Surface Quasi-Elastic Light Scattering: A Probe of Interfacial Rheology

By Ian Hopkinson

POLYMERS AND COLLOIDS GROUP, CAVENDISH LABORATORY, MADINGLEY ROAD, CAMBRIDGE CB3 0HE, UK

1 Introduction

Interfacial rheological properties can be probed using a wide range of techniques, including direct mechanical surface viscometers, pendant droplet and oscillating bubble methods, and mechanically excited wave methods. A further member of this family of techniques is surface quasi-elastic light scattering (SQELS). Langevin has provided an excellent review of this technique.

Capillary waves with amplitudes of a few angstroms and wavelengths of the order of 100 µm are found at all fluid interfaces, and they scatter light very efficiently. Most of this light is scattered elastically, but a component of it is scattered inelastically through an exchange of momentum between photons and the interfacial waves. The inelastically scattered light is found in a cone surrounding the specular reflection, and this cone represents inelastically scattered light with a range of q values. The power spectrum of the inelastically scattered light contains information on the interfacial properties.

SQELS complements other interfacial analysis techniques in a number of ways, but perhaps the key feature is that it becomes more sensitive as the interfacial tension is reduced because the amplitude of the thermal fluctuations in the interface is increased. The capillary wave frequency is of order 10–100 kHz, and so the interfacial properties are probed at a much higher frequency than is usual. This corresponds to short time-scales, and this feature may be significant when applying the results to the understanding of fast industrial processes such as emulsification.

Data are normally analysed in terms of a model which treats the interfacial layer as a thin flat elastic sheet. More recently, Buzza et al. have proposed a model that explicitly incorporates features of a polymer brush into the representation of the interfacial layer. The dispersion relation D(ω) for waves at an air-liquid interface is given by:

D(ω) = εq2 + iων(q + m)] [γq2 + iων(q + m) - ρω2/q] - (iων(m - q))2. (1)

The parameter m is defined by

m = [square root of (q2 + i ωρ/ν, Re(m) > 0,] (2)

where ITLνITL is the subphase viscosity and ITLρITL is the subphase density. The quantity ITLγITL is the surface tension (or transverse modulus) and ε is the dilational modulus. Solving equation (1) for D(ω) = 0 gives an expression for the wave frequency ω as a function of the scattering vector q. While the solutions describe both dilational and transverse waves, in a light scattering experiment it is only the transverse waves that scatter light. Their power spectrum Pq(ω) is given by:

(3) Pq(ω) = kT/πω Im [iων(m + q) + εq2/D(ω)]].

The behaviour of the dilational waves can be inferred because there is coupling between the dilational and the transverse waves. A fluid-fluid interface can be modelled using a trivial modification of equation (1). In the experiments carried out here, a photon correlation spectrum is acquired and this is simply the Fourier transform of the power spectrum Pq (ω). The surface moduli can be expanded to take into account viscous effects:

γ = γ0 + iγγ', ε = ε0 + iωε'. (4)

In this work data are analysed by directly fitting the correlation function with a theoretical curve calculated from the interfacial properties. An alternative is to fit the correlation function with a damped cosine, and then either work directly with the frequency and damping thus obtained, or find values of the interfacial properties that are consistent with the values of the frequency and damping. Since there are four interfacial properties and only two determined parameters, it is necessary to make some assumptions.

The milk protein β-casein has been extensively studied. The key feature of particular interest here is its substantially random coil structure which makes it comparable to synthetic polymers. In addition, its well-defined sequence leads to a well-defined polyelectrolyte behaviour and a well-defined molecular weight; these features are difficult to obtain in a synthetic polyelectrolyte. Here, we examine β-casein spread at air-buffer interfaces, and in addition to doing SQELS experiments at pH 7.0 we also measure conventional surface pressure-area isotherms over a range of pH.

We make use here of the Aguie-Beghin model to analyse the isotherm data. This model treats the protein as a multiblock copolymer, and calculates a scaling exponent y (where Π ~ Γy using conventional methods of polymers scaling theory. The value of the scaling exponent varies with the solvent quality. If, in a particular solvent, a polymer chain obeys the scaling relationship between radius of gyration and monomer number predicted by the Flory–Huggins model, then the polymer solution is described as being 'ideal' and the solvent as being a 'theta' solvent. A solvent that leads to a more compact configuration is known as a 'poor' solvent, and one that produces a more expanded configuration is known as a 'good' solvent.

2 Experimental Methods

A surface quasi-elastic light scattering apparatus has been constructed based on the design proposed by Earnshaw and Hård. This is illustrated schematically in Figure 1. The goal of such an apparatus is to measure the power spectrum of light scattered inelastically from the capillary waves at the fluid interface as a function of scattering vector q. Photon correlation spectroscopy (PCS) is a convenient means by which to measure the small shifts in frequency that this entails. The photon correlation is done in heterodyne mode, and so it is necessary to provide a coherent source of light of the original frequency at the appropriate q value. This light is provided using a weak diffraction grating. In order for the heterodyne signal to dominate the correlation function, the ratio of the intensity of the inelastically scattered light to the 'reference' light must be adjusted to a value of the order of 10-3.

Figure 1 shows the experimental arrangement. Light of wavelength 532 nm is provided by a 150 mW single-mode diode-pumped solid-state laser (Laser Quantum, Manchester). Polarization and intensity are controlled using the combination of the half-wave plate (λ/2) and the prism polariser (P). The beam size, profile and collimation are controlled using the spatial filter (S). The grating (G) provides a fan of diffracted 'reference' beams. The lenses L1 (f = 150 mm) and L2 (f = 350 mm) perform two tasks; they converge the reference beams and the main beam to a single point at the fluid interface, and they focus the reference beams and the main beam at the front of the photomultiplier. The relative intensity of the reference beams is adjusted by moving the neutral density filter (NDF) such that it intercepts the diffracted spots, but not the main beam. The mirrors M1–M4 direct light from the laser onto the surface and from there into the detector. The light is detected using a photomultiplier (PMT) and processed using a PC-card-based photon correlator (Brookhaven Instruments, Worcester). The pulse discriminator used in the PMT is modified to allow the use of the 'multi-photon' mode originally described by Earnshaw. At the detector the laser light appears as a bright central spot with a series of focused reference spots at 2–3 mm intervals away from the central spot. Each of these spots is composed of the reference beam originating from the diffraction grating and inelastically scattered light from the main beam. The reference beams are sufficiently weak that inelastic scatter from them can be ignored. Each spot corresponds to light being scattered to a different q value, and the mirror M4 is adjusted in order that the appropriate reference beam falls on the detector.

The liquid surface is maintained in a Langmuir trough (Nima Technology, Coventry) mounted on an active anti-vibration table (Halycion, Germany), both of which are enclosed in a draught-proof enclosure. These steps are necessary because the liquid interface is highly sensitive to perturbation by drafts and vibration. Conventional isotherms were measured simultaneously with the light scattering data, using a Wilhelmy plate.

The SQELS data presented here were obtained at a range of surface concentrations for a layer of β-casein spread at the surface of a 0.01 M phosphate buffer of pH 7.0 at a temperature of 23 °C. The data were all collected at scattering vector, q = 34 654m-1 from the specular reflection. At each point a set of 10 correlation functions was acquired, each of which corresponded to accumulated data for one minute.

The β-casein (Sigma, C-6905, 90% pure) was used as supplied. A 1mg ml-1 solution in water was prepared and then the appropriate volume (typically 50 µL) of this solution was dispensed dropwise onto the buffer surface using a micropipette. In addition to data collected on the pH 7.0 buffer, conventional isotherms were acquired for pH = 5.8 and pH = 8.5 (phosphate buffer), pH = 4.8 (citrate buffer), and pH = 10.0 (tris-HCl buffer).

3 Results and Discussion

Figure 2 shows an example of a typical correlation function, along with a fit using the full spectral expression. The inset shows the residuals between the experimental data and the fitted curve (scaled by a factor of 10).

Figure 3 shows a comparison between the surface pressure as a function of surface concentration as measured by Wilhelmy plate and the surface pressure derived from the SQELS experiments. The SQELS data are consistently slightly lower (by about 1mN m-1) than those measured conventionally. A non-zero value for the transverse viscosity, γ', could lead to an increase in the value of the SQELS determined value of n, but not a decrease. The most likely explanation for this discrepancy is an error in the measured q value, an approximate solution for the capillary wave being

γ = ρω2/q3 (5)

Therefore the uncertainty in γ, and hence in the surface pressure, is around three times the uncertainty in the q value. To account for the discrepancy of 1mN m-1, an error in q of only 0.6% is required.

The dilational modulus ε can be calculated from the conventionally measured surface pressure (Π) versus (A) isotherm using the expression

ε = -dΠ/d ln A. (6)

This was done by interpolating the isotherm data to uniform increments in area and then carrying out a numerical differentiation using an 11-point second-order Savitsky-Golay filter. It is this set of data which is compared with the values of the dilational modulus obtained using SQELS in Figure 4. At surface concentrations below 1mg m-2 there is good agreement between the moduli measured using SQELS and those measured conventionally. However, at higher surface pressures, the value of the SQELS measured ε is somewhat higher than that measured conventionally. This type of behaviour has been observed previously, and the discrepancy can be attributed to the exchange of β-casein between surface and bulk. This process has the effect of reducing the dilational modulus at low frequencies (i.e. corresponding to the conventional measurement), but not at the high frequencies used in the SQELS experiment. The variation of ε with the frequency of measurement has been discussed by Lucassen and van den Tempel.