Polymers are remarkable molecules with a particularly rich behavior and a wealth of interesting properties. Statistical mechanical arguments may be used to understand these properties. In this chapter, we present an elementary theory of polymer configurations and polymer dynamics. We also offer a brief exposition of the theory of Brownian dynamics. This is a powerful theoretical formalism for studying the motion of molecules in a solution.

Polymers

The study of conformations and conformational motions of flexible polymer chains in solution is of great scientific and technological importance. Understanding the physics of macromolecules at the molecular level helps the synthesis and design of commercial products. It also provides insight into the structure and functions of biological systems. Flexible polymers have therefore been the subject of extensive theoretical treatments, a wide variety of experiments, and computer simulations (see Further reading at the end of the chapter).

Historically, theoretical treatments have resorted to simple phenomenological models of polymeric materials. In the framework of statistical mechanics, polymeric chains are at a first stage considered to consist of independent elements or segments. The principal property of macromolecular behavior taken into account with this representation is the flexibility of the chains. With non-interacting monomeric units having uncorrelated directions, it is straightforward to show that the chains acquire random-walk behavior.

Molecular dynamics (MD) simulations generate trajectories of microstates in phase space. They simulate the time evolution of classical systems by numerically integrating the equations of motion.

With their strength tied to available computer speed, simulations continue to become a more powerful tool. A letter to the Journal of Chemical Physics by B. J. Alder and T. E. Wainwright in 1957 was the first work that reported results from molecular dynamics simulations. The Lawrence Radiation Laboratory scientists studied two different sized systems with 32 and 108 hard spheres. They modeled bulk fluid phases using periodic boundary conditions. In the paper they mention that they counted 7000 and 2000 particle collisions for 32 and 108 particle systems, respectively. This required one hour on a UNIVAC computer. Incidentally, this was the fifth such commercial computer delivered out of the 46 ever produced. The computer cost around $200 000 in 1952 and each of ten memory units held 100 words or bytes. Nowadays, a $300 personal computer with a memory of approximately 500 000 000 bytes can complete this simulation in less than 1 second. And Moore's empirical law that computer power doubles every 18 months still holds.

Alder and Wain wright computed the pressure of the system and their results compared favorably to results of Monte Carlo simulations of the same system. In the paper, they state in an unduly reserved manner that “this agreement provides an interesting confirmation of the postulates of statistical mechanics for this system.”

The output of simulations is a list of microscopic states in phase space. These are either a sample of points generated with Metropolis Monte Carlo, or a succession of points in time generated by molecular dynamics. Particle velocities, as well as higher time derivatives of particle positions, can also be saved during a molecular dynamics simulation.

This information is periodically recorded on computer disks during the simulation for subsequent analysis. Minimal analysis can be conducted during the actual simulation, primarily in order to ensure that equilibrium is reached and the simulation is numerically stable. Usually most of the analysis is conducted off-line, after the simulation is completed. The drawback is that large files must be stored and processed. Furthermore, saving positions and velocities does not occur for every integration time step. Instead, positions and velocities are saved infrequently, which results in some information being thrown away. It is more convenient, however, to analyze results off-line, developing and using analysis algorithms as needed. It is also important to maintain an archival record of simulation results.

Structural properties

The pair distribution function, g(r), can be computed from stored particle positions. A program to compute it is given below and can be found in http://statthermo.sourceforge.net/. It computes the distance between all pairs of particles, and counts the number of particle pairs that are in a specific distance range, defined by a parameter delr.

Tractable exploration of phase space

Statistical mechanics is a powerful and elegant theory, at whose core lie the concepts of phase space, probability distributions, and partition functions. With these concepts, statistical mechanics explains physical properties, such as entropy, and physical phenomena, such as irreversibility, all in terms of microscopic properties.

The stimulating nature of statistical mechanical theories notwithstanding, calculation of partition functions requires increasingly severe approximations and idealizations as the density of systems increases and particles begin interacting. Even with unrealistic approximations, the partition function is not calculable for systems at high densities. The difficulty lies with the functional dependence of the partition function on the system volume. This dependence cannot be analytically determined, because of the impossible calculation of the configurational integral when the potential energy is not zero. For five decades then, after Gibbs posited the foundation of statistical mechanics, scientists were limited to solving only problems that accepted analytical solutions.

The invention of digital computers ushered in a new era of computational methods that numerically determine the partition function, rendering feasible the connection between microscopic Hamiltonians and thermodynamic behavior for complex systems.

Two large classes of computer simulation method were created in the 1940s and 1950s:

1. Monte Carlo simulations, and

2. Molecular dynamics simulations.

Both classes generate ensembles of points in the phase space of a well defined system. Computer simulations start with amicroscopicmodel of the Hamiltonian.

Monte Carlo methods are computational techniques that use random sampling. Nicholas Metropolis and Stanislaw Ulam, both working for the Manhattan Project at the Los Alamos National Laboratory in the 1940s, first developed and used these methods. Ulam, who is known for designing the hydrogen bomb with Edward Teller, invented the method inspired, in his own words, “… by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully?”. This was a new era of digital computers, and the answer Ulam gave involved generating many random numbers in a digital computer.

Metropolis and Ulam soon realized they could apply this method of successive random operations to physical problems, such as the one of neutron diffusion or the statistical calculation of volumetric properties of matter. Metropolis coined the term “Monte Carlo” in reference to the famous casino in Monte Carlo, Monaco, and the random processes in card games. Arguably, this is the most successful name ever given to a mathematical algorithm.

Nowadays Monte Carlo refers to very many different methods with a wide spectrum of applications. We present the Metropolis Monte Carlo method for sampling the phase space to compute ensemble averages, although often we only use the term “Monte Carlo” in subsequent discussions.

Systems, properties, and states in thermodynamics

A system is the part of the universe we are interested in. The rest of the universe is called surroundings. The system volume, V, is well defined and the system boundary is clearly identified with a surface.

Systems are described by their mass M and energy E. Instead of using mass M, a system may be defined by the number of moles, N/NA, where N is the number of molecules and NA is Avogadro's number.

There are three kinds of system in thermodynamics:

1. Isolated systems. In these systems, there is no mass or energy exchange with the surroundings.

2. Closed systems. In these systems, there is no mass exchange with the surroundings. Energy can flow between the system and the surroundings as heat, Q, or work, W. Heat is the transfer of energy as a result of a temperature difference. Work is the transfer of energy by any other mechanism.

3. Open systems. In these systems, mass and energy may be exchanged with the surroundings.

A thermodynamic state is a macroscopic condition of a system prescribed by specific values of thermodynamic properties.

There are important engineering and physical processes that appear random. A canonical example is the motion of Brownian particles, discussed in Chapter 11. In random or stochastic processes, in contrast to deterministic ones, there is not one single outcome in the time evolution of a system, even when initial conditions remain identical. Instead there may be different outcomes, each with a certain probability. In this chapter we present stochastic processes and derive a general framework for determining the probability of outcomes as a function of time.

We are starting the discussion with reacting systems away from the thermodynamic limit. Typically, reacting systems are modeled with ordinary differential equations that express the change of concentrations in time as a function of reaction rates. This continuous and deterministic modeling formalism is valid at the thermodynamic limit. Only when the number of molecules of reacting species is large enough can the concentration be considered a continuously changing variable. Importantly, the reaction events are considered to occur deterministically at the thermodynamic limit. This means that there is certainty about the number of reaction events per unit time and unit volume in the system, given the concentration of reactants.

On the other hand, if the numbers of reacting molecules are very small, for example in the order of O(10-22NA), then integer numbers of molecules must be modeled along with discrete changes upon reaction.