Callister Chapter 4: Imperfections in Solids

Candidacy season is upon us, so this chapter summary will just be a very quick one. In this post, I will run through the different kinds of point defects, dislocations, and interfacial defects. Please refer to the textbook for additional information on this topic, such as calculating the number of vacancies, imaging defects, and determining grain size.

Point defects
-this type of defect is associated with one or two atomic positions
-vacancies occur when atoms are missing from their lattice positions
-interstitials occur when atoms are found in lattice sites that are normally unoccupied for a given crystal structure
-impurities are foreign atoms that either take the place of lattice atoms or squeeze into interstitial sites
-a solid solution may form when the original structure of a crystal is maintained despite the addition of impurity atoms
-substitutional solid solutions form when an impurity atoms substitutes a host atom (this happens when their atomic radii and electronegatives are similar and they have the same crystal structure)
-interstitial solid solutions form when the impurity atom is small and occupies interstitial sites

Dislocations
-this type of defect is linear/one-dimensional
-types of dislocations include edge, screw, or a combination of the two
-in an edge dislocation, the Burgers vector is perpendicular to the dislocation line
-in a screw dislocation, the Burgers vector is parallel to the dislocation line
-the following figure depicts these types of defects (point A is pure screw and point B is pure edge)

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print.

Interfacial defects

-this type of defect is two-dimensional

-external surfaces are considered interfacial defects because surface atoms don't bond to surrounding atoms the same way as atoms throughout the crystal do (also, surface atoms have higher energy)

-grains boundaries are boundaries separating grains that are oriented in different crystallographic directions

-twin boundaries are grain boundaries that have mirror symmetry

-stacking faults occur when there is an interruption in stacking sequence

-phase boundaries occur when there are changes in physical/chemical composition

A (very simplified) review of Flory-Huggins theory

This post is a simple and qualitative review of regular solution theory and how it extends into the Flory-Huggins theory. Regular solution theory is a statistical model that gives an expression for the free energy of mixing of a binary system. Though simple, this theory provides a useful picture for the phase behavior between two components and lays the framework for the Flory-Huggins theory.

Regular solution theory considers two contributions to the free energy of mixing; namely, the entropy of mixing and the enthalpy of mixing. For a binary system with components 1 and 2 having mole fractions x₁ and x₂, Boltzmann’s definition of entropy gives us the following expression for entropy of mixing per mole of lattice sites (I will omit the math here, but this comes from applying Stirling's equation to Boltmann's entropy):

In other words, the entropy of mixing describes the number of ways in which objects 1 and 2 can be arranged on a lattice and is thus related to intermolecular interactions. On the other hand, the enthalpy of mixing considers interactional contributions - that is, the exchange energies between pairwise nearest neighbors. By defining the interaction parameter χ as the exchange energy per molecule normalized by kT, we obtain the following expression for enthalpy of mixing per mole of lattice sites:

Combining these two expressions using ΔG_m = ΔH_m – TΔS_m, we obtain the well-known expression for the free energy of mixing per site:

By extending this, we can understand the Flory-Huggins theory, which describes the interactions between a polymer and solvent. In this model, we consider each lattice site to be the volume of one solvent molecule and we assume that each polymer occupies N lattice sites (such that the volume of one solvent molecule = the volume of one polymer segment = the volume of one lattice site). For volume fractions φ₁ and φ₂, the free energy of mixing per site would be:

In the above equation, the entropic terms favor mixing (note that as N increases, entropy decreases) and the enthalpic term opposes mixing (when chi is positive). The Flory-Huggins interaction parameter chi can be determined by the Hildebrand solubility parameters using:

where V₀ is the molar volume of the solvent and δ₁ and δ₂ are the solubility parameters of the polymer and solvent. 

Finally, it is worthwhile to mention some of the simplifying assumptions made in this theory. For example, it is a mean-field theory (treats local interactions the same way as bulk interactions), which is inaccurate for polymer solutions because crosslinking inevitably causes non-uniformity in the solution. This model also assumes that there is no change in free volume during mixing and that entropic contributions only come from translational configurations.

Callister Chapter 14: Polymer Structures

There are many types of polymers: natural polymers such as wood, rubber, cotton, wool, leather, and silk; biological polymers such as proteins, enzymes, starch, and cellulose; and synthetic polymers such as polypropylene (PP), polyvinyl chloride (PVC), and polystyrene (PS). In this chapter, we will discuss the various structural elements of polymers, such as their chemistry and crystallinity. The effect of structure on polymer properties will be discussed in the next chapter.

Chemistry of polymers

The term polymer means “many mers,” where mer originates from the Greek word meros, which means “part.” Thus, polymers are chains of repeat units. The chain forms by the sequential addition of monomer units, as illustrated in the figure below:

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

In this example, polyethylene is formed as an unpaired electron is transferred to each ethylene monomer that is added to the chain. When all the repeat units are the same, like in this case, the resulting chain is called a homopolymer. When two or more different repeat units are involved, the result is called a copolymer. The table below shows the repeat units of some of the most common polymers.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Molecular weight

Since the lengths of polymer chains will all be different, molecular weights of polymers are described as average molecular weights. There are two ways to do this: number-average molecular weight and weight-average molecular weight. The number-average molecular weight can be computed using:

where M_i is the mean molecular weight in the interval i and x_i is the fraction of the total number of chains within that interval. On the other hand, weight-average molecular weight can be calculated using:

where M_i is the again the mean molecular weight in the interval i and w_i is the weight fraction of molecules in that interval. Number-average molecular weight will be more sensitive to low molecular weight species while the weight-average molecular weight will be more sensitive to higher molecular weight molecules. The polydispersity index, which is a measure of the breadth of molecular weight distribution, is the weight-average molecular weight divided by the number-average molecular weight. Another parameter that is used to describe polymer size it the degree of polymerization. This can be calculated by:

where m is the molecular weight of the repeat unit.

Molecular shape

Polymer chains can take on different configurations depending on the degree of twisting and bending that occurs along the backbone. As depicted in the carbon backbone examples below, each successive carbon atom can lie on any point of the cone of revolution that has a 109 degree angle with the previous bond. In the middle example, the chain is fairly linear, whereas in the example on the right, the rotation of the carbon atoms into other positions results in a more twisted shape. Rotational flexibility depends on factors such as the repeat unit structure or the existence of bulky side chains. Quantitatively, chain conformations can be described using models such as the free-jointed chain, the freely-rotating chain, or the hindered rotating chain.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Some definitions related to molecular shape are the end-to-end distance and the persistence length. For a polymer made up of monomer “vectors” l_i, the end-to-end distance h is the sum of l_i.

The persistence length, on the other hand, is the characteristic length scale for the exponential decay of the correlations of backbone tangents. Essentially, the persistence length is how long it takes for the backbone to bend, on average, at a right angle. 

Molecular structure

There are different types of molecular structures found in polymers, such as linear, branched, crosslinked, and network. They are shown below:

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Linear polymers can be described as a “mass of spaghetti.” They are fairly flexible and the chains are held together by Van der Waals forces and hydrogen bonding. Branched polymers are harder to pack together and thus have lower density. Crosslinked polymers are formed when chains are joined together by covalent bonds. Rubbers are often crosslinked through the process of vulcanization. Finally, network polymers are three-dimensional and are the result of multiple covalent bonds.

Thermoplastics are linear or branched and soften when heated and harden when cooled. Thermosets are crosslinked or network and, once hardened, will not soften upon heating.

Isomers

Stereoisomers can be isotactic, syndiotactic, or atactic. An isotactic configuration is when R groups are found on the same side of the chain. A syndiotactic configuration is when the R groups alternate sides. An atactic configuration is when the R groups are randomly positioned.

There are also geometrical isomers, which occur within polymers with double bonds in the backbone. When side groups are on the same side of the double bond, the polymer is said to be in the cis structure. When the side groups are diagonally across from each other, the polymer is in the trans structure. 

Copolymers

When chains are composed of more than one type of repeat unit, the resulting polymer is called a copolymer. Copolymers can be advantageous because they allow polymer chemists to combine properties from different homopolymers. Below are the different types of copolymers. For a more detailed discussion on block copolymers, check out this post.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Crystallinity in polymers

A polymer is said to be crystalline when its molecular chains are aligned and packed in an ordered arrangement. On the other hand, it is amorphous when the chains are misaligned and disordered. Generally, it is easier to achieve crystallinity in polymers that have simple and/or regular chemical structures. Often, polymers will have crystalline regions interspersed within an amorphous matrix. The degree of crystallinity can be determined using the following equation: 

where ρ_s is the density of a specimen for which the percent crystallinity is to be determined, ρ_a is the density of the amorphous part, and ρ_c is the density of the perfectly crystalline part.

Crystalline regions can be described using the chain-folded model, depicted below. In this model, chains within a platelet are aligned and fold back and forth on themselves.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Many semicrystalline polymers form spherulites. As shown in the figure below, spherulites grow radially outwards and consist of lamellar chain-folded fibers with amorphous regions between them.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Defects in polymers

Due to the chainlike nature of polymers, the concept of defects is different than in metals and ceramics—though they still exist. As shown in the figure below, defects such as impurities and screw dislocations follow very similar definitions as they did in other materials. However, besides traditional defects, chain ends are also considered defects (usually associated with vacancies). Also, the surfaces of chain-folded layers and boundaries between crystalline regions are considered interfacial defects.

Callister, William D. Materials Science and Engineering: An Introduction. New York: John Wiley & Sons, 2007. Print

Diffusion in polymers

Small foreign molecules can diffuse between the molecular chains of polymers. The mechanism for this diffusion is similar to the interstitial diffusion that occurs in metals, although in polymers diffusion occurs through small voids between chains, from one open region to another. Thus, rates of diffusion are greater in more “open” amorphous regions than through crystalline regions. Smaller molecules also diffuse faster than larger ones.

Diffusion in polymers is often characterized by the permeability coefficient, P_M. Steady-state diffusion through a polymer membrane can be described using a modified Fick’s first law:

where J is the diffusion flux , ΔP is the difference in pressure of the gas across the membrane, and Δx is the membrane thickness.