The discussions in the previous chapters show that already the spontaneous emission, i.e., the simplest manifestation of the quantum-optical fluctuations, exhibits highly nontrivial features as soon as the quantum light is coupled to interacting many-body systems. We have seen, for example, that pronounced resonances in the semiconductor luminescence originate from a nontrivial mixture of exciton and plasma contributions in contrast to the simple transitions between the eigenstates of isolated atomic systems. As a consequence of the conservation laws inherent to the light–matter coupling, the photon emission induces rearrangements in the entire many-body system leading, e.g., to pronounced hole burning in the exciton distribution. As discussed in Chapter 29, this depletion of the optically active excitons leads to a reduction of the total radiative recombination and the appearance of nonthermal luminescence even when the electron–hole system is in quasiequilibrium. Already these observations show that the coupled quantum-optical and many-body interactions induce new intriguing phenomena that are not explainable by the concepts of traditional quantum optics or classical semiconductor physics alone.

The foundations of quantum optics are based on systematic investigations of simple systems interacting with few quantized light modes. In this context, one can evaluate and even measure the exact eigenstates or the density matrix with respect to both the photonic and the atomic degrees of freedom. In semiconductor systems, currently the investigations using one or a few quantum dots (QD) are closest to the atomic studies because, due to their discrete eigenstates, one can treat strongly confined quantum dots to some extent like artificial atoms.

In the early 1900s more and more experimental evidence was accumulated indicating that microscopic particles show wave-like properties in certain situations. These particle-wave features are very evident, e.g., in measurements where electrons are diffracted from a double slit to propagate toward a screen where they are detected. Based on the classical averaging of particles discussed in Section 1.2.2, one expects that the double slit only modulates the overall intensity, not the spatial distribution. Experimentally, however, one observes a nearly perfect interference pattern at the screen implying that the electrons exhibit wave averaging features such as discussed in Section 2.1.2. This behavior, originally unexpected for particle beams, persists even if the experiment is repeated such that only one electron at a time passes the double slit before it propagates to the detection screen. Thus, the wave aspect must be an inherent property of individual electrons and not an ensemble effect.

Another, independent argument for the failure of classical physics is that the electromagnetic analysis of atoms leads to the conclusion that the negatively charged electron(s) should collapse into the positively charged ion because the electron–ion system loses its energy due to the emission of radiation. As we will see, this problem can be solved by including the wave aspects of particles into the analysis. In particular, as discussed in Section 2.3.3, waves can never be localized to a point without increasing their momentum and energy beyond bounds.

In this book, we encounter the hierarchy problem when we apply the equation-of-motion technique to analyze the quantum dynamics of the coupled light–matter system. To obtain systematic approximations, we use the so-called cluster-expansion method where many-body quantities are systematically grouped into cluster classes based on how important they are to the overall quantum dynamics. With increasing complexity, the clusters contain

1. independent single particles (singlets),

2. interacting pairs (doublets),

3. three-particle terms (triplets),

4. coupled four-particle contributions (quadruplets),

5. as well as higher-order correlations.

In this context, the N-particle concept is somewhat formal because it refers to generic N-particle expectation values 〈N〉, which may consist of an arbitrary mixture of carrier, photon, and phonon operators, as discussed in Section 14.2. In order to truncate the hierarchy problem at a given level, 〈N〉 is approximated through a functional structure that includes all clusters up to the predetermined level while all remaining clusters with a higher rank are omitted. It is natural that the corresponding approximations can be systematically improved by increasing the number of clusters included.

To the best of our knowledge, the idea of coupled-clusters approaches was first formulated by Fritz Coester and Hermann Kümmel in the 1950s to describe nuclear many-body phenomena. The approach was then modified for the needs of quantum chemistry by Jiri Cizek 1966 to deal with many-body phenomena in atoms and molecules. Currently, it is one of the most accurate methods to compute molecular eigenstates.

Historically, the scientific exploration of new phenomena has often been guided by systematic studies of observations, i.e., experimentally verifiable facts, which can be used as the basis to construct the underlying physical laws. As the apex of the investigations, one tries to identify the minimal set of fundamental assumptions – referred to as the axioms – needed to describe correctly the experimental observations. Even though the axioms form the basis to predict the system's behavior completely, they themselves have no rigorous derivation or interpretation. Thus, the axioms must be viewed as the elementary postulates that allow us to formulate a systematic description of the studied system based on well-defined logical reasoning. Even though it might seem unsatisfactory that axioms cannot be “derived,” one has to acknowledge the paramount power of well-postulated axioms to predict even the most exotic effects. As is well known, the theory of classical mechanics can be constructed using only the three Newtonian axioms. On this basis, an infinite variety of phenomena can be explained, ranging from the cyclic planetary motion all the way to the classical chaos.

In this book, we are mainly interested in understanding how the axioms of classical and quantum mechanics can be applied to obtain a systematic description for the phenomena of interest. Especially, we want to understand how many-particle systems can be modeled, how quantum features of light emerge, and how these two aspects can be combined and utilized to explore new intriguing phenomena in semiconductor quantum optics.

In the previous chapter, we have already discussed that the microscopic origin of spontaneous light emission in semiconductors is significantly more complicated than for the atomistic situation where a single entity, i.e., an isolated electron–ion pair, emits a photon. In the artificial situation of a purely excitonic population without any Fermionic substructure, all the plasma contributions vanish and the semiconductor luminescence equations (SLE) predict that, e.g., the emission at the 1s energy stems only from the 1s-exciton population. The very same conclusion follows from the simplified atomic picture analyzed in Chapters 16–23 because the isolated atomic entities are uniquely defined by their eigenstates |φλ〉 and the eigenenergies Eλ. For example, the two-level luminescence Eq. (23.44) yields an emission that is proportional to the population of the excited state.

In reality, the semiconductor excitations are hardly ever dilute enough for the electron–hole pairs to be treated as isolated entities. Instead, the excited quasiparticles interact collectively with the emitted photons. Consequently, it is not justified to omit the Fermionic aspects from Eq. (28.47). In addition, semiconductors with a continuous band structure always have a much greater number of available plasma than exciton states and both of them usually contribute to the excitonic luminescence.

Introduction

We now consider measurements of the dynamic structure factor S(q,t) of substantially monodisperse polymers. The work here represents a sixth application of light scattering spectroscopy, as discussed in Chapter 4, to solution dynamics. Prior Chapters 6, 7, 8, and 9 included light scattering determinations of segmental motion, self- and tracer diffusion, rotational diffusion, and probe diffusion. The immediately prior chapter included light scattering spectra of colloidal systems, in which the underlying forces (hydrodynamic, volume exclusion) are much the same as in polymer solutions.

Monodisperse solutions of flexible coils present new complications not seen in earlier discussions of light scattering spectroscopy. Light scattering measurements of rotational and segmental diffusion are only sensitive to a single “internal” variable, an orientation vector or tensor. Self- and tracer diffusion use dilute scattering chains in the presence of a nonscattering matrix, at small scattering vector q; neither chain internal modes nor interference between scattering from pairs of tracer chains affects the scattering spectrum. Scattering from colloids and optical probes examines center-of-mass motion of rigid particles that have no significant internal modes. Here the scattering polymer coils are flexible and often are nondilute, so light scattering spectra include relaxations arising from single-chain center-of-mass displacements, relative motions of chain segments on a single chain, and correlations between positions and motions of chain segments on pairs of chains.

This volume presents a systematic analysis of experimental studies on the dynamics of polymers in solution. I cover not only classical methods, e.g., rheology, and more modern techniques, e.g., self-diffusion, optical probe diffusion, but also radically innovative methods not generally recognized as giving information on polymer dynamics, e.g., capillary zone electrophoresis. Actual knowledge comes from experiment. The intent is to allow the data to speak for themselves, not to force them into a particular theoretical model in which they do not fit; freed of the Procrustean bed of model-driven analysis, the data do speak, loudly and clearly.

The Phenomenology examines what we actually know about polymer motion in solution. The objective has been to include every significant physical property and experimental method, and what each method shows about polymer motion. The list of methods includes several that have not heretofore been widely recognized as revealing the dynamics of polymer solutions. Undoubtedly there are omissions and oversights, for which I apologize. The reader will note occasional discussions that speak to particular models, but experiment comes first, while comparison with various hypotheses is postponed.

The following dozen chapters demonstrate that the vast majority of measurements on polymer dynamics can be reduced to a very modest number of parameters. These parameters have simple relationships with underlying polymer properties such as polymer molecular weight. The relationships in turn speak to the validity of several possiblemodels for polymer dynamics, models whose validity is also tested by a number of more qualitative observations on how polymers move in solution.

This very short chapter sketches a theoretical scheme – the hydrodynamic scaling model – that is consistent with the results in the previous chapter, and that predicts aspects of the observed behavior of polymers in nondilute solution. The model is incomplete; it does not predict everything. However, where it has been applied, its predictions agree with experiment. Here the model and its developments as of date of writing are described qualitatively, the reader being referred to the literature for extended calculations.

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(1). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2, 3). The other collective coordinates of the Kirkwood–Riseman model are lumped as “internal coordinates” whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to establish and maintain the coil's bead density and radius of gyration. However, the spring constant of the links only affects the time evolution of the internal coordinates; it has no effect on translation or rotation of the coil as a whole.

When a coil moves with respect to the solvent, each bead sets up a wake, a fluid flow described in first approximation by the Oseen tensor.

Introduction

The early electrophoresis experiments of Tiselius, first published in 1930, examined the motions of proteins in bulk solution as driven by an applied electrical field(1). In the original method, a mixture of proteins began at a fixed location. Under the infiuence of the field, different protein species migrated through solution at different speeds. In time, the separable species moved to distinct locations (“bands”). Electrophoresis is now a primary technique for biological separations(2, 3). Two improvements were critical to establishing the central importance of electrophoresis in biochemistry: First, thin cells and capillary tubes replaced bulk solutions. Second, gels and polymer solutions replaced the simple liquids used by Tiselius as support media. These two improvements greatly increased the resolution of an electrophoretic apparatus. Electrophoresis in true gels is a long-established experimental method. The use of polymer solutions as support media is more recent. An earlymotivation for their use was the suppression of convection, but electrophoretic media that enhance selectivity via physical or chemical interaction with migrating species are now an important biochemical tool.

Electrophoresis and sedimentation have a fundamental similarity: in each method, one observes how particular molecules move when an external force is applied to them. In sedimentation, the enhancement of buoyant forces by the ultracentrifuge causes macromolecules to settle or rise. In electrophoresis, the applied electrical field causes charged macromolecules to migrate. The experimental observable is the drift velocity of the probe as one changes the molecular weight and concentration of the matrix, the size or shape of the probe, or the strength of the external force.

There are already vast numbers of reviews, monographs, edited collections, conference proceedings, and web pages on polymer diffusion, light scattering, electrophoresis, rheology, and almost every topic I cover, other than optical probe diffusion.Why does the world need another book about polymers in solution?

On one hand, the chosen topic has reached a certain degree of maturity. Over the past decade the spate of new research papers on polymer dynamics has greatly slackened, so in the half-decade I needed to write this volume the first-written chapters did not date badly.On the other hand, there are some radically new methods and results whose significance for polymer physics does not seem to be widely recognized.

What do I offer that has not been said many times before?

First and foremost, my focus is phenomenology. There are bits of theoretical discussion hither and thither throughout the volume, but most chapters discuss experiment. If you want to read about models for polymer motion or the formal basis of particular experimental methods, you must for the most part look elsewhere. Except for light scattering spectroscopy, I give very little background on experimental methods and interpretation. The extremely extensive theoretical literature on polymer dynamics in solution is not reviewed. For such reviews see, for example, Graessley(1), Tirrell(2), Pearson(3), Skolnick and Kolinski(4), Lodge, et al.(5), and (more recently but less directly) McLeish(6). Recent papers by Schweizer and collaborators include extensive background references(7–9).

Introduction

This chapter considers polymer segmental diffusion, the motions of small portions of a polymer relative to the chain as a whole. Studies of motions of short sections of polymer molecules should be seen as being complementary to studies of the motions of small molecules through polymer solutions, as described in Chapter 5. Localmotions of polymer chains on distance scales comparable to q−1 contribute to the polymer dynamic structure factor S(q, t), as discussed in Chapter 11. Important experimental techniques sensitive to segmental motion include depolarized light scattering, time-resolved optical scattering, and nuclear magnetic resonance, as treated in the next three sections.

Depolarized light scattering

The molecular basis for depolarized light scattering by model molecules that resemble polymers was described by Patterson and Carroll(1,2), who discuss intensities and linewidths for depolarized light scattering modes and their relationship to orientation fluctuations. Depolarized light scattering is in part sensitive to local chain motions and in part sensitive to whole-chain motions, as shown by the behavior of the VH spectrum.

The molecular weight dependence of the VH spectrum for low-molecular-weight polymers was found by Lai, et al.(3), who examined concentrated (550 g/l) polystyrenes (9.1 and 18.1 kDa) in cyclohexane. The VH spectrum was dominated by a singlemodewhose relaxation time is independent of polymermolecularweight and scattering angle. Lai, et al. also saw the tail of a very fast mode whose relaxation time distribution was not fully resolved. A dominant mode with these properties is reasonably interpreted as being created by segmental motions that reorient modest portions of a much longer chain.

So far we have considered linear transport phenomena, in which the response is directly proportional to the circumstance causing the response. Polymer solutions, however, are fundamentally nonlinear, and show a wide variety of additional behaviors not expected from simple linear descriptions. These behaviors may be divided, somewhat crudely, into unusual flow behaviors arising from nonzero normal stress differences, time-dependent phenomena in which the system shows memory, so that the response to a series of forces depends on when they were applied, and several modern discoveries not discussed in more classical references.

At some point, the constraints of time and space insist that the discussion be curtailed, so we here present a taxonomy of nonlinear viscoelastic phenomena, without the considerable quantitative analyses seen in prior chapters. The objective is to represent the range of observed phenomena and provide references that give entrées into the literature. No effort has been made to give a thorough collection of published results. If Lord Rayleigh's critique – science is divided between quantitative measurement and stamp collecting – is invoked on this chapter, the stamps are indeed beautiful, but are likely to be more thoroughly quantitatively examined when readers become aware of their existence.

Normal stress differences

A variety of the classical unusual flow effects seen with polymer solutions can be traced back to the normal stress differences N1 and N2. A full mathematical description becomes quite lengthy, and may be found in Graessley(1).The following remarks provide an exceedingly compressed description.