Light Scatterings

Light scattering is also nsed extensively for the study of polymers in solntion, one application of which is the determination of molar masses (see section 3.2.2). 

Light scattering is one of the most widespread characterization techniques for polymers. Therefore, technical and methodical details will not be explained here - please see Refs.  

The determinant in the denominator is to be calculated at constant temperature and pressure. It reduces to the single second derivative (3 G/ 3 9 )p.p = (3p., / 39 )p. for the case of a strictly binary monodisperse polymer solution. The average particle scattering factor is of primary importance in studies of the size and shape of the macromolecules, but it is merely a constant for thermodynamic considerations. 

Conventionally, the so-called Rayleigh factor (or ratio) is applied  

Vi partial molar volume of the solvent in the polymer solution at temperature T 

The optical constant for unpolarized light summarizes the optical parameters of the experiment  

Light-scattering methods can be divided into two major categories methods which measure time-averaged scattering (static methods) or methods which ob- 

Lai et al. [242] used dynamic light scattering to measure the size of nanometer-scale water droplets in reversed micelles of perfluoroheptanoic acid (PFHA) in 1,1,2-trichlorotrifluoroethane. 

Huglin, Light Scattering from Polymer Solutions , Academic Press, London, 1972. 2 M. B. Huglin, in Macromolecular Microsymposia—16 , ed. B. Sedl4 ek, Pergamon Press, London, 1978. See also Pure Appl. Chem., 1977, 49, 929. 

Burchard in Applied Fibre Science, vol. 1 , ed. F. Happey, Academic Press, London, 1978. 

22 Chromatix Inc., 1145 Terra Bella Av., Mountain View, California, U.S.A. U.K. agents, Rofin Ltd., Winslade House, Egham Hill, Egham, Surrey. 

In the region where intramolecular fluctuations are unimportant, polydispersity broadens the Rayleigh line and the broadening can be used to characterize the extent of the polydispersity. Burchard et have discussed the theory and 

Rayleigh line-broadening methods have been applied increasingly to highly swollen gel systems although there is much less agreement about the interpretation 

We return to Equation 8.9 for Rayleigh scattering in scalar form  

Note that an average is being defined and, in general, angnlar brackets will be used to denote averages. Generally autocorrelation functions decay with time exponentially, u(t) oc exp (tlx), and it is x that provides information on the dynamics. Note that u(0) provides the mean squared magnitude of A. 

Instead of proceeding directly to obtain the autocorrelation using Equation 8.67, in one common application (heterodyne) the scattered field is mixed with incident field/light and the resulting field is E = + Ef. Now the square of 

is the irradiance corresponding to the incident radiation, /f(0) is the irradiance of the scattered fight, and if(t) is proportional to 6c(0)6c(/) . What follows in most derivations is a lengthy disconrse in statistics and the many justifications made on statistical gronnds. The flnctnations are assnmed to be snmmable ovct contribntions from individnal molecnles, which requires that the Systran be dilute. This correlation function for an individual molecule is related to another funetion G, called the van Hove (1954) correlation function (who also gave the proof)  

In summary, the intensity of the scattered beam mixed with the incident beam is measured and autocorrelated (heterodyne). This autocorrelation fimction has an exponential decay. The time constant can be used to calculate the mumal diffusion coefficient. Berne and Pecora (1976) provide more detailed derivations and further treatments of all special cases. 

This means that the intensity of scattered light is much higher for blue light than for red. This is the same effect that gives us a blue sky and remarkably colorful sunsets over Los Angeles and other polluted cities around the world. 

When optical radiation of intensity /q is incident on a particle, if the particle is smaller than the incident wavelength X, then an oscillating electric dipole is induced in the particle. This dipole radiates with the following intensity / distribution  

Both Rayleigh scattering and Mie scattering can be observed clearly in dilute solutions, but at higher colloid concentrations additional effects 

Macromolecular solutions and colloidal suspensions show light dispersion phenomena, which is found to be essentially dependent on the molecular weight of the dispersing substance, according to the Rayleigh-Gans-Debye approximation (Striegel et al. 2009)  

By measuring the excess ratio R d) at different concentrations, the molecular weight can be determined. Because in a humic sample (as with most macromolecular materials), a distribution of molecnlar weights is present, and because light dispersion is a property essentially dependent on M, the measuranent yields the weight average molecular weight  

However, other parameters are required to fully characterize a distribution (Striegel et al. 2009). 

For a given mode of vibration of angular frequency u and scattering vector q we can assume 

We now proceed to obtain the dynamic equations using the theory and notation summarised in Section 4.2.5 on page 150. First note that n n = 1 to first order in these quantities and that the constraint V v = 0 leads to the relation 

This relation essentially allows us to follow the results of the Orsay Group [212] and choose to write all quantities involving Vi in terms of V3. It should also be pointed out that the Orsay Group use the form exp[i(q x -h ojt)] in their perturbations (cf. [212, Eqns. (III.11)-(III.12)]) instead of the form appearing in equations (5.515) and (5.516). Of course, the final results are identical irrespective of which form is used, the results of the Orsay Group being obtained simply by replacing u by —u. 

From the above comments the constraints (4.118) are satisfied and it remains to derive the d30iamic equations (4.119) and (4.120). We begin by considering the balance law for linear momentum in (4.119). The non-zero components of the symmetric rate of strain tensor Aij and the skew-symmetric vorticity tensor Wij can be given in terms of V2 and V3 by using the relation (5.519) to find 

It then follows from equations (5.530) to (5.541) that the linear momentum balance equations (5.527) and (5.529) finally reduce to, respectively, the two simultaneous equations 

For gases, Rayleigh showed that the reduced intensity of the scattered light Rg at any angle 0 to the incident beam of wavelength X could be related to the molar mass of the gas M, its concentration c, and the refractive index increment (dnidc) by 

The quantity Rg is often referred to as the Rayleigh ratio and is equal to (igr Ho), where /g is the intensity of the incident beam, ig is the quantity of light scattered per unit volume by one center at an angle 0 to the incident beam, and r is the distance 

When a solute is dissolved in a liquid, scattering from a volume element again arises from liquid inhomogeneities, but now an additional contribution from flucm-ations in the solute concentration is present, and for polymer solutions the problem is to isolate and measure these additional effects. This was achieved by Debye in 1944, who showed that for a solute whose molecules are small compared with the wavelength of the light used, the reduced angular scattering intensity of the solute is 

Here q and n are the refractive indices of the solvent and solution, respectively, and N is the number of polymer molecules. Differentiation of the virial expartsion for Tt with respect to c, followed by substitution in Equation 9.17 and rearrangranait leads to 

These equations are vahd for molecules smaller than ( 720) when the angular scattering is syrmnetrical, X being the wavelength of light in solution, i.e., A, =(Ay o). 

FlGURE 8.6 Schematic diagram of static light scattering. 

Equation (8.4.2) is valid only at infinite dilution. For finite concentrations, the use of a virial expansion of the type introduced in Eq. (8.3.22) leads to 

It is common practice to define a quantity Rg (called the Rayleigh ratio) as follows  

On measuring Rg as a function of concentration at a low 6 value, K /Rg is plotted versus c. The intercept of such a plot represents extrapolation to zero concentration, and from Eq. (8.4.5), we have the following  

If the polymer sample is polydisperse, then Rg can be written as a sum /f, over all of the molecular-weight fractions so, Eq. (8.4.7) becomes 

A simple way to determine the relationship between molecular weight and intensity of scattered light is to recast Equation (4.4) by defining an optical constant K using Equation (4.5) and Ro, the Rayleigh factor (Equation (4.6)). 

In the simplest case Equation (4.6) relates the Rayleigh factor to the molecular weight in the absence of solvent-macromolecule and macromolecule-macromolecule interactions. The molecular weight is determined from the slope of a plot of the Rayleigh factor versus weight concentration of macromolecules, c. 

In the case of macromolecule-macromolecule interactions, the relationship between the Rayleigh factor and weight concentration of macromolecules is complicated by the fact that the coefficient B is needed to account for this effect (Equation (4.7)). 

The determination of molecular weights by the light scattering method is based on the principle that the intensity of light scattered at some angle, 6, is a 

Springer Series in Wood Science Methods in Lignin Chemistry (Edited by S.Y. Lin and C.W. Dence) 

According to the fluctuation theory of light scattering (Flory 1953), the relationship between the Rayleigh factor and the physical characteristics of the macromolecules in solution is expressed in the following general form  

The term (0) describes the angular variation of light scattered at constant concentration (Kratochvil 1972). The form of P(0) 1 is dependent upon the size and shape of the scattering particles and provides the basis for their determination. Under the limiting condition of small angles, P(0) 1 may be expressed in the form of Eq. (5)  

This equation shows that for particles such as lignin molecules, generally much smaller than the wavelength ((s2)z 2), P(6) l approaches unity. Thus, for dilute solutions, Eq. (3) becomes  

One of the newest particle sizing techniques is light scattering. This technique is used to measure particle size distribution, colloid behavior, particle size growth, aerosol research, clean room monitoring, and pollution monitoring. 

is the refractive index, is the wavelength of light in a vacuum. The first scattering function, P(Q) approaches zero as Q approaches one, and the second function, S(Q) is a second scattering function. We see that the second term, which is the contribution from intermolecular effects, approaches zero as the concentration vanishes. For the special case, P = 1 (ft 0) and c 0, we have  

It can be shown [69] that for a polydisperse polymer, this equation is still valid with M replaced by M . Thus, light scattering provides a measure of the weight-avere e molecular weight. In order to determine M , it is necessary to extrapolate data obtained for various combinations of c and 0 to c = 0 and 0=0. This double-extrapolation can be accomplished by means of a Zimm plot, which is a plot of cIR versus Xc + sin 0/2, where X is a multiple of ten, often 1000. 

Great care is necessary in making light scattering measurements, as the presence of minute amoimts of contaminant will lead to large errors. However, if sufficient care is taken, the following information can be obtained from the intercept, Aj from the limiting slope of 

All materials can scatter light and this phenomenon is known as the Tyndall effect. However, colloids are capable of intense light scattering and therefore exhibit extreme turbidity. The white appearance of milk is a classic example. 

Light is scattered in all directions, when it meets an object (e.g. a colloid particle) of dimensions within an order of magnitude (i.e. a factor of 10) or so of its own wavelength. The intensity, angular distribution and polarization of the light scattered depend on the size and shape of the scattering particles, the interactions between them, and the difference in refractive indices of the particles and the dispersion medium. This is exploited in lightscattering measurements to measure particle size distributions and has found wide applications. The method is fast, can use both dry and wet samples and requires very little preparation. 

Scattering increases with particle size and therefore this property can be used to follow the progress of particle aggregation (coagulation) and thus study the stability of a colloidal dispersion. 

Particles, and structures in general, can also scatter X-ray and neutrons and several techniques are based on those principles (Table 9.2). 

Pigments are used in plastics, coatings, tires and cosmetics for several reasons. They can, for example, add colour, improve mechanical properties, or act as biocides. Very often it is of interest to know the particle size distribution of pigments directly in a product (e.g. a coating). 

A light beam propagating in matter interacts with the electrons of the medium. In insulators, the electrons are bound and the incident radiation induces a local polarization. Therefore, a scattering centre is a small polarizable element with the size of a monomer this element can be assimilated to a dipole in forced oscillation regime. The radiation produced by this dipole is the scattered radiation. This is Rayleigh scattering. 

We shall now present a brief resume of the formulas necessary for the computation of activity coefficients from light-scattering measurements. The method is practicable only for species of high molecular weight. 

We consider a binary solution and take into account only Rayleigh scattering. Using Boer s law, we have 

I = intensity of transmitted radiation X = thickness of sample T = turbidity of sample 

The turbidity due to composition fluctuations, with neglect of density fluctuations, as calculated by statistical theory, is 

A typical plot of experimental data is given in Fig. (11-2). In this graph, /fC2/r is plotted against C2. The curve intersects the ordinate at l/Mj since 

Radiation interacts with matter through the effects of the electric field vector on the electron distributions in molecules. Absorption of radiation involves raising a system from one energy level to a higher level by the absorption of a quantum of energy (a photon). Elastic scattering of radiaLion involves no such quantum jumps and can be discussed in classical terms.  

The intensity of a scattered wave at a distance r from its source is proportional to the square of the polarisability of the particle and inversely proportional to r1. According to electromagnetic theory, the ratio of the scattered intensity (I) to the incident intensity (/0) in the plane normal to the direction of polarisation 

The above calculations refer to light scattered by a single particle. Provided that a colloidal dispersion is not too concentrated, the scattering is simply the sum of contributions from individual particles. Thus the intensity of light scattered at an angle 6 by unit volume of a dilute suspension of particle concentration c is 

The intensity of scattering is seen to depend, through R, on the relative refractive index, and it falls to zero if n = n,). Conversely, the greater the ratio of the indices the stronger the scattering. Since R is a function of the particle size, one can also in principle obtain the particle volume, or radius if the particles are spherical, by means of light-scattering measurements. Because 

The total scattered intensity (the turbidity, r) is obtained by integrating the scattered intensity over the surface of a sphere to give 

The domains exhibiting nematic order in the isotropic phase (in general smaller than the optical wavelength) give rise to a scattered intensity when illuminated by light. It has been shown that this scattered intensity is quasi-independent of the scattering angle [5, 6] and should vary as T-T ) in the vicinity of the N-I transition [4], as shown in Fig. 2 for the isotropic phase of MBBA [4]. 

The extension of the local nematic order in the isotropic phase can be characterized by a coherence length, By precise measurements, it has been shown that varies according to 

On apporaching the I/N transition, the coherence length, increases to reach limited values of about 10-12 nm at the tran- 

Although in the absence of an externally applied field the equilibrium value of. s in the isotropic phase is zero, there can occur fluctuations in the order parameter about the zero value. This gives rise to an anomalous scattering of light. 

We write the free energy expression (2.5.1) in a more general form in terms of the tensor order parameter (2.3.5), with i=j = 3 corresponding to the long molecular axis  

If the incident and scattered radiations are both polarized along z, 

Thus the intensity of scattering should vary essentially as T — T ) (fig. 2.5.5) and the ratio of the scattered light polarized along z to that polarized along X should be 4/3. Both these predictions have been verified quantitatively for 

If the order parameter varies gradually from point to point, the free energy expression should include gradient terms as well, which can be written in the form 

Rayleigh s results do not apply fully to solutions. He had assumed that each particle acted as a point source independent of all others, which is equivalent to assuming that the relative positions of the particles are random. This is true in the gases with which he worked, but is not true in liquids. Hence, for solutions, the scattered light is less intense by a factor of about 50 due to interference of the light scattering from different particles. 

Rayleigh showed that for light scattering, the basic relationship is 

In a dilute gas with molecules significantly smaller than the wavelength of light, the individual molecules act as point scatterers. For a gas consisting of N molecules in a volume of V m, the total scattering is A/V times the scattering from a single molecule. 

The polarisability, a, of the molecule is proportional to the refractive index increment dn/dc, and to the relative molar mass of the molecule in question. The full relationship is  

the value of I/Iq is dependent on relative molar mass of the molecules involved in the light scattering. The Rayleigh ratio, Rg, may be defined as  

Viscosity is a method similar to GPC in terms of MWt determination of CPs, in that a reference polymer whose structure and void volume can be assumed to be similar to the CP being measured must be used, and a suitable solvent must be found. The most successful viscosity determinations for MWt purposes have been those carried out on P(ANi) in cone. (96%) H2SO4, using a standard viscometer [415, 416]. 

Light scattering is possibly one of the most accurate and absolute methods of MWt determination. However, several problems remain with the use of this method for CP MWt measurement. Firstly, a solvent that yields a true solution of the CP, rather than a colloidal suspension, is preferred but not always found. Secondly, the high absorption of materials such as P(ANi) allow for only low concentrations to be 

In the absence of absorption, x is related to the primary intensities of a beam before and after it has passed through a thickness l of the medium, by the equation 

In solutions, part of the light scattering arises from fluctuations in refractive index caused by fluctuations in composition. The well-known equation for light scattering from solutions is based on these considerations 

Na = Avogadro s number (mol)-1 = wave-length of the light in vacuum (m) II = 

This equation forms the basis of the determination of polymer molecular masses by light scattering, which is one of the few absolute methods. 

however, is correct only for optically isotropic particles, which are small compared to the wave-length. If the particle size exceeds 2/20 (as in polymer solutions), scattered light waves, coming from different parts of the same particle, will interfere with one another, which will cause a reduction of the intensity of the scattered light to a fraction P(6) given by 

Bauer et al. [237] and Alms et al. [238] have studied a wide range of organic molecules (benzene, mesitylene, methyl iodide, nitrobenzene, etc.) in solution. They have compared the measured long-time rotational relaxation times with both Perrin s ellipsoid rotational times with stick boundary conditions [223] and with those from Hu and Zwanzig s similar calculation based on slip boundary conditions [227]. There is close agreement between experiment and the slip boundary condition model of Hu and Zwanzig. Typical rotational times could be expressed as 

The success of eqn. (113) in reproducing experimental data was investigated by Fury and Jones [240] using experimental results of Jones and co-workers [241] from Raman scattering and NMR studies. There was moderately good agreement and r0 0.2—1.8ps. Recently, Tanabe [235] has studied the temperature and pressure dependence of the Raman scattering spectrum of benzene. Experimental results were most closely reproduced by the Hynes et al. theory [221, 222], 

Dielectric relaxation measurements define an operational correlation time for the decay of the correlation function (P cosO)). For alcohols, the monomer rotation time, r2, increases from 18ps for n-propanol at 40°C to 44 ps for n-dodecanol at 40°C [83], A small measure of saturation in the dielectric relaxation time of alkyl bromides with increasing chain length has been noted by Pinnow et al. [242] and attributed to chain folding. 

Jones and Schwartz [243] have discussed the possibility of using electron spin resonance as a means of monitoring rotational relaxation events. 

Washburn (ed.), International Critical Tables, vol. Ill, p. 28, McGraw-Hill, New York (1929). 

Cotton, G. Wilkinson, C. A. Murillo, and M. Bochmann, Advanced Inorganic Chemistry, 6th ed., Wiley, New York (1999). 

Physical Methods for Chemists, 2d ed., chaps. 10-12, Saunders, New York (1992). D. F. Shriver and R W. Atkins, Inorganic Chemistry, 3d ed.. Freeman, New York (1999). 

One of the practical applications of dynamic light scattering involves the determination of particle sizes in media dispersed as dilute suspensions in a liquid phase. This aspect of dynamic light scattering is the focus here. Analysis of the scattering data will yield the translational diffusion constant D for a dilute aqueous suspension of polystyrene spheres, and this is directly related to the radius of the spheres. In addition, scattering will be studied from dilute skim milk, which reveals that a distribution of particle sizes exists for this system. 

Brillouin scattemg theory Brillouin(1922) experiment Gross(1930) 

Cconcentration by weight j radius of gyration A 2 second Virial coefficient Afmolecular weight 

II iBnllouin line intensity I Rayleigh line intensity 

Polarization induced in a molecule P=a0EoCos2rt i/0t + (EoQo/2)(i /iQ) [cos2rr (u, )t +cos2rr (vt-v, )t] Qnormal coordinate dielectric field a tpolarizability 

Frequency difference between scattered and incident light Doppler effect  

The basic equation used for molecular weight and molecular size can be written as [3,5]  

the key equation at the limit of ero angle and zero concentration, respectively, relating light scattering intensity to and the z-average radius of gyration (R ) may be written as  

Variation of central maximum of laser beam due to Phase Separation 

The reactive MAA/PMAA/water system was analyzed during in situ NMR spectroscopy (Wang, 1997). Similar studies with nonreactive MAA/PMAA/water analogs have resulted in nothing unusual in terms of chemical shift and nuclear spin relaxation behavior. 

The NMR sample that resulted in Fig. 2.1.9 was then cooled to room temperature for 8 h in order to terminate the polymer radicals and redistribute the polymer. Then, it was placed in the NMR probe chamber that was preheated to 80°C and allowed to heat for 10 min. After the proton NMR spectrum was taken (Fig. 2.1.10(a)), the sample was removed from the NMR probe chamber and allowed to cool again. 

When the NMR chamber has been preheated to 90 C, the sample was inserted into the preheated NMR probe chamber and the proton spectrum (Fig. 2.1.10(b)) taken after 10 min. Again, the sample was removed and allowed to cool to room temperature, while the NMR probe chamber was preheated to lOO C. Finally, the sample was reinserted in the preheated NMR probe chamber and the spectrum is taken (Fig. 2.1.10(c)) after 10 min. What can be observed from the water sub-spectra is that at higher temperatures, similar peak broadening was observed as that in the reactive FRRPP system at 80 C. 

These NMR smdies show that polymer domains in the reactive FRRPP system are at higher temperatures than the average, validating a basic foundation of the FRRPP concept. The question to be answered now is how hot are these polymer domains compared to systems under solution polymerization processes. 

Laser Chemistry Spectroscopy, Dynamics and Applications Helmut H. Telle, Angel Gonzalez Ureiia Robert J. Donovan 2007 John Wiley Sons, Ltd ISBN 978-0-471-48570-4 (HB) ISBN 978-0-471-48571-1 (PB) 

CHS QGHT SCATTERING METHODS RAMAN SPECTROSCOPY AND OTHER PROCESSES 

As stated, all these scattering processes do not rely on the photon being resonant to a molecular transition thus, the wavelength of the incident light does not matter much, so that light sources from UV to IR can be utilized with no need to match the observed molecules. However, to a certain extent, the choice of 

Of the scattering processes, only Raman scattering provides easily accessible molecular-specific information, which is important when investigating chemical reactions thus, only this process will be discussed further in this chapter. Nevertheless, both Mie and Rayleigh scattering serve their purpose, and analytical instmmentation based on these processes is marketed commercially (e.g. particle image veloci-metry or particle sizing). 

Another interesting development in the field of crystalline polymer blends was initiated by the Toyota Motor Corporation, which recently introduced a super-olefin polymer. This material was designed on a lamellar level (on a nanometer scale), and it has been speculated that this material is a blend of low-molecular-weight poly (propylene)(PP) with a new olefinic copolymer. This material, which has excellent hardness and toughness, as well as good flow properties, was designed to replace the expensive reaction injection molding (RIM) polyurethane [3]. 

1 and 6 are the wavelength of light and the scattering angle measured in the medium, respectively. y r) is the correlation function, defined as 

The intensity of scattered radiation f is given by the statistical average of the multiple of the conjugate of scattered radiation field (Rayleigh s 4th power law)  

Under a methodological point of view, in order to obtain significant data, utmost care must be taken in getting rid of any contaminant, such as dust or impurities. 

Static light scattering, in which the scattered intensity is measured as a function of the angle around the sample, is a mainstream technique for the determination of the molecular weight of polymers. By acquiring the scattered intensity at several angles on solutions at different concentrations, a Zimm plot can be drawn, from which the molecular mass and the radius of gyration of the polymer can be obtained [247,248]. 

Methods similar to those illustrated for X-ray or neutron scattering can be applied to light scattering, such as the Guinier relationship [249] or the fitting by appropriate theoretical models developed formalizing a shape and structure factors [250]. 

When particles are small enough to undergo Brownian motion, there is a continuous variation in the distance between the particles. As a consequence of this motion, constructive and destructive interference of the light scattered by neighboring particles yields intensity fluctuations. Following the intensity fluctuations as a function of time, the diffusion coefficient of the particles can be measured, and consequently, via the Stokes-Einstein equation, if the viscosity of the medium is known, the hydrodynamic radius or diameter of the particles can be calculated. Dynamic light scattering is therefore a very efficient method to determine the 

Intensity fluctuations are usually analyzed by determining the intensity autocorrelation function, which can be described as the ensemble average of the product of the signals (i.e., the number of photons in a given sampling interval) at time t and at time t + r, where r is the delay time, as a function of the delay time itself, over a delay range 100 ns to several seconds depending on the particle size and viscosity of the medium. 

Developments and modifications of Rayleigh s original theory allow Re to be related to the molar mass of a homodisperse polymer M through the equation 

The equation can be simplified since differentiating Equation (3.72) with respect to c for a monodisperse polymer sample gives 

If the solvent is not too good the higher terms in the expansion can be ignored and so substituting Equation (3.86) into Equation (3.85) gives 

It is customary to group the parameters which are constant for a given scattering experiment in one term K which is given by 

Equation (3.87) can also be generalized for a polydisperse polymer. In this case NM /V can be replaced by E NjM- /V but since, by definition. 

It is also possible to use spectroscopic detectors such as infrared and ultraviolet. One practical use of the latter is to determine if chemical modifiers (e.g., silane coupling agents) are bound to polymer molecules. This is done by measuring the molecular weight distribution of the modified plastic at a UV wavelength where the polymer itself does not contribute to the absorbance and comparing this to the distribution of the unmodified plastic. 

This is a standard procedure for molecular weight determinations and involves the use of specially designed viscometers to accurately measure the viscosity of a polymer solution. From this the intrinsic viscosity is determined and hence the molecular weight. The time taken for the polymer solution to pass between two marks on the viscometer is compared to that of pure solvent and the ratio is the viscosity of the solution. Successive dilutions give a range of concentrations and times from which the intrinsic viscosity can be calculated. The value for this is then entered into the Mark-Houwink equation  

Depending on the source of the Mark-Houwink parameters the molecular weight can be expressed as either the number or weight average. 

There are two principal osmometry techniques vapour pressure osmometry and membrane osmometry. 

Vapour pressure osmometry involves the indirect measuring of the lowering of the vapour pressure of a solvent due to the presence of a solute. It is based on the measurement of the temperature difference between droplets of pure solvent and of polymer solution maintained in an isothermal atmosphere saturated with the solvent vapour. Calibration is by the analysis of standards of known molecular weight and should be over the entire range of molecular weights of interest to ensure the best results. The technique is useful for polymers that have molecular weights in the 500-50,000 range. 

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Figure 5 shows tluee patterns of light scattered from tlrree different wires. Three graphics of (7) of increasing T/a are shown in figure 6. 

Figure i Light scattered from three different wires. 

The scattering techniques, dynamic light scattering or photon correlation spectroscopy involve measurement of the fluctuations in light intensity due to density fluctuations in the sample, in this case from the capillary wave motion. The light scattered from thermal capillary waves contains two observables. The Doppler-shifted peak propagates at a rate such that its frequency follows Eq. IV-28 and... 

B. J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 1976. 

Foam rheology has been a challenging area of research of interest for the yield behavior and stick-slip flow behavior (see the review by Kraynik [229]). Recent studies by Durian and co-workers combine simulations [230] and a dynamic light scattering technique suited to turbid systems [231], diffusing wave spectroscopy (DWS), to characterize coarsening and shear-induced rearrangements in foams. The dynamics follow stick-slip behavior similar to that found in earthquake faults and friction (see Section XU-2D). 

The dynamics of polymers at surfaces can be studied via dynamic light scattering (DLS), as described in Section IV-3C. A modification of surface DLS using an evanescent wave to probe the solution in a region near the interface has... 

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

Berne B J and Pecora R 1976 Dynamic Light Scattering (New York Wiley) ch 10... 

Kirkpatrick T R, Cohen E G D and Dorfman J R 1982 Light scattering by a fluid in a nonequilibrium steady state. II. Large gradients Phys. Rev. A 26 995... 

In the next section we discuss linear hydrodynamics and its role in understanding the inelastic light scattering experiments from liquids, by calculating the density-density correlation fiinction,. Spp. 

Out of the five hydrodynamic modes, the polarized inelastic light scattering experiment can probe only the tliree modes represented by equation (A3.3.18), equation (A3.3.19) and equation (A3.3.20). The other two modes, which are in equation (A3.3.17), decouple from the density fluctuations diese are due to transverse... 

Figure A3.3.3 Time-dependent structure factor as measured tlnough light scattering experiments from a phase...

The phase shift is measured by comparing the phase of the fluorescence with the phase of light scattered by a cloudy but non-fluorescent solution. 

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