Introduction

In previous chapters, a range of experiments on aspects of polymer solution dynamics, from electrophoretic mobility to single-chain diffusion to linear viscoelasticity, has been treated(1). The previous chapter described results that were found with each method. What do these types of measurement tell us about how polymer molecules move through solution? The answers to this question come in a substantial number of parts and pieces, best treated separately before being assembled into final conclusions. There are undoubtedly other parts and pieces that might have been discussed, such as the consequences of changing the relative size of matrix and probe polymers, or the consequences of polymer topology. This chapter stays with answers most central to our purpose.

Comparison with scaling and exponential models

We began in Section 1.2 by observing that the large number of theoretical models could with a modest number of exceptions be partitioned into two major phenomenological classes, based on whether themodels predicted scaling (power-law) or exponential dependences of transport coefficients on polymer concentration, molecular weight, or other properties. What do the data say about the relative merit of these classes of theoretical model?

An obvious first question is whether the precision of experimental measurement, as viewed through the lens of our data analysis methods, is adequate to say whichmodels are acceptable. Can we distinguish between power laws and stretched exponentials? The answer is unambiguously in the affirmative.

Introduction

A characteristic feature of many molecules and chemical bonds is an electric dipole moment whose orientation is substantially fixed as viewed in internal molecular coordinates. Dielectric spectroscopy observes the temporal evolution of net polarization induced in these dipole moments by an applied electrical field. It was first noted by Stockmayer(1) that polymer chain dipoles may be grouped into three classes, namely: (A) dipoles that point along the polymer backbone, so that in the simplest cases the dipoles are relaxed by (i) reorientation of the polymer end-to-end vector r, and (ii) breathing modes that changes the vector's length; (B) dipoles that point perpendicular to the polymeric backbone, so that they are relaxed via crankshaft-like motions of the backbone; and (C) dipoles associated with polymeric side groups that are relaxed via rotation of the side groups around the bond axis linking the side groups to the backbone. Type-A dipoles may be in turn divided into two classes, namely (i) polymers in which the dipolar units are linked head to tail, so that the total dipole vector and the end-to-end vector are necessarily linearly proportional, and (ii) polymers in which the dipole is associated with only part of the polymeric repeat unit, so that the total dipole vector and the end-to-end vector may fluctuate with respect to each other. Class C dipoles generally relax the most quickly, while class A dipoles relax the most slowly.

Introduction

For the most part, this volume does not discuss experimental methods. Other works provide sound descriptions of techniques and theoretical principles. How-ever, light scattering spectroscopy gave the earliest phenomenological findings that led me to writing this book. Also, the method was the basis of my career's experimental work. Here we consider optical techniques, and the correlation functions and diffusion coefficients that they determine. The emphasis is on understanding the physical quantities that are measured by light scattering spectrometers and related instruments. Actual measurements on complex fiuids appear in later chapters.

Quasielastic light scattering spectroscopy (QELSS) is an optical technique for observing transient local fiuctuations in the concentration and orientation of molecules in a liquid(1). Inelastic neutron scattering is sensitive to the same fluctuations. The wavelengths and time scales to which light and neutron scattering are sensitive are quite different, but a single theoretical picture describes the physical quantities observed by the two methods. Related optical techniques, notably Fluorescence RecoveryAfter Photobleaching (FRAP), Forced Rayleigh Scattering (FRS), Fluorescence Correlation Spectroscopy (FCS), and Depolarized Dynamic Light Scattering (DDLS) each give somewhat different information about molecularmotions. QELSS is also called Dynamic Light Scattering, “dynamic” in contrast to the “static” of “Static Light Scattering,” in which the experimenter determines the average intensity of the scattered light.

We first consider howlight scattering spectra are related to positions and motions of the dissolved scattering centers. We then present a nomenclature for diffusion coefficients. At least three paths have been used to calculate light scattering spectra of macromolecule solutions; each is described below.

Remarks

This chapter presents a phenomenological description of viscoelastic properties of polymer solutions. While aspects of the description will appear familiar, this chapter is fundamentally unlike other chapters in this book. In Chapter 2 we discussed sedimentation. Much of the literature appeared before younger readers were born, but the sedimentation coefficient s is the coefficient familiar to everyone who has ever been interested in the method. In Chapter 3 we discussed capillary electrophoresis in polymer solutions. The notion that this method gives information about the polymer solutions being used as support media is nearly novel, but the electrophoretic mobility μ is the coefficient familiar to everyone who uses the technique. Similar statements apply to each of the other chapters. The perspective in prior chapters on a solution property may not be the same as seen elsewhere, but the parameters used to characterize the property have been familiar.

To treat viscoelasticity we need to do something different.

The classical viscoelastic properties are the dynamic shearmoduli, written in the frequency domain as the storage modulus G′(ω) and the loss modulus G″(ω), the shear stress relaxation function G(t), and the shear-dependent viscosity η(κ). Optical flow birefringence and analogousmethods determine related solution properties. Nonlinear viscoelastic phenomena are treated briefly in Chapter 14.

Solution properties depend on polymer concentration and molecular weight, originally leading to the hope that one could apply reduction schemes and transform measurements of the shear moduli at different c and M to a few master curves.

The major portion of this monograph discusses how given pulses of laser radiation affect individual atoms. This chapter inverts that relationship, describing how the matter alters the fields. Those field changes provide quantitative measures of the quantum-state changes produced in the atoms. Their description therefore is an adjunct to Chap. 19.

Basically the incident radiation produces excitation which, in turn, alters the various multipole moments of the atoms. When viewed as a macroscopic sample of matter, such changes alter the electric polarization field P and the magnetization field M of the matter through which the radiation must pass. The Maxwell equations, see App. C.1, provide the needed description of how the P and M fields alter the electric and magnetic fields E and B. The combination of the Maxwell equations for the fields and the Schrödinger, Bloch, or Liouville equations for the atoms provide the tools needed to construct a self-consistent description of radiation passing through matter – atoms responding to a pulsed field and traveling waves being modified by the resulting atomic changes [All87; Ale92; Muk99; Die06; Vit01a]. The present chapter, drawing on [Sho90, Chap. 12] and App. C, discusses this theory.

Incoherent radiation passing through matter typically undergoes exponential attenuation in accord with eqn. (6.9). A measure of the incremental change of intensity in distance L by absorption coefficient κ is the optical depth κL. This parameter appears in rateequation treatments of incoherent light attenuation.

The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space.

General remarks

General system. At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3.

Pure state. If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2N –1 real numbers are needed. Out of these 2N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).

As light passes an atom, it exerts forces on the charges, electrons, and nuclei that alter the atomic structure. These may be slight distortions (perturbations) of the electron cloud or they may be more severe, as described in subsequent sections of this monograph. We wish to determine those changes, given the radiation field, or to devise a radiation field that will produce specified changes.

The changes to the atomic structure also affect the radiation that subsequently passes the atom. To describe those effects we must consider wave equations for radiation in the presence of altered atomic structure. More generally we must find self-consistent equations for the atoms and the field together, as discussed in Chaps. 21 and 22. Here we consider mathematical descriptions of the influence of coherent radiation on individual atoms, molecules, or other single quantum systems.

Individual atoms

Traditional sources of emission and absorption spectra, though revealing energy states of the constituent atoms and molecules, are macroscopic samples. One observes averaged characteristics of many individual particles, see Chap. 16. Quantum theory offers the basic formalism for dealing with individual atoms exposed to controlled radiation fields. Several experimental techniques provide acceptable approximations to this ideal. The following paragraphs note some of these examples.

Vapors

Neutral atoms or molecules in a vapor move freely along straight-line paths, interrupted by brief collisions that redirect the two collision partners. When the kinetic energies of the two partners are small, there can be no transfer of kinetic energy into internal energy of either particle – the collision is elastic.

Quantum changes of three-state systems have some similarities with those of two-state systems. When subject to steady illumination the populations may undergo oscillations similar to the Rabi cycling of two-state systems, and various forms of adiabatic following are possible. Analytic solutions to the relevant TDSE exist [Sho90, Chap. 23]. The additional degree of freedom, typically allowing controllable parameters of a second laser pulse, allows a wider variety of controlled excitation. The resulting differences and similarities to two-state systems have been discussed at length [Whi76; Sho77; Rad82b; Yoo85; Car87].

Three-state linkages

Two-field linkages. The simplest extension of two-state excitation allows two laser fields, here identified by letters P (for pump) and S (for Stokes), as befits the stimulated Raman process discussed in Chap. 14. The carrier frequencies of the two fields, ωP and ωS, are each assumed to be close to resonance with one, and only one, Bohr frequency, so that each field can be uniquely identified with a particular transition (failure of this restriction, and the resulting linkage ambiguity, is discussed in [Una00]). I will assume that the P field is (near) resonance only with the 1–2 transition, while the S field is (near) resonant only with the 2–3 transition; these interactions thereby form a two-step linkage chain. This system has three possible linkage patterns, shown in Fig. 13.1.

The linkage patterns (sometimes called configurations) differ by the ordering of the energies of the linked states.With the assumption that population initially occupies state 1, as in Fig. 13.1, the definitions are:

Ladder: The ladder system has the energy ordering E1 < E2 < E3.