To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Suppose we have an object under some prescribed load (i.e., an applied force).We can describe how the object will deform in response to that load with the results of elasticity theory. For the purposes of this textbook, we will restrict the discussion to the regime of small displacements, in which we can use a linear version of elasticity theory. The fundamental assumptions of linear elasticity are that (1) the displacements (strains) are small and (2) there are linear relationships between the strains and their associated stresses (we define stress and strain hereinafter). The assumption of linear elasticity is reasonable for many applications and is used extensively in structural analysis.
We note that there is a further restriction to linear elasticity. The applied stress must be low enough so that yielding does not occur, i.e., so that the material does not undergo permanent deformation. Consider a thin metal rod, for example. If one applies a small force to the rod, it deforms but springs back to its original state when the force is removed. If you keep increasing the force, eventually the rod bends and does not return to the original state when the force is removed. That deformation is caused by the movement of linear defects called dislocations, which are described in Appendix B.5.We also describe a basic model of plastic deformation in terms of dislocation motion in that section. Later in this chapter, in Appendix H.5, we discuss the relationship between elastic and plastic strain.
This chapter introduces some basic concepts used in the computations in this text. It is not a programming guide, as each software system has its own language and defined functions. We discuss some general methodologies that are common among programming languages and that crop up in a number of the methods in this text, for example the calculation of random numbers. We also discuss a few numerical methods. Specific implementations of the various methods are presented online at http://www.cambridge.org/lesar.
SOME BASIC CONCEPTS
Computers are discrete and thus all problems, whether discrete or continuous in space or time, must be converted to discrete methods on a computer. The requirement of having discrete methods presents challenges and guides the development of most of the models seen in this text. Some methods, such as molecular dynamics in Chapter 6, may be continuous in one dimension (space), but are solved with discrete time steps. Others, such as the Potts model of grain growth in Chapter 10, are discrete in both space and time.
RANDOM-NUMBER GENERATORS
A common need in essentially all of the methods discussed in this text is for random numbers. It is in many ways odd to discuss random numbers when talking about computers, which are precise and anything but random. Algorithms have been developed, however, that yield series of numbers that look random, at least relative to certain statistical measures of randomness. These algorithms are generally referred to as random-number generators. The challenge is that generators are not all of the same quality.
In Chapter 7 we introduced the Monte Carlo method, with the focus being on calculating equilibrium properties based on sampling of the degrees of freedom in a Hamiltonian (energy) function. In this chapter, the focus is on the Monte Carlo method applied to rates. We shall see that for certain classes of problems, we can find an association between the Monte Carlo “time” and actual time, opening the door to a new class of simulation methods that can model time-dependent processes at time scales far beyond what is possible with standard molecular dynamics.
The fundamental input to the kinetic Monte Carlo method is a list of possible events such as a jump from one site to another in a diffusion problem, a chemical reaction, etc. Associated with each event will be a rate, which will be related to a probability that the event will occur. An understanding of rates is thus very important, so in Appendix G.8 we give a brief review of kinetic rate theory.
A number of researchers independently developed what has come to be known as the kinetic Monte Carlo method. The N-fold way as a methodology for accelerating the simulations of the Ising model is probably the first example [38], which will be discussed in Chapter 10. Voter introduced a similar approach as a way to study the dynamics of cluster diffusion on surfaces [327]. We will discuss his calculation later in this chapter as one of two examples of the complexities, and limitations, of the kinetic Monte Carlo approach.
THE KINETIC MONTE CARLO METHOD
Consider a system whose properties are dominated by thermally activated processes, such as diffusion.
We introduce a novel method for producing polystyrene (PS)-grafted multiwalled carbon nanotubes (MWCNTs), which provides a direct route to composites where carbon nanotubes (CNTs) are the major component. Infrared and Raman spectroscopies confirmed that the MWCNTs were functionalized with PS. Thermogravimetric analysis showed that CNTs increase thermal stability of the composite up to a critical loading (∼40 wt%) beyond which high nanotube loadings decrease the polymer degradation temperature, as a consequence of the thermal properties of CNTs and the composite morphology. Even at loadings as high as 80 wt% MWCNTs, the composite is an effective masterbatch material for both solution- and melt-processing. These results show that in situ polymerizations can be flexible and robust techniques for nanocomposite processing, overcoming limitations of conventional processing techniques to produce nanocomposites with very high nanotube loadings, not achieved hitherto.
Nanoindentation was performed on amorphous silicon nitride films of different thicknesses deposited on gallium arsenide (GaAs) (001) substrates using a conical indenter. Both “pop-in” and ‘pop-out’ were observed from the load-displacement curves when the indentation load exceeded a critical value. Pop-in occurring during loading is associated with plane-slip in the GaAs substrate, and pop-out during unloading is attributed to the interfacial delamination between the film and the substrate. Finite element modeling (FEM) was used to analyze the stress evolution during unloading. The FEM results showed that the stress at the interface evolved from compressive to tensile status during the withdrawal of indentation load, and the interfacial debonding was induced at a critical tensile stress, which is consistent with the pop-out observed. A deformation model for interpreting the pop-in and pop-out events is thereby proposed.
The probability for amorphous silicon (a-Si) to phase transform under indentation testing is statistically determined as a function of annealing temperature from the probability of a pop-out event occurring on the unloading curve. Raman microspectroscopy is used to confirm that the presence of a pop-out event during indentation is a clear signature that a-Si undergoes phase transformation. The probability for such a phase transformation increases with annealing temperature and reaches 100% at a temperature of 340 °C, a temperature well before the temperature where the average bond-angle distortion is fully minimized. This suggests that multiple processes are occurring during full relaxation.
Nano-structured graphene has recently attracted extraordinary attention due to its potential use as an electronic or spintronic material. We investigated the electrical conductivities of antidot and Ar-sputtered graphene samples under a magnetic field in terms of the carrier density. Antidot samples exhibit conductivity that is well explained by charged impurity scattering, which is associated with intravalley scattering. This suggestion is supported by the low intensity of the Raman D band, which is related to intervalley scattering induced by structural defects. In contrast, Ar-sputtered samples show a strong D band and conductivity that is affected by defect scattering. The difference in the main scattering mechanism between the two types of samples appears as Shubnikov-de Haas oscillations at high magnetic fields, which are observed in antidot samples but not in Ar-sputtered samples. Furthermore, an analysis of weak localization effects in both samples at low fields reveals that intra- and intervalley scatterings play significant roles in antidot and Ar-sputtered samples, respectively.
We have investigated the microstructure, the quasistatic and high-rate mechanical properties of magnesium (Mg)-based metal-matrix composites (MMCs) reinforced with nanoparticles, also termed as metal-matrix nanocomposites (MMNCs), in this case reinforced with nanoparticles of β-phase silicon carbide (β-SiC) the volume fraction ranging from 5 to 15 vol%. The yield and the ultimate strength increase with reinforcement volume fraction up to 10 vol% nanoparticles. MMCs with micrometer-sized SiC particles have higher yield strength than their MMNC counterparts, whereas the ultimate strength shows the opposing trend, suggesting greater strain hardening in the MMNCs. Transmission electron microscopy shows that the average interparticle distance decreases with increasing SiC vol%. Recrystallization was reported as completed during sintering at 575 °C [R.D. Doherty et al., Mater. Sci. Eng. A, 238, 219 (1997)], but dislocations might be generated due to thermal expansion mismatch of Mg/SiC during cooling. The majority of Mg-grains below 20 nm remain around the nanoparticles. As such a reverse volume fraction effect takes place in 15 vol% nanoparticle-reinforced MMNCs, which off sets the strengthening advantage induced by the nanoparticles.
“Nanostructured” germanium (Ge; also known as “voided,” “porous,” “nanoporous,” “cratered,” and “honeycomb” Ge) created via ion beam modification has been studied for many years. This work reviews the progress made in studying and characterizing the nanostructured morphology, particularly via the use of experimental techniques such as scanning electron microscopy, atomic force microscopy, and transmission electron microscopy. Specifically, the empirical observations of the structural evolution of Ge as a function of ion beam modification conditions are discussed with added emphasis placed on quantification of the microstructure. The experimental observations and microstructure quantification are further discussed in terms of the implications for proposed formation mechanisms of the nanostructured morphology. Potential uses of the nanostructured morphology in chemical sensor and energy storage applications and suggested future lines of research to further the fundamental understanding of nanostructuring in Ge using ion beam modification are also presented.
We have prepared stable ultrafine narrow dispersed copper nanoparticles (Cu-NPs) using a facile chemical reduction technique below room temperature (300 K), without any template. X-ray diffraction and high-resolution transmission electron microscopy studies reveal the growth of highly crystalline Cu-NPs with an average diameter of 2.2 nm. Interestingly, these Cu-NPs demonstrate both interband electronic transitions along with usual surface plasmon resonance, a unique phenomenon previously unobserved in any noble metal nanoparticles. These Cu-NPs do not get oxidized easily and could be suitable candidates for different optical devices, heat transfer liquids, and biological applications.
Nickel-yttrium nanocrystalline alloys with an as-milled grain size of approximately 6.5 nm were synthesized using high-energy cryogenic mechanical alloying. The microstructural changes due to annealing were characterized using x-ray line broadening, microhardness, focused ion beam channeling contrast imaging, and transmission electron microscopy. Experiments demonstrated that increasing yttrium content led to stabilization of the nanocrystalline grain size at elevated homologous annealing temperatures. Additionally, it was found that inadvertent contamination with nitrogen during the milling process caused the formation of yttrium nitride (YN) precipitates, which, in turn, resulted in an additional nonlinear hardening effect beyond the expected hardening due to grain-size reduction. Results reveal that kinetic pinning by YN particles is effective in retaining a nanostructure to relatively high temperatures.
The structures of γ- and δ-K4P2O7 are solved by X-ray powder diffraction (conventional laboratory X-ray and synchrotron data, respectively), both in hexagonal symmetry (aγ = 5.9645(3) Å, cγ = 14.4972(8) Å, Vγ = 446.64(4) Å3 at 300 °C, Zγ = 2, space group P63/mmc; aδ = 10.211 45(7) Å, cδ = 42.6958(4) Å, Vδ = 3855.59(7) Å3 at room temperature, Zδ = 18, space group P61) with cell–supercell relations $a_\delta \approx a_\gamma \sqrt{3}$ and cδ ≈ 3 cγ. In the experimental conditions, the expected β/γ transition previously announced at 486 °C is not observed; the γ-form is stable at least up to the maximum temperature of our measurements (700 °C). In the γ-form, similar to the orthorhombic form of Na4P2O7, idealized, the pyrophosphate group is in eclipsed conformation, the K+ cations occupying three different coordinations. In the δδ-form, two of the three different [P2O7]4− groups are staggered and one eclipsed, the K+ cations occupying 12 independent sites.