chains. It is analogous to running through a field and then suddenly encountering a forest - the onset of fixed obstacles necessitate that you slow down to avoid intersecting the constraints (my other crazy example is reptation is like a Plinko machine, google it for hilarious Price is Right results). We called diffusion in the presence of entanglements reptation. The key point that we recognized was that entanglements are more likely to form when polymer chains are long, and thus the molecular weight is higher; we thus define a reptation regime where the molecular weight of the polymer is above a certain critical threshold and the diffusive motion transitions from being primarily Rouse-like to primarily reptation.

To quantitatively describe reptation, we invoked the idea of a ”tube” of free volume surrounding a polymer chain, where the size of the tube reflects the position of constraints with respect to the chain. Inside the tube, we imagined the motion of each section of the chain as relatively unconstrained, and therefore would be described using the Rouse model since the polymer is highly elongated. We thus picture the chain diffusing in 1 dimension, either forward or backward along the tube. As the polymer gradually diffuses forward, we imagine the front of the tube growing into the available free volume between constraints while the back of the tube is ”destroyed” to maintain volume conservation. Note that the growth of the tube itself is effectively a random walk - over a large length scale, the tube looks coiled up as it navigates in between obstacles. However, if we zoom-in on the tube and look at the segments of polymer inside the tube itself, there the constraints of the tube restrict the motion to effectively 1D and the polymer looks elongated. We thus have to think of the motion of both the segments of the chain inside the tube and the tube itself separately, as their properties essentially exist over different length scales (and time scales).

We called the time scale over which a given tube is completely reconstructed the longest relaxation time. Note what this time means physically - the polymer chain has travelled a distance inside the tube equivalent to the length of the tube, which is itself equivalent to the contour length of the chain reflecting the 1D motion of the chain inside the tube. If we look at the center of mass of the chain, however, it is has travelled a distance equivalent to its radius of gyration, since the tube itself has moved in a random walk. Therefore, the time necessary to reptate a distance equivalent to the size of the polymer chain can be inferred from the amount of time it takes to diffuse the contour length of the polymer inside the tube. Qualitatively, since the contour length of the tube is much larger than the radius of gyration, this time scale is much longer than the time scale to travel the radius of gyration in the Rouse regime and reptation is much slower than diffusion in the Rouse regime. Finally, we should note that this time scale is also the time scale necessary for entanglements themselves to form and disentangle, so the assumption of fixed constraints is a bit simplistic; however, you can argue that the Rouse motion INSIDE the tube is still very fast compared to the reptation time, so the assumption is justified.

Thus, we are able to relate the amount of time taken for a polymer to diffuse a distance its own size to the molecular weight of that polymer, since at low molecular weights its motion will be described by the Rouse (freely-draining) model while at high molecular weights its motion will be described by reptation. We also related the time necessary to travel this distance to the relaxation time of the polymer, which again as we said above governs the ability of the polymer to return to equilibrium. Connecting these concepts leads to an interesting observation - because this analysis effectively says that the relaxation time is much longer in the reptation regime, and we saw earlier that long relaxation times are characteristic of solids, does this mean that in the reptation regime the polymers act like solids?

The answer to this question is a qualified yes. As the relaxation time of a material increases from a viscous liquid-like state to a solid elastic-like state, it will enter a regime where the material is viscoelastic and demonstrates qualities of both phases. Viscoelastic behavior is highly characteristic of rubbery materials which we will discuss in the next lecture. In the reptation regime, we do indeed observe rubbery behavior even in the absence of physical or chemical cross-links. The molecular reason is that entanglements act as cross-links; however, in the context of time scales, we can also say that the relaxation time of an entangled melt is much higher than an unentangled melt and thus the melt enters the viscoelastic regime.

One final point that bears mention is the idea of the molecular weight of a polydisperse sample. If we say that there is a single critical molecular weight that determines the difference between the Rouse regime and reptation regime, but we have a sample that consists of many polymers of different molecular weights, how do we determine what regime we are in? More specifically, what molecular weight average would be appropriate to describe this sample. We can think of this in the behavior of small and large chains mixed together. Small chains in the Rouse regime will have a much higher diffusion coefficient than the larger chains, and as a result will diffuse effectively instantaneously compared to the large chains. Diffusion of the large chains is thus ”rate-limiting”, and as a result the weight-average molecular weight will provide a better average than the number-average molecular weight because it better characterizes the molecular weight of larger polymers in the sample.