The harmonic approximation

Condensed matter physics made great progress by concentrating on excitations and avoiding the problem of ab initio solution of the conformation of crystals. The conformation or ground state problem is the determination of the static equilibrium structure of a system and is a difficult problem. For most crystals the ground state problem is still an unsolved theoretical problem. An example of progress without first solving for the ground state was the development of phonon theories for determining the motion of the atoms of a solid. Phonon is the term used to describe quantized propagating normal mode vibrational excitations and is used in analogy to photon, which describes the quantized propagating excitations of the electromagnetic field. Propagation of excitations is an essential element of large systems. Phonon theories apply not only to equilibrium phenomena such as the study of internal energy and specific heats but also to dynamic problems such as electrical and thermal conductivity. The study of excitations with the ability to propagate is an improvement over simple vibrational mode studies that only deal with static equilibrated systems. As we will discuss in later chapters, the propagation of energy along the helix is relevant to a number of biological processes and the distinction between propagating excitations and static excitations will be of importance.

Phonons are the principal excitations found in insulators and are important excitations in all types of condensed matter.

Anomalous stability of poly(dA)–poly(dT)

The four DNA bases come in two sizes. The larger bases, adenine and guanine, are variants of the purine structure which is a double ring structure (Saenger, 1984). The smaller two bases, thymine and cytosine, are pyrimidine variants with a single ring. The base pairing scheme requires one purine and one pyrimidine in each pair making the total span the same for all complementary base pairs. The total molecular mass of the two pairs is almost identical as well. The difference in size makes the overlap of neighbor bases different in homopolymers compared to copolymers. In homopolymers a purine is stacked above another purine and pyrimidines above pyrimidines. The interbase gap undergoes helical twist but tends to be continuous, as seen in Figure 9–6 in Saenger (1984). In the copolymers the gap moves from one side of the twisting center line to the other depending on which side is the purine or which the pyrimidine. This shifting coupled with the helical twist in the copolymers causes considerable overlap of atoms of the large bases from one level to the next in every second base pair. The overlap is between bases on different strands in the copolymer and the van der Waals stacking interactions at the overlap becomes a cross strand interaction. The copolymer has these large interstrand interactions but the homopolymer does not. One would therefore expect that copolymers with this added cross strand interaction would be more stable against strand separation melting than homopolymers.

Dynamics of macromolecules

Biological macromolecules carry out functions that require them to have very complex physical and chemical dynamics. Theoretical analysis of that dynamics has proven very difficult because the macromolecules are large systems whose dynamics is very nonlinear. The large size arises because the systems have many atoms which cannot be simply related by symmetry operations and the dynamics has to be analyzed down to motions on the atomic level to fully appreciate what is going on. This microscopic approach is important because macromolecules often function by changing their conformation on the atomic level. The nonlinearity results because many bonds are weak relative to physiological temperatures, in many biological processes bonds are broken and bonding rearranged. In this book we detail a method for studying macromolecular dynamics that is particularly well suited to large systems that are also highly nonlinear. With an additional operator included the method is particularly useful in studying the dissociation of chemical bonds in large nonlinear systems. We know of no other approach that can efficiently study the melting or bond disruption problem in such large systems on a microscopic scale. To date the majority of the applications of this method, and those presented in this book, are to problems of base pair separation in the DNA double helix. The advantages and disadvantages of the method and how it may be extended to other systems can, however, be seen in these applications.

Significant understanding of biological processes has been made by studying the dynamics of macromolecules at the microscopic level. The bulk of the researchers interested in the results are biochemists, chemists, pharmacists, and biologists. The MSPA approach developed in this book is based on methods used in condensed matter physics which are not familiar outside that discipline. The method is useful in solving long time scale problems in molecular biology that should be of interest to the biochemists etc. working in the field. This book is therefore aimed at presenting both a coherent development of the MSPA methodology and some of the background needed to understand it by persons not coming from a physics background. Biophysicists may find the results and the physics background of interest. To increase the usefulness of the book to the different readers I have, to the extent possible, concentrated on concepts and description in the main text and kept the mathematical formalism in appendices. On the other hand physicists may be interested in the way that physics methodologies have to be altered to be applied to be useful in this new situation. The complexity of biological systems is greater than that usually dealt with in condensed matter investigation and changes in approach are necessary. Physicists may also be interested in the development of a new approach to cooperative transitions that seems to work very well, particularly in large complex systems.

Changes in the ground state problem

As pointed out in Chapter 3, there is a calculational advantage to working in an effective harmonic approximation. All the work discussed so far has used this approximation, i.e. assumed a ground state for each polymer determined from experimental observation. Control variables that could alter that ground state have not been changed, the only control variable allowed to vary is the temperature. Other factors affect DNA melting, such as salt concentration, hydration level, and hydrostatic pressure, and it would be useful to extend the methods developed to study effects of changes in these other control variables. This chapter is about ways to introduce changes in control variables that would normally be thought of as elements affecting the ground state solution. We show that one can selectively put back into the problem particular static or ground state elements, without carrying out a full ground state solution, and determine their effects on the dynamics and melting.

For example, consider increasing the hydrostatic pressure on a system containing double helices. The atoms would be pushed closer together, altering the interatom distances in the ground state. Another example would be changes in salt concentration in the environment of the helix. Altering salt concentration changes the shielding of the Coulomb interaction between the highly charged phosphate groups, which would alter the static equilibrium positions in the helix.

The main focus of our previous discussion of MBE has been the identification of various universality classes. The models we discussed are expected to be valid on a coarse-grained level, at which the exact structure and form of an island does not matter. However, with the perfection of experimental tools, it is possible to observe the interface morphology at the atomic scale – leading to the discovery of rich island morphologies. In this chapter we focus on this early-time morphology, for which the coverage is less than one monolayer; this regime is usually referred to as submonolayer epitaxy.

The phenomenology is quite simple. Start with a flat interface, and deposit atoms with a constant flux. The deposited atoms diffuse on the surface until they meet another atom or the edge of an island, whereupon they stick. Thus if at a given moment we would photograph the surface, we would observe a number of clusters – called islands – with monomers diffusing between them. What is the typical size and number of the islands? What is their morphology? How do these quantities change with the coverage and with the flux? These are among the questions we address.

Model

Let us consider in more detail the deposition process outlined above. Consider a perfectly flat crystal surface with no atoms on it. At time zero we begin to deposit atoms with a constant flux F. Atoms arrive on the surface and diffuse (the deposition and diffusion processes take place simultaneously).

In Chapters 12–15, we discussed in detail the properties of growth processes dominated by deposition and surface diffusion. We saw that one origin of randomness is the stochastic nature of the deposition flux, which generates a nonconservative noise in the growth equations. A second component of randomness on a crystal surface comes from the activated character of the diffusion process. As we show in this chapter, this type of randomness generates a conservative component to the noise, leading to different exponents and universality classes.

While diffusive noise is certainly expected to be present in MBE, there is no experimental evidence for the universality class generated by it. For this reason, we separate it from the discussion of the simple MBE models. This chapter can be considered to be a theoretical undertaking of interest in its own right, investigating the effect of the conservation laws on the universality class.

Conservative noise

An important contribution to randomness on a crystal surface arises from the activated character of the diffusion process. Equation (13.5) results from a deterministic current that contributes to interface smoothing – i.e., only particle motion that aids smoothing is included. But, as discussed in Chapter 12, particle diffusion is an activated process in which all possible moves – each with its own probability – are allowed. Because the nature of the diffusion process is probabilistic, an inherent randomness is always present.

The notion of universality suggests that the exponents determined in the previous chapters are unique, since they belong to the only possible growth equation with a set of given symmetries. Any model or experiment will show these exponents if the hydrodynamic limit has been reached. But there is one term in all these equations that we have largely ignored thus far, one whose form is not fixed by symmetry principles: the noise η. We have assumed that this noise is uncorrelated – i.e., that it has a Gaussian distribution. But is this the only type of noise possible in a physical system?

What if the magnitude of the noise in a given point is not independent of the magnitude of the noise in a different point – i.e., what if there are spatial correlations in the noise? What if the magnitude of the noise at a given time is not independent of the noise at a different time – i.e., what if there are temporal correlations in the noise? In many experimental situations, we know little about the nature of any noise that may be present, so to consider it Gaussian is perhaps an unrealistically simple assumption.

One possibility is that the correlation length of the noise is finite, i.e., that the different events ‘know’ about each other only if they are within a finite spatial separation ξ or temporal separation τ.

In previous chapters, we focused on interfaces that grow and roughen due to thermal fluctuations, the origin of the randomness arising from the random nature of the deposition process. For a class of interface phenomena, however, we do not have deposition, but rather we have an interface that moves in a disordered medium. The experiment described in Chapter 1 in which a fluid interface propagates through a paper towel is one example. The velocity of the interface is affected by the inhomogeneities of the medium: the resistance of the medium against the flow is different from point to point; we call this quenched noise, because it does not change with time. Fluid pressure and capillary force drive the fluid, and disorder in the medium slows its propagation. If the disorder ‘wins’ the competition, the interface becomes ‘pinned.’ Conversely, if the driving forces win, the interface stays ‘depinned.’ This transition from a pinned to a moving interface – obtained by changing the driving force – is called the depinning transition.

In the following three chapters, we discuss how quenched disorder leads to interface pinning – and depinning. We will show that the same theoretical ideas describe interface motion in a random field Ising model, which is relevant to the problem of domain growth in a disordered magnetic material.

The problem of a moving interface in the presence of quenched noise is a new type of critical phenomena, arising from ‘quenched randomness.’

The primary focus of this book is on interface roughening generated by various nonequilibrium deposition processes. However, crystal surfaces may be rough even under equilibrium conditions – with no atom deposition. Consider, e.g., a flat surface in equilibrium at a very low temperature. Thermal fluctuations do not have an observable effect on the shape of the crystal, and all atoms remain in their appropriate lattice positions. As temperature increases, the probability that an atom will break its bonds with its neighbors increases. Some atoms hop onto neighboring sites, thereby generating roughness on the atomic scale. At first glance, one might expect a gradual transition to a rough morphology, since the higher the temperature, the more the atoms wander on the surface – until the surface melts. Indeed this is a correct description of the short-ranged correlations between neighboring atoms. However, as we shall see, on much longer length scales there is a distinct (higher order) thermodynamic phase transition that takes place at a critical temperature TR. For T < TR the crystal is smooth, corresponding to a flat facet, while for T > TR it is rough, implying a rounded crystal shape analogous to a liquid drop. This ‘roughening transition’ can be successfully described using ideas of statistical mechanics, the formalism being analogous to that developed in previous chapters.

Equilibrium fluctuations

The lowest energy state of the crystal corresponds to a flat surface.

In the previous chapter, we introduced a simple growth model that exhibits generic scaling behavior. In particular, the interface width w increases as a power of time [Eq. (2.4)], and the saturated roughness displays a power law dependence on the system size [Eq. (2.5)]. There exists a natural language for describing and interpreting such scaling behavior, and this is the language of fractals. In this chapter, we introduce the concepts of fractal geometry, which provide a language in terms of which to better understand the meaning of power laws. Isotropic fractals are self-similar: they are invariant under isotropic scale transformations. In contrast, surfaces are generally invariant under anisotropic transformations, and belong to the broader class of self-affine fractals. We will therefore also discuss the basic properties of self-affine fractals, as well as numerical methods for calculating the critical exponents α, β, and z.

Self-similarity

An object can be self-similar if it is formed by parts that are ‘similar’ to the whole. One of the simplest self-similar objects is the Cantor set, whose iterative construction at successive ‘generations’ is shown in Fig. 3.1. If we enlarge the box of generation 3 by a factor of three, we obtain a set of intervals that is identical to the generation 2 object. In general, at generation k we can enlarge part of the object by a factor of three and obtain the object of generation (k – 1).

Most of our life takes place on the surface of something. Sitting on a rock means contact with its surface. We all walk on the surface of the Earth and most of us don't care that the center of the Earth is molten. Even when we care about the interior, we cannot reach it without first crossing a surface. For a biological cell, the surface membrane acts not only as a highly selective barrier, but many important processes take place directly on the surface itself.

We become accustomed to the shapes of the interfaces we encounter, so it can be surprising that their morphologies can appear to be quite different depending on the scale with which we observe them. For example, an astronaut in space sees Earth as a smooth ball. However Earth appears to be anything but smooth when climbing a mountain, as we encounter a seemingly endless hierarchy of ups and downs along our way.

We can already draw one conclusion: surfaces can be smooth, such as the Himalayas viewed from space, but the same surface can also be rough, such as the same mountains viewed from earth. In general the morphology depends on the length scale of observation!

How can we describe the morphology of something that is smooth to the eye, but rough under a microscope? This is one question we shall try to answer in this book. To this end, we will develop methods to characterize quantitatively the morphology of an arbitrary interface.