Studying the friction law for colloids at the modified water-air interface
The investigation of large interfacial area systems such as foams entails the study of the flow and deformation behavior exhibited by films of surface active molecules formed at the water-air interface (Fuller 2012). This study enables understanding the factors controlling the physical behavior exhibited by thin liquid films, which are the basic building blocks of foams and consist of a liquid layer enclosed by two layers of surface active molecules (Langevin 2014).
The flow and deformation behavior of interfacial films can be addressed by experimentally measuring surface rheological quantities such as the surface viscosity (Fuller 2012). Diverse experimental techniques allow measuring the surface viscosity of surface films but they are limited by a lack of interfacial sensitivity (Sickert 2003). Overcoming that limitation in interfacial rheology involves considering in detail the concept of the dimensionless Boussineq number, which measures the relative contribution of the interfacial and bulk stresses on the hydrodynamic drag the probe experiences (Samaniuk 2014). An experimental rheological measurement will be interface sensitive if the Boussineq number is large. This fact sets a limit for the lower value of surface viscosity that can be measured experimentally, which bring many kind of interfacial monolayers out from the scope of interfacial rheology (Sickert 2003). However, owing that the Boussineq number depends on the aspect ratio of the employed interfacial probe (Samaniuk 2014), it enables increasing the sensitivity of experimental surface rheology techniques by employing smaller interfacial probes. Reducing the size of the probe enables measuring lower values of interfacial viscosity while keeping large the Boussineq number. This fact drives researchers into considering the employment of interfacial probes as small as colloids (Prasad 2006).
The employment of colloids as rheology probes has been successful for the case of bulk liquids, where the viscosity of the liquid can be determined from the friction factor in the Einstein equation, which is given by the Stokes friction law for a sphere in translational motion (Dhont 1996). This technique is termed as microrheology (Cicuta 2007), owing to the fact that a sample volume in the range of micro-liters can be evaluated. Speaking in proper rheological terms, microrheology is a creep measurement, where the mean square displacement of the diffusing colloids is related to the compliance, and this last in turn is related to the complex shear modulus (Evans 2009).
In the case of colloids attached to a fluid-fluid interface the Stokes friction law does not describes properly the hydrodynamic friction the colloids experience at the interface, which is given by the friction coefficient in the Einstein equation. Diverse interfacial phenomena make the hydrodynamic friction colloids experience at the interface to differ from the one experienced at bulk liquids. An early derivation of the friction law for diffusing objects at the fluid interface was done for Saffman and Delbruck (Saffman 1975, Saffman 1976) for the case of a cylinder translating inside a lipid monolayer at high Boussinesq number. This derivation was employed to determine the diameter of labeled structures diffusing in a lipid membrane of known surface viscosity. The diffusing structures are modeled like a cylinder with length equals to the membrane thickness. The same approach is followed by Prasad et al. (Prasad 2006) to measure the surface viscosity of a protein film at the surface of water. In this case, the diffusing objects are spherical polymer colloids of known diameter. Since the Boussinesq number for the evaluated system is considered large, the viscous drag coming from the protruding of the spherical colloids into the water sub-phase can be neglected.
A similar derivation of the Saffman-Delbruck equation was obtained by Hughes et al. (Hughes 1981) for the case of smaller Boussinesq number. While Fischer et al. (Fischer 2006) extended the work of Saffman and Delbruck by determining the friction law for spherical-shaped particles diffusing at an interface modified by a surface film for near zero and zero Boussinesq number.
Other authors have intended to improve the Saffman-Delbruck equation by determining the friction law in the case of spherical particles, and its dependence with the contact angle of the particle at the interface. This is the case of Fischer et al. (Fischer 2006) who determined it for the case of large Boussinesq number, and considering the hydrodynamic flow the particles experience at the interface as an incompressible one, and of Danov et al. (Danov 1995, Danov 2000) who instead considered an compressible hydrodynamic flow. However, the predicted dependence of the hydrodynamic friction with the contact angle of the diffusion particle has been questioned by Boniello et al. (Boniello 2015) based on experimental results contradicting both models. However, the results were obtained for particles diffusing at the free surface of water, which is not the case of the calculation by Fischer et al. (Fischer 2006), which only addresses the modified interface. Likewise, it is currently not clear if the hydrodynamic flow colloids experience at the water-air interface is purely incompressible in all cases due to the Marangoni effect or instead it may exhibit some compressibility in some cases as suggested by Samaniuk et al (Samaniuk 2014), or in all cases as suggested by Zell et al. (Zell 2016).
In the case of colloidal particles diffusing at the free surface of water the hydrodynamic flow can be considered as a compressible one, and a calculation of the friction law by Danov et al. (Danov 1995, Danov 2000) is available. However, the model does not account for the hydrodynamic drag linked to fluctuations of the three phases contact line as suggested by Boniello et al. (Boniello 2015).
While experimental reports mainly address the dependence between the friction coefficient of the colloids at the interface and the surface concentration of the film (Hilles 2009, Lee 2010, Maestro 2011, Samaniuk 2014, Sickert 2007, Sickert 2003, Bonales 2007), which is assumed as being related to the surface shear viscosity, the experimental evaluation of the dependence of the diffusion coefficient of the colloids with other factors, for instance, its contact angle, at a free or modified interface has demonstrated being a more direct way to understand the friction law of colloids at interfaces (Boniello 2015).
In the present work we follow this line of research, and the experimental friction coefficient of polymer colloids at the water-air interface is evaluated as a function of the viscosity of the bulk phase, which is tuned by variation of the temperature of the fluid. Diverse experimental situations are considered, including the presence of an interfacial film, and at different surface concentrations. This approach is original in its formulation, and enables a direct comparison between experimental results and numerical calculations of the friction law for colloids at interfaces.
We employ polystyrene colloids with a diameter of \(250nm\). The colloids are stabilized by sulphate groups. A suspension of these colloids was purchased from Molecular Probes, USA. The spreading solution is prepared by diluting a small volume of the stock solution in ultra-pure water, and mixing with \(50\%\) of \(1-propanol\). A small volume of the spreading solution is added at the surface of the liquid. The spreading solution of interfacial material, namely, polymers, phospholipids or surfactants is prepared by dissolving the material in a proper solvent. A small volume of the interfacial material solution is added to the interface, and at least one hour is left for solvent evaporation before starting measurements. In the case of water soluble materials, such as SDS or PF127, the material is dissolved in the sub-phase directly. Ultra-pure water from Sigma-Aldrich is employed for preparing the sub-phase. The glass cell containing the sample is previously cleaned with a diluted solution of Hellmanex and rinsed with ultra-pure water. Furthermore, an important step in the sample preparation is checking that a flat auto-correlation function is measured for the free surface of water, which indicates there are not impurities at the interface being able to generate dynamic light scattering. The temperature in the cell is regulated employing a MINISTAT temperature control system, and one hour is left before starting measurements to allow the sample reaching the thermal equilibrium.
The illumination source for the light scattering experiment is a laser spot created at the surface of the water by setting the laser beam in a total reflection configuration. The instrument consists of a rotating goniometer containing the laser and detector arms, and in the center a cell holder (Stocco 2011). The goniometer can be tilted at any angle, which enables detecting light in the three dimensional space. The scattered light is collected at a single-mode optical fiber, which is connected to a photon counting modulus and an avalanche photo-diode. The auto-correlation function is build up employing an ALV correlator, and visualized with the ALV-5000 software. The whole set-up is placed on a anti-vibration table which isolate the liquid interface from external mechanical perturbations.
The auto-correlation function is exponentially fitted using the ALV-5000 software. The multi-angle plot is build up from the fitting of the auto-correlation function at several scattering vectors. The data can be also analysed employing the polydisperse multi-angle plot as will be show later.