Competitive adsorption of proteins and low-molecular-weight surfactants: computer simulation and microscopic imaging

https://doi.org/10.1016/j.cis.2003.08.003Get rights and content

Abstract

Proteins and low-molecular-weight (LMW) surfactants are used in the food industry as emulsifying (and foaming) ingredients and as stabilizers. These attributes are related to their ability to adsorb at fluid–fluid (and gas–fluid) interfaces lowering the interfacial (and surface) tension of liquids. Hence, the study of the properties of adsorbed layers of these molecules can be expected to lead to a better understanding of their effect on food products. Direct proof of the validity of mesoscopic models of systems of proteins and LMW surfactants can only be achieved by quantitative theoretical predictions being tested against both macroscopic and mesoscopic experiments. Computer simulation constitutes one of the few available tools to predict mathematically the behaviour of models of realistic complexity. Furthermore, experimental techniques such as atomic force microscopy (AFM) now allow high resolution imaging of these systems, providing the mesoscopic scale measurements to compare with the simulations. In this review, we bring together a number of related findings that have been generated at this mesoscopic level over the past few years. A useful simple model consisting of spherical particles interacting via bonded and unbonded forces is described, and the derived computer simulation results are compared against those from the imaging experiments. Special attention is paid to the adsorption of binary mixtures of proteins, mixtures of LMW surfactants, and also protein+surfactant mixed systems. We believe that further development of these mathematically well-defined physical models is necessary in order to achieve a proper understanding of the key physico–chemical processes involved.

Introduction

Proteins and low-molecular-weight (LMW) surfactants are key components of many foodstuffs [1]. Some dairy products, for example ice cream, contain both proteins and LMW surfactants in their formulation. Both types of molecules can adsorb at fluid interfaces, reducing the interfacial tension and so facilitating the formation of emulsions and foams and providing stability to droplets and bubbles. However, their molecular properties are very different [2].

LMW surfactants are small molecules each consisting of a hydrophilic head group and one or several hydrophobic tails. When such molecules reach an air–water or oil–water interface, they tend to adsorb by arranging the hydrophobic tails within the non-aqueous phase and the hydrophilic head in the water phase (see Fig. 1a). LMW surfactants are very mobile and they are particularly efficient at reducing the interfacial tension. As a result, they rapidly coat the newly created oil–water and air–water interface during emulsification and foaming.

Proteins are high-molecular-weight molecules each consisting of a chain of amino acids. As there are polar, non-polar and ionic amino acids, proteins contain a mixture of hydrophilic and hydrophobic groups. In aqueous solution, a protein molecule will tend to fold in a coil-like structure in order to expose the most hydrophilic groups to the water and hide the most hydrophobic segments in the centre of the coil (see Fig. 1a). However, when a protein molecule reaches an air–water or oil–water interface, the molecule will partially unfold orientating its hydrophobic groups towards the non-aqueous phase (Fig. 1a). Proteins are very slow at diffusing and adsorbing as compared with LMW surfactants; and they do not normally lower the interfacial tension so efficiently. However, proteins form thick protective layers at the surface of oil droplets and gas bubbles which, under appropriate conditions, can prevent coalescence after an emulsion or foam has been formed thereby conferring long-term stability to the system.

When a mixture of LMW surfactants or a mixture of proteins or a surfactant+protein mixed system is exposed to an interface, the different species compete to adsorb and lower the interfacial tension [3]. During the equilibration of the interface—this may take from seconds in the case of pure LMW surfactants to up to several hours for protein-containing systems—the molecules adsorb and desorb dynamically. If one of the components of the mixture is presented to the interface first, then the second component will tend to replace the adsorbed molecules of the first type partially or totally depending on the relative surface-activity of the species and their mutual cross interaction. Interestingly, in some cases, the mixture does not adsorb homogeneously over the surface. On the contrary, interfacial regions rich in one or the other species are present either during the equilibration process or in metastable states.

It is a common practice in the study of such systems (and in general of most physico–chemical systems) to generate a mesoscopic model of the molecules involved in order to explain the behaviour of the system in a qualitative fashion. These kinds of models, which are often not very precisely defined, take into account a few essential characteristics of the molecules under study—such as molecular weight, flexibility, hydrophilic–lipophilic balance, presence of reactive groups, and charge distribution—instead of the detailed chemical structure. This provides a useful way of elucidating the important factors affecting a particular phenomenon. However, the majority of these models are expressed in a qualitative language during the interpretation of experimental results, but if these types of models are amenable to be expressed in a physico–mathematical language describing quantitatively the molecules and their interactions, the calculation of the behaviour of an ensemble of molecules can be made by solving the basic equations of motion through the use of a suitable computer simulation algorithm [4], [5].

Atomic force microscopy (AFM) is a powerful experimental tool to probe very small length scales on a surface [6], hence the usefulness of this technique to inspect the structure of adsorbed protein films at interfaces [7]. It is particularly interesting that with AFM we are able to reach resolutions almost down to molecular length scales. Therefore, any mesoscopic model can be assessed at the ultimate level of description, i.e. down to the small-scale structures formed by the molecules themselves. Additionally, fluorescence microscopy provides a method to distinguish different species in a protein mixture [8]. This can be particularly useful, for example, if one is interested in studying the homogeneity of the layer formed at the interface.

In this paper, we review a number of experiments and computer simulations of model systems that have been carried out in recent years. We pay special attention to the description of the structure of the interface. We describe a general but simple mesoscopic model for adsorbing proteins and surfactants, and several examples of computer simulation results on the competitive adsorption of these types of molecules are shown. Atomic force microscopy and fluorescence microscopy experiments are presented and compared with the simulation results. Additionally, we present some rheological properties of adsorbed protein layers that can be extracted from the simulations.

In this section we give a concise overview of the most common computer simulation techniques relevant to this topic. We recommend the reader to refer to the book by Allen and Tildesley [5] for a thorough introduction to the subject.

Every physico–chemical system comprises a number of basic interacting entities. Depending on the intended level of description, these entities can be atoms, molecules or molecular complexes. We give the name particle to any of these basic entities. A theoretical model for these particles consists in a mathematical definition of the way they interact with each other and with external forces. When such a model is complex, as is usually the case when dealing with complex fluids, the physical equations that describe the behaviour of a system composed of these model particles are unattainable analytically. In a few cases, further approximations can be used to simplify the equations. Mainly, however, a numerical solution of the set of fundamental physical equations is normally the practical alternative. This numerical approach is referred to as ‘computer simulation’. The two basic techniques for performing computer simulations consist in either solving the equations of motion of the involved entities (particles) that form the system—this is termed molecular dynamics (MD) simulation—or in generating ensembles of representative configurations of the system at random—what is called Monte Carlo (MC) simulation.

It is important to mention here that there exist other numerical techniques to study the properties of different molecular models. Self-consistent field theory for example, is a very useful technique to study the equilibrium properties of molecular systems (see for example Ref. [9], [10], [11], [12]). However, such types of studies are not computer simulations in the sense that the fundamental physical equations are solved indirectly. In self-consistent field theory, the unknown function to be calculated numerically is the density distribution of molecular segments at thermodynamic equilibrium rather than the positions and/or velocities of each segment.

In a classical MD simulation, Newton's equations are solved for a set of particles. The computation involves the solution of a system of N coupled differential equations (one for each particle) of the formmid2ritdt2=Fiextrit+j≠iNFi,jrit,rjt,where mi is the mass of particle i, ri(t) is its position at time t, and Fiext and Fi,j are the forces applied on i by any external field and by particle j, respectively. Of course, solution of such a system of differential equations requires initial conditions for the positions of the particles ri(t=0) and their velocities vi(t=0). Then, any appropriate finite difference technique can be used to solve the equations numerically and so obtain the position ri(t) and velocities vi(t) of the particles at discrete intervals of time Δt. Once the trajectories of the particles have been calculated for a long enough period of time—millions of units of Δt, typically—the macroscopic properties (specific heat, viscosity, etc.) and mesoscopic properties (structure factor, clustering, etc.) can be extracted through the appropriate time averages.

Classical MC simulation relies on the ensemble statistical theory. According to this theory, an average over all possible configurations of the system is equivalent to an average over the configurations that the system visits during its time evolution, as long as it is in thermodynamic equilibrium. Therefore, a representative set of MC configurations can be generated at random and then used to obtain mesoscopic and macroscopic averages. The simulations carried out under the MC scheme are normally simpler and more efficient than the ones based on MD. However, since the real trajectories of the particles are not calculated, no direct dynamic information can be extracted from MC simulations. Thus, only equilibrium properties of the systems of interest can be studied via MC simulation. Unfortunately, many of the most interesting properties of real surface-active molecules are related to their slow dynamics, which evolves over periods of hours, days or even weeks, without reaching proper thermodynamic equilibrium.

There is also a third, more coarse-grained simulation approach, namely Brownian dynamics (BD). In studying systems of surface-active molecules (and complex fluids in general), a mixture of solvent molecules (typically water) and surface-active molecules (proteins or LMW surfactants) has to be simulated. Normally, such solutions contain a few surface-active molecule for every tens of million of water molecules. Therefore, it is clear that the simulation of a few hundred surface-active molecules by the MD or MC approach would involve the calculation of trajectories or configurations of a prohibitively large number of solvent molecules. However, if we are not actually interested in the behaviour of the solvent, it is more sensible to avoid such a waste of computing resources by focusing just on the surface-active species alone, and including the solvent contribution as an effective external perturbation. This is the basis of Brownian dynamics.

In a BD simulation, as in MD, the equations of motion of the particles are solved numerically. However, no solvent particles are included explicitly. Instead, every other particle is subjected to two external forces that mimic the effect of the solvent on them. These two forces are the drag force and the random buffeting that generates the Brownian motion of the solute. In other words, BD is concerned with the numerical solution of the Langevin equation. For an isolated spherical particle the drag force is taken as −3π η σ vi, where η is the solvent viscosity, σ is the particle diameter, and vi is the particle velocity. The random buffeting is simulated through a random Gaussian distribution with zero mean and variance 2kBT/(3π η σ), which is consistent with the fluctuation–dissipation theorem and leads to Einstein's relation for the diffusion coefficient [D0=kBT/(3π ησ)]. Consequently, we have to solve a system of N coupled differential equations—one for each surface-active particle—of the formmid2ritdt2=Fiextrit+j≠iNFi,jrit,rjt−3πησdritdt+FiRt.Here, FiRt represents the random force. Again, as in MD simulation, finite difference algorithms are used to solve these equations numerically.

An important point has to be made with respect to all types of computer simulation. Because the systems studied are invariably much smaller than real systems, an unrealistically large proportion of particles are close to the edges of the system boundaries. Therefore, one expects that the presence of any container wall should affect significantly the behaviour of the entire simulated system. A way to lessen this effect is to use periodic boundaries conditions. That is, we allow particles on one side of the system to interact with particles on the opposite side, as if copies of the virtual container were placed all around the system to create borders of the same nature as the system itself. Within the same scheme, a particle that crosses the boundaries of the system (say by diffusion) is introduced through the opposite edge to create a smooth motion across the border. Although this computational trick is very useful and efficient, it cannot eliminate other small-size effects: for example, in the calculation of long-ranged particle–particle forces or in the analysis of density fluctuations larger than the basic simulation system size. In practice, the only way of ensuring that size effects do not influence a particular result is to perform independent simulations with different system sizes and then extrapolate the results to infinitely big systems.

It is appropriate also here to mention briefly the non-equilibrium simulation techniques used to analyse the response of a system to an external perturbation [5], [13]. It is the case that the linear response of the system to small perturbations can be obtained from an equilibrium MD or BD simulation through the fluctuation–dissipation theorem [5]. However, non-equilibrium simulation techniques require less computer effort to obtain a similar degree of accuracy, and they also allow us to study the response of the system to large perturbations, where the response is likely to be highly non-linear.

The response of the system to an external perturbation can be measured through the interparticle stress tensor σ. For a pairwise-additive interaction it is given by the Kirkwood formula [14]σαβ=ρkBαβ1Vj>iNi=1N−1rαijFβij,where V is the volume of the system. Here, α and β indicate the different Cartesian components of the stress tensor σ, the interparticle distance rij, and the interparticle force Fij, respectively. Normally, the macroscopic rheological quantities of interest are related to the symmetric part of σ, i.e.,σαβ=12σαββα.

Shear flow can be imposed on a simulated ensemble of particles in much the same fashion as for a real system in the laboratory. The most widely used method involves the creation of a shear flow profile by the addition of a position and time-dependent external force quantified by the affine strain γxy(t) applied on the xy-plane (or any other appropriate plane). The periodic boundary conditions have then to be modified to account for a linear flow profile across the infinite replicas [15]. For a sine wave input at frequency f=ω/2π, we haveγxyt0sinωt,and the time domain response isσxytγ0=G′ωsinωt+G″ωcosωt,where G′(ω) and G″(ω) are the storage and loss moduli, respectively. These can be extracted by integrating over a sufficiently large integral number of shear cycles n asG′ω=ωnπγ002nπωσxytsinωtdt,G″ω=ωnπγ002nπωσxytcosωtdt.Several initial cycles of the oscillation have to be discarded from the analysis so as to ensure that only the steady state response to the perturbation is accounted for.

The dilatational rheology of a system can be extracted in a completely analogous fashion to the shear rheology [16], [17]. In this case, the system is subjected to a small oscillatory change in volume. This is achieved by creating an extensional flow profile on application of a position- and time-dependent external force quantified by the dilatational strain-rate. Again, the periodic boundary conditions have to be adapted in order to account for a volume-changing system. After an integral number of dilatational cycles, the dilatational storage modulus ε′(ω) and loss modulus ε″(ω) can be extracted through equations analogous to , . Additionally, we can investigate large deformation-dilatational rheology by compressing (expanding) the system at a constant strain-rate down to (up to) few times the original size and analysing the response of the system though the stress tensor [18].

It is worth mentioning here that in what follows we concentrate on interfacial systems. Therefore, the rheological properties discussed above should be adapted to interfacial shear and interfacial dilatational quantities by using the appropriate interfacial stress tensor. A detailed discussion and application of interfacial shear and dilatational rheology in computer simulation can be found in Ref. [17].

In order to study the competitive adsorption of different surface-active components, a simple simulation approach has been developed by Wijmans and Dickinson [18], [19]. The model is a straightforward variation of a previous one designed to describe interacting protein systems in the bulk (i.e. far from interfaces) [20]. In its simplest version, the model consists of two types of spherical particles (of diameters σ1 and σ2) interacting via a steeply repulsive spherical core potential:φcrijσrij36,where σ=1/2σij, σi and σj are the diameters of the interacting particles, rij=|rirj| is the centre-to-centre distance of the particle pair, and ε is an energy parameter that is set equal to kBT. As usual, kB is the Boltzmann constant and T is the absolute temperature. All quantities are then expressed in units of ε, σ1 and σ1m1 for energy, length and time, respectively, with σ1 and m1 being the diameter and mass of the particles of type 1.

To mimic adsorption, each particle of type α (α=1 or 2) interacts with an external potential, acting in the z-direction:φsαzi=εσ1σ1+wziσα236σ1σ1+w+ziσα236ziσα2≤zcαεσ1σ1+wzcασα236σ1σ1+w+zcασα236ziσα2>zcαThis potential has a square-well-like shape with one infinite wall and one finite wall. The infinite wall prevents particles from escaping to the phase in which they are not soluble—typically air or oil in the case of proteins. Conversely, the finite wall allows for interchange of particles between the interface and the phase where they dissolve—typically an aqueous solution in an experimental setup. It is important to note that some surfactants can also dissolve in a non-aqueous medium. The adaptation of Eq. (10) to account for oil-soluble surfactants is straightforward but will not be considered here. The parameter w in Eq. (10) defines a minimum width for the potential well, and it is usually set to 0.05σ1. Then, zcα should be greater than w. The parameter zcα defines simultaneously the total width of the potential well and the particle adsorption energy for each type, namely Eadsα≡φsαzcα−φsα0[18]. A particle i of type α is said to be adsorbed if zizcα. Notice that the potential representing the presence of the interface is positioned in such a way that particles adsorb by ‘touching’ the interface, which is at z=0. Therefore, particles of different sizes adsorbed at the interface will not have their centres on the same z-plane, as exemplified in Fig. 1b.

In order to account for the intermolecular cross-linking behaviour of some proteins, the adsorbing particles can also interact through flexible bonds (see Fig. 1b). Nodes are created on the nominal surface of the spherical particles (σi/2 from the centre). The bond interaction acts along the straight line that joins the corresponding nodes, and it depends on the node-to-node distance bij only:φBbij=0bij≤b1εBbij−b1b02b1<bij≤bmaxεBbmax−b1b02bij>bmax.

We note that, as these forces do not need to operate in purely radial directions, they can give rise to torques acting on each particle. Consequently, a rotational contribution to the equation of motion has to be applied in addition to Eq. (1) or Eq. (2).

The interparticle bonds can be created or destroyed through different protocols. Normally, a bond would be created with the given probability PBαβ when two particles—one of type α and one of type β—approach within a distance b1. Initially, nodes are created on the line that joins the particle centres. After bond formation, the nodes that define the ends of the bond are fixed within the corresponding particle reference system. That is, the nodes remain fixed at the initial position on the surface of each particle. If the particle moves such that the length of a bond exceeds bmax, the bond is deemed to be broken. By setting bmax=∞, irreversible (unbreakable) bonds can be simulated.

In some cases, to allow for various kinds of repulsive and attractive interactions, an additional non-binding force between the particles can be introduced. The simplest case is dictated by the interaction pair potentialφUBαβrij=εUBαβrcαβ−rijrcαβ−σrij≤rcαβ0rij>rcαβ,where rcαβ defines the range of the interaction, εUBαβ accounts for its strength, and again σ=1/2αβ); σα and σβ are the diameters of the interacting particles. The indices α and β indicate the type of particle i and j, respectively.

Adding together all the above contributions, the force acting on particle i of type α at time t is given byFit=−riφsαzit+j≠iφcrijtBbijtUBαβrijt,where β identifies the corresponding type for particle j. The two terms on the right hand side of Eq. (13) correspond, respectively, to the two first terms on the right hand side of either Eq. (1) or Eq. (2). The torque around the centre of particle i isτit=jbj×bjφBbijt.This last sum runs over all the bonds of particle i, where bj is the position of node j with respect to the centre of the particle.

The system is not homogeneous in the z-direction since the interfacial external potential depends on the z-coordinate of the particles. Therefore, the periodic boundary conditions can be applied in the x- and y-directions only. Particles cannot escape in the negative z-direction (upwards in Fig. 1b), as they are trapped by the infinite wall of the interface, but they can move away (and become ‘lost’) in the positive z-direction (downwards in Fig. 1b). In order to avoid this effect, which would otherwise not be present if periodic boundary conditions were applicable in the z-direction, an additional ‘wall’ needs to be added at the given z-position. One way of achieving that is by introducing an external potential of the form:φwαziσαzw−zi36,where zw is the position of the restricting ‘wall’. In this way particles initially placed between z=0 and z=zw can only move within this range in the z-direction.

The properties of the present model can, in principle, be studied by means of MD, MC or BD. However, the first two techniques would require, in principle, the introduction of an extra species in the system to account for the solvent. Therefore, BD simulations are normally more efficient, as they introduce the solvent effect by just assuming a bulk solvent viscosity, η. In this case, the unit of time is generally changed to a more convenient scale: this is the average time taken for a particle to diffuse a distance equal to its radius in an infinitely diluted system, i.e.τr=σ/226D0=πσ3η8kBT.

The model defined so far consists of a number of parameters that characterize the system. These parameters are: the sizes of the particles σ1 and σ2, the corresponding adsorption energies Eads1 and Eads2 (determined by zc1 and zc2, respectively), the maximum length of the bonds bmax, the strength of the bonds defined by εBb02, the equilibrium length of the bonds b1, the reaction probabilities PB11, PB22andPB12, the strengths of the long-range forces defined by εUB11, εUB22andεUB12 and, finally, the ranges of the non-bonded forces rc11, rc22andrc12. By selecting different sets of parameters, we are able to model various sorts of systems like binary mixtures of proteins, binary mixtures of LMW surfactants or protein+LMW surfactant mixtures. In later sections, we show how this can be done and how results from the simulated systems compare to some recent experimental results from AFM and fluorescence microscopy.

It is worth emphasising that the model presented in this section does not consider the internal structure and flexibility of the molecules. This is particularly inadequate for very flexible protein molecules, as the model cannot account for the unfolding of the molecule upon adsorption. However, some globular proteins such as β-lactoglobulin may be reasonably well represented by the model at mesoscopic resolution, since unfolding takes place over long time scales, especially with strongly interacting proteins. A molecular dynamics simulation of the entire structure of a single β-casein molecule adsorbing to an oil–water interface over a time period of 1 ns has been carried out recently [21]. Naturally, the computational cost of simulating an entire protein film using this scheme for experimentally significant time scales is prohibitive for today's computer resources.

As has already been stated, interfacial films formed by the adsorption of surface-active species to a gas–liquid (or liquid–liquid) interface can be simulated by BD. In order to make a comparison between the patterns formed by these simulations and experimental systems we require methods of probing similar length scales. This is complicated by the fact that the film sits at a liquid boundary, making it inherently unstable. The problem of imaging such interfacial films can be approached in two different ways. Either the sample can be viewed non-invasively in-situ using optical techniques or the interfacial film can be transferred, unchanged, onto a solid support where it becomes amenable to imaging at high resolution by probe microscopy. Both approaches have advantages and disadvantages, which make them useful under different circumstances.

The foremost probe microscopy is atomic force microscopy (AFM), which uses a sharpened silicon nitride cantilever to scan over the surface of the sample. In its normal mode of operation the deflection of the cantilever is maintained at a predetermined value as the scan progresses in order to keep the forces acting between tip and sample constant. By monitoring the movements of the scanner that are required to maintain constant deflection of the cantilever, the sample topography is revealed as a three-dimensional image. AFM has the advantage over electron microscopy of being able to operate in liquids at ambient temperatures and pressures, and so it is an ideal instrument for biological studies. Its resolution is limited not by diffraction but by the sharpness of the tip. In practice, AFM can routinely achieve molecular resolution and in ideal circumstances sub-molecular resolution [22]. However, in order to obtain these high levels of resolution, the sample needs to be mounted on an atomically flat solid–substrate such as mica. This is achieved by a technique known as Langmuir–Blodgett (LB) transfer where the solid support is dipped, at low speed through the interface into the sub-phase, and then removed. The transfer of the interfacial film is controlled by the surface properties of the substrate. With a hydrophilic substrate the contact angle between the liquid and the substrate dictates that the film will only transfer on the upward stroke, removing a single section of interfacial film. Although the film may be altered during the transfer, comparisons with Brewster angle microscopy [23] and specially designed in-situ AFM experiments [24] have shown the same behaviour and morphology as those using the LB transfer technique. Despite the high spatial resolution, because the technique is essentially topographical, it cannot be used to distinguish between molecules with only small differences. Most protein molecules for example look very similar to one another in AFM images. In Section 5, we give some examples of AFM images of LB films drawn from fluid–fluid interfaces where mixtures of proteins and LMW surfactants were adsorbed competitively.

Optical techniques that can interrogate an interfacial film in-situ such as Brewster angle microscopy (BAM) suffer from limited resolution (typically 1 μm). Despite the limited spatial resolution of BAM it can be extremely sensitive to surface composition. The two requirements for obtaining good BAM images are phase regions of sufficient size, typically tens of microns, and significant variation of the optical properties between the two phases. For example, one common use of BAM is to look at the complex interfacial phase behaviour of lipids [25]. In such systems small changes in the orientation (tilt) of the molecule between liquid-expanded and liquid-condensed phases are readily apparent if the different domains are large enough. The limited spatial resolution means that this technique is not suited to looking at the formation of structures in protein films or mixed protein+LMW surfactant films unless (or until) the phase separated regions become sufficiently large. BAM is certainly not suitable for looking at phase separation in mixed protein systems because of the similarity in optical properties.

Although neither of the two techniques discussed above can distinguish between different proteins, the use of a combination of LB transfer to a transparent solid–substrate and fluorescence microscopy can be used to provide images with a resolution of a few hundred nanometres—which is the diffraction limit for optical methods. In this method proteins are covalently linked to various fluorescent probes. The labelled protein solutions are then used to form an interfacial film that is transferred to a solid support for imaging. An additional disadvantage of this approach is the effect the fluorescent moiety might impart to the functionality of the protein. Examples from two groups who have used variations on this technique are given in Section 4.

When surface-active molecules adsorb at an interface they achieve a higher local concentration than in the corresponding bulk solution. Consequently, the pair interactions between the molecules become more important at the interface in determining the overall behaviour of a mixture of surface-active components than they do in solution. One of the most interesting features of more concentrated mixtures is the greater likelihood of phase separation. It seems relevant, therefore, to speculate on situations under which surface phase separation might occur. In the following, we discuss the possible cases that may arise in this respect during the adsorption of binary mixtures of surface-active molecules.

A crucial factor in determining the possible occurrence of phase separation at an interface relates to the time scale over which the interface equilibrates with its corresponding bulk solution. For highly non-soluble surfactants or high-molecular-weight proteins, present on surfaces, such processes can be particularly slow [26]. Thus, it is easy to encounter situations where the duration of an experiment is far shorter than the necessary equilibration times. In such cases, the overall composition of the interface remains virtually constant during the experiment, having initially been set by the amount of each surface-active species that is introduced onto the interface. For this very slow exchange dynamics, then, the interfacial region may be considered as a separate phase isolated from the bulk. Associated with such an interface, covered by a binary mixture of two different surface active molecules, is an excess free energy per unit area fs1, Γ2), where Γ1 and Γ2 denote the coverage of each species in the mix. It must be noted that in general, should phase separation occur at a certain range of values of Γ1 and Γ2, then fs1, Γ2) is not experimentally accessible in this range. Rather, its behaviour has to be inferred, albeit qualitatively, from a suitable theoretical model; a situation not dissimilar to bulk systems. Outside this range, however, information regarding fs1, Γ2) is obtained from experimental data which, for example, might involve surface pressure and interfacial tension measurements. The phase separation behaviour at the interface is dictated by the form of fs1, Γ2). In particular, if the matrix of second derivatives (∂fs/∂Γi∂Γj) (Hessian matrix) is positive definite everywhere, then the interface remains stable against fluctuations in composition at all values of coverage and will show no tendency for surface phase separation. For an interface to begin to exhibit such behaviour, the Hessian matrix must cease to be positive definite at least over a certain range of Γ1 and Γ2. In this range of values of surface coverage, the mechanism of phase separation is spinodal decomposition, while outside the range phase separation can occur through a nucleation and growth process.

From a more physical point of view, conditions that lead to the ‘convex’ form of fs1, Γ2) and the onset of phase separation at air–water interfaces, are only realised if certain degree of unfavourable interactions are present between the two surface-active species or between these and the water molecules. While the former set of interactions can arise in certain circumstances, the latter are highly unlikely. Even for highly insoluble surfactants at the interfaces, it is the polar or ionic hydrophilic sections of the molecules that remain in the aqueous phase. Such sections have a strong preference to be in contact with the solvent molecules and in the absence of the hydrophobic parts would, by themselves, be soluble in water. Thus, for amphiphilic molecules at air–water interfaces, the only remaining interaction that can realistically lead to phase separation behaviour is that existing between the two surface-active species. This in turn is dominated by the lateral interactions between the hydrophobic parts of the two different sets of molecules. For high-molecular-weight synthetic co-polymers, consisting of large hydrophilic and hydrophobic blocks, the required unfavourable interactions are relatively easy to obtain. Small incompatibilities between different monomers, comprising the hydrophobic parts of the two species, become far more prominent because of the substantial size of the interacting blocks concerned, as well as the relatively higher concentration of the molecules on the interface. Achieving the same level of unfavourable interactions between LMW surfactant species is more difficult and requires the hydrophobic parts of the two sets of molecules to be chemically very different. An interesting example of such a system, which has received some attention in the literature (see for example Ref. [27]), involves binary mixtures of fluorocarbon–hydrocarbon-based insoluble surfactants placed on air–water interfaces.

In contrast to synthetic block co-polymers, proteins are made from a variety of hydrophilic and hydrophobic amino acids that tend to be more uniformly distributed along the length of the chains. In other words, such molecules rarely possess large blocks, which happen to consist almost entirely of the same strand of amino acid monomers. Furthermore, the same variety of amino acid groups is to be found in different protein species. Thus, even if a certain degree of incompatibility between different hydrophobic amino acid types exists, the structure and composition of proteins, as discussed above, makes the possibility of large enough direct unfavourable lateral interactions between such molecules unlikely.

At first sight then, the above discussion suggests that mixtures of different proteins should not exhibit interfacial phase separation behaviour, at least not in the strict thermodynamic sense. Any observed patterns resembling onset of phase separation are to be interpreted as transients, which should decay out over sufficiently long periods of time. However, in a previous publication [51], we have argued that this picture might not always be true. Protein molecules often contain groups, which can associate or even form inter-molecular bonds. An extreme and rather well studied case [28] involves proteins which contain thiol groups (–SH), as say with β-lactoglobulin. In bulk solutions, such groups are hidden from each other within the interior of the molecules. However, on the air–water interface, β-lactoglobulin molecules unfold and expose these reactive groups, thus giving rise to the possibility of formation of covalent bonds between the chains. Though in the case of covalent bonds these bonds tend to be irreversible, in other circumstances weaker bonds (e.g. hydrogen bonding) might form, which can break and reform. It has been speculated [51] that the presence of such bonds between one set of proteins, but not the other, can provide an effective attraction, which can overcome the entropy of mixing, causing formation of separate phases on the interface. Certainly, examples of such behaviour have been found in bulk solutions for water soluble polymers. For example, incorporation of a few small hydrophobic groups, comprising no more than 2% w/w of the polymer to polyethylene glycol (PEG) chains, is known to cause phase separation in solutions consisting of mixtures of such modified and identical non-modified PEG molecules [29]. Both resulting phases have been reported to be clear solutions, with roughly the same amount of PEG present in each. However, the viscosities of the two phases are found to be five orders of magnitude different [54]. This is thought to be due to the presence of predominately modified PEG in one phase and that of non-modified chains in the other. The hydrophobic groups on the modified chains are known to associate reversibly [30] in water, forming spanning networks that lead to the observed high viscosity. It has been shown theoretically that such association results in a net attraction between modified PEG molecules sufficient to explain the observed behaviour of such mixtures [54]. As discussed in the later section, following the presentation of a number of simulation results, it is proposed that a similar mechanism can operate between mixtures of two different proteins residing on interfaces. However, an important point to emphasise is that such association or bonds have to be transient to allow sufficient mobility on the interface. Permanent bonds lead to a system that becomes kinetically trapped, arresting the subsequent phase separation dynamics. While bonds have to be weak enough to be reversible, they have to nevertheless have sufficient strength to lead to enough effective attraction to induce phase separation. An important question is, therefore, whether these two opposing requirements can simultaneously be met in real or even in model simulated systems. We shall attempt to provide some clues to the answer in Section 4.

Our discussion so far has been entirely concerned with systems where the dynamic of exchange between the bulk and the interface is assumed to be very slow. It is also useful to briefly consider the opposite case where the bulk solution and the interface are at equilibrium. For such systems, just as before, there is an excess free energy per unit area fs1, Γ2), arising from the presence of the interface. However, unlike the situation discussed above, the overall composition of the interfacial layer is no longer constrained. Since molecules of both species can freely exchange with the bulk, it is appropriate now to consider the interfacial thermodynamic potential (per unit area), i.e.AsΓ1,Γ2=fsΓ1,Γ2−μ1bΓ1−μ2bΓ2where μib is the chemical potential of the species i in the bulk solution. The composition of the interface is determined by the values of Γ1 and Γ2, which happen to minimise the function defined in Eq. (17). At this lowest value, As is also equal to the interfacial tension for the interface [31]. In most circumstances one expects a single global minimum for the function in Eq. (17). Nevertheless, one might argue that for some particular set of values of bulk concentration of the two surface-active species, As1, Γ2) might develop two minima, at different interfacial compositions, which just happen to have the same minimum value. In principle then, two such phases can co-exist on the interface. In practice, however, achieving this situation is highly unlikely. First of all, the situation will only arise at some precisely related values of bulk concentration of the two species. Controlling the amount of two sets of molecules to such a degree would be difficult. Even if such a situation could be realised, there are no constraints on the overall composition of the interface. Therefore, in time, one of the phases will tend to dominate at the expense of the other. This will be determined by the kinetic of adsorption of the two species and the initial conditions of the experiment.

Our motivation, in discussing these cases involving free exchange of surface-active species with the bulk, is to emphasise the importance of irreversible adsorption as a prerequisite to having true phase separation processes at the surface. We finish this section by drawing attention to a situation intermediate between the two cases discussed above, where one of the surface-active species is irreversibly adsorbed, whilst the other is in equilibrium with the bulk solution. This is an interesting possibility that can arise when a high-molecular-weight protein is very slowly displaced by relatively soluble LMW surfactant molecules, adsorbing from bulk. Once again the simulation results (presented in Section 5) can provide some clues as to the likelihood of phase separation behaviour of such systems.

Section snippets

Adsorption of one-component systems

To understand the simulations of the competitive adsorption of proteins and LMW surfactants we have first to consider the behaviour of each separate type of molecule adsorbing to an interface. Although no AFM imaging can be done for pure surfactant layers—mainly because of the high mobility of these molecules—some images have been obtained for β-lactoglobulin adsorbed at oil–water interfaces [3]. Furthermore, interfacial rheological properties of adsorbed proteins have been investigated

Competitive adsorption of mixed LMW surfactants

When a mixture of two different LMW surfactants (or any mixture of surface-active species) is allowed to adsorb at an interface, the two kinds of molecules have to ‘compete’ for space: hence, the name ‘competitive adsorption.’ Naively, we might expect the adsorbate with the larger value of Eads to dominate the interface if the mixture bulk concentration ratio is approximately 1:1. For species with equal binding capacity for the surface, it would be expected that the one present at higher bulk

Competitive adsorption of mixed proteins

The competitive adsorption of sticky hard spheres has been studied in the framework of the Percus–Yevick integral equation theory, both in general [45] and in relation to protein mixtures [46]. The model system [47] consists of a binary mixture of particles interacting through a hard sphere particle–particle potential and a sticky surface–particle potential. It has been shown [46] that for binary mixture of equally sized adsorbing hard spheres with no extra interactions, the isotherms

Competitive adsorption of proteins and LMW surfactants

When a solution containing protein and LMW surfactant is adsorbed, the surfactant is likely to prevail at the interface, after equilibration, if both species are present at high enough bulk concentrations [37], [52], [59]. This is due to the fact that LMW surfactants are much smaller in size than proteins, and so they can reduce the interfacial tension more efficiently by adsorbing a much larger number of molecules within the same surface area (see the discussion in Section 3 about the relative

Concluding remarks

We have shown in this review that the combination of imaging experiments and computer simulation based on simple models has brought about important new understanding of the mechanisms involved in the competitive adsorption of mixtures of different adsorbates. Both experiments and simulations show that the displacement of protein layers by LMW surfactants normally occurs in an inhomogeneous way throughout the interface—with the LMW surfactant forming separated domains. This type of situation is

Acknowledgements

This research was supported by the BBSRC (UK). Computing was done on the Leeds Grid Node 1 facility, funded under the 2001 HEFCE Science Research Investment Fund initiative at the University of Leeds, which is one of the partners in the White Rose Grid project.

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