Chapter 10: Amorphous and Semi-Crystalline Polymers

Review of Crystallization:

Now we are going to zoom about a bit and describe the structure of polymers in terms of short range and long range order, hopefully concepts you remember from Materials. We will start by talking about semi-crystalline polymers, meaning polymers that are capable of obtaining long ranged ordered morphologies at low temperature. As is characteristic of all crystals, polymers in a crystalline state are highly ordered, leading to the alignment of polymer chains stabilized by secondary intermolecular bonding interactions. The ability of a polymer to crystallize is thus highly dependent on the molecular structure of its constituent monomers, as we could imagine bulky sidegroups may inhibit ordered chain packing. Hence, we generally distinguish between amorphous polymers, which undergo a glass transition (second order transition) at low temperatures and generally only have short-range order, and semi-crystalline polymers, which actually crystallize at low temperatures and may have a melting temperature (first order transition). Finally, note that semi-crystalline polymers also undergo a glass transition (just like any material can in principle), but the glass transition temperature is below the melting temperature and as a result is only observed when the polymer is cooled down quickly such that crystallization does not have enough time to occur. We will discuss the properties of the glass transition and amorphous polymers in a bit but first we will discuss semi-crystalline polymers as they are a bit simpler to deal with.

In general, crystallization is an ordering process associated with a loss of entropy in the system in exchange for achieving stronger enthalpic interactions. The crystalline state is favored at low temperatures since the free energy is written as

$$\Delta G= \Delta H - T\Delta S \nonumber$$

so the penalty for reducing the entropy is less significant at lower temperatures. In the case of polymers, we know that stretching out polymer chains is associated with a significant decrease in conformational entropy, but the resulting extended morphology also allows chains to become extremely close to each other on the atomic level, leading to favorable non-specific (van der Waals, dipole interactions) and specific (hydrogen bonding, electrostatic) enthalpic interactions. At a sufficiently low temperature, we can thus imagine that the equilibrium morphology would consist of fully extended polymer chains that are perfectly aligned against each other to maximize bonding (like uncooked spaghetti in its package).

The process of crystallization depends on an interplay between both kinetics and thermodynamics; the same is true of polymer crystallization. For polymers in the high-temperature melt state the polymer chains will be highly intertwined due to the flexibility of the chains. We can imagine, then, that there would be a significant kinetic barrier for polymer chains to rearrange these highly intertwined chains into an aligned structure characteristic of a ordered state. Furthermore, we also discussed that polymers pay a significant entropic cost to fully extend, providing a further kinetic barrier against completely ordered chains. Conceptually then, we can imagine a kinetically trapped state where some chains or some parts of chains are capable of packing against each other, but overall there are still regions of disorder because the chains cannot appropriately diffuse (or more accurately reptate, a term we will discuss in future lectures) to an equilibrium ordered state. While this frustrated state is meta-stable, it might be stable for a sufficiently long period of time that the fully-aligned equilibrium state is not reached. This state is then called semi-crystalline because there will be large regions that are still disordered mixed in between regions of crystalline, well-ordered chains. Often the crystalline state is thus described in terms of a degree of crystallinity, which attempts to measure the fraction of the polymer that is crystalline versus the fraction that is disordered. In reality there are virtually no polymers that are 100% crystalline they are all either semi-crystalline or amorphous.

Polymer Crystal Morphology/Structure:

It is effectively impossible to generate a completely aligned polymer crystal, but there is still a thermodynamic driving force for the alignment of polymer chains below the crystallization temperature T_C. To gain partial crystalline character even in the presence of topological constraints that prevent complete ordering, the chains will tend to form a folded lamella structure (not to be confused with the lamellae observed in microphase-separated diblock copolymers). This lamella structure is defined by a series of polymer chains that fold back and forth, such that chains line up and pack against each other but at a fraction of their fully extend length, allowing chain ends to still be free and disordered. The lamella structure allows monomers to interact via non-specific or specific secondary intermolecular bonding. This state is preferred because for polymers to pack in a completely extended state, the polymer would essentially have to fully extend spontaneously first, then pack next to another fully extended polymer, etc, which is highly unlikely.

We can then define a crystalline unit cell for the polymer as found in other crystalline materials. Typically, the unit cell appears as a hexagonally close packed array of cylinders. The unit cell dimensions can then be derived based on bond lengths, etc, since in the crystalline state the all-trans state will tend to be preferred. As in other materials, the unit cell allows us to understand translational and rotational symmetries of the extended chains.

Finally, note that we described these polymers as semi-crystalline above. The folded, close packed surfaces would thus be the crystalline phase, while any parts of the chain that extend beyond the folds before repacking would be the disordered phase (this is a bit easier to see visually than described here, so please look at the lecture slides). Since chains are so long, it is also possible for a single chain to have multiple regions which are folded and disordered respectively, allowing different ordered regions to be effectively linked together by a disordered region in between. This will lead to hierarchical organization of ordered structures.

Branching Defects Depress the Melting Temperature:

The driving force for crystallinity is the ability for chains to pack closely together at the atomic level in order to maximize non-specific (and possibly specific) bonding. As a result, any structural changes in a polymer that would inhibit packing tends to oppose the formation of crystals. For example, polyethylene (PE) can exist in a branched state, where several alkyl side groups extend away from the main polyethylene backbone. This branching greatly changes the properties of the PE chain, and actually leads to different engineering applications. When crystallizing, the branches can be handled in two different ways. If the crystallization process occurs very quickly (fast quench/high undercooling), typically branches will be incorporated/frozen into the crystalline morphology, leading to the polymer equivalent of defects in the crystal structure. If crystallization occurs slowly, then the branches will be excluded from the perfect crystal structure and tend to form a disordered fringe. This is again the balance between kinetics and thermodynamics.

If a branches do organize as defects in the crystal structure, we should also note that the melting point changes as a function of these defect sites, since these essentially act as an impurity which depresses the melting point. We can write out the change as

$$\frac{1}{T_M(x)} - \frac{1}{T_M^0} = - \frac{R}{\Delta H}\ln(1-x) \nonumber$$

where x is the mole fraction of non-crystallizable units, i.e. the number of branches or defects. T_M⁰ is the melting point in the limit of no side chains or defects, and ∆H is the latent heat at the melting point. Qualitatively, this expression says that the melting temperature decreases as more branches are introduced (x increases), which makes sense because ordering is more difficult in the presence of branches, leading to a smaller crystallite size, smaller number of secondary intermolecular interactions, and thus a lower temperature is necessary for crystallization. Similarly, molecular weight should also affect the melting point since chain ends can also act as impurities.

Spherulites: Hierarchical Structure of Semi-Crystalline Polymers

We’ve discussed that a single polymer chain will fold and form lamalla upon crystallization. However, typically we deal with crystallizing a large number of entangled polymer chains at once, and hence have to consider the large scale morphology of the resulting structure. When you crystallize a polymer melt, many polymer chains will tend to form lamella which has a sandwich like structure with a thickness l defined as the length of the folded polymer chains along the surface of the lamella. l is typically on the order of 100s of ˚As and the fully extended polymer length is typically thousands of ˚As. Now in the case of a very dilute polymer solution the polymers are typically well separated and are not entangled. This system will crystallize as isolated chain folded single crystals. As the polymer concentration increases the chains will overlap and entangle and polymers will fold onto multiple lamella. This linking of ordered regions disordered amorphous polymer regions creates a structure we call a spherulite where multiple lamella grow outward from a single nucleation site. In this case, growth consists of chains folding onto the surface of previously crystallized sections of polymer. Spherulites consist of lamella (ordered region) and large disordered regions of polymer chains filling space between the growing lamella. Note that the lamella do not necessarily consist of a single chain, but could consist of different polymer chains that also wind through the disordered sections, effectively joining together ordered regions with what are called tie molecules. The term spherulite comes from the tendency of growth to proceed outward with spherical symmetry. Spherical symmetry is possible because lamella can branch off where chains fold onto a surface at a slightly different angle, allowing lamella to continue growing in a different direction. Thus,we can think of a complete spherulite as a number of chains folded into the lamella surrounded by a matrix of amorphous chains, some of which act as tie molecules joining the growing lamella together. The relative proportions of the disordered and crystalline regions will determine the degree of crystallinity of the sample. Finally, it should be noted that due to the presence of an amorphous region, a semi-crystalline polymer will still undergo a glass transition associated with the glass transition of JUST the amorphous regions.

In a large melt, many spherulites will nucleate independently, and as a result their eventual size is restricted by the presence of other spherulites which impinge upon each other. Effectively spherulites can reach sizes as large as 100 µm if they do not impinge upon each other. The distribution of sizes will depend on how quickly nucleation occurs which in turn depends on the degree of undercooling of the sample. Remember that undercooling is the amount of temperature we cool the sample below the melting temperature or

$$\Delta T = T_M -T \nonumber$$

where ∆T is the amount of undercooling, T_M is the melting temperature, and T is the actual temperature.

At a high degree of undercooling, nucleation will be strongly preferred thermodynamically so many spherulites will nucleate quickly. As a result, we can imagine all of the spherulites in a sample nucleating at about the same time and growing at about the same rate, so that when they begin to impinge upon each other they will be around the same size. Hence, high degrees of undercooling lead to an even distribution of spherical sizes. Physically, the boundaries between spherulites that are about the same size will be flat reflecting identical nucleus sizes and growth rates. Conversely, if the degree of undercooling is low, there will be a lower thermodynamic driving force for nucleation and hence nucleation will occur slowly. We can thus imagine nuclei appearing in a sample sporadically and at different times, such that there will be a large distribution of sizes when spherulites impinge upon each other. In this case, the boundaries between spherulites will appear as curved. problem on this

Melting Temperature for Semi-Crystalline Polymers:

Now that we have a picture of the structure we can dive back into the thermodynamics of chain folding and crystallization in detail and you can bet that a competition of entropy and enthalpy will be involved. Right now we will start with a simple case and only consider the thermodynamics of melting a chain-folded regime. Let’s start by defining a few parameters

• σ = surface free energy along the growing edge of the lamella
• σ_E = surface free energy along the disordered folded surface of the lamella
• x = edge length of lamella
• l = thickness of lamella
• T_M⁰ = Melting temperature of infinitely long polymer crystal (i.e. no folds)

The goal here is to understand the free energy change for forming lamella near the melting temperature as a function of the lamella thickness l. One important thing to keep in mind is that the surface free energy along the well-ordered lamella edges and the surface free energy around the disordered folded surface are different.

The total free energy change will consist of two terms

1. ∆G_S - which is the free energy change at the surface of the crystal where the ordered region interacts with the disordered region. This contribution should be positive (penalty) as creating a surface where ordered and disordered regions meet is very unfavorable for packing.
2. ∆G_V - the free energy change in the volume or bulk of the crystal. This contribution should be negative as at temperatures T < T_M as the bulk will want to order to maximize secondary intermolecular interactions.

Now at the melting temperature, the change in free energy between liquid and solid is 0 so we can write that

$$\Delta G_V = \Delta H - T_M^0 \Delta S = 0 \nonumber$$

$$T_M^0 = \frac{\Delta H}{\Delta S} \nonumber$$

At temperatures near the transition, then, we can approximate the change in entropy using the expression above for the change in entropy at the melting point. So for some general temperature T that is near T_M⁰ , we can write

$$\Delta G_V(T) = \Delta H(T) - T\Delta S(T) \nonumber$$

$$\approx \Delta H(T) - T\frac{\Delta H(T)}{T_M^0} \nonumber$$

$$\approx \Delta H(T) \left (1 - \frac{T}{T_M^0} \right ) \nonumber$$

$$\approx \Delta H(T) \left (\frac{T_M^0 - T}{T_M^0} \right ) \nonumber$$

We can then rewrite this equation slightly by defining the degree of undercooling as ∆T = T_m⁰ − T. Rearranging gives:

$$\Delta G_V = \Delta H \Delta T/T_m^0 \nonumber$$

Note that we did not include any terms related to the number of chains, so we can consider this a free energy change per unit volume, assuming ∆H and ∆S are also measured per volume.

For the surface free energy

$$\Delta G_S = 2\sigma_E x^2 + 4\sigma x l \nonumber$$

where the first term considers the top and bottom surface and the second term accounts for the four sides. The total free energy for the sample is

$$\Delta G = -\Delta G_V x^2 l + \Delta G_S = -\Delta G_V x^2 l + 2\sigma_E x^2 + 4\sigma x l \nonumber$$

We can simplify this expression by assuming that x >> l as the growing lamalla tend to elongate much more than they thicken. So the second term of the surface free energy is very small and we can ignore the term of 4σxl. Now, to find the melting point for finite l, we can set the free energy change to 0 and solve

$$\Delta G_V x^2 l = 2\sigma_E x^2 \nonumber$$

$$\Delta H(T) \left (\frac{T_M^0 - T}{T_M^0} \right ) l = 2\sigma_E \nonumber$$

$$T_m(l) = T_M^0\left (1 - \frac{2\sigma_E}{l\Delta H} \right ) \nonumber$$

Hence we see that as the thickness of the lamella, l, increases, the resulting melting temperature also increases to a maximum of T_M⁰ when l = ∞. This makes sense since at infinite lamella thickness the chains are maximally elongated and the crystalline state is highly preferred preferred, so as this thickness reduces the temperature necessary to melt the lamella decreases.

Finally, note that using similar reasoning, this same approach can be employed to find the critical lamella thickness necessary for stable nucleation at a given temperature.

Kinetic Consideration of Crystallization: Crystallization Rate

The process of crystallization involves understanding the competition between both kinetics and thermodynamics. Qualitatively, we know that at lower temperatures, the entropic contribution to the total free energy of a system will be lowered, and as a result ordered morphologies to maximize enthalpic interactions are preferred. However, as the temperature of a system is lowered, both fluctuations and transport in that system will also be reduced, which may hinder the ability of ordered regions to grow. We thus see that there is a delicate balance between the thermodynamics and kinetics of crystallization. The rate of crystallization is defined as the rate at which a sample becomes crystalline, which depends on both the nucleation of new spherulites and the growth of existing spherulites.

Polymer crystallization proceeds by this two-step process of nucleation and growth, which is qualitatively identical to the nucleation and growth of small molecule crystals. In the nucleation step, a small crystal of polymer spontaneously forms due to thermal fluctuations in the system. For a polymer, we can imagine this initial nucleus as simply a section of polymer that is elongated and then folds onto itself (or a neighbor). The initial nucleus will have some surface area and some volume; the surface area will have an associated surface energy which will be due to relatively unfavorable interactions between the surface of the crystal and the surrounding polymers in the bulk. Conversely, nucleus formation will be stabilized by favorable interactions between molecules in the volume of the crystal (i.e. favorable enthalpic interactions from aligned chains). The thermodynamic tendency for a stable nucleus to form will thus depend on the surface-to-volume ratio of the initial nucleus. For small nuclei, the surface-to-volume ratio is high, opposing nucleus formation, but as a nucleus gets bigger the ratio decreases. At a certain critical size (r^∗) the free energy change for forming a nucleus achieves a maximum, so that any increase in the nucleus size beyond this point decreases the free energy change.

Thus, once a nucleus of the critical size is achieved, growth becomes thermodynamically favorable in order to continue reducing the free energy of the system. Furthermore, we can imagine the peak in this curve as something of an energy barrier for nucleation, since the formation of a cluster of a given size will depend on the fluctuations in the system that allows for the spontaneous formation of a cluster. For a polymer crystal, the critical size effectively refers to the elongation of a single chain necessary for nucleation; since we know that elongation is opposed by an increase in entropy, we can imagine that at high temperatures it is very hard to spontaneously elongate a chain to the critical size. Thus, as the temperature of a system is lowered, nucleation becomes easier.

Growth

First, consider the rate of spherulite growth, meaning the rate at which additional polymers pack on the faces of existing lamella leading to their outward expansion. This rate is dependent largely on the amount of time it takes for polymer chains to diffuse to the growing face. Qualitatively, diffusion is a temperature-activated process, which can be thought of as a bunch of molecular jumps through space with some activation energy Q_A. If we consider diffusion as an Arrhenius process, then we can write the diffusion coefficient as proportional to

$$D \approx \exp \left ( -\frac{Q_A}{k T} \right ) \nonumber$$

We see that the lower the temperature - that is, the greater the degree of undercooling - the slower diffusion will occur, as should be expected. Hence, lowering the temperature below the melting temperature slows down the transport of polymer to growing spherulites.

Nucleation

Next, though, consider the rate of nucleation. In general, we expect nucleation to occur more frequently at lower temperatures, since then nucleation will be thermodynamically preferred, and the thermodynamic driving force for nucleation will increase the lower the temperature gets. We can approximate the rate of nucleation as

$$I = \exp \left (\frac{\Delta G}{kT } \right ) \nonumber$$

Thus, we expect the rate of nucleation to increase as the temperature lowers. The optimal crystallization rate thus results from some intermediate amount of undercooling to maximize contributions from both nucleation and growth. Near the melting temperature, the likelihood of forming a nucleus is very small, so the crystallization rate is slow. Far below the melting temperature, the rate of diffusion is very low so spherulites nucleate very frequently but are quite small and hardly grow. Crystallization rate is thus maximized at some small amount of undercooling, again underlining the importance of both kinetics and thermodynamics in crystallization.

Modifying Melting Characteristics

As engineers we want to be able to tune and manipulate the crystallization and melting characteristics of our polymers and other soft materials. And one of the great things about soft matter is that we have a lot of knobs that we can tune and tweak to make sure our soft material is ideal for our particular application of interest. There are a couple of strategies that we can utilize

1. Add branches: adding branches will lower the melting point same behavior if you have a lower MW.
2. Modify the chemical structure of backbone: stiffer polymers easier to crystallize and bulky groups along the main chain will have higher melting points as rotation will be sterically unfavorable. Highly flexible groups like Si or O will decrease melting point. Any groups that participate in H-bonding, electrostatic interactions, etc will increase melting point.
3. Side chains: Adding long flexible side chains will decrease the melting temperature as they will act as steric barrier to oppose ordered packing while shorter bulky side chains can enhance elongation by restricting the rotation of the main chain.
4. Environmental conditions: temperature, applied stress, pressure, MW (small change)

Some of this behavior can be seen below and how it affects melting point and lamella thickness

$$\begin{array}{|c|c|c|} \hline \textbf{System change} & \textbf{Melting point} & \textbf{\(l\)} \\ \hline \text{Undercooling } \uparrow & - & \downarrow \\ \hline \text{Molar mass } \uparrow & \uparrow & - \\ \hline \text{Pressure } \uparrow & \uparrow\uparrow & \uparrow\uparrow \\ \hline \text{Stress} & \uparrow\uparrow & \uparrow\uparrow \\ \hline \text{Add branches} & \downarrow & - \\ \hline \text{Bulky groups in backbone} & \uparrow\uparrow & - \\ \hline \text{Flexible groups in backbone} & \downarrow\downarrow & - \\ \hline \text{Bulky groups in side chain} & \uparrow\uparrow & - \\ \hline \text{Flexible groups in side chain} & \downarrow\downarrow & - \\ \hline \text{Specific bonding interactions} & \uparrow\uparrow & - \\ \hline \end{array} \nonumber$$

Non-Crystalline/Amorphous Polymers:

Now that we have exhausted our discussion of semi-crystalline polymers we can move on to discussing non-crystalline/amorphous polymers which are polymers that do not undergo a melting transition but instead a glass transition when the temperature is lowered. These polymers then form amorphous but solid structures that lack the long range orientational and translational order of crystals.

Short Range and Long Range Order

Non-crystalline materials are characterized as having short-range order (SRO) but not long range orientational or long range translational order (LRO). This means that the probability of locating another atom within some distance r is more probable at certain short distances of r than at long distances, where the probability essentially becomes constant. Short-range order (SRO) arises because atoms prefer to pack together is characteristic of liquids in general due to a combination of bonding interactions and weak non-specific bonding. In polymers, SRO is due to local chemistry (such as polymer connectivity), excluded volume and the restricted conformational states due to the finite rotations of intramolecular covalent bonds. Long-range order arises in crystalline solids where the presence of translational symmetry (i.e. a well-defined unit cell) is due to highly specific, strong binding interactions (covalent bonds) or a strong non-specific intermolecular bonding. Polymers that exhibit short-range order but not long-range order are called amorphous; non-crystalline polymers are amorphous at all temperatures, while semi-crystalline polymers exhibit both amorphous regions and crystalline regions.

Pair/Radial Distribution Function:

One technique that we can utilize to get a quantitative measure of the short and long range order of both semi-crystalline and amorphous polymers is a pair distribution function g(r). To calculate g(r), you take a single atom i as a reference and calculate the number of atoms dn in a spherical shell of volume dv = 4πr²dr centered around atom i, where dr is the thickness of the shell and r is the radius of the shell. This counts how many atoms are some distance r from the atom i we are looking at. We can then repeat this measurement for all such atoms i in the sample, and take the average. g(r) is thus defined as:

$$g(r) = \frac{1}{\langle \rho \rangle}\frac{dn(r, r+dr)}{dv(r,r+dr)} \nonumber$$

averaged over all atoms in a system. Note that this equation boils down to counting the number of atoms at some distance r, dividing by the size of the shell since this increases as r increases, then normalizing by the average density to make this a dimensionless quantity. You can also think of g(r) as being the relative probability of finding an atom at any given distance r - relative meaning that the total integral of g(r) is not 1. Plotting g(r) then provides an illustration of order in different phases of material. In a gas, there is no order inherent to the material, so at any given distance the same relative number of atoms should be expected, and so g(r) is just 1 (i.e. the average number at any distance is uniform and equal to the density). In a crystal, g(r) will have a large number of sharp peaks at different distances, corresponding to the known fixed distances of atoms in the crystalline lattice. In a glass or liquid, we see short range order - there is a broad peak at a characteristic distance slightly larger than the diameter of an atom, implying that atoms tend to cluster close together, and other broad peaks at distances related to probable atomic separations, but sharp peaks at absolutely fixed distances are not observed. In glassy polymers, again, this feature can be related to the excluded volume of the monomers, and hence we expect a peak at some distance corresponding to two times some effective radius that reflects the excluded volume of an individual monomer.

Glass Transition Temperature:

We have just discussed how non-crystalline amorphous polymers can be distinguished from semi-crystalline polymers based on the presence of long range order. However, another critical parameter that is utilized to distinguish/describe amorphous polymers is the glass transition temperature (T_g. Typically, non-crystalline polymers can be broadly classified as either rubbery or glassy, both of which exist as highly interpenetrated/entangled Gaussian coils at relatively high temperatures above the glass transition temperature. When we say interpenetrated, the physical picture is of polymer coils that overlap with each other such that separate

coils intertwine with each other. In this highly interpenetrated state, without solvent, the polymer acts as if it is unperturbed - that is, in the melt state the polymer is at the θ condition, and can be treated as an ideal chain. You can think of melt polymers as essentially feeling some pressure due to surrounding polymers that overcomes excluded volume effects, yielding an ideal state. The ideal nature of polymers in the melt make them much easier to think about theoretically.

The glass transition temperature is the temperature at which a polymer transitions from its fluid-like state to a glassy state. When we say a glass, we mean a material with only short-range order but lacking the translational fluctuations associated with liquids. Hence it is like a solid, but with randomly positioned atoms rather than ordered ones as we expect in a crystal. We can identify the physical origin of the glass transition temperature in two different ways - first, you can think of increasing the temperature from below the glass transition, or you can think of decreasing the temperature from above the glass transition temperature. In either case, the glass transition is a competition between the available thermal energy kT and the strength of intermolecular bonds _ij.

If we think of increasing temperature, then the glass transition temperature is the point at which thermal energy is sufficient to break local intermolecular bonds, enabling fluid-like motions - that is, the point where $kT > \epsilon_{ij}$. If we think of decreasing temperature, then the glass transition temperature is the point at which the viscosity of the polymer essentially becomes infinite, eliminating molecular motion. In either case, the key property of a glass is that the rearrangement of atoms is hindered, limiting the ability of the system to relax to equilibrium when a stress/perturbation is applied. This is intimately related to the concept of a characteristic relaxation time which we will take a quick aside right now to discuss.

Relaxation Time τ^∗:

The physical properties of amorphous polymers (and materials in general) are influenced by τ^∗, the characteristic relaxation time of the polymer, and how large this relaxation time is relative to a relevant experimental time (or the interaction time) t. The relaxation time is essentially a measure of how long it takes for a material to return to equilibrium after the application of some perturbation. The ratio between the relaxation time and experimental time is called the Deborah number - i.e. $De = \frac{\tau^*}{t}$. It is easiest to think of the importance of the relaxation time in terms of known material behavior.

Let’s consider as an example a system consisting of some material in a container such that the material is magically attached to the container walls. We impose a perturbation on the system consisting of moving the walls of the container apart such that the material in the container is deformed/stretched. First consider the case that the relaxation time of a material is much smaller than the experimental time, such that τ^∗ << t and De << 1. Because the relaxation time is so much lower than the experimental time, the material effectively relaxes instantaneously to the new system dimensions as the perturbation is applied; in other words, the system adjusts to the new constraint by relaxing to equilibrium immediately. Physically, we could imagine this as the material rearranging its constituent molecules to instantaneously fill the new volume of the container - we would say that the material flows, and call the material in the container a liquid.

Now consider the opposite case, where the relaxation time is much greater than the experimental time such that τ^∗ >> t and De >> 1. Now when we adjust the walls of the container, the material effectively never relaxes to equilibrium as the perturbation occurs, instead being driven very far from equilibrium into a high energy state. In our physical example, we would imagine the material in the container being stretched but being unable to adjust the positions of its atoms because the relaxation time is so long, so that the material instead builds up a large amount of strain energy; we would consider this an elastic response and call the material an elastic solid. Note that the only distinction we are drawing here between the liquid and solid case is the experimental time - this implies that if we apply a stress/strain to a solid and wait long enough, it would appear to flow like a liquid (in the case of crystalline solids this would be due to the gradual movement of defects throughout the material to change the solid dimensions). In an intermediate regime where the relaxation time is roughly the same as the experimental time, the material will exhibit behavior consistent with both viscous liquids and elastic solids; we call these materials viscoelastic and will discuss their properties much more in future lectures.

We can gain some understanding of relaxation times from the molecular structure of a given material. For example, for small molecules like water, the relatively free motion of water due to its small size would lead to a small relaxation time, and hence water is only solid at low temperatures. Polymers tend to exhibit a longer relaxation time due to the connectivity monomers, requiring collective motion to adjust to a perturbation. As we will discuss shortly, at lower temperatures the energetic cost for this motion is too great to allow the polymer to flow, leading to glassy behavior. We might also imagine, then, that polymers with more rigid backbones have longer relaxation times due to the lessened flexibility of the chain.

Characterizing a polymer as a rubber, liquid, or solid (glass) is impossible without further information on the overall environment, as the response of the polymer depends on the time scales involved. We will also see that the relaxation time in polymers is a function of temperature, yielding many of the characteristic mechanical properties which we will discuss in future lectures. To wrap up if the molecular motion is slow/hindered, the relaxation time of the polymer is very long and thus the polymer exhibits solid-like behavior.

Back to Glass Transition Temperatures:

A key point about the glass transition temperature is that it is not strictly a thermodynamic transition. The melting temperature, for example, results because there is some specific temperature for materials where the free energy of the liquid phase becomes lower than the free energy of the solid phase, leading to melting behavior (and more specifically there is an abrupt change in thermodynamic quantities reflecting a first-order phase transition). Hence, the melting point is thermodynamic in nature, resulting from the competition between the higher entropy liquid phase and lower enthalpy solid phase. The glass transition is not strictly thermodynamic, as a glass is not a stable thermodynamic phase but rather a kinetically-trapped phase resulting from the large energy barriers faced by a glass when it must adjust to a perturbation. The glass transition thus reflects the kinetics and time scales of a system and as such the transition temperature is not easily defined - in fact,the exact measurement depends on cooling rate and typically a range of glass transition temperatures are reported for specific samples.

Free volume theory of T_g

An explanation for the origin of the glass transition temperature comes from free volume theory. Free volume (not be confused with excluded volume) is the space in a polymer sample in excess of the volume present in a random, densely packed glass. Free volume is the volume that monomers in a particular sample are able to access via thermal fluctuations. In a liquid state, we would imagine that the free volume is high, allowing large fluctuations and easy response to perturbations due to the availability of free volume through which monomers can move. By contrast, in the glassy state the free volume is very low and as a result molecular motion is hindered by steric/excluded volume considerations. The free volume is thus

$$V_F(T)= V(T) - V_0(T) \nonumber$$

where V₀ is the volume of a close-packed random glass (0.63), V is the actual volume of the sample, and V_F is the free volume. Note the temperature dependence of each term, which comes from thermal expansion governed by the thermal expansion coefficient

$$\alpha = \frac{1}{V}\frac{\partial V}{\partial T} \nonumber$$

As we gradually increase the temperature the volume available to the system will increase, reflecting increased fluctuations in monomer positions due to more available thermal energy.

Since the thermal expansion coefficient will be different between the glass and fluid states, V_F will increase with increasing temperature because the thermal expansion coefficient in the liquid state α_l is greater than the thermal expansion coefficient α_g in the glassy state (this controls the increase in V₀ with T). We can now write

$$V_F(T) = V_F(T_g) + (T-T_g)\frac{dV_F}{dT} \text{ for } T>T_g \nonumber$$

Dividing by V puts the right side in terms of the difference in thermal expansion coefficients, and reduces all volumes to fractions for the total volume

$$f_F(T) = f_F(T_g) +(T-T_g)(\alpha_l - \alpha_g) \nonumber$$

T_g is then defined as the point where this fractional free volume falls below some critical value, and practically speaking can be found by looking at where the slope of the sample volume changes as a function of temperature

(reflecting the change in expansion coefficient). Below the critical value, we can imagine the positions of the monomers are frozen due to the low volume available for monomer movements. Note that by this definition, polymers with greater free volume in the fluid state will experience a lower glass transition temperature, since you will have to decrease the temperature more to reduce the free volume fraction to the critical threshold. Qualitatively, this makes sense since if there is more free volume originally, it requires less thermal energy to access that volume, and so the glass transition temperature will lower. This observation implies that the glass transition temperature will depend significantly on the molecular structure of the polymer, we will talk about this in a second. Finally, it has been shown experimentally that the critical fraction for the glass transition is fairly constant across many different molecules, making this analysis appropriate.

The glass transition temperature is also dependent on the rate of cooling of the sample. The cooling rate dependence of the glass transition temperature can be explained by what’s called percolation. If we imagine the polymer sample superimposed on a lattice, and each lattice point can be thought of as either in the glassy state or fluid state, then the network is percolated when a connected network of glassy points spans the entire lattice. In other words, it is not necessary for the entire polymer sample to be glassy; it is only necessary for connected regions that cross the network to be glassy, since these connections essentially cut off the fluid parts from each other, leading to glassy behavior on the macroscale. We can think of each lattice point as transitioning between fluid and glassy with some characteristic probability, and can also imagine some other characteristic probability for the point to transition from glassy back to fluid. Physically, this reflects the ability of monomers to move through the available free volume, and hence at lower temperatures we expect the probability of a fluid-toglass transition to increase, and the probability for a glass-to-fluid transition to decrease as the decreasing free volume favors the glassy state. As we cool the sample, then, the probability of forming a glass progressively becomes higher, and the time taken for glass points to transition back to fluid takes progressively longer. If we cool at a very fast rate, lattice points will have sufficient time to transition from fluid to glass but not back (since as we cool it takes longer and longer for the glass-to-fluid transition), leading quickly to a percolated network at a high temperature since glassy behavior will be observed as soon as connected glassy regions span the sample. If we cool at a slower rate, a percolated network is more difficult to form because at higher temperatures the relatively short time associated with the glass-to-fluid transition will disrupt the network as it forms. This means that the glass transition will be observed at a lower overall temperature.

T_g and Chemical Structure of Monomers:

Trends in T_g can be explained by looking at the chemical structure of monomers due to the influence of monomer structure on both chain flexibility and free volume. Recall that T_g is determined by the onset of long range cooperative molecular motion, meaning the motion of 10-30 connected chain monomers at once. The cooperative motion requires both sufficient thermal energy to induce the movement of these monomers and sufficient free volume for the monomers to move into. Both of these requirements are influenced by monomer structure. There are essentially 3 elements of monomer structure that can influence these motions:

1. Chain interactions - energetic interactions between monomers
2. Ease of rotation about main chain bonds - whether there is significant steric hindrance to rotation
3. Amount of free volume available - how densely the monomers pack

We can think of polymers as essentially divided into their backbone and sidechains coming from that backbone. In polymers with flexible backbones, like polyethylene (lots of C-C single bonds) and PDMS (Si-O bonds, which are flexible due to the 4 electrons on the oxygen atoms in place of hydrogens), the hindrance to rotation is low and hence less thermal energy is necessary to induce molecular motion, leading to a low T_g. On the other hand, polymers with phenyl molecules along the backbone (e.g. polycarbonate) tend to have a much higher T_g, since the bulky phenyl constituent greatly increases the amount of energy necessary for rotation (i.e. the rigidity). Note that we can generally relate C_∞ to the rigidity of a backbone, and hence expect T_g to increase with C_∞.

Similarly, large bulky sidechains, such as phenyl groups, also oppose rotation due to steric hindrance, and in addition decrease the amount of free volume available since they occupy a greater excluded volume, leading to a higher glass transition temperature. Finally, intermolecular interactions, such as hydrogen bonds or ionic interactions, which are typically seen between sidechains, will tend to greatly increase the glass transition temperature since thermal energy will also have to break these bonds to induce rotation. The lecture notes provide some examples of polymers,

sidechains, and the related T_gs. The main principles to keep in mind are that chain flexibility allows easier cooperative movement of monomers, decreasing the T_g, and increased free volume around the chain backbone lowers the barrier to cooperative movement and hence decreases the T_g.

We can also relate these observations back to the idea of crystallinity, as well, as the factors that influence the glass transition temperature will also influence the ability of polymers to crystallize. In fact, there is a correlation between T_g and T_m for semi-crystalline polymers - typically T_g is about 0.5 to 0.8 T_m in Kelvin.

Molecular Weight Dependence of T_g

The free volume explanation of the glass transition temperature can also be used to explain an observed molecular weight dependence of T_g. In general, the free volume of chain ends should be much greater than the average free volume along the main part of the polymer chain (as expected since the chain ends are only connected to one, rather than two, monomers). As a result, more chain ends in a sample tend to increase the overall amount of free volume, and hence decrease the glass transition temperature. In two samples of the same mass, the sample with polymers of lower molecular weight will tend to have more chain ends, and hence a higher free volume and lower glass transition. We can thus write:

$$T_g(\overline{M}_n) = T^\infty_g - \frac{c}{\overline{M}_n} \nonumber$$

where $T^\infty_g$ is the T_g of a sample with infinite molecular weight and c is the concentration of the sample. This basically says that as the average molecular weight in a sample increases, the glass transition temperature also increases, reflecting the decreasing number of chain ends. Again, note the similarity between this argument and the change in the melting point of semi-crystalline polymers with molecular weight, where chain ends effectively acted as defects in the crystalline matrix.

T_g of Mixtures

We can again refer to the free volume concept to approximate the glass transition temperature of mixtures. If it is assumed that both components of a mixture have the same free volume when mixed as they would separately, we can sum the total free volumes of the mixture based on the weight fraction of each species in the mixture. Note that to avoid phase separation behavior, the mixture should be either two miscible polymers blended together, or two polymer species that form a random copolymer and hence are ”mixed” by the virtue of covalent bonds. Without going through a full derivation, we can find that because free volumes are additive, we can equivalently add glass transition temperatures of the two components to find:

$$T_{g,co} = T_{g,A}w_A + T_{g,B}w_B \nonumber$$

where w_A and w_B are the weight fractions of components A and B.

Tuning T_g with Small Molecule Additives

We just saw that the glass transition temperature of a miscible blend can be easily predicted, we can thus figure out ways to arbitrarily increase of decrease the glass transition temperature of a polymer species by adding small molecules. If we wish to decrease the T_g, we can add small molecular plasticizers, which essentially add free volume to the system. Similarly, if we add small molecules that fit in nicely with the polymer chain of interest and hence decrease the free volume, we can raise the T_g. These molecules are called anti-plasticizers. An interesting example of an anti-plasticizer from biology is cholesterol, which induces the formation of a gel phase in lipids by reducing the free volume of lipid chain ends. We can thus use the knowledge of mixtures of polymers and small molecules to control the glass transition temperature of a mixture; this might be useful for decreasing the glass transition temperature of a system to enable better processing control, for example.

XRD Analysis:

We have talked about XRD analysis and have done our XRD lab which included a polymer. We know that the peaks observed in the diffractogram will be more broad than that for metals due to the lack of long range translational and orientational order for polymers. However, we can obtain some key information from XRD analysis of polymers to allow us to deduce some information about the structure of polymers and how they might be arranged. Let’s take for example the case of the family of polymethacrylates, specifically PMMA, PPMA, PEMA, and PBMA.

When we examine the XRD profile of these polymers we can see that there are some XRD peaks that do not shift for the entire family of polymers. Additionally, the peaks that appear to be consistent between the family are all located at large Bragg angles (2θ). At lower Bragg angles there appears a considerable amount of peak shifting and perhaps even some creation or destruction of peaks in the XRD curve. How can we explain what is happening here?

Well again remember that we observe peaks in the XRD curves when there is constructive interference which occurs when at that particularly incident angle there is some local order perhaps even long-grange order of some type. We also know from Bragg’s law that the Bragg Angle $\theta \propto \frac{1}{d}$. So the characteristic distance where this local order appears is inversely proportional to the incident angle. Let’s

think about the structure of these polymers at very short distances like the C-C bond distance or the distance between the methyl groups. This distance will not change depending on the polymer we are looking at in this family. These small distances correspond to large angles and so this explains why the peaks are conserved for the different polymers.

Now what is happening for the other polymers. Well there are other characteristic distances for these polymers. For example the distance from the chain backbone to the end of the R functional group. Or even the inter-chain distance within the polymer. These distances will change depending on whether the polymer is PMMA, PBMA, etc. If the distance changes there will be a corresponding shift in the location of the peaks. And we see that on average as the side group R increases in length for these polymers the peaks tend to shift to the left which is consistent from what we know about diffraction! This is critical information as it gives us a schematic of the structural characteristics of an unknown polymer chain.

DSC Analysis:

Differential Scanning Calorimetry (DSC) is a thermal analysis technique useful for measuring thermodynamic properties of materials such as specific heat, melting point, boiling point, glass transition temperature (in amorphous/semi-crystalline materials), heat of fusion, reaction kinetics, etc. The technique measures the temperature and the heat flow, corresponding to the thermal performance of materials, both as a function of time and temperature.

Typically a DSC will utilize a heat flux type system in which the differential heat flux between a reference (e.g. sealed empty aluminum pan) and a sample (encapsulated in a similar pan) is measured. The reference and the sample pans are placed on separate, but identical, stages on a thermoelectric sensor platform surrounded by a furnace. As the temperature of the furnace is changed (usually by heating at a linear rate), heat is transferred to the sample and reference through the thermoelectric platform. The heat flow difference between the sample and the reference is then measured by measuring the temperature difference between them by using thermocouples attached to the respective stages. The DSC provides qualitative and quantitative information on endothermic heat absorption (e.g. melting) and exothermic heat release (e.g. solidification or fusion). These processes display sharp deviation from the steady state thermal profile, and exhibit peaks and valleys in a DSC thermogram (heat flow vs. temperature profile). The latent heat of melting or fusion can then be obtained from the area enclosed within the peak or valley.

DSC analysis is an extremely critical and useful tool in characterizing the thermodynamic quantities of polymer, specifically in identifying T_g and T_m as well as determining if a polymer is amorphous or crystalline.

As you can see below if we are working a polymer that is semi-crystalline we can typically observe 2 peaks in the DSC curve as well as a change in the slope of the curve of heat flow vs temperature. Let’s take this analysis step by step. If we remember back to Materials Science we know that a first order phase transition will result in a discontinuity in the first order derivative of the free energy as a function of temperature (at the temperature of the transition). Whereas second order transitions will exhibit a discontinuity in the second derivative of the free energy and will only exhibit a change in slope for the first derivative as a function of temperature. Now the plot that we are looking at is heat flow vs temperature which is essentially δH vs T so we are looking at a plot of the first derivative of free energy.

Now the first feature that we come across (increasing in temperature) is a change in slope. Well we know this must be a second order transition. Additionally we know that this transition occurred at fairly low temperatures. Since our sample is a polymer this transition most likely signifies the glass transition temperature, T_g. This should make sense as we know the T_g is a second order transition. Now the next feature that we see is an exothermic peak. Well we know that this peak indicates a first order transition and if this is an exothermic reaction this should signify some solidification or re-crystallization, T_c. This might at first seem counter-intuitive, why would increasing temperature cause the polymer to crystallize? Well it is because as we increase temperature we increase the mobility of the previously glass amorphous regions. With this increased mobility there is a higher likelihood of encountering other chains the enthalpic interactions accessed here might favor the formation of crystalline regions. This curve might not show up in every polymer. The last peak is an endothermic one and we will recognize this as the melting point, T_m. Now remember this does not break the carbon-carbon backbone bonds but instead the secondary intermolecular interactions.

How would this curve change for a purely amorphous polymer? Well we would simply see a change in the slope to indicate the glass transition and the would be it. Now what if you had a thermoset polymer which has to cure and form crosslinks. Well, we would see a T_g as this is still a polymer. Now would you see a melting temperature? No a thermoset does not undergo melting due to the permanent crosslinks! They would exhibit an exothermic peak that would indicate the energy required to cure/crosslink the polymer!