Callister Chapter 3: The Structure of Crystalline Solids

In the last chapter, we talked about the different types of atomic bonding. Here, we’ll go one step further and discuss how these atoms are arranged—that is, what types of crystal structures they can have. This Callister chapter puts emphasis on the most common metallic crystal structures (FCC, BCC, and HCP), but I will try to give a broader overview of all Bravais lattices followed by a review of X-ray diffraction and Bragg’s law.

Bravais lattices and symmetry

A Bravais lattice is as an infinite set of discrete points with an arrangement and orientation that appears exactly the same from whichever of the points the array is viewed. In other words, it is an array of points with translational symmetry. The basis is the actual atoms that are positioned on these lattice points. Thus, a crystal = lattice + basis. Crystals are built out of unit cells, which are the smallest repeating units that show the full symmetry of the crystal.

Symmetries can be described using space groups, which are the lattice’s translational symmetry plus other symmetry elements which are called point groups. This can be summarized by the graphic below:

There are seven crystal systems, each with their own symmetries. We can begin with the cubic system, which has four 3-fold axes and three 4-fold axes. By stretching or compressing the cubic system along one body diagonal, we obtain the trigonal system. Since we lose all the previous symmetries except along the axis we deformed, the trigonal system has only one 3-fold axis. Similarly, by stretching or compressing the cubic system along one axis, we arrive at the tetragonal crystal system, which has one 4-fold axis (along the axis we stretched/compressed). By deforming this system along a second axis, we get an orthorhombic crystal system. This will have three 2-fold axes perpendicular to each of the faces. Next, if we shear one face with respect to the opposite face, we will have a monoclinic system. The monoclinic system will have one 2-fold symmetry, since there is only one face around which you can do a 2-fold rotation. Lastly, by shearing a second face relative to the opposite face, we get a triclinic system, which unsurprisingly has no symmetries. These crystal systems and their parameters are summarized in the table below:

http://saravanamoorthy-physics.blogspot.com/2013/12/introduction-bonding-in-solids-major.html

Callister Chapter 2: Atomic Structure and Interatomic Bonding

The way atoms are arranged and how they interact with each other within a material directly affect the material’s properties. The most classic example of this is the comparison between graphite and diamond, both of which are made of carbon but which have very different properties. In graphite, each carbon atom is bonded to three other carbons, forming sheets that easily slide past each other. In diamond, each atom is bonded to four other carbons, forming strong tetrahedra throughout the crystal, making diamond the hardest known material. In this chapter, we will discuss atomic structure, bonding forces and energies, and types of bonds.

Atomic structure

-an atom consists of a tightly bound nucleus of protons and neutrons that are surrounded by an electron cloud

-towards the end of the nineteenth century, it became clear than many phenomena involving electrons could not be explained with classical mechanics, leading to the birth of quantum mechanics

-the main stipulation of quantum mechanics is that electrons have quantized energies (they can only have specific values of energy)

-two of the main models used to describe atoms are the Bohr atomic model and the wave-mechanical model

-the Bohr model assumes that electrons revolve around the nucleus in discrete orbitals, as seen in the figure below

Callister Chapter 1: Introduction

Now that it’s summer, I am finally making good on my promise to post chapter summaries of Materials Science and Engineering: An Introduction, better known as the Callister textbook. There are 22 chapters in total and approximately 11 weeks until Hell Month aka the candidacy exam, so my goal is to cover about two chapters a week. That being said, if there is any topic/chapter that you find especially interesting—or if you just feel like being a super awesome friend—feel free to talk to me about writing your own summary that I can add to this blog!

Materials science and engineering plays an integral role in life as we know it—indeed, it not only influences our everyday lives, but has governed the advancement of humankind so much so that early civilizations are now described by their materials development (Stone Age, Bronze Age, Iron Age). This chapter describes the purpose of materials science and engineering and classifies materials into several main categories.

The purpose of materials science and engineering

-materials science is the study of the relationship between a material’s structure and its properties

-materials engineering is the design of a material’s structure to produce desired properties

-from small scale to large scale, a material’s structure—that is, its internal arrangement—includes subatomic, atomic, microscopic, and macroscopic structure

-a material’s properties fall into the classifications of mechanical, electrical, thermal, magnetic, optical, and deteriorative

-the way a material is processed influences its structure, which in turn influences its properties, and ultimately determines its performance

Tecnai Standard Operating Procedure

What I've learned from the few sessions I've had on the Tecnai at Penn State's Materials Characterization Lab so far is that using a TEM is really, really confusing. And being told again and again that the instrument is very expensive and that you must be very careful only adds to the anxiety!

That being said, I've decided to write up a procedure for alignment on the Tecnai in as much detail as I could. I am not an experienced user by any stretch of the word, and I do suggest that you use what I've provided here in conjunction with your own notes from your training session, but I hope that these steps can help you in some way.

Please note that this procedure only applies to basic imaging on the Tecnai (I know, I know, these steps look anything but "basic.") Other techniques, such as electron diffraction or electron energy-loss spectroscopy, may require additional steps. I'll be sure to update this post when I eventually learn how to do those!

What are block copolymers and how do they self-assemble?

The following post is a modified version of a short paper I wrote for MatSE 542 last semester. It is essentially a summary of the paper “Self-assembly of block copolymers” by Yiyong Mai and Adi Eisenberg published in The Royal Society of Chemistry in 2012. 

Introduction
Block copolymers (BCPs) are a fascinating class of materials that have recently attracted significant attention due to their ability to self-assemble into a variety of morphologies, such as spheres, cylinders, gyroids, and lamellae. This post will discuss what block copolymers are, the thermodynamics behind microphase separation, and their theoretical and experimental phase diagrams.

Simply put, block copolymers consist of two or more chemically dissimilar polymer blocks that are thermodynamically immiscible yet covalently bonded. Figure 1 below illustrates the most popularly studied structures of block copolymers which can be formed with two types of blocks, A and B. Such a material is referred to as a diblock copolymer, while structures consisting of three blocks are called triblock copolymers, and so on. The chemically distinct blocks will separate into different domains while the covalent bonds restrict this demixing to local length scales, resulting in what is called microphase separation and giving rise to the aforementioned myriad of morphologies. 

Figure 1: Schematic of different structures of diblock copolymers. Reproduced with permission from The Royal Society of Chemistry.

Microphase separation

Microphase separation in BCPs is driven by a combination of the unfavorable mixing enthalpy and a small mixing entropy, with the covalent bonds holding blocks together to prevent macroscopic phase separation. For a diblock copolymer consisting of blocks A and B, microphase separation is influenced by three parameters: the volume fractions of the A and B blocks (f_A and f_B), the total degree of polymerization of the two blocks (N = N_A + N_B), and the Flory-Huggins interaction parameter (χ_AB). The Flory-Huggins parameter is dependent upon several factors and can be described with the equation below:

where z is the number of nearest neighbors per repeat unit, k_B is the Boltzmann constant, T is the temperature, and ε is the interaction energy of the respective block pairs. The degree of microphase separation is determined by the product of the interaction parameter χ_AB and the total degree of polymerization N. Thus, as χN decreases (or as temperature increases), the blocks become increasingly miscible while the combinatorial entropy increases and the copolymers become disordered. This behavior is referred to as an order-to-disorder transition (ODT).

Relationship between image contrast and electron dose

As a means of offering more background on my radiation damage study, this quick post will highlight some relevant information from the paper Experimental high-resolution electron microscopy of polymers by David C. Martin and Edwin L. Thomas.

The main takeaway is that high resolution electron microscopy of organic materials is limited because of their sensitivity to radiation damage. In order to understand the relationship between image contrast and radiation damage more quantitatively, let us take a look at a few simple equations. The number of electrons Q which are incident on an area d² is

where J is the total electron dose (electrons per unit of area) and d is the smallest resolvable feature size of the object of interest. Since the standard deviation in the intensity of an object illuminated with Q electrons is the square root of Q (this is simply the square root of the variance, which is the number of electrons Q), the noise of an electron microscopy image is given by

From this equation we can see that the noise is minimized as the electron dose J is increased – and herein lies our problem. Since organic materials, such as the polymers that I am studying, are especially sensitive to radiation damage, my current goal is to determine a critical electron dose at which high contrast can be achieved without destroying the material’s structure.

According to this paper, the critical dose J_c can be obtained by fitting the intensity of diffraction peaks as a function of electron dose to the following exponential function:

Thus, as explained in my research update post, the critical dose can be obtained by taking the inverse of the decay rate. As for the extra background intensity term, I am still working on whether that should be included in the fit because I’m not sure if I can assume the decay goes to zero.

Radiation damage study of P3HT and P3HT-b-PFTBT

A couple of weeks ago, I obtained my first set of TEM electron diffraction data with the help of another lab member, Thinh Le. Much like how electrons passing through a grating will produce an interference pattern due to particle-wave duality, electrons accelerated through a TEM sample’s periodic structure will result in a diffraction pattern. As seen in the examples below, a crystalline specimen will produce a spot pattern whereas a semi-crystalline or amorphous specimen will produce diffraction rings. Ring formation occurs because each randomly oriented domain produces its own diffraction pattern, and the superposition of these patterns forms rings.

http://www.ammrf.org.au/myscope/tem/background/concepts/imagegeneration/diffractionimages.php

Since the samples I was studying were polymers (P3HT and P3HT-b-PFTBT), their electron diffraction patterns were rings. However, they are very sensitive to electron beam damage – in fact, every time I moved the electron beam to a new location on the sample, I could see the diffraction ring fading away right before my eyes the longer it was exposed to the beam. For this very reason, the first milestone that I hope to achieve is to calculate a critical dose D_c at which soft materials such as P3HT and P3HT-b-PFTBT can be imaged with maximum contrast without destroying their structure.

To characterize this radiation damage, I took 10 consecutive electron diffraction images at one sample location at regular time intervals for both P3HT and P3HT-b-PFTBT (with increasing time, the electron dose also increases proportionally). By extracting the intensity of the diffraction rings using the software Digital Micrograph, I was then able to plot the intensity of the rings against the electron dose. The resulting data can be fitted to an exponential curve and D_c can be calculated as the inverse of the decay rate.

Unfortunately, the values for D_c that I calculated did not agree with values previously calculated by my lab. Possibilities for this discrepancy could be an inaccurate electron dose calibration or imprecise intensities. The first of these issues is out of my hand, but I will attempt to remedy the second by calculating radially integrated intensities using Mathematica.