Biophysics is not a single entity. It is a congregate of a plethora of disciplines such as molecular biology, physics, chemistry, biochemistry, bioinformatics, nanotechnology and even mathematics, to name a few. It answers a lot of questions quite similar to the ones researched in molecular biology but uses very unconventional methods to do so. To put it simply, it studies the interaction of various systems of a cell within the cell itself and outside it and can cover an astonishingly wide range of topics. More specifically, biophysics is the study of life phenomenon.

The living organism and the biosphere are not isolated; they exchange matter and energy continuously. From the thermodynamic perspective, “A living organism feeds on negative entropy”. Several alternative definitions exist. For example, the postulate of life states that “Life itself should be looked upon as a basic postulate of biology that does not lend itself to further analysis”. According to Bohr's uncertainty principle, “Physico-chemical properties of living organisms and the life phenomenon cannot be studied simultaneously”. This implies that cognition of one excludes the other. According to Schrödinger, “An organism is an aperiodic crystal”. This is a very well-defined conceptualization of any organism. An organism is a complex many-body system of numerous biomolecules interacting through innumerable physico-chemical reactions in an orderly, coordinated and regulated manner. In physical science, a crystal exhibits spatial order which allows a comprehensive description of this material through statistical means. However, no such order is observed in living organisms. Nonetheless, millions of biochemical reactions are carried out with excellent accuracy and reproducibility inside the numerous cells constituting the organism.

Settling down of heterogeneous suspensions over a period of time is a common phenomenon in everyday life. Such processes are very slow and completely governed by the uniform gravitational field of the earth. The importance of sedimentation as an analytic method to examine differential molecular weight of particles dispersed in a solvent medium was realized by Mason and Weaver (1924). The method was further developed into a novel branch of molecular transport theory by Svedberg (Svedberg and Pederson 1940). Determination of the molecular weight of synthetic polymers, proteins, nucleic acids and polysaccharides is of prime importance to both physical and organic chemists. Sedimentation methods have enjoyed remarkable popularity in analytic chemistry as reliable and robust tools. It must be realized that similar to molecular diffusion, sedimentation is a purely transport process. In fact, diffusion and sedimentation are competing processes in any given polymer–solvent system. Further any treatment of molecular transport in the dispersion medium, the flow equations are constituted following irreversible thermodynamic concepts. Thus, sedimentation equilibrium behaviour of polymer molecules in a solvent is significantly dependent on conformation, concentration, molecular weight and molecular charge density of the polymer. This makes the data interpretation of sedimentation experiments tedious. At the same time, one of the compelling reasons why the experiments to determine the molecular weight of proteins have been successful is because, for relatively homogeneous globular protein dispersions, the thermodynamic non-ideal terms are negligible and experimental data analysis is not cumbersome. In this chapter, some basic and essential features of sedimentation equilibrium will be discussed.

Basic concepts

Polymer solutions are complex liquids at any given temperature and require specialized thermodynamic treatment. The phase stability of polymer solutions is a pre-requisite for any potential application. In general, the theoretical calculation of the thermodynamic properties of liquids and solutions involves determination of their configurational properties (those that depend only on intermolecular interaction) ignoring the internal movement of molecules. As a result, we can define configurational or intermolecular energy of a solution as the energy of a liquid minus the energy of the same substance in the state of an ideal gas at the same temperature. Thus, as is evident, configurational thermodynamic properties can have combinatorial and/or non-combinatorial properties. This attribute of polymer solutions has attracted much attention in the past (Flory 1953; Hildebrand 1953; Huggins 1941, 1942).

Thermodynamics demands that entropy be the deciding factor that governs solution stability. Entropy of mixing arising due to the rearrangement of different molecules is called the geometrical or combinatorial entropy of mixing. The non-geometrical (non-combinatorial) contribution of the entropy of mixing results from the energy of interaction between the components present in the solution, resulting in contraction of the solvent and the formation of oriented solvation layers (hydration sheathes). This involves a decrease in entropy of the solvent. The former contribution(∆Scomb > 0) favours dissolution (∆G = ∆H – T∆S becomes more negative), the latter contribution (∆Snon-comb < 0) does not favour dissolution. We find that under specific conditions, in some systems, the first contribution may dominate over the second and then the total entropy of mixing becomes negative. This concept of polymer solutions has been discussed in excellent detail by Flory (1953).

Coacervation

Coacervation is usually defined as a process during which a homogenous solution of charged macromolecules, undergoes liquid–liquid phase separation, giving rise to a polyelectrolyte rich dense phase. It is the spontaneous formation of a dense liquid phase of poor solvent affinity. The loss of salvation arises from interaction of complementary macromolecular species. The formation of such fluids is well known in mixtures of complementary polyelectrolytes. It can also occur when mixing polyelectrolytes with colloidal particles.

Following the pioneering work of Bungenberg De Jong (1949), coacervates are either categorized as simple or complex based on the process that leads to coacervation. In simple coacervation, the addition of salt promotes coacervation. In complex coecarvation, oppositely charged polyelectrolytes can undergo coacervation through associative interactions. The other liquid phase, the supernatant, remains in equilibrium with the coacervate phase. These two liquid phases are immiscible and therefore, incompatible. Complex coacervation of polyelectrolytes can be achieved through electrostatic interaction with oppositely charged proteins and polymers. The charges on the polyelectrolytes must be large enough to cause significant electrostatic interactions, but not precipitation.

Potential applications of coacervates are many starting from protein purification, drug encapsulation to treatment of organic plumes. This calls for better understanding of the coacervate structure and the transport of biomolecules inside this phase. Several questions pertaining to the structure of coacervates can arise. The foremost of these is: is it a gel-like or a solution-like phase?

We have already discussed the global motion of long chain molecules in different thermodynamic and hydrodynamic environments. It has also been realized that in practice the probe length scale determines the physical parameter accessible in a measurement. In neutron scattering experiments, this is typically ~ a few nm, in X-ray scattering and diffraction techniques this is ~ 0.1–1 nm, electron microscopes use length scales of < 0.1 nm whereas for light scattering this is ~ 500 nm, for ultrasonics this is ~1 mm and classical gradient diffusion (CGD) uses probing length scales of several centimeters. The significance of these different measurement techniques is that if a polymer chain has a characteristic length of say ~100 nm, light scattering and CGD will measure its centre of mass translational diffusion, neutron scattering will probe its internal relaxation modes of segments and X-ray and electron microscopic techniques can be used to study the dynamics of the bond structures in individual monomers. On the other hand, if the chain has a physical dimension of ~ 1 μm, even light scattering can probe its internal modes. It must be realized that there is a whole class of polymer properties that involve mass transfer. Thus, the issue of polymer dynamics becomes relevant.

In this chapter, we shall be concerned with the dynamics of the internal modes in a long chain polymer. There are two different models for this—one is due to Rouse (1953) and the second one is due to Zimm (1956).

To get a clear idea of the phase state of matter it is necessary to understand the concept of phase. The term ‘phase’ can be defined structurally and thermodynamically. It is part of a system separated from other parts by interfaces and differing from them in thermodynamic properties. A phase must possess sufficient spatial extension for the concepts of pressure, temperature and other thermodynamic properties to be valid. Structurally, phases differ in the order of mutual arrangement of their molecules. Depending on this order, there are three phase states, namely: crystalline, liquid and gaseous.

Polymer substances possess high molecular mass and hence their boiling points must be very high. They decompose when heated, and their decomposition temperatures are always far below their boiling points. Due to this, polymeric substances cannot be converted to the gaseous state and exist only in the condensed state—liquid or solid. A study of the phase states and ordering of polymers reveals a number of specific features related to the large size of their molecules.

It is pertinent to discuss the possibility of formation of an ordered state in a polymer system. In any ordering process, the existence of short-range and long-range orders are defined by the distance over which the order extends, vis-a-vis, the dimensions of the monomers. A polymer is associated with two types of structural elements: monomers and chains. Hence, while discussing short-range or long-range order, it is informative to assign which of these elements is ordered. In practice, the existence of long-range order may comprise the arrangement of both structural elements. It is clear that long-range order of monomers in one dimension can generate a linear chain of polymers.

The physics of nucleic acids deals with the study of molecular structure–property relationship to describe life phenomena, in particular heredity and variability. The origin and development of molecular biophysics is associated with the genetic role of nucleic acids and with their interpretation. Physics has played a vital role in providing a foundation to molecular biology. For instance, the discovery of the DNA duplex structure was facilitated by data obtained from the X-ray diffraction studies by Watson and Crick (1953). They proposed a structure which has two helical chains each coiled around the same axis. The bases are located inside the helix whereas the phosphates on the outside. Schrödinger (1944) has discussed these issues in his book What is Life? In biomolecules, the relation between the molecular structure and its biological function is not trivially correlated. Due to high linear charge density, the DNA molecule acts as a strong polyelectrolyte. It is twisted into a very loose coil in its single strand conformation. Such a coil is associated with a persistence length of 50 nm in a 0.15M NaCl solution whereas it is 80 nm in a 0.0015M NaCl dispersion. We shall discuss some structural as well as functional aspects of these informational molecules in the following sections.

DNA stacking

Let us look at some examples of simple models that describe base pair stacking. We already know that for DNA the matching base pairs are A-T and G-C, while for RNA, it is A-U and G-C.

In analyzing various phenomena in TM compounds in the previous chapter, we have already several times come across the situation when a material, depending on conditions, can be in an insulating or in a metallic state. Such metal–insulator transitions can be caused either by doping (a change in band filling) or by temperature, pressure, magnetic field, etc. The topic of metal–insulator transitions is one of the most interesting in the physics of systems with correlated electrons. Such metal–insulator transitions often lead to dramaticffects and a drastic change in all properties of the system; and the large sensitivity of materials close to such transitions to external perturbations can be used in many practical applications.

In principle, metal–insulator transitions are not restricted to systems with correlated electrons. They are often observed in more conventional solids, well described by the one-electron picture and standard band theory. However the most interesting such transitions, often significantly different from those in “band” systems, are indeed met in systems with strongly correlated electrons, in particular in transition metal compounds – see for example Mott (1990) or Gebhard (1997).

Different types of metal–insulator transitions

One can divide all metal–insulator transitions into three big groups; these are discussed in the sections below.

Metal–insulator transitions in the band picture

The first group of metal–insulator transitions are transitions which can be understood on the one-electron level in the framework of band theory – although, of course, interactions of some type are always necessary for such transitions.

Charge-transfer insulators

Until now, when considering systems with strongly correlated electrons, we mostly discussed the properties of d-electrons themselves. However most often we are dealing not with systems with only TM elements (pure TM metals), but with different compounds containing, besides TM ions with their d-electrons, also other ions and electrons. These may be itinerant or band electrons, for example in many intermetallic compounds; some of these will be considered below, in Chapter 11. But more often we are dealing with compounds such as TM oxides, fluorides, etc., which are insulators. Still, even in this case we have in principle to include in our discussion not only the correlated d-electrons of transition metals, but also the valence s- and p-electrons of say O or F. This we have already done to some extent when we were considering the crystal field splitting of d levels in Chapter 3, in particular the p–d hybridization contribution to it, see Section 3.1 and Figs 3.5–3.8.

In some cases we can project out these other electrons and reduce the description to that containing only d-electrons, but with effective parameters determined by their interplay with say p-electrons of oxygens. In other cases, however, we have to include these electrons explicitly. This, in particular, is the case when the energy of oxygen 2p levels is close to that of d levels.

After having presented briefly in Chapter 1 the general approach to the description of correlated electrons in solids using a simplified model – the nondegenerate Hubbard model (1.6) – from this chapter on we turn toward a more detailed treatment of the physics of transition metal compounds, which will take into account the specific features of d-electrons. The well-known saying is that “the devil is in the details.” Thus if we want to make our description realistic, we have to include all the main features of the d states, the most important interactions of d-electrons, etc. We begin by summarizing briefly in this chapter the basic notions of atomic physics, with specific applications to d-electrons in isolated transition metal ions. For more details, see the many books on atomic physics; specifically for application to transition metals, see Ballhausen (1962), Abragam and Bleaney (1970), Griffith (1971), Cox (1992), and Bersuker (2010).

Elements of atomic physics

Here we recall some basic facts from atomic physics, which will be important later on. We give here only a very sketchy presentation; one can find the details in many specialized books on atomic physics, for example the works cited above and Slater (1960, 1968).

The state of an electron in an atom is characterized by several quantum numbers.