How to deal with the Coulomb interactions between particles is one of the more difficult questions in physics. Without their presence, N-particle problems can generally be turned into N one-particle problems. An exact treatment of the Coulomb interaction requires knowledge of the positions of the particles in space. This would not be a problem if the particles were not moving or the velocities of particles differed by orders of magnitudes. The last assumption is generally valid for the Coulomb interaction between electrons and the nuclei. Since in most systems the nuclei are more or less fixed (say, in a solid or a molecule), we can replace the effect of the nuclei on the electrons by an effective potential. For the interaction between the electrons, we can separate the electrons into two types. Core electrons are strongly bound to the nuclei and generally have a binding energy of tens to thousands of electronvolts. If the atomic shells of these electrons are full, we can include their effect in the effective potential of the nucleus. On the other hand, this approach often fails for the valence electrons, a term that we loosely use to describe electrons in states that have a relatively low binding energy, such as the highest-occupied and lowest-unoccupied molecular orbitals (HOMO and LUMO, respectively) in a molecule, the states close to the Fermi level in a metal, or the valence and conduction bands in a semiconductor. However, even for these electrons, treating their interaction in terms of an effective potential often works surprisingly well.

The most commonly-used approach is the local-density approximation used in density-functional theory. However, for many systems, this theory has serious deficiencies. In Chapter 5, we looked at Coulomb multiplets for an atom/ion, which are often clearly visible in X-ray spectroscopy. These effects cannot be described within an effective independent-particle framework. For solids, materials that are known to be insulating are often predicted to be metallic when the interactions between particles are described in terms of an effective potential.

The journey of a thousand miles begins with a single step. I have been offering a course on molecular biophysics to advanced master level students since 1992 (the students have a background of physics, physical chemistry, chemical engineering, etc.). These students have little exposure to biology and organic chemistry. However, research focus is shifting towards soft matter science which is highly interdisciplinary, and holds a promise of generating customized, smart and biocompatible materials. Therefore, the need for learning physics of polymers and biopolymers has increased many folds. This course is taught with the objective to provide a robust background in these topics to students. I have converted my lecture notes into this publication. There are no textbooks in the market till date that cover the topics discussed herein in a single volume. The content has been used in a one semester course that I teach to MSc Physics students. The mathematical prerequisites for this book are modest.

Macromolecules in solutions can be distinctly characterized from their transport behaviour in the solution phase. The study of the transport processes yields coefficients like the diffusion coefficient, sedimentation coefficient, intrinsic viscosity, friction constant, etc. of the dissolved solute particles. These coefficients are dependent on two parameters. First is the size and shape of the solute particle. Second is the type of the solvent medium and its environment (pH, temperature, pressure, ionic strength, etc.).

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).