Using Self-Consistent-Field Theory to Understand Enhanced Steric

Biomacromolecules 2005, 6, 3018-3029

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Using Self-Consistent-Field Theory to Understand Enhanced Steric Stabilization by Casein-Like Copolymers at Low Surface Coverage in Mixed Protein Layers Emma L. Parkinson, Rammile Ettelaie, and Eric Dickinson* Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, U.K.

We present a statistical mechanical approach to predicting the properties of mixed copolymer layers using the Scheutjens-Fleer self-consistent-field theory. Our model copolymers are based on the primary structures of the major bovine casein monomers, Rs1-casein and β-casein. Numerical calculations have been carried out to determine the polymer segment density profiles at an isolated hydrophobic surface and the interaction forces as a pair of polymer-coated surfaces is brought to close interlayer separation. For a copolymer model containing hydrophilic and hydrophobic segments, we show how the steric stabilizing capacity of a caseinlike macromolecule at very low surface coverage is enhanced in the presence of a thin dense layer of shorter tethered amphiphilic chains. Using a more refined protein model, which also distinguishes between the charged and uncharged hydrophilic segments along the chain, we clearly demonstrate that the enhanced steric repulsion from β-casein exceeds that from Rs1-casein. These calculations explain how the replacement of just a few percent of β-lactoglobulin by casein can inhibit the heat-induced thickening and flocculation behavior observed experimentally with some whey protein-stabilized oil-in-water emulsions. Introduction On the basis of a range of combined experimental and theoretical evidence, it is well-established that, under conditions close to saturation surface coverage, the disordered protein casein is an excellent steric stabilizer of food colloids and especially oil-in-water emulsion systems.1-3 The reasonable possibility arises, therefore, that, in colloidal systems containing mixed protein ingredients, a casein surface concentration well below saturation surface coverage might lead to enhanced steric interaction and hence improved stability with respect to aggregation. Experiments in support of this hypothesis were reported recently4,5 involving the positive influence of small amounts of casein on the stability of heat-treated emulsions made with whey protein isolate or β-lactoglobulin as the primary emulsifying agent. The present paper aims to provide a deeper theoretical understanding of the stabilizing role of adsorbing copolymers at moderately low concentrations in the presence of other adsorbed material. We use as the basis for our calculations the self-consistent-field (SCF) theory of Scheutjens and Fleer.6,7 The chain lengths and segment type distributions in our model polymers are loosely based on the known primary structures8 of the two major caseins of cow’s milk, Rs1-casein and β-casein. The driving force for adsorption of these latticebased polymers at a plane hydrophobic surface is the attractive interaction energy for the surface by the nonpolar chain segments. For specified sets of interactions (segmentsegment, segment-solvent, etc.), this theory can give equilibrium segment density profiles and interaction free energies between pairs of closely approaching surfaces. Mixed layers * To whom correspondence should be addressed (E-mail: E.Dickinson@ leeds.ac.uk).

can also be treated by the SCF theory.9 Here, we investigate the effect of the presence of another major adsorbed protein component (loosely representing β-lactoglobulin) by incorporating into the model a high coverage of a second idealized copolymer of shorter chain length. By far the most extensively studied of the casein monomers is β-casein. This macromolecule has all the essential molecular characteristics of an excellent emulsifying agent and stabilizer. It is a flexible linear amphiphilic polyelectrolyte of not too high molecular weight (24 kDa) carrying a moderate net charge (-15 e) at neutral pH. The protein has little ordered secondary structure and no intramolecular covalent cross-links. The highly nonuniform distribution of hydrophilic and hydrophobic residues produces a distinctly amphiphilic structure resembling a simple water-soluble surfactant or block copolymer. On the basis of experimental studies at various hydrophobic surfaces,10-16 it is now reasonably well accepted1-3,17 that the β-casein molecule adsorbs with its extensive hydrophobic region (160-170 segments) anchored at the surface (trains and small loops) and its hydrophilic tail (40-50 segments) protruding extensively into the aqueous phase. SCF theory calculations have confirmed this representation.18,19 The predicted segment density profile at neutral pH and low ionic strength has a dense inner layer (1-2 nm) and an extended outer region. This is in agreement with neutron reflectivity experiments on β-casein at both airwater and oil-water interfaces.11-14 The previous SCF calculations have demonstrated18,19 the crucial role of the cluster of five charged phosphoserine residues in β-casein in maintaining the steric stabilizing layer, while also inhibiting any interfacial precipitation (multilayer formation). Bovine Rs1-casein is a slightly smaller molecule than β-casein, and more highly charged (-22 e at neutral pH). It

10.1021/bm050329i CCC: $30.25 © 2005 American Chemical Society Published on Web 09/02/2005

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Enhanced Steric Stabilization in Mixed Protein Layers

also is amphiphilic, but the distribution of hydrophilic and hydrophobic residues along the chain is more random than for β-casein.20 Emulsion droplets stabilized by Rs1-casein are more highly charged than those coated with β-casein.21 In contrast to the dangling tail predicted for adsorbed β-casein, our previous SCF calculations suggested19 more of a looplike conformation for the adsorbed Rs1-casein. The major whey protein in milk, β-lactoglobulin (β-lg), produces an adsorption layer at hydrophobic surfaces that is rather dense and thin (ca. 2 nm) at neutral pH.12,22 The globular structure is retained on adsorption. The saturated adsorbed β-lg layer can be regarded as a close-packed monolayer of deformable particles.23 As the globular protein molecules partially unfold, there is conversion to a twodimensional gel-like layer24-26 with strengthening of nonbonded intermolecular interactions and some slow covalent cross-linking.27 It is the combination of the compactness of the adsorbed β-lg layer and its gel-like viscoelasticity that mainly distinguishes it from adsorbed casein. An important difference between these two kinds of proteins in food processing is that, whereas the caseins are extremely stable to heat, the globular whey proteins (especially β-lg) are susceptible to thermal denaturation at temperatures above ca. 70 °C.28 This means that a whey protein-stabilized emulsion is sensitive to heat-induced flocculation and thickening.4,29,30 In practice, of course, the caseins and the whey proteins are typically present together at the interface in many dairy emulsions, and experiments on model systems by Dalgleish and co-workers have shown31-33 that their competitive adsorption behavior is rather complicated and highly temperature-dependent. Against this background, we are interested in the possible “protective effect” of casein in whey protein-containing systems. What we have recently discovered4,5 is that replacing just a small proportion (a few percent) of whey protein isolate4 or pure β-lg5 by casein can confer stability toward emulsion flocculation and thickening induced by heating (e.g., 90 °C for 3 min). Moreover, we find β-casein to be a more effective protective agent than Rs1-casein. We interpreted our results inituitively4,5 in terms of the known steric stabilizing effect of Rs1-casein loops and, more especially, of β-casein tails, even at very low fractional surface coverages (down to casein/β-lg ratios on the order of 1:100). The purpose of this paper is to place our interpretation on a more rigorous statistical mechanical foundation. In what follows, we represent the protein polymers in an idealized waysa representation even simpler than we used in previous SCF calculations13 on adsorbed β-casein to compare with neutron reflectivity experiments.11,12 The aim here is not to get quantitative agreement between theory and experiment, but to explore the fundamental phenomena involved. We shall determine the influence of copolymer surface concentration and composition on density distributions at single hydrophobic surfaces and between pairs of such surfaces. It is already recognized1-3 that the SCF theory is capable of distinguishing between the net attractive potential of a pair of Rs1-casein-coated surfaces (under certain conditions) and the net repulsion for the pair of equivalent β-caseincoated surfaces.34 Here, we explore similar potential differ-

ences at low casein surface coverages in the presence of a second polymeric adsorbate species representing the space occupied close to the surface by the adsorbed whey protein. Theoretical Background and Calculation Methodology We divide the full set of protein amino acid residues into a small number of classes, based on their degree of hydrophobicity and their ionic and polar nature, as well as their affinity for adsorption at a hydrophobic surface. For simplicity, all segments within the same class are considered identical; that is, they are assumed to have the same interaction parameters with the solvent, the surface, and the segments belonging to other classes. Similarly, we ignore any size differences between amino acid types, taking all polymer segments and solvent molecules as being of the same size, hereafter denoted a0. Because of their interactions with the solid surface and the restrictions that the presence of such an interface imposes on the number of allowed chain configurations, the concentration profiles of the protein molecules in the vicinity of the surface will tend to be substantially different from those in bulk solution. The primary aim of the SCF calculation is to determine the variation of the density profile φRi (r) for every monomer residue of kind R, belonging to protein species i, as a function of distance r away from the interface. Since we are predominately interested in the effective interactions that are mediated by the adsorption of protein layers between approaching particles, it is more useful to consider the density profile in the gap between two such solid surfaces. The contribution to the free energy of the system, per unit surface area, arising from an arbitrary variation of the concentrations of polymer and solvent in the gap between two infinite flat solid surfaces, a distance L apart, is given by35,36 ∆F kT

)

{

1

L L ∑i ∑a -∫0 N [φRi (r) - ΦRi ] dr - ∫0 φRi (r) ψR(r) dr

1

i

∑∑∫ 2 ij Rβ 0

L

}

+

χRβ[φRi (r) - ΦRi ][φβj (r) - Φβj ] dr +

∑i ∑R χRs[φRi (0) + φRi (L)]

(1)

where T is the temperature and k is Boltzmann’s constant. The free energy change ∆F is calculated with reference to a state where all the concentration profiles are uniform throughout the system. In writing eq 1, it is assumed that the two solid surfaces are homogeneous, so that all density variations in directions parallel to the plates have already been averaged out. The summations involving indices i and j are taken over all the different molecular species present in the system, including the solvent. Those taken over indices R and β are summed with respect to different monomer groupings, as explained above. The length of the chains of type i is denoted here by Ni, and the bulk concentration of monomers of kind R on such chains is represented by ΦRi . For the solvent molecules, we have N ) 1. In the remainder of this article, unless stated otherwise, it is more convenient

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Parkinson et al.

to express L and all the other distances in terms of the monomer size (a0). Similarly, the free energy contribution ∆F is now expressed as energy per unit area (a02), and the concentration profiles φRi (r) become identical to volume fractions for the corresponding monomers. The final term in eq 1 represents the interaction energy of each monomer segment type with the flat solid interface as characterized by the interaction parameter χRs for that group. For nonpolar amino acid residues, the hydrophobic interaction strength is such that we typically assume χRs ≈ 1-2 (in units of kT). Interactions between different monomer segment groupings, as well as the segment-solvent interactions, are similarly characterized by the usual Flory-Huggins parameters χRβ . In this sense, all the intermolecular forces are taken as being of short range, i.e., extending only to the immediate locality of each monomer residue. It can be pointed out that eq 1 is not an explicit expression in φRi (r), as it also involves a set of auxiliary fields, ψR(r), for each monomer segment type R. These can be thought of as a set of external fields, each acting on all the monomers belonging to their respective groups. When applied to the system and in the absence of any internal interactions, these fields lead to the corresponding density profiles specified in eq 1. It is possible to expand ψR(r) to low orders in φRi (r) and ∇φRi (r) in order to obtain an approximate (but explicit) expression for the free energy in terms of the density profile variation. A frequently encountered example of such an expression is the one appearing in the random phase approximation (RPA) theory.37 Since all the results reported here are based on numerical calculations, we shall refrain from introducing such additional approximations. While the knowledge of the appropriate set of fields, ψR(r), for an arbitrary set of density profiles allows one to calculate the free energy change associated with that profile, it is the concentration profile resulting in the lowest value of ∆F which is of primary interest. In common with other types of mean field approach, the SCF theory assumes that it is the most probable density profile that dominates the thermodynamic properties of the system and that fluctuations around this most probable profile are sufficiently small to be ignored. Such an assumption is known to work well for concentrated polymeric systems,7 like the relatively dense layers of adsorbed protein considered here. As minimization of ∆F is carried out numerically, it is normally required to consider a discretized form of eq 1. Although the grid size need not necessarily correspond to the size of monomer segments,38 this is normally chosen to be so in the Scheutjens-Fleer scheme.39,40 This has the advantage that now the scheme has a physical interpretation in terms of a model of polymers on a lattice.6,39,40 Thus, the region between the two solid surfaces is now divided into L layers, with the values of the variables ψR(k) and φRi (k) to be determined for each layer from k ) 1 to k ) L. The minimization of the free energy is achieved when the condition R

ψ (k) - χRs(δk0 + δkL) -

∑i ∑γ

χRγ[φγi (k)

ψβ(k) - χβs(δk0 + δkL) -

-

Φγi ]

)

∑i ∑γ χβγ[φγi (k) - Φγi ]

(2)

is satisfied for each layer and for all sets of values of the indices R and β.6,7 The symbol δkj in the above expression denotes the usual Kronecker δ function (δkj ) 1 for k ) j, and δkj ) 0 otherwise). In arriving at eq 2, the rather obvious but important requirement that the sum of the volume fractions of all species within each layer should be equal to unity, i.e., ∑i∑R φRi (k) ) 1, is also taken into account. In lattice models, it is customary to allow for interactions between different monomers to extend between adjacent layers. This can easily be achieved by replacing the φγi (k) in eq 2 with their values averaged over three consecutive layers, 〈φγi (k)〉, as defined by 〈φγi (k)〉 ) λ-1φγi (k - 1) + λ0 φγi (k) + λ+1φγi (k + 1) (3) The values of the constants λ-1, λ+1, and λ0 depend on the underlying lattice chosen for the purpose of the calculations. For the simple cubic lattice adopted here, the numerical values are λ-1 ) λ+1 ) 1/6 and λ0 ) 4/6. The calculations proceed by considering a trial set of values ψR(k) for each layer and each type of monomer. To calculate the various volume fractions associated with this initial choice of values of the auxiliary fields, one needs to construct the s-segment weight factors for every species i present in the system. Additionally, for the model proteins considered here, the s-segments can be chosen from either end of the chain. Unless a molecule is perfectly symmetrical, the two choices will be different from each other, and both are required. We shall use the index j ) 1, 2 below to distinguish between the two. The s-segment weight factor is obtained using the recurrence relation (j) G(j) i (k, s) ) Gti(s)(k)[λ-1 Gi (k - 1, s - 1) + (j) λ0 G(j) i (k, s - 1) + λ+1 Gi (k + 1, s - 1)] (4)

The function ti(s) denotes the type of monomer in segment position s on the sequence of residues that makes up the chains of species i. The segment number s is counted from the appropriate end of the chain, in each case, j ) 1, 2. For each type of monomer, including the solvent, we have GR(k) ) exp[-ψa(k)]. The single-segment weight function is set (2) to G(1) i (k, 1) ) Gti(1)(k) or Gi (k, 1) ) Gti(Ni)(k) for j ) 1, 2, depending on the end-point distribution function that is being calculated, to start the recurrence relations in eq 4. With the s-segment weight functions determined, one can invoke the so-called compositional law6,7 to calculate the values of various volume fractions at different positions in the gap between the two surfaces. Here, the compositional law takes the form

φRi (k)

)

∑β Φβi Ni

Ni

∑ s)1

δR,ti(s)Gi(1)(k, s) Gi(2)(k, Ni - s + 1) Gti(s)

(5)

The volume fractions of different monomer types, obtained according to eq 5, are next checked to see if they satisfy the incompressibility condition, ∑i∑R φRi (k) ) 1, as well as the criteria described in eq 2 (suitably modified, as described above, using eq 3). By consideration of the degree of


mismatch, the values of the fields ψR(k) are systematically altered, and the calculations are repeated with the new values. The iterative process continues until all the necessary conditions are simultaneously met and convergence is obtained. The final choice of ψR(k) and the resulting values of φai (k) provide us with the required equilibrium density profiles and the corresponding free energy change for the system. It is well-known that protein molecules have large adsorption energies and strong affinities for hydrophobic interfaces. As a result, in colloid or emulsion systems it can be the case that the protein is overwhelmingly located at the surface of particles or oil droplets, leading to a significant depletion of the bulk concentration which may reach very low values.41 Furthermore, the adsorption-desorption kinetics for these macromolecules are usually slow, so that the protein surface coverage, during the relatively short collision time between the particles, has no time to change. In such cases, it is clearly more convenient to specify the number of chains per unit surface area, rather than the bulk volume fraction of the protein molecules. In the calculations, this can be readily achieved by considering chains to be anchored to the surface at a suitable point along their sequence (preferably a strongly hydrophobic segment at one end of the molecule), with the bulk concentration ΦRi of such species then set to zero. This procedure in turn requires some modification to the form of eq 1 for the free energy. The first term in that equation, involving the integral of the excess concentration, has to be replaced by -2Γi ln(Zi) for all the anchored species.7 Here, Zi ) 1/2∑kGi(1)(k, Ni) is the single-chain partition function for molecules of type i, and Γi is the number of chains per unit area on each surface. It is assumed throughout this work that Γi is identical for both surfaces. Similarly, the singlesegment weight function, required to initiate the recurrence relation in eq 4, is modified to read Gi(1)(k, 1) ) Gti(1)(k)(δk,1 + δk,L), once again reflecting the fact that one end of each chain is now anchored at one or the other of the two approaching surfaces. The computer programs used to implement the SCF calculations were written in Fortran. They were run on a PC under the Linux operating system. Mixed Interfacial Layers of Copolymers: A Simple Model In modeling the individual casein chains (see below), we shall divide the set of amino acid residues making up each protein into five distinct groupssnamely, hydrophobic, polar, and positively and negatively charged types (with the latter further subdivided into two groups according to the magnitude of the negative charge carried). First, however, to introduce some of the key issues involved, it seems appropriate to present results for an even simpler polymeric model where all of the hydrophilic residues (polar or charged) are lumped together in a single group, irrespective of their polarity. So, we study copolymers with just two kinds of segmentsshydrophobic and hydrophilic. While it is not suggested that such a crude model will be able to reproduce all the experimentally observed features we wish to explain here, we consider that it is instructive and worthwhile to identify the dominant relevant effects that do emerge even


Figure 1. Three simple model copolymers used in the preliminary SCF calculations. Dark and light circles represent hydrophilic and hydrophobic segments, respectively. The sequences of the two kinds of segments along the chains in polymers A and B are based, respectively, on the primary sequences of Rs1-casein and β-casein.

at this level of simplification. One would expect such generic effects to relate to factors arising largely as a consequence of the unique primary sequence of the caseins, rather than from any specific details of the segment-segment or segment-solvent interactions. In what follows in this section, we take the solvent as being athermal (χ ) 0) with respect to the hydrophilic segments and a bad solvent (χ ) 1) with respect to the hydrophobic segments (in units of kT). Similarly, we set χ ) 1 for the Flory-Huggins interaction parameter between the two classes of segment, implying an unfavorable interaction between the two. Furthermore, we assume that the hydrophilic groups have no affinity for adsorption at a hydrophobic surface, whereas the adsorption energy for each hydrophobic segment is taken to be χs ) -1 (in units of kT). The latter is on the order of the energy involved in the formation of a typical hydrophobic bond.42 Figure 1 shows the precise distributions of hydrophobic and hydrophilic segments along our model copolymers A and B, with the two sequences being based on the primary structures of Rs1-casein and β-casein, respectively. The steric interaction potential per unit area between a pair of flat surfaces is given by the change in the free energy of the system, V(r) ) F(r) - F(∞), when two isolated plates are brought from infinite separation to within a distance r of each other. This potential is also simply related to the force, fp(r), exerted by two spherical particles of radii R on each other, through the so-called Deryaguin approximation42 fp(r) ) πRV(r)

(6)

3022


Figure 2. Steric forces mediated between two colloidal particles of radius R covered with layers of polymer A (dashed line) or polymer B (solid line). The normalized force is plotted against the separation in units of monomer size (a0). The numbers of chains per unit monomer area (a02 ) are 0.003 for polymer A and 0.005 for polymer B, representing the approximate saturation surface coverage for each species. It is assumed that the surface coverage remains constant as the particles approach.

Figure 2 shows the calculated force (in units of kTπR/a02) plotted against the surface-to-surface separation of two colloidal particles. The dashed line represents the force for a pair of particles covered with polymer A at fixed surface coverage of 0.003 chains per unit monomer area (size a02). The solid line displays the same results for polymer B at 0.005 coverage. These values of surface coverage are chosen to conform with the equilibrium adsorption for each polymer, calculated using the same simple model for each copolymer. Of course, the adsorbed amount is somewhat dependent on the bulk concentration of the protein present. However, in common with most other amphiphilic macromolecules, these polymers attain their saturation coverage at very low bulk concentrations. For polymer B, we find that the surface coverage changes from 0.0045 to 0.0059 chains per unit monomer area as the bulk volume fraction in the solution varies by some 5 orders of magnitude from 10-7 to 10-2. The corresponding values for polymer A are 0.0021 and 0.0036, respectively. It can be seen from Figure 2 that, at the “saturation” surface coverage, a considerable steric repulsion between the particles arises because of the presence of polymer B chains on the particle surface. Polymer A, on the other hand, does not reproduce the same qualitative result. Instead, we find the mediated interaction between the particles to be attractive at intermediate separations. This we consider to be a reflection of the fact that the primary structure of β-casein causes it to behave more like a diblock copolymer, whereas the behavior of Rs1-casein resembles more closely that of a triblock copolymer.43 The existence of attractive forces in systems involving triblock copolymers has been well-studied; their origin is attributed to the so-called bridging phenomenon.44 This involves simultaneous adsorption of chains onto the surfaces of two neighboring particles, at small separations. In contrast to Figure 2, the plot in Figure 3 for polymer B corresponds to the situation where the surface coverage (0.001 chains per unit monomer area) is significantly below the saturation value. Not only is there no repulsive force

Parkinson et al.

Figure 3. As in Figure 2, except that there is a lower surface coverage of 0.001 chains per unit monomer area for polymer A or polymer B. Note the very different ordinate scale as compared with Figure 2.

evident, but instead, there is a strong close-range attraction between the particles. This is consistent with the inability of very low concentrations of (sodium) casein(ate) present alone as an emulsifying agent to stabilize fine oil-in-water emulsions.45 Our recent experiments4,5 on heat-treated emulsions containing mixed milk proteins also involve droplets with a small surface coverage of casein on the surface. But, in this case as well as casein molecules, there are also whey proteins predominantly covering the oil-water interface. On the basis of existing knowledge,12,26 one could reasonably infer4,5 that the adsorbed whey protein forms a thin interfacial layer and that this layer alone is incapable of sustaining colloidal stability in a heated system containing denatured whey proteinsboth adsorbed and nonadsorbed. Nevertheless, we discovered4,5 that the presence of just a very small amount of casein in the system was sufficient to prevent this heatinduced destabilization. We hypothesize that the presence of a thin, but dense, polymeric layer close to the oil-water interface enhances the steric stabilizing capability of the casein(ate), especially the β-casein component. We suggest that it does this by pushing and stretching the flexible casein chains away from the surface. This in turn gives rise to a strong repulsive interaction, at casein surface coverages for which such a force is not predicted in the absence of the globular protein layer. As a preliminary test of this hypothesis, we use the SCF theory with polymer B representing β-casein. In keeping with our simplified approach, we construct a crude representation of the relatively dense (but otherwise thin) interfacial whey protein layer using tethered short chains of polymer C (see Figure 1), each consisting of 25 monomers composed of alternating hydrophobic and hydrophilic segments. We stress that polymer C is definitely not meant, either in terms of its chain length or its segment distribution, as any sort of realistic representation of an individual β-lg molecule. Rather, the presence of polymer C is simply a convenient contrivance, within the context of the SCF theory, to introduce a densely packed polymeric environment close to the surface, with the effect of simulating a thin layer occupying a considerable fraction of the available segments near the surface and therefore exerting an excluded volume effect on other species


Figure 4. Steric forces for a pair of mixed layers containing polymer C at a surface coverage of 0.01 chains per unit monomer area (a02) and polymer B at a surface coverage of 0.001 (solid line). The normalized force is plotted against the separation in units of monomer size (a0). Included for comparison are results for the cases where polymer B is not present (dashed line) and where it is replaced by the same amount of polymer C (dashed-dotted line).

that might be present in that vicinity, i.e., polymer A or polymer B at low surface coverage. Figure 4 shows the result of calculating the interparticle force mediated by a mixed layer of polymer B + polymer C. The amount of polymer B at the interface is the same as that in Figure 3, while that of polymer C, representing the whey protein layer, is fixed at 0.01 chains per unit monomer area (a02). In contrast to the corresponding curve in Figure 3, the presence of a modest repulsive force at intermediate particle separations can now clearly be observed. While we estimate that this force is still too small by an order of magnitude to overcome the attractive van der Waals interactions in oil-in-water emulsions (see below), the principle in which the presence of polymer C enhances the steric repulsion of polymer B is demonstrated. For comparison in Figure 4, we also include the calculated separation-dependent force for the case of the polymer C layer alone (when polymer B is removed) and for the case when removed polymer B is replaced by additional polymer C such that the overall polymer content at the interface (segment volume fraction) remains unchanged. In both these latter cases, the interactions are predicted to be purely attractive, showing that it is the presence of polymer B in the mixed layer that is responsible for the development of the steric repulsion. The magnitude of the steric repulsion might be expected to become stronger by increasing the density of the inner layer. Figure 5 tests this prediction by increasing the surface coverage of polymer C to 0.015 chains per unit monomer area, while keeping the polymer B content constant at 0.001 chains per unit monomer area. Once again, the solid line shows the results of the variation of the force with separation distance, calculated for the mixed layer. By comparing this with the case where polymer B is absent (dashed line) and where polymer B is replaced with polymer C while maintaining the same overall copolymer coverage (dashed-dotted line), the clear role of the low concentration of polymer B in inducing the repulsion is evident. Furthermore, it is also seen that the magnitude of the steric repulsion is now comparable in magnitude to that of the predicted attractive van der Waals force between the particles, also included in


Figure 5. As Figure 4, except that the surface coverage of polymer C is increased to 0.015 chains per unit monomer area. Also plotted (lower dashed line) is the estimated force due to van der Waals interactions between oil-in-water emulsion droplets (see text).

Figure 6. Influence of polymer C on segment density profile of polymer B at the isolated surface. The segment volume fraction φ is plotted as a function of distance from the surface in units of monomer size (a0): s, mixed layer of composition as in Figure 5; - - -, polymer B present alone.

Figure 5 (the short dashed line). To estimate the latter, we use the usual expression -AH/(12r2) for the van der Waals interaction between spherical particles of radius R (.r), with the value of the Hamaker constant AH taken as 1 kT, typical of that found for oil-in-water emulsion droplets.46 It is instructive to compare the average distribution of polymer B at the interface when present as part of the mixed layer with that when present alone. The solid line in Figure 6 shows the density profile of polymer B plotted against the distance away from the isolated surface, for the same mixed layer as in Figure 5. The dashed line displays the density profile when polymer C is removed. It can be seen that there is a substantially greater extension of polymer B into the bulk solution in the mixed copolymer system. In Figure 7, we depict results for the same system as in Figure 5, but now at an even higher surface coverage of 0.02 for polymer C, while maintaining the same number density of polymer B, i.e., 0.001 chains per unit monomer area. Once again, in the absence of polymer B, the interaction between the particles (dashed line) is short-ranged and attractive. The force does become repulsive if the removed type-B polymers are replaced with a sufficient number of short type-C chains to keep the total amount of polymer at the interface the same

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Parkinson et al.

Figure 7. Interaction forces as a function of particle separation for a pair of mixed layers with surface coverages of 0.001 and 0.02 chains per unit monomer area (a02) for polymers B and C, respectively (solid line). Included for comparison are results for the cases where polymer B is not present (dashed line) and where it is replaced by the same amount of polymer C (dashed-dotted line).

as in the mixed film (dashed-dotted line). But even here, the repulsion is still not as strong as that achieved with the mixed protein layer, and more crucially, it can be seen that the repulsive force only comes into operation at smaller separation distances, where the magnitude of the van der Waals interaction is already large. In contrast, the range and magnitude of the repulsion induced for the mixed adsorbed layer is such that it can easily overcome the interparticle van der Waals force, with no need for any additional electrostatic (or other) stabilizing contributions. The numerical results described above provide support from statistical mechanics for the concept that the addition of a small amount of casein-like copolymer to a thin, but relatively dense, whey protein layer, can give rise to sufficient steric repulsion to protect emulsion droplets against heatinduced aggregation. The inferred stabilization mechanism requires the presence of both protein components in the interfacial layer. According to our SCF calculations, in the absence of the other adsorbed species, each of the single copolymer components alone is unable to impart sufficient steric stability to the system. In the next section, by considering a more refined model of the casein polymers, we shall explore the extent to which the theory can distinguish between the relative stabilizing effects of the two main monomeric caseins. Adsorbed rs1-Casein or β-Casein in the Presence of Whey Protein Influenced by previous casein SCF calculations,18 initiated originally to compare with neutron reflectivity experiments,11-13 we now classify the amino acid residues into five different categories: (1) hydrophobic (nonpolar), (2) hydrophilic (polar but uncharged), (3) potentially positively charged, (4) potentially negatively charged, and (5) phosphoserine (net charge of -1.5 e at pH ) 7).41 With this segment categorization scheme, Figure 8 gives a diagrammatic representation of the proteins Rs1-casein (199 residues + 2 end groups ) 201 segments) and β-casein (209 residues + 2 end groups ) 211 segments) based on the primary

Figure 8. Linear polymer structures representing Rs1-casein and β-casein in the refined SCF calculations. The five different segment categories are (1) hydrophobic (grey), (2) hydrophilic uncharged (white), (3) positive charge (thick left diagonal stripe), (4) negative charge (thin right diagonal stripe, light gray), and (5) phosphoserine (black). The N-terminal and C-terminal groups are designated as segments of types (3) and (4), respectively. Table 1. Values of Flory-Huggins χRβ Parameters (units of kT) for Each Pair Interaction Typea type

w

1

2

3b

4c

5

s

w 1 2 3b 4c 5 s

0 +1 0 -1.5 -1.5 -2.0 0

+1 0 +1 +1.5 +1.5 +2 -1.5

0 +1 0 0 0 0 0

-1.5 +1.5 0 0 -1.5 -2 0

-1.5 +1.5 0 -1.5 0 +2 0

-2.0 +2 0 -2 +2 0 0

0 -1.5 0 0 0 0 0

a Key: (w) solvent, (1) hydrophobic, (2) hydrophilic, (3) potentially positively charged, (4) potentially negatively charged, (5) phosphoserine, and (s) surface. b Including end amino group. c Including end carboxyl group.

structures of Swaisgood.8 Each casein chain is grafted to the surface at its C-terminal end, which is where the higher proportion of hydrophobic residues is located, especially for β-casein. We do not attempt here to represent the realistic structure of an individual whey protein molecule. Again, as described above, we simply mimic the effect of a thin globular protein layer by incorporating tethered short chains of polymer C (see Figure 1). The key structural feature being included here is the substantially increased local polymer segment content close to the surface. Table 1 gives the complete set of assumed Flory-Huggins parameters for the various molecular interactions involving surface (s), solvent (w), and segments (1-5). Negative and positive values imply attraction and repulsion, respectively. We assume that only the nonpolar residues have a positive affinity for the surface (χ1s ) -1.5); all the other segments are athermal with respect to the surface. The nonpolar segments are repelled by the solvent (χ1w ) 1) and also by the other segments to varying degrees. By contrast, the charged residues have an affinity for the solvent (χ3s ) χ4s ) -1.5), which is taken to be greater for the more highly


Enhanced Steric Stabilization in Mixed Protein Layers Table 2. Theoretical Surface Coverage Fraction of Polymers in SCF Calculations Related to the Protein Content in Emulsion Experiments4,5 total protein content in emulsion (wt %)

casein content as proportion of total protein present (%)

estimated surface coverage fraction (β-casein)

estimated surface coverage fraction (Rs1-casein)

estimated surface coverage fraction of β-lactoglobulin

3 3 3 3 3

0 1 2.5 5 100

0.0 0.000098 0.000245 0.00049 0.0098

0.0 0.0001 0.00025 0.0005 0.01

0.01 0.00990 0.00975 0.0095 0.0

charged phosphoserines (χ5s ) -2). The polar (uncharged) segments are assumed to be athermal with respect to the solvent and the charged segments. The casein concentrations and protein surface compositions assumed in the SCF calculations are chosen to correspond approximately to conditions pertaining to our recently reported emulsion experiments.4,5 The surface coverage fraction is given by φp )

Γp × NA × a02 Mw

(7)

where Γp represents the surface coverage (g m-2), Mw is the molecular weight of the protein (Da), NA is Avogadro’s number, and a0 is the monomer size (0.3 nm). The value of φp represents the number of chains per unit monomer size square, much in the same way as was defined in the previous section. The experimental system typically contains 3 wt % protein. When just 1% of that protein is β-casein, Γp can be calculated by dividing the mass of protein adsorbed by the total surface area of oil present. For example, in 10 g of emulsion containing 0.03% β-casein (ca. 0.003 g of β-casein) and a surface area of oil of 68.95 m2 (as determined experimentally), we have Γp ) 0.003/68.95 ) 0.000 043 5 g m-2. Taking Mw ) 23.988 kDa and a0 ) 0.3 nm, we then have φp ) 0.000 098 for the surface coverage fraction of β-casein in a 2.97 wt % β-lg + 0.03 wt % β-casein emulsion. The same calculation for Rs1-casein (Mw ) 23.623 kDa) gives an estimated surface coverage fraction of 0.000 100. So, if 1% of protein present gives a surface coverage fraction of ca. 10-4, then the surface coverage fraction of the total protein present is ca. 0.01. We assume that the surface coverage fraction of β-lg at the surface makes up this difference (i.e., 0.0099). To mirror the experiments, then, the addition of the casein is accompanied by a corresponding small loss of the pseudo-β-lg (polymer C) from the surface. For the convenience of the reader, the experimental protein concentrations and their corresponding surface coverage fractions in the SCF calculations are summarized in Table 2. The following compositions (expressed here in terms of the experimental parameters) are specifically considered in these refined SCF calculations: (i) 3 wt % β-lg (i.e., in the absence of casein to confirm that some net attraction is predicted, consistent with flocculation on heating in the

Figure 9. Interaction energy A(D) in units of kT (site)-1 as a function of interlayer separation D for model systems expressed in terms of protein surface coverage fractions: (a) 0.01 β-lg, (b) 0.000 49 β-casein, (c) 0.000 245 β-casein + 0.009 755 β-lg, and (d) 0.000 49 β-casein + 0.009 51 β-lg.

experiments); (ii) 2.97 wt % β-lg + 0.03 wt % β-casein or Rs1-casein (i.e., 1% of the protein is casein); (iii) 2.85 wt % β-lg + 0.15 wt % β-casein or Rs1-casein (i.e., 5% of the protein is casein); (iv) 0.0 wt % β-lg + 0.15 wt % β-casein or Rs1-casein (i.e., β-lg is absent and only a small amount of casein is present). Figure 9 shows the calculated interaction potential A(D) (units of kT) for the β-casein + β-lg system as a function of the distance D between the planar surfaces (now measured in nanometers). The mixed system consists of polymer C chains (Figure 1c, representing adsorbed β-lg) tethered to the surface and variable amounts of the β-casein look-alike (Figure 8) also tethered to the surface. We observe that, for a pair of plates coated with β-lg alone (φp ) 0.01), in the absence of any casein, the SCF theory predicts a surfacesurface attraction of range D ≈ 2 nm (curve (a)). Similarly, for only a low density of grafted β-casein chains (φp ) 0.000 49), in the absence of any β-lg, the theory also predicts attraction (curve (b)), and this attraction is of even longer range (3-4 nm) than the pure β-lg alone. At this very low coverage in the absence of β-lg, there is insufficient β-casein to stabilize the system sterically, and attraction arises from polymer bridging. However, when the whey protein and β-casein are both present together, the theory predicts some stabilizing interlayer repulsion and a maximum in A(D) at D ≈ 1-2 nm. The height of the predicted “primary” maximum increases from ca. 1 kT for substitution of 2.5% of the β-lg by β-casein (φp ) 0.000 245) (curve (c)) to more than 15 kT for substitution of 5% of the β-lg by β-casein (φp ) 0.000 49) (curve (d)). These predictions confirm that, according to our model, the replacement of just a few percent of whey protein by β-casein tails can produce enough steric repulsion to transform the system from an unstable one to a stable one (colloidal potential barrier height > 15 kT).46 Figure 10 presents results of some equivalent calculations for the β-lg + Rs1-casein system. Curve (a) for the β-lg alone (φp ) 0.01) is exactly the same as in Figure 9 (but note the different vertical scale), and curve (b) for Rs1-casein alone (φp ) 0.0005) is rather similar to that for β-casein alone, i.e., attractive and of longer range than β-lg alone. However,

3026


(site)-1

Figure 10. Interaction energy A(D) in units of kT as a function of interlayer separation D for model systems expressed in terms of protein surface coverage fractions: (a) 0.01 β-lg, (b) 0.0005 Rs1-casein, (c) 0.000 25 Rs1-casein + 0.009 75 β-lg, and (d) 0.0005 Rs1-casein + 0.0095 β-lg.

Parkinson et al.

Figure 12. Average distance of each segment of β-casein from the isolated surface as a function of its position along the chain: (a) 0.000 49 β-casein, (b) 0.000 098 β-casein + 0.009 902 β-lg, and (c) 0.000 49 β-casein + 0.009 51 β-lg. The C-terminal group (designated segment number 1) is tethered to the surface.

indication of the magnitude of these DLVO contributions relative to the steric contributions calculated from SCF theory for Rs1-casein and β-casein can be found in Figure 11. The van der Waals interaction potential is calculated assuming a Hamaker constant of AH ) 5 × 10-21 J and the double-layer repulsive potential from UR(h) ) 2πr0aψ2[κ exp(-κh)][1 + exp(-κh)]-1 (8)

Figure 11. Direct comparison of the effects of (a) Rs1-casein and (b) β-casein on the interaction potential A(D) in mixed layers containing β-lg (data from Figures 11 and 12): (a) protein layer contains 5% Rs1-casein + 95% β-lg; (b) protein layer contains 5% β-casein + 95% β-lg. Also plotted for comparison are (c) the electrostatic double-layer repulsion at ionic strength 40 mM and (d) the van der Waals attraction for particles of diameter 0.42 µm (see text for further details).

the potentials for the mixed layers containing Rs1-casein (curves (c) and (d)) look quite different from those with β-casein. In particular, the presence of whey protein and Rs1-casein together at the surface leads to a predicted enhanced attraction of range D ≈ 2-6 nm. Substitution of 5% of the β-lg by Rs1-casein (φp ) 0.0005) does lead to some repulsion at short separations (D < 2 nm), but the height of the potential barrier in A(D) is less than half that in the equivalent system containing β-casein (see Figure 9), and in addition, there is an unfavorable “secondary” minimum of ca. 5 kT at D ≈ 3 nm. Hence, the SCF theory demonstrates that a small percentage of Rs1-casein, like that of β-casein, can in principle have a sterically stabilizing influence on an otherwise unstable β-lg-coated surface. Furthermore, the direct comparison in Figure 11 confirms our expectation that Rs1-casein should be less effective in this regard than β-casein, as was found experimentally.5 In addition to the steric interaction between protein-coated surfaces, the net colloidal interaction between a pair of emulsion droplets will involve other contributions. The most important of these are the van der Waals and electrostatic contributions of the DLVO theory.42,46 The colloid is deemed stable if the combined “electrosteric” repulsion (steric + electrostatic) dominates the van der Waals attraction. Some

where h is the interparticle separation, a is the particle radius (0.21 µm), r is the relative dielectric constant (78.4), 0 is the permittivity of free space, ψ is the surface potential (taken as -20 mV), and κ is the Debye parameter (κ-1 ) 1.5 nm at ionic strength ) 40 mM). We resist the temptation to add together these various contributions algebraically. We simply observe from Figure 11 that the van der Waals and doublelayer repulsions are of roughly similar ranges and opposite signs and that the additional steric repulsion introduced by the β-casein is very definitely significant relative to the net DLVO contribution. Furthermore, the difference between the SCF theory predictions for Rs1-casein and β-casein is highly significant compared with UR(h) at h ≈ 2-3 nm. A powerful feature of the SCF technique is that it can provide information on the equilibrium distribution of individual segments along the polymer chains, as well as the overall polymer density distribution. In the present context, useful insight into the relative stabilizing abilities of Rs1-casein and β-casein can be gained from monitoring the effect of system conditions on the average locations of the individual protein segments within the interfacial film. Figure 12 shows the effect of the presence of β-lg on the average distance from the surface as a function of the β-casein segment number. When present alone at low surface coverage (φp ) 0.000 49), the hydrophilic region (the end 40-50 residues) lies relatively flat and close to the surface (

Using Self-Consistent-Field Theory to Understand Enhanced Steric

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