Photon Correlation Spectroscopy: Technique and Scope

7 Photon Correlation Spectroscopy: Technique and Scope TERENCE A. KING University of Manchester, UK 7.1

911

INTRODUCTION

7.2 PHOTON CORRELATION SPECTROSCOPY TECHNIQUES 7.2.1 Summary of Light Scattering 7.2.2 Dynamic Light Scattering 7.2.2.1 Translational diffusion 7.2.2.2 Larger particles 7.2.2.3 Depolarized scattering 7.2.3 Linewidth Measurement 7.2.4 Photon Correlation Spectrometers 7.2.5 Photon Correlation Spectroscopy Data Analysis 7.2.6 Related Techniques

912 912 913 915 915 916 917 919 922 924

7.3 APPLICATIONS OF PHOTON CORRELATION SPECTROSCOPY TO POLYMERS 7.3.1 Scope 7.3.2 Polymer and Colloid Characterization 7.3.3 Polymer Dynamics 7.3.4 Gels 7.3.5 Bulk Polymers 7.3.6 Liquid Crystals 7.3.7 Polymer Adsorption 7.3.8 Phase Transitions 7.3.9 Electrophoretic Scattering and Application to Poly electrolytes

924 924 924 925 927 928 928 929 930 930

7.4

931

7.1

REFERENCES

INTRODUCTION

The light-scattering technique of photon correlation spectroscopy is now firmly established as a valuable tool for the study of the structure and dynamics of macromolecules. Over the last 25 years there has been steady development of the technique and a growing, and now extensive, range of applications to polymer systems. The determination of translational and rotational diffusion coefficients and internal molecular flexing provides a means of polymer characterization for molecular weight and molecular weight distributions, giving size and structural information. Study of polymer dynamics in isolated or weakly interacting molecules in dilute solutions, and strongly interacting molecules in semidilute and concentrated solutions, has provided experimental data concurrent with the extensive development of theoretical models of polymers using renormalization group and scaling theories. Photon correlation spectroscopy is also an important technique in the study and characterization of crosslinked and physical gels, solid and melt bulk polymers, colloids, liquid crystals, aspects of polymer phase transitions and polyelectrolytes. As well as the application to synthetic polymers the technique has found broad application to biopolymers; these applications are not addressed here. Molecular scattering of light has been investigated over the last 100 years, and scattering at the incident wavelength, Rayleigh and Rayleigh-Debye scattering, has been used extensively in more recent times for characterization of molecular weight, size, shape and interactions. The treatment of total intensity light scattering and its application to static molecular and particle characterization is CPC-DD

911

Solution Methods

.912

well documented. 1 " 5 Photon correlation spectroscopy is a more recently developed inelastic lightscattering process, involving frequency broadening of the central Rayleigh line about the incident light wave frequency. The earlier discoveries of the inelastic light-scattering processes of Raman and Brillouin scattering, in which there is a frequency shift between the incident and scattered light, have also been applied extensively to polymer systems. Combined Rayleigh-Brillouin scattering has proven fruitful in the study of bulk systems. We concentrate here only on the technique and application of photon correlation spectroscopy. As well as the name 'photon correlation spectroscopy' many other names have been used in the literature to describe the method, in which broadening of the central Rayleigh line is measured, dependent on the way in which the scattered signal is detected or processed. These names include: light-beating spectroscopy, self-beat spectroscopy, quasielastic light scattering, Rayleigh linewidth spectroscopy, laser Doppler spectroscopy, intensity fluctuation spectroscopy and optical mixing spectroscopy. For standardization we adopt here the general name of photon correlation spectroscopy (PCS). This name has recently come into more general use since processing of the scattered signal is usually done by digital time autocorrelation of detected photons. Light incident on a medium induces an oscillatory polarization of the medium which in turn acts as a secondary source of radiation in the form of scattering. Fluctuations in the dielectric constant of the medium induce net scattering of radiation. The intensity, angular distribution, polarization and frequency shift of the scattered light are determined by the size, shape and molecular interactions of the scattering centres. With this process information on the structure, interactions and molecular dynamics within the scattering medium can be derived. The frequency distribution of light scattered from macromolecules was investigated by Pecora (1964 and subsequently) 6-8 who showed that the macromolecular translational diffusion coefficient, the rotational diffusion coefficient and internal motion dynamics may be derived. Also in 1964, the previously demonstrated 9 technique of optical mixing was used 10 to measure the small frequency broadening of the Rayleigh-scattered peak from dilute suspensions of polystyrene latex particles. The subsequent growth of photon correlation spectroscopy and its application to polymers and biopolymers is described in a number of books and reviews. 11-30 For early and general introduc­ tions see the books of Berne and Pecora 17 and Chu 12 and to two review proceedings. 1 3 1 8 , 2 0 - 2 2 Attention is also drawn to Chu's account of the application of PCS to polymer solutions in Chapter 8 of this volume. 7.2 7.2.1

PHOTON CORRELATION SPECTROSCOPY TECHNIQUES Summary of Light Scattering

Light is scattered by a molecule in solution if the molecule has a polarizability different from its surroundings. 31-33 The molecular polarizability difference gives a spatial inhomogeneity to the dielectric constant of the medium or equivalently a refractive index difference. The classical mechanism of light scattering involves the electric field of the incident light inducing an oscillating dipole moment in the molecule which re-radiates to form the scattered radiation. Inhomogeneities of the medium may arise from spontaneous thermal fluctuations or concentration fluctuations, for example for a polymer solution or colloidal dispersion. The intensity of the scattered light depends on the intensity of the incident light, the scattering angle, the solution parameters and the light polarization. Figure 1(a) illustrates a form of the scattering geometry. A linearly polarized, monochromatic plane wave of wavelength A is incident on the scattering medium and an optical detector is at the point P. For Rayleigh scattering the molecules have dimensions <^ A, so that the molecules sense a uniform field E0 and the scattering is not sensitive to the shape of the molecule. Conservation of momentum in the scattering process leads to a wavevector diagram as shown in Figure 1(b). The quantity q = ks — k{\s the difference between the wave vectors of the scattered and incident waves. Since in Rayleigh scattering the wavelength of the scattered light is very near to that of the incident light, ks~k{= 2njX>x. Then 2/c;sin

l_2j

— — sin

-

(1)

A, l_2j where 6 is the angle of scattering, n0 is the medium refractive index and Lx the vacuum wavelength of the light. If a is the largest dimension of the scattering molecules, when qa <^ 1 they behave as point scatterers.

Photon Correlation Spectroscopy: Technique and Scope

q

913

Ifl-^sin(f)

Figure 1 (a) Light-scattering geometry; and (b) wavevectors and conservation of momentum diagram with definition of wavevector q

The electric field of the scattered light is in general Es{R,t)

=

ks A ( * s A E 0 ) a —

4neR

^-Qi(co0t-ks.R)eiq.r

(2)

for a molecule of polarizability a and where e is the relative dielectric constant of the medium surrounding the molecule. For molecules of size <^ X (Rayleigh scattering) the polarized scattered light intensity / s at position distance R is

- - r-i

I4

/0

V2

Ln 0 J

sin 2 0

[n 2

16TT 2 K 2 L

-

n2]2 °

(3)

where V is the molecular volume and n0 ( = y/s) is the refractive index of the surrounding medium. When the polarizability of the molecule a is isotropic such that the molecule a is homogeneous with refractive index n, then a = (n2 — n%) V. The scattered light intensity is seen to be proportional to /cj (i.e. \/Xf), to I0 and inversely proportional to R2. The total intensity of scattered light contains information on the static properties of the scattering medium, i.e. the size and shape of the scattering molecules and the thermodynamic quantities. The other basic quantity which can be measured by light scattering is the frequency distribution of the scattered light; this carries information on dynamical quantities, such as diffusion coefficients, internal motion and molecular velocity. There light is scattered from refractive index fluctuations, which for a polymer solution arise from the polarizability difference between the solute and solvent.

7.2.2

Dynamic Light Scattering

The phase ^ s of the scattered field £ s is made up of the phase of the incident field at position r, co0t — k{r plus a phase shift due to propagation of the scattered field from position r to position /?, —ks(R — r\ so that \j/s = (co0t — ks'R) + (&s — k^-r. The scattered electric field from a single molecule is Esj(t)

Aj(t)e-i
=

(4)

where Aj(t) is the amplitude of the scattered light from molecule j and \l/j(t) is the phase difference from the optical path of light scattered from position r} compared to that at the origin, ij/j(t) = q*r(t\ i.e. the phase of the scattered field depends on the position r of the scattering molecule. The total scattered field is N

EMt)

=

I

Aj(t)Qi
(5)

j

Since the molecules are moving, r is a function of time and ij/s has a dependence determined by the molecular dynamics. The scattered field varies in time due to translational diffusional motion or to changes in Aj(t) induced by rotational or internal motion. A less common case is for smaller values of molecular occupational number N, where fluctuations in N lead to scattered light fluctuations;34 polymer solutions at typical concentrations provide many molecules in the scattering volume. For

Solution Methods

914

example a laser beam focused into a polymer solution illuminates a sample volume defined by a 100 jam focal spot diameter, with detection optics collecting from that volume; the sample volume is ~ 10" 6 cm 3 . A polymer of molecular weight M ~ 105 at a concentration of 1 mgcm" 3 is equivalent to about 10 10 molecules in the sample volume. When the scattering molecule is undergoing Brownian motion, r is a random variable and Es has a randomly modulated phase. The scattered light is broadened in frequency with an optical frequency distribution S(a>) as illustrated in Figure 2(a). Since the particle motion contains no preferred direction, the spectrum contains a continuous distribution of frequencies centred around co0. The correlation function of the electric field G (1) (T) is also a measure of the frequency distribution and contains information on the molecular motion. 3 5 - 4 0 It is the Fourier transform of the power spectrum S(a>) G^d)

=


+

T)>

(6)

where < > denotes a time or ensemble average and T is the correlation time S(co)

1

=

— 271

f00

G ( 1 ) (T)e i < a t di

(7)

J0

Discussions of the nature of correlation functions and molecular dynamics can be found in refs. 41 and 42. (a)

Df- W

Figure 2 (a) Illustration of an optical spectrum of scattered light; and (b) electric field correlation function

An illustration of a scattered optical spectrum and its normalized field correlation function is shown in Figure 2. The broadening of the Rayleigh-scattered light spectrum contains information on the motion of the scattering molecules. We are concerned here only with the broadening of the central Rayleigh component of the scattered spectrum; the other types of inelastic scattering, the Brillouin doublet and Stokes and anti-Stokes Raman scattering, both occur at much greater frequency shifts. The normalized electric field correlation illustrated in Figure 2(b) is g(1)W =

G(1)(T)

(8)

G ( 1 ) (0)

In terms of amplitude and phase time dependences


(

\

Q{-iq-lr(T)-r(0U}

e<-o<>C A (T)Q(T)

(9)

where A(x) is the scattering amplitude per molecule and C A ( T ) and C^(T) are the amplitude and phase correlation functions. For small molecules {a <^ q~1) or spherical molecules the amplitude part

Photon Correlation Spectroscopy: Technique and Scope

915

of the autocorrelation function becomes <>4*(0M(T)>

=

CA(T)

\±_^L



=

x

(10)

Then g (1) (t) carries information on the translational diffusion coefficient DT through C^x). This is related to the intermediate structure factor Gs(r,x), which is the probability of finding a particle at position r at time T if it was at the origin at T = 0. C,(T)

=



G s (r,T)e(-'*- r )d 3

=

(11)

Thus for spherical identical scatterers undergoing Brownian motion in solution g»(x) =

e-

DT 2t

" e",'a,0T

(12)

The associated optical spectrum is
=

(a>-co0)2

+

(DTq2)2

(13)

which is a Lorentzian function centred at co0 with a halfwidth DTq2.

7.2.2.1 Translational diffusion An estimate ofthe degree of broadening and correlation decay time, assuming a system of polystyrene with M w = 670000 in cyclohexane with DT = 1.66 x 10 " 7 cm 2 s" * and observed at 90°, leads to a linewidth of about 2 x 10 4 Hz and a characteristic decay time of the correlation function t c = l/DTq2 = 5x 10" 5 s. The diffusion coefficient D may be related to the molecular friction factor/through the Einstein relation kT

DT = j For a spherical molecule of radius a,f=6nrja9 DT

=

(14)

where rj is the dynamic viscosity of the solvent, then kT 6nt}a

=

kT 6nrjRh

(15)

where Rh is the hydrodynamic radius of the molecule.

7.2.2.2 Larger particles When the molecular size is not much less than q~x, the scattering intensity is reduced by intramolecular interference. For molecules with size ~ 1/q, the scattered intensity Is is given as

where P(9) is the molecular interference form factor. The most general form of P(0) is described by Mie scattering theory. 2 " 5 The Rayleigh-Debye approximation is valid when {n — n0)a< X/An. Then for an isotropic molecule P(0) = |l/KjKe^*rdz;|2 where V is the volume of the scattering molecule. Values of P(0) have been calculated for many scattering particle shapes.4 For any shape of particle at small scattering angles, P(0) = 1 —
Solution Methods

916

dependent scattering amplitude; also for flexible molecules intramolecular dynamics gives a similar contribution. The form of the correlation function G (1) can be calculated for model systems. Here we consider rod-like molecules and flexible coils. For a rod-like molecule of length L, the dynamics of the molecule will contain contributions from translation and rotation. The form of G ( 1 ) ( T ) for a molecule having a rotational diffusion coefficient DR i s 7 4 3 ' 4 4 |G (1) |

=

/se-«2DTt[S0 + Sie~6l)Rt

+

....]

where S0,SX . . . are weighting factors as shown in Figure 3(b). For qL<3,S0 for qL = 6, 5X = 0.1 and is significant compared with 5 0 . In a similar way the form of G (1) for a flexible coil is 8 G ( 1 ) (T)

=

IsQ-q2DTtlP0

+

P^"2^1

+

...]

(17)

= 1 and Sx = 0, while

(18)

where P0,P1 . . . are weighting factors for the translational and internal modes of motion (Figure 3a) and TX is the characteristic time for the longest intramolecular relaxation.

Figure 3 (a) Relative intensity functions P(x) for a flexible polymer, x=%q2 < I2 >e with < I2 >e = equilibrium mean square endto-end distance of the polymer.7 (b) Relative intensity functions for an optically rigid rod of length L. S0: pure translation, Sx: translation-rotation, Sh: sum of all higher terms.8

7.2.2.3 Depolarized scattering For the incident field E{ shown in Figure 1, linearly polarized with E{ perpendicular to the scattering plane, optically isotropic point scatterers give a scattered field intensity Jw, also linearly polarized perpendicular to the scattering plane. Scatterers which are optically anisotropic or geometrically anisotropic give a second depolarized scattered component, 44 " 49 intensity JHV, polarized in the scattering plane. The depolarized scattered electric field has its amplitude modulated by rotation of the scatterer. For a rigid particle with rotational symmetry the correlation function for the depolarized field |G^>(T)|

=

Isde-lDTq2

+ 6D^x

(19)

At small angle Dq2 -► 0 and then the correlation function is determined by the rotational diffusion coefficient D R . DK has values typically within the range DR ~ 10 2 -10 6 s~x corresponding respectively to large rod-like molecules and small spherical molecules.

Photon Correlation Spectroscopy: Technique and Scope 7.2.3

917

Linewidth Measurement

The scattered light intensity is proportional to the square of the time average of the electric field = < Is > oc < | Es | >2

Scattered intensity

(20)

where < > denotes the time average. In order to measure the very small optical linewidths indicated in Section 7.2.1 optical mixing techniques are employed since dispersing optical instruments, such as diffraction gratings and Fabry-Perot interferometers, do not have sufficient resolving power. There is a large body of work on optical mixing, from which PCS has emerged, in the field of optical coherence, the quantum theory of light and intensity fluctuation interferometry. 50-52 Since photon correlation instruments generally have greater efficiency than real time spectrum analyzers these are usually preferred over power spectrum measurements in PCS, while power spectrum analysis is used in laser velocimetry. There are two basic forms of optical mixing: heterodyne and intensity fluctuation (self-beat). A note of clarification is essential here since the literature contains conflicting nomenclature. 16 By hetero­ dyne mixing we refer to mixing the scattered light with a reference light wave (local oscillator) unshifted or shifted in frequency from the incident light beam — this is usually and most con­ veniently derived from the laser source providing the incident beam.* In self-beat optical mixing the scattered wave is not mixed with a reference signal but is directly detected. The nomenclature adopted here is taken as a consensus of that appearing in the literature. The fluctuating scattered light intensity is illustrated together with related quantities in Figure 4. The normalized electric field autocorrelation function is gd)(T)

^^-f-

=

-

(21)

This quantity can be measured by heterodyne detection. With a reference beam £ r , the total field at the detector is Em(t)

=

Es(t)

+

ET(t)

(22)

+

(23)

whose intensity is Aot

l^totl2

=

=

/,«

+

/rW

2Re{£ s (t)£ r *(t)}

where Re indicates the real part of the expression in brackets. Then the total intensity correlation function is given by Gg>

=

G<2>

G<2>

+

+

2

2Re{G<1>G<1>}

+

(24)

with E%(t) =

£0s(£)e'l>os' +
(25)

and Er(t)

= E0t{t) &<»*' +MM

(26)

Then G^d)

^0sXgil)(T)

=

G ( S 2) (T)

=

2[l

g(2)(T)]

+

If

+

(27)

g(2)(x)]

(28)

From which Gff

=

2[l

+

+

2/ r {l

+

008(0)0.-<»or)Tg(1)M}

<29)

For Ir > , the first term may be neglected and G ^ oc Re{G (1) (r)} shifted to be around the frequency co0r. As described later heterodyne mixing and detection has some valuable advantages in polymer studies with PCS. * When mixing the scattered light with a reference signal which is unshifted in frequency, this is termed homodyne detection in other areas.

CPC—DD*

Solution Methods

918 (a)

(b)

(c)

(d)

!

I

Figure 4

(a) Spectrum and correlation function of ideal laser source; (b) fluctuating scattered light intensity; (c) standardized detected photon counts; and (d) optical spectrum and g (2) (i) of scattered light under heterodyne detection

In self-beat detection the intensity autocorrelation function is determined as lim r ^

G(2)(T)

— IT

Is(t)Is(t

+

i)dt

(30)

It is the Fourier transform of the power spectrum and is readily measured by digital techniques. The normalized form of G ( 2 ) ( T ) is g (2) (T)

=


+


2

r)Es(t

+ T)>

(31)

With some restrictions (such that the scattered field is a Gaussian random process), the correlation functions g(1)(?) and g(2)(f) are connected through the Siegert relation g<2>(T)

=

1

+

|g C1) (T)| 2

(32)

Experimentally in self-beat PCS the intensity autocorrelation function is measured as C(T)

=

g(2)(T)

=

B [ l 4- 6|g ( 1 ) (T)| 2 ]

(33)

Here B is a constant and b is a geometric factor dependent on the detector area. 53 For g (1) (r) 2 be~2DTq2x]. = e "o 0 r e -D T ^ t w e find g( >(T) = J B[l + The detector has an average photocurrent < i > which is proportional to the average light intensity: oc < / s > . Since the scattered light is normally at low level and in the form of discrete photon pulses, the scattered signal and hence correlation function is most usefully recorded using digital photon detection. In terms of photon counts g(2)W =



Z

n n

i t+p

(34)

Photon Correlation Spectroscopy:

Technique and Scope

919

where r — p AT, AT = channel width, Nc = number of correlation channels and = average number of protons counted in time A T. Heterodyne optical mixing has several features which are valuable in application to polymer With heterodyne detection the autocorrelation function is linearly proportional to systems. g ( 1 ) (i), the first order scattered electric field correlation function, and leads to useful simplification in multiexponential data analysis. A double exponential first order correlation function of the form g(1)W

AQ~

(35)

BQ-

with constants A and B and decay frequencies a and b converts by use of the relation g (2) (t)= 1 + |g (1) (i)| 2 to a current autocorrelation function in self-beat optical mixing i(T)ocl

+

+

2ABQ-{a

+ b)x

(36)

In heterodyne mixing the equation for analysis is simpler, being i(i)

oc

1

+

Ae~at

+

Be~bt

(37)

This is valuable in polydispersity analysis and the derivation of molecular weight distributions using multiexponential algorithms. With self-beat detection (i) the presence of the background term A needs to be derived before the function can be simplified by taking the square root, and (ii) application of self-beat detection is restricted to applications giving scattered fields with Gaussian statistics. Practical considerations are that heterodyne mixing provides (i) improved signal-to-noise under conditions of equivalent measurement, and (ii) inadvertent partial heterodyning, by stray light acting as a reference when operating in the direct detection configuration, can be avoided. 7.2.4

Photon Correlation Spectrometers

The main components and layout of a photon correlation spectrometer in the intensity fluctuation (self-beat) configuration to measure the angular distribution of scattered light are shown in Figure 5. We describe here the usual form of PCS using 5 7 - 6 0 digital autocorrelation. Light scattered by the sample, held in a special cell, is detected normally as a function of angle 0, but also sometimes at a fixed angle, and analyzed by a time autocorrelator or spectrum analyzer. The light source is usually a laser, although non-laser sources can be used under special circumstances,61 and the detector is a Focussing lens

Sample cell

Laser

Coherent optical mixing Detector photomultiplier

Correlator

Stepper motor drive Display

It(t)

/

/

Keyboard

Polarizer .^ Aperture / /-^Lens iris Field stop-define sample volume Coherence area aperture Filter Detector photomultiplier

Figure 5

Schematic of self-beat photon correlation spectrometer and detection optics

920

Solution Methods

photomultiplier or sometimes a sensitive photodiode. The signal from the correlator is stored and analyzed by a computer. The laser provides continuous high intensity in a single radiation mode which can be focused to a small volume with a dimension ~ 2FX, where F is the f-number of the focusing lens. The most common lasers used are He-Ne (X = 632.8 nm or 543.2 nm), Ar ion (X = 488.0 nm or 514.5 nm) and He-Cd (X = 441.6 nm). Sample fluorescence or stray light is removed by a line filter in front of the detector. For strongly scattering samples the He-Ne laser is used at a typical power level of 3-10 mW. The argon ion laser provides higher stable powers up to 1 W for weakly scattering systems but requires to be used within the power limitations of the sample and avoiding sample heating which induces convection. The laser is required to operate with minimal intensity fluctuations with an instability of < 1 %; for the heterodyne operation a single longitudinal mode of the laser is usually selected by the use of an etalon in the laser cavity. Detection photomultipliers must have the required characteristics of sensitivity, wavelength response, speed and low noise and are used with a high speed pulse amplifier-discriminator to convert the photon signal into a form suitable for the correlator. The photomultiplier can also be obtained with low noise and free of self-correlation, induced by after-pulsing in the detector process, by testing and selecting the photomultipliers; detector self-correlation would distort the measured correlation function at short correlation times. Several modern and complete commercial spectrometers are available, satisfying the requirements for a general purpose instrument. Since PCS can be used to carry out a wide range of different types of measurement or may be applied to particularly difficult systems the need may arise for a purposemade instrument. Vibrations of the spectrometer are required to be avoided by the use of stable optical construction and a vibration isolation system to prevent coupling of vibrations to the instrument. Relative motion of the laser with the sample should be avoided by mounting laser source, scattering chamber and detector on a common baseplate. Efficient operation requires as much light as possible to be scattered into a single coherence area at the detector. The input focusing lens is selected to condense light into the scattering volume but without adding an unacceptably large spread of input angles leading to error definition of wavevector q. The radius of the laser beam at the focus of a lens of focal length/is/a, where a is the divergence of the incident beam and is typically 0.5 mrad. The preferred small focal length of the lens conflicts with the error induced in q of Aq ~ (2n/d)cos(6/2) where d is the diameter of the scattering volume. In optical mixing to study fluctuations in the scattered field a collection aperture is used to limit the detection area near to one coherence area. 38 The coherence area Ac is Ac = X2R2/wd sin 9 where w is the laser beam diameter in the scattering volume of dimension d and R is the scattering volume to detector distance. For large Ac the laser beam diameter w and scattering volume size should be small. The sample is normally held in a round or square cell similar to a spectrometer cuvette and in a temperature-controlled index-matched enclosure. Careful temperature control is required to avoid convective effects. Measurements can be made up to high temperatures; in the author's laboratory a light-scattering cell for operation up to 1000 °C for use with inorganic glass samples has been constructed. Sample volumes can be very small (e.g. ~0.1 cm 3 ) or larger dependent on require­ ments. For accurate measurement great care is required on sample preparation to avoid contami­ nation from dust. Sample cleaning by micropore filtering and centrifugation is generally used and solvent distillation may also be employed. Centrifugation of the sample in specially constructed combined centrifuge-sample cells is effective in avoiding contamination. Special signal processing has also been proposed to counter contamination. 62,63 The autocorrelator receives a continuous signal and records the correlation function r

1 f

G(T) = lim —— I(t) I(t — z)dt. With digital photon detection and the time range divided into T^oo 2T J _ r JV- 1

finite channels of time width T, the correlator computes the function G(kT) = £ nk ni+k where k is i=0

the channel number and n, is the number of counts in channel i. A correlation function and associated residuals from data analysis are shown in Figure 6. A full multibit correlator performs the product operation with the number of bits in each channel corresponding to the sample counts. To speed up computation the signal can be converted to single bit by limiting with a clipping level nc such that nt = 0 if nt < nc and n{ = 1 for nt>nc. The clipped correlation function is a good approximation to the true correlation function if the signal obeys Gaussian random statistics, i.e. the signal is from a large number of independent scatterers undergoing random Brownian motion. If the

Photon Correlation Spectroscopy: Technique and Scope

10

20

30

921

40

Channel number n

Figure 6 (a) Autocorrelation function for two-component polystyrene, M w = (1.1 +0.498) x 106 in benzene at 25 °C. Sampling time 200 /is, heterodyne mode, (b) Data residuals from S-exponential analysis; see Section 7.2.5

sample contains only a few scatterers or the scatterers have dynamically coupled scattering then the clipped correlation function will be distorted. A number of correlation systems have been developed. 57-61 A wide range of correlator sample times can be provided typically from 10" 7 s up to minutes; sample times less than 100ns can be provided often with single bit operation or in burst-mode operation. An alternative method of measuring the correlation for fast times has been explored using pulse interval timing between pairs of scattered photon pulses. 64 The correlator channels may be distributed in time in various ways as well as with a linear time range. Delaying a group of channels at the end of the time range by a long delay enables the baseline of the correlation function to be obtained more accurately by measure­ ment rather than by computer analysis. A more recent and very valuable development 58 ' 65 for polymer applications is the use of channels distributed exponentially in time (logarithmic correlator) or with several linear groups of channels with linear time ranges and distributed to cover different time regions. These latter instruments are able to record the correlation function simultaneously over a very broad range of time. In many polymer systems the correlation function is characterized by a distribution of relaxation times, for example in polydisperse systems or in highly concentrated or bulk samples.66 The PCS correlation function can then be highly non-exponential and great care is required in the collection and analysis of the data. In these cases collection of the data is required over a wide range of delay times to derive the relaxation times adequately. The use of exponential sampling provides a wide range of effective correlation times. The limiting precision of PCS is set by sample considerations and noise error sources. The signalto-noise (S-N-R) ratio is influenced by light intensity, data collection time and stray light. The concentration of the sample affects the scattered intensity and for a polymer-solvent system has a minimum value set by the ratio of signal photons from the solute compared to stray photons from the solvent. For a correlation function of the form G(T) = 1 4- a e ~ r \ the number of decay times in a measurement run time of Te is TTe, leading to a S-N-R of (T r e )*. The S-N-R can be improved by increasing Tc9 the duration of the measurement, or by increasing T by scattering at longer angles, since T oc Dq2. Stray light from contamination and dust scattering, flare off cell walls and optical components and impurity scattering leads to distortion of the correlation function and may cause mixed self-beat-heterodyne detection to occur.

Solution Methods

922 7.2.5

Photon Correlation Spectroscopy Data Analysis

Fitting of the correlation function C(T) = g(r) by a sum of exponential components is often required. In many cases there may be no prior knowledge of the number or weighting of the exponentials. In general the time correlation function can be written as tAi*~xlti

C(T) =

(38)

i= 1

where p is the required number of exponentials. One of the most common methods employed is that of non-linear least squares fitting, such as the Marquardt routine. 67 These routines are not reliable in fitting more than two exponentials with background. In the case of poly disperse systems the general form of the time correlation function is the Laplace transform of a distribution line width function G(T) C(i)

=

g^>(x)

-r =

G(r)e"rtdr

Jo

(39)

In principle, knowledge of G(T) allows the particle size distribution to be derived, when the scattering intensity as a function of particle size is known. In recent years a number of methods have been developed 68-84 to extract G(r) from C(T). For spherical monodisperse particles the normalized self-beat photon correlation function is g(2)(t)

=

1 + C|g(1)(x)|2

=

1

Ce- 22,T " 2t

+

(40)

where C is a constant. In the heterodyne mode the correlation function may be reduced to g(2)(t)

=

1

+

Cg (1) (i)

=

1

+

Ce- DT « 2t

(41)

With a polydisperse system, the distribution of molecular sizes gives a distribution in DT and the correlation function is made up of a distribution of exponentials. Then from equation (39) g(1)(T)

=

e" rT C(r)p(r)dr

(42)

with G(r) = C(r)p(T) in which p(T) takes account of the variation of the scattering power with particle size. Equation (42) is of the form of a Laplace transform and to derive the linewidth distribution requires Laplace inversion of the data. This presents substantial difficulty as in the inversion small changes in the data can give rise to large errors in the inverted information. A general purpose and flexible method of inverting PCS data has been developed 74 under the name CONTIN. This has been widely applied in PCS studies with excellent results for PCS data having low noise. The program contains safeguarding constraints to avoid the ill-posed nature of the inversion. An early method of analysis was based on a cumulant expansion 85,86 of the correlation function ln|g(1)(r)| =

-ft

+ i^ 2 T 2 1.

-

where km is the cumulant and fit =

i M 3* 3 J.

+ Up* 4.

-

3^ 2 ]T*

+

...

=

X U n * - ^ - - (43) k=

j

m.

r oo _ (T - T)1 G(r) d r . Equation (43) may be fitted by a least

squares routine to the correlation function and values for \i2, /i 3 , . . . obtained. The average width r = T G ( r ) d r = mean relaxation time. The variance is /^ 2 /r with \i2 (r-r)2G(r)dr. - \ : o For low q, qR < 1 and F = DTq2 where DT is the z-average DT. Practically it is difficult to obtain accurate values of the moments greater than the second and the method is restricted to relatively low unimodal distributions. When the moments have been obtained G(T) may be reconstructed using Pearson's method. 87 Cumulant analysis is discussed further in Chapter 8. An eigenvalue expansion multiexponential method has been developed 69 ' 71 of particular utility in application to polymer systems. The linewidth distribution is represented as a series of exponentially

Photon Correlation Spectroscopy: Technique and Scope

923

spaced linewidths such that G(r)dr

=

G*(lnr)d(lnr)

G*(lnD

=

£a„.5(lnr

-

(44)

tor.)

(45)

where n is the number of functions whose upper value is limited by the noise level. The closest points that can be resolved are given by r.-,

=

exp

[

[-]

(46)

LcomaxJ 2 "1 f noise level ~|

(47)

This defines a resolution limit such that two linewidths Tn and Tn_ t cannot be resolved unless they are separated by at least the factor exp(7r/ctfmax) which is dependent on the noise level of the correlation function. The theoretical Hnewidth resolution when noise-limited from equation (47) is shown in Figure 7(a), with the maximum noise level permitting resolution of two decay components shown in Figure 7(b).

Figure 7 (a) Noise-limited theoretical Hnewidth resolution comax = (— 2/tt) In (noise level/y/n); and (b) maximum noise level for resolution of two decay components

Using the eigenvalue analysis it has been shown 6 5 , 7 2 ' 7 6 ' 7 8 that the Laplace inversion can be reduced to a linear fitting problem using the defined linewidths. When correlation function data points are placed with exponentially spaced delay times T, re At , t e 2 A t . . . xn . . . interpolation between these points enables the correlation function to be reconstructed for all values of T from the values at T„, within the limits set by the noise bandwidth^ A method of histograms has been used 88 " 90 in which G(T j) is assumed to be constant over a range [Tj — AT/2] to [Tj + AT/2] and one histogram element is approximated as a single exponential. Then Ar

g(1)W

■ - r~

ECU,]

exp[— r,T]dr

(48)

The analysis reduces to the fitting of exponentials to the data and is an approximation to the eigenvalue expansion method. This is related to a more general method of splines. A method of linear programming (SIMPLEX) or non-negative least squares has been assessed91 for PCS. It has been shown that even finely structured G(T) functions can be recovered from the Laplace transform, avoiding convergence problems and negative amplitudes. The program incor­ porates a technique of sequences of residuals defining the choice of the maximum retrievable resolution.

924

Solution Methods

Analysis of PCS decay curves by a new S-exponential sum method has been demonstrated. 92 This has been applied to the extraction of mean linewidths and moments of unimodal polymer distributions, mixtures of monodisperse polymers, and bimodal polydisperse systems; an example is shown in Figure 6. 93 " 95 The use of a profiled function incorporating the measured mean and pQlydispersity index has been shown 8 1 - 8 3 to improve the inversion and resolution of closely spaced peaks; this also provides a rapid method for computation. The progress made in recent times in photon correlation instrumentation and in methods of data analysis now enables several exponential components to be reliably extracted from PCS data having low noise content. The interactive operation of the measurement with the data analysis, 96 ' 97 aided by an on-line computer, enables the best parameters for each measurement to be set up.

7.2.6

Related Techniques

There are a number of promising developments in photon correlation spectroscopy techniques which add greater versatility to application of the method to polymers. It is valuable to arrange the spectrometer such that both the PCS linewidth-scattered light (for dynamic information) and also the total intensity angular distribution (for static information) can be measured. We have seen that depolarized 7HV scattering can provide additional discrimination where more than one dynamical process contributes to the linewidth. 44-49 New compact spectrometers can be designed using fibre-optic links; 98 this is now routinely used in the application of PCS to biomedical blood flow measurements. 99,10 ° The technique of fluorescence correlation spectroscopy 101-105 enables weakly scattering systems to be investigated and with the use of attached chromophores allows specific features of molecular dynamics to be emphasized. Similarly the use of resonance-enhanced scattering techniques in Rayleigh scattering, 106 ' 107 for example by using tuned lasers, may be used to increase the scattering sensitivity. Combination of the PCS technique with a microscope attachment for detection enables very small samples to be studied and has been successfully applied to biological systems. 108 Development of the PCS method for use in surface scattering 109 and with fibre-optic evanescent wave methods 110 would lead to novel configurations. Several other PCS configurations have been investigated which have potential application to polymers; these include forced Rayleigh scat­ tering, 111 a differential scattering technique to observe small conformational changes in macromolecules, 112 the use of multidetectors 113 and anticorrelations, 114 and application of oscillatory electric fields to the scattering system. 1 1 5 - 1 1 8 Finally, future technical developments on correlation methods with microprocessor and transputer-based systems may provide faster and more compact instruments.

7.3 7.3.1

APPLICATIONS OF PHOTON CORRELATION SPECTROSCOPY TO POLYMERS* Scope

PCS was applied to the study of polymer characterization and polymer dynamics from the inception of the technique. Early studies were on the diffusion and sizing of colloidal particles and on the translational diffusion of polymer coils in solution under variation of conditions, molecular weight, solvent and concentration. This work was extended to the study of a variety of polymer systems which included gels, liquid crystals, bulk polymers, biopolymers, micelles, polymer adsorption and polyelectrolytes. More recently PCS has been applied to a systematic study of polymer dynamics in association with new theoretical studies using scaling and renormalization techniques and to a wide range of polymer systems, particularly semidilute and concentrated systems, gels and bulk polymers to provide information on collective dynamics. Generally the study of polymer dynamics by PCS has provided a complementary technique to investigation of static properties by small angle X-ray scattering and neutron diffraction. * See Chapter 8 for the application of PCS to the characterization of polymers in solution.

Photon Correlation Spectroscopy: Technique and Scope

925

201-

/

1

r*

151—

X

/ 0

U-l

2

I

4

L

6 8 10 *2/ v inWr«-2\

12

14

Figure_ 8 Linewidth variation with q2 for polystyrene in benzene Af = 488^nm, r = 2 5 ° C , heterodyne detection. 6 4 3 93 4 55 2 _ Mm : 1.1 x 10 , c = 4.16 x 10" gcm" . The solid line is calculated from (D0)2 = 2.18 x 1 0 ~ M z - ° c m s

7.3.2

Polymer and Colloid Characterization

The use of PCS for the determination of translational diffusion coefficients of polymers in solution is well established. Examples covering a wide range of systems are contained in refs. 119-138. For monodisperse polymer solutions which can be well cleaned, values of DT can be obtained to an accuracy of better than 1%. An example of the q2 plot of the linewidth is shown in Figure 8 for a sample of polystyrene in benzene. For a polydisperse polymer solution, each contributing narrow molecular weight range in the distribution has an associated diffusion coefficient Dt. The PCS autocorrelation function is then a sum of exponentials with relaxation time [q2Di)~1, each weighted by the scattering power. Fitting the correlation function to a single exponential gives only an illdefined average value. Where the correlation times are distinct and well separated multiexponential fitting can provide the separate values up to a number of about five, dependent on experimental conditions. When the exponential components are close together only recently developed data analysis as described in Section 7.2.5 enables meaningful data to be extracted. The cumulant analysis* in which the logarithm of the correlation function is fitted to a power series, gives as the first linear cumulant kx = q2Dz providing an apparent z-average of the diffusion coefficient, defined as Dz = ED?M i C i P i (q)/ 1 LM i C i P i (q). This quantity differs from the true z-average diffusion coefficient unless the scattering factors Pt{q) = 1, i.e. unless the molecules are small or 6 -* 0°. The Dz has been combined with the weight average sedimentation coefficient to give, by the Svedberg equation, the weight average molecular weight.139 The second cumulant k2 gives the variance in D and is a measure of the polymer polydispersity. The cumulant analysis has been applied to the characterization of many polymers (for example refs. 140-161), and is often used as a rapid assay of the polydispersity, although it is restricted to unimodal distributions of relatively narrow polydispersity. When the molecules are sufficiently large for form factors to be important, the angular dependence of k1 and k2 provides additional information on the moments of the size distribution. The cumulant analysis can also be used to obtain M n and M w for polymers for which D as a function of M is known. Study of DT has been made to investigate polymers in many situations such as temperature variation, 150 at large M w , 1 5 1 for chain stiffness and statistical lengths, 152 and for comparison of good and poor solvent conditions. 153 7.3.3

Polymer Dynamics

Photon correlation spectroscopy has been applied to the study of polymer dynamics since the early 1960s. It was soon found that accurate values of the self-diffusion coefficient could be obtained * See Chapter 8.

926

Solution Methods

under a wide variation of polymer type, size, concentration and solution conditions. Of course care was needed in the sample preparation and in the measurement technique. It is, in a way, unfortunately too easy in PCS to make a measurement and obtain a result, but it requires some diligence in sample preparation and measurement to make such a result a reliable one. PCS has been used extensively to extract rotational and intramolecular dynamics in the presence of translational diffusion. The use of depolarized scattering reviewed in Section 7.2.2 has been applied 1 6 2 " 1 6 5 to polymers, biopolymers and colloids. In recent years the understanding of polymer chain dynamics has advanced significantly in dilute, semi-dilute and concentrated ranges. PCS is a valuable technique for measurement in each of those regions. The high level of interest in the application of light scattering to polymer dynamics is heightened by the advances in understanding from new theoretical approaches of scaling and renormalization group, leading to new insights and novel predictions. 166 " 177 The application of PCS also complements the techniques of small angle X-ray scattering and neutron diffraction, which are sensitive to spatial dimensions of polymers up to 10 nm, whereas the q values relevant to visible light scattering probe over dimensions of 30 < q'1 < 3000 nm. Thus PCS provides a probe of polymer dynamics on a scale matching overall polymer dimensions. In PCS, information from the intensity correlation function provides the intermediate scattering function S(g, t)

-



Sfa.0 = -Z«p{if[r,(t)

= ps2(q,t) -

(49)

r/0)]}

(50)

where V is the scattering volume, rifj(t) is the position of the i,j monomers at time t and /} is a geometrical factor. Polymer models in exact forms have been developed for S(q9t\ in order to interpret scattering data; these include models for isolated chains under theta conditions and for chains with hydrodynamic interaction through the preaveraged Oseen tensor. Calculations of S(g, t) for a flexible polymer have been made by Pecora 6 " 8 using a Green function solution of the Fokker-Planck equation for the Rouse-Zimm bead-and-spring model. All the dynamical regions shown in Figure 9 are accessible by PCS. Here the dimension R is a characteristic polymer size, Rh or Rg. For qR<4l, PCS is sensitive to fluctuations greater than the size of the polymer, the correlation linewidth is T = DTq2, and the translation diffusion coefficient DT is determined. For qR ~ 1 both translational diffusion and internal polymer dynamics are sensed and as the condition is emphasized, qR > 1, internal dynamics dominate. With qR <^ 1 the recent work in renormalization group theory and the 'blob' model has predicted polymer dynamical dependences and has been thoroughly investigated by PCS polymer translation diffusion. Under theta conditions D(c->0) ~ M _ i for molecular weight M. For good solvent conditions D(c -► 0) oc M~a with 0.5 < a < 0.6; for smaller values of M smaller values of a

4

3

(2) Single chain translational and internal dynamics

2

I

I

2

3

4

5

6

c/c* Figure 9 Polymer dynamical regimes accessible by PCS for variation of polymer radius R or concentration c. The overlap concentration c* ^ 3N/mRl> where N = degree of polymerization

Photon Correlation Spectroscopy: Technique and Scope

927

than predicted may be explained by the hydrodynamic radius Rh not reaching the asymptotic limit. 178 Under the condition qR > 1 the PCS correlation function enables the fundamental mode of internal dynamics to be measured 1 7 8 - 1 8 9 and to be related to polymer models 1 9 0 - 1 9 9 such as the simple or extended bead-and-spring model and dynamic scaling exponents. This mode is controlled by the balance between the elastic and viscous forces on the chain. Within the asymptotic limit, predictions of the large scale intramolecular relaxation rate have given T ~ (kT/rj)q3 within the Zimm model with hydrodynamic interaction, and T ~(kT/rj)q4' for the Rouse model with no hydrodynamic interaction. The concentration dependence of the translational diffusion coefficient has been investigated for several systems. For dilute solutions the diffusion coefficient can be written DT

=

kT —

T

NK~\

1

-

-—

[1

+

2A2 Mc

+

3A3Mc2

+

...]

(51)

where/is the friction coefficient of the polymer in solution, A2 and A3 are the second and third virial coefficients of osmotic pressure and v is the partial specific volume. At the theta temperature DT depends on concentration only by the concentration dependence o f / a s / = ^ (1 + kf c + k'{ c2 + . . . ) and the concentration dependence of D for dilute solutions is first order in c D(c)

=

D 0 (l

+

kDc)

(52)

where D0 = kT/f0 is the translational diffusion coefficient at infinite dilution. The thermodynamic and hydrodynamic dependence of kD is given as kD

=

2A2M

-

k{

-

Nv — M

(53)

This relation has been used to determine the second virial coefficient A2 by PCS. 1 2 4 In the semidilute region c > c* the polymer coils overlap while the polymer volume fraction is still relatively low. This region has been of considerable interest recently in the application of the new theoretical methods mainly based on scaling techniques. 200 " 204 PCS provides a measure of the cooperative diffusion coefficient Dc which is associated with a hydrodynamic correlation length £h

where rj0 is the viscosity of the solvent. The concentration dependence of Dc and £h depends on the quality of the solvent. For good solvents £h is identified with the average distance between nearest chain contacts £p and scales with monomer concentration p as £h ~ p~*. For theta solvents the pair interaction is absent and the hydrodynamic correlation length is associated with ternary contacts, i.e. £h ~ p " 1 . A detailed study of the semidilute region by PCS has been made 2 0 5 - 2 x 5 and also of the cross-over between good and theta solvents. 206 " 210 PCS studies have shown that the concept of a sharp cross-over between good and theta conditions to be untenable. Following de Gennes, 200 polymers in the semidilute region have physical entanglements with lifetime TT and a mean distance between entanglements £. For frequencies co > 1/TT the entangle­ ments do not relax and the solution behaves as a network with permanent crosslinks, termed a pseudo-gel. The following sequence of scaling laws has been proposed: good solvent: £ h ~ P ~ * , marginal solvent: £ h ~ P - i and theta solvent: £h~ p~x. A body of data has been drawn up supporting this analysis. In PCS the autocorrelation function is found to be strongly nonexponential in the semidilute region with evidence of slow decay modes. 216 The concept of physical entanglements is unresolved and requires further investigation. Recent studies 2 1 7 , 2 1 8 have been made by PCS on high molecular weight polystyrene in an ethanol/ethyl acetate mixed solvent system in the semidilute regions. Using multiexponential analysis two components were resolved: a fast gel mode and a slow hydrodynamic mode. PCS has been shown to be a valuable probe of the semidilute region and further study aiming for a detailed understanding of the general area of condensed polymer systems is merited.

Solution Methods

928 7.3.4

Gels

The random collective dynamics of a network is describable by a diffusive process; crosslinked polymer gels have been shown to be amenable to study by P C S . 2 1 9 - 2 3 1 Since the wavelength of light is much larger than the average distance between crosslinks, in one of the first analyses 2 1 9 ' 2 2 5 ' 2 2 9 a continum model has been developed for the process. Fluctuations of the polymer network around its equilibrium position are driven by an osmotic force tending to equalize the concentrations and an elastic force maintaining the position of the network; the fluctuations are damped by the frictional force between the polymer network and solvent. Concentration fluctuations in the network may be described by a displacement vector i#(r, t) giving the displacement of a point r at time t from its average position. Small deformations of the network follow the equation d2u

du

where p is the average density and 5 is a stress tensor. Solution of the equation of motion leads to the normal modes in the gel following a diffusion equation with diffusion coefficient D D

=

elastic modulus —-m —-— friction coefficient

4 longitudinal mode: E = L = K+ -\i\ 3

=

E /

(56)

transverse mode: E = fi

where L, K and ju refer to the longitudinal, bulk and shear moduli. From PCS measurements the quantities L/f and fi/f can be obtained and in principle light scattering enables the viscoelastic parameters of the gel to be determined. Scaling arguments may be used to relate K, f and D to the network concentration or to the number of polymer units between crosslinks N, i.e. K ~ 0 1 / 4 ~ M 1 ' 5 , / ~ "1/2 ~ N2'5, D ~ 0 3 / 4 ~ AT 3 / 5 . Polyacrylamide-water gels, 219 crosslinked networks 2 2 0 ' 2 2 1 and pseudo-gels 222 have been investigated, including PS-benzene, PS-ethyl acetate and PDMS-toluene. It has been shown 2 2 1 ' 2 2 3 ' 2 2 9 that the dynamical behaviour of gels is analogous to that of semidilute solutions, although experimental data fit better to scaling laws for semidilute solutions than for gels, probably due to the effects of pendant chains in networks of variable density. In PCS experiments stray light scattered by inhomogeneities in gels or semidilute solutions gives a reference signal unshifted in frequency and inducing heterodyne optical mixing. Then the correlation function is C(T) = A = Q~D(I2X + B where A is an amplitude and B a background. As for semidilute solutions, recent developments in exponential sampling in PCS measurements and multiexponential analysis should provide a powerful method to identify relaxation processes and separate informative signals from spurious scattering. 7.3.5

Bulk Polymers

Light scattering in bulk polymers is due to fluctuations in the local dielectric constant. Theoretical approaches predict frequency-shifted Brillouin peaks and a dynamic Rayleigh peak, centred around the incident frequency and associated with relaxation of the moduli, and slowly relaxing adiabatic and isothermal density fluctuations. Theoretical treatments have been made by Mountain 232 and Rytov 233 and more recently by Wang and co-workers. 234 Near the glass transition temperature Tg relaxation of the longitudinal modulus is slow enough to be observed by PCS, the density fluctuations are isothermal and the scattering is weak. In polymers there may also be a strong central peak due to anisotropy fluctuations. Considerable progress has been made in the application of PCS to bulk polymers, particularly in the work of Patterson, 235 " 237 Wang 2 3 8 " 2 4 0 and co-workers. A pure sample of poly(ethyl methacrylate) (PEMA) has been prepared 241 by thermal polymerization from the monomer, providing a weakly scattering anisotropic polymer near Tg (65 °C). The scattering from this sample was attributed to thermal density fluctuations. The observed average relaxation time changed by a factor of ten times for a 5 °C change near Tg. This is characteristic of the primary glass transition. A distribution of relaxation times has been determined 242 for poly(methyl methacrylate) (PMMA) with two peaks in the distribution being observed near Tg; this is a general phenomenon in the poly(alkyl methacrylate)s. PCS has been applied 238 to melt samples of poly(vinyl acetate) which

Photon Correlation Spectroscopy: Technique and Scope

929

indicates that the dynamical properties of melts can be explored by this technique. Use of the PCS technique with exponential sampling enables a very wide range of relaxation times to be simulta­ neously recorded and heterodyne detection may be beneficial in this study.

7.3.6

Liquid Crystals

Nematic liquid crystals are strong scatterers of light resulting from thermal fluctuations associated with the liquid crystal director. The dynamic fluctuations may be ascribed to three elastic and six viscotic constants. Lorentzian broadening with linewidths Tx and T 2 results from two overdamped modes within the nematic liquid crystal due to splay-bend and twist-bend combinations. PCS has been used by several groups 2 4 3 " 2 4 6 to determine the ratio of elastic to viscotic constants in nematic liquid crystals. Self-beat and heterodyne optical mixing have been used to measure the viscoelastic ratios [k22lyi~\, w n e r e ^22 *s a Franck elastic constant and yx the twist viscosity. The scattered intensity relates to ratios of the elastic constants and PCS gives elastic/viscotic constant ratios. As for scattering from other bulk media, stray elastically scattered light is unavoidable and operation with partial heterodyning can be circumvented by purposely working in the heterodyne mode. Partial heterodyning is undesirable since it can lead to multiple exponential decay constants and make data analysis intractable. Application of electric or magnetic fields to the nematic liquid crystal enables the elastic/viscotic constants to be separated into individual elastic and viscotic constants. 247 " 250 PCS has been used 251 to measure k22/y1 for the polymer-monomer liquid crystalline mixture polysiloxanepentylcyanobiphenyl and with an applied electric field to determine k22 and y1.

7.3.7

Polymer Adsorption

Several studies have been made of polymer adsorption to determine particle size and adsorbed polymer layer thickness. 252 " 256 The Stokes-Einstein equation DT = kT/6nr]Rh is used to derive the hydrodynamic radius Rh from the translational diffusion coefficient DT. Accurate values of the adsorbed layer thickness can be obtained if attention is paid to experimental conditions with averaging over a full angular distribution. Accuracy in the measurement of layer thickness is improved by making the measurements as a function of particle concentration, particularly for aqueous dispersions at low ionic strength, to eliminate structural effects from interparticle interactions. Where the adsorbed layer thickness is to be derived from the difference in hydrodynamic radii of the coated and uncoated particles then accurate measurement of the radii is essential since the adsorbed layer is usually only a small fraction of the particle radii. PCS studies have been made on the system poly(ethylene oxide)/polystyrene latex/water for narrow fractions of PEO on surfactant freelatices. 257 The hydrodynamic thickness of the adsorbed layer 5h increases more rapidly with M w than does the free coil radius Rh. For constant polymer concentration it is predicted that <5h ~ M 0 , 8 ; this is in good agreement with experimental data, an example 258 of which is shown in Figure 10. To make this application more suitable for PCS a differential scattering technique is required. A spectrometer designed for differential measurements has been described.112

7.3.8

Phase Transitions

In recent years there has been great interest in the coil-globule transition of a flexible polymer from the random coil state at the theta temperature to a collapsed globule at reduced temperatures. 259 ' 260 The transition may be induced by reducing the solvency as well as the temperature over the region between the theta temperature and the coexistence curve at which phase separation takes place. With reduction in temperature from the theta temperature, the hydrodynamic radius Rh decreases from its theta value, Rh~N1,29 to the collapsed value, Rh~N1/3, where N is the degree of polymerization. Within the theories of classical mean field and renormalization group, in the temperature region in which the polymer is collapsing, Rh ~ N113 |T| " 1 / 3 with x being the reduced temperature T = (7"— 0)1 T.

Solution Methods

930

A

10° I04

i i i 11 11 I05

i

t

i i i i ii 1 I0 6

Figure 10 Adsorbed layer thickness S determined by PCS.258 Measurements of S for polyethylene oxide) fractions (M w /M n = 1.03 to 1.85) at different molecular, weights adsorbed at 7=25°C onto surfactant free polystyrene latex (Rh = 116 nm). Gradient = 0.85 ± 0.05

Various experimental studies of the coil-globule transition have been made by small angle neutron scattering, viscosity, elastic light scattering and PCS. There are stringent experimental requirements to be satisfied to observe the transition. Very high dilution is required in order to observe isolated chain collapse without interchain overlap aggregation. Large molecular weight polymers with narrow molecular weight distributions are also required. Measurements 261,262 have been made on polystyrene of M w = 26 x 106 in cyclohexane. Figure 11 shows the collapse transition for a highly diluted sample of polystyrene (Mw = 21 x 106) in cyclohexane as the temperature is reduced below the theta temperature. 262 It has been confir­ med 260 that the molecular weight dependence is Rh~MU3 | T | " 1 / 3 , with a smooth crossover between the theta and collapse regimes, though studies made on a very pure high molecular weight sample M w = 41 x 106 with observation of the transition. Some dramatic phase transition effects have been observed by PCS by Tanaka (reviewed in ref. 263) in polyacrylamide/acetone/water gel caused by changes in solvent composition and temperature. The gel undergoes collapse when the acetone concentration is increased or the temperature is lowered. Ionization of the network has been shown to play a role in the transition. A non-ionized gel undergoes a continuous change in equilibrium volume when the acetone concentration or temperature is changed. When a small number of the polyacrylamide chain units are hydrolyzed, to form ionizable acid groups, a sharp reversible volume collapse occurs, with up to a 500-fold volume reduction. This volume collapse can also be induced by change in the pH of the gel fluid. The large

130

I 120 (A

=

HO

2 0

IOO

E c

90

■D

1 80

>*

I

70 60 24

26

28

30

32

34

36

Temperature (°C)

Figure 11 Hydrodynamic radius for high molecular weight polystyrene Mw = 27 x 106 in cyclohexane,262 showing collapse of the polymer for reduction in temperature below the theta temperature (T~ 35 °C)

Photon Correlation Spectroscopy: Technique and Scope

931

volume collapse has been attributed to the change in osmotic pressure of hydrogen ions dissociated from the ionizable groups.

7.3.9

Electrophoretic Scattering and Application to Polyelectrolytes

Macroions in solution in an electric field acquire a terminal velocity v

=

\iE

(57)

where /i is termed the electrophoretic mobility. Light scattered from the moving ions undergoes a Doppler shift with frequency change vF

=

2fiE

A

[91 sin - cos a

l_2j

(58)

By heterodyne mixing with unscattered laser light the frequency shift vE is derived and may be analyzed with a spectrum analyzer or autocorrelator giving g(r)

=

A

+

Be-I)T*2Tcos(27rvET)

(59)

The light-scattering cell in electrophoretic light scattering requires careful design to provide a homogeneous electric field and to ensure that extraneous effects such as bubble formation at the electrodes, electro-osmosis and Joule heating induced convection are avoided. A few studies of polyelectrolytes by PCS have been initiated. In a study of Sodium-polystyrenesulfonate of M = 105 in water, PCS was used to obtain information on the conformation of the polyions. 264 The cooperative diffusion coefficient in the semidilute region (c > c*) was found to follow a power law, Deff ~ c°-52±0 ° 2 ; at high concentration a constant Deff was observed. These results indicated that the conformation of the polyion was rod-like at low concentration and stayed extended between entanglements in the semidilute region. PCS has been used to demonstrate a conformational change of a weak acid polyelectrolyte upon ionization, 265 and a few other studies on charged polymers can be cited. 266 " 269 Theories of light scattering for polyelectrolyte solutions predict a non-exponential autocorrelation function in which there is a coupling between ionic relaxation and hydrodynamic self-diffusion. These aspects are relevant to several topics of techno­ logical interest, such as polyelectrolyte steric stabilization, and justify further exploration.

7.4

REFERENCES

1. H. C. Van de Hulst, 'Light Scattering by Small Particles', Wiley, New York, 1962. 2. D. Mclntyre and F. Gornick (eds.), 'Light Scattering from Dilute Polymer Solutions', Gordon and Breach, New York, 1964. 3. I. L. Fabelinskii, 'Molecular Scattering of Light', Plenum Press, New York, 1967. 4. M. Kerker, 'The Scattering of Light and Other Electromagnetic Radiation', Academic Press, New York, 1969. 5. M. B. Huglin, 'Light Scattering from Polymer Solutions', Academic Press, New York, 1972. 6. R. Pecora, J. Chem. Phys., 1964, 40, 1604. 7. R. Pecora, J. Chem. Phys., 1965, 43, 1562. 8. R. Pecora, J. Chem. Phys., 1968, 49, 1032. 9. A. T Forrester, R. A. Gudmunsden and P. O. Johnson, Phys. Rev., 1955, 99, 1691. 10. H. Z. Cummins, N. Knable and Y. Yeh, Phys. Rev. Lett., 1964, 12, 150. 11. H. Z Cummins and L. H. Swinney, Prog. Opt., 1970, 8, 133. 12. B. Chu, 'Laser Light Scattering', Academic Press, New York, 1974. 13. H. Z. Cummins and E. R. Pike (eds.), 'Photon Correlation and Light Beating Spectroscopy', Plenum Press, New York, 1974. 14. S. H. Chen and S. Yip (eds.) 'Spectroscopy in Biology and Chemistry — Neutrons, X-Rays and Lasers', Academic Press, New York, 1974. 15. B. Crosignani, P. Di Porto and M. Bertolotti, 'Statistical Properties of Scattered Light', Academic Press, New York, 1975. 16. P. N. Pusey and J. M. Vaughan, Dielectr. Relat. Mol. Processes, 1975, 2, 48. 17. B. J. Berne and R. Pecora, 'Dynamic Light Scattering with Applications to Chemistry, Biology and Physics', Wiley, New York, 1976. 18. H. Z. Cummins and E. R. Pike (eds.), 'Photon Correlation Spectroscopy and Velocimetry', Plenum Press, New York, 1977.

932

Solution Methods

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