Chapter 5
Interfacial Phenomena
Intermolecular Forces
All atoms and molecules attract one another due to van der Waals forces which arise because of an uneven distribution of electrons around an atom or a group of atoms so that the positive and negative centres do not coincide. The result is the production of a dipole whose behaviour can accurately be described only by recourse to quantum mechanics. The van der Waals forces may be classified into three types:
1. Keesom attraction between polar groups, that is between permanent dipoles. This results from the preponderance of attractive over repulsive orientations since, during rapid thermal motion, dipoles are closer when under attractive orientation.
2. Debye attraction between a permanent dipole and the dipole which it induces in another group.
3. London attraction between individual atoms whereby even nonpolar molecules attract each other. Although such atoms or molecules do not possess a dipole moment, they do possess a dipole which is rapidly fluctuating due to the motion of the electrons and whose frequency is about 1015 to 1016 s−1. Since the time average of the electron distribution is symmetrical, the effect of these transitory dipoles would be self-cancelling were it not for the fact that they induce an in-phase relationship with neighbouring transitory dipoles. It is this effect which enables even the inert gases to liquefy and is responsible for the refractive dispersion of light. Because of the latter phenomenon, the London force is frequently referred to as the dispersion force.
All three components therefore have an electromagnetic origin which gives rise to a potential energy field U around the atoms. Because the attraction (negative by convention) potential varies inversely as the sixth power of the distance r between the interacting atoms or groups, the attractive force varies inversely as the seventh power since the force F is the gradient (Fig. 5.1)
Fig. 5.1 Potential energy curve for a pair of hydrogen molecules
At close distances of approach, interpenetration of the electron clouds produces a very large repulsive energy (Born repulsion) which varies approximately with r−12. The potential energy function may therefore be written (Lennard–Jones 6–12 potential)
where A and B depend on the nature of the molecules. Because the force of attraction varies with r−7, the influence is felt over only a fraction of a nanometre. The van der Waals forces are therefore short-ranged.
The relative contributions of the three components are listed for a few simple molecules in Table 5.1. Generally, the London component predominates but the Keesom component is excessive with highly polar molecules such as water. In such cases the Keesom force is particularly strong because the hydrogen atom shares only two electrons in the molecule so that it can approach very close to the electronegative atom of a neighbouring molecule thereby forming a hydrogen bond. The close approach of the hydrogen proton allows it to pull closer to itself the electron cloud around the electronegative atom of the other molecule. The resulting hydrogen bond strength will depend on the electronegative atoms (generally oxygen, nitrogen or fluorine) on either side of the hydrogen proton and upon the inductive effect of other groups. Its magnitude is usually between 10 and 30 kJ.mol−1 and is therefore appreciable although still only a fraction of the strength of covalent bonds which are mostly 200–800 kJ.mol−1
Table 5.1 Relative magnitudes of intermolecular interactions
Hydrogen bonding produces molecular association which, in the case of hydrogen fluoride and formic acid, is strong enough to persist even in the vapour state if the temperature is not too high. Water would boil at about 203K were it not for its association into aggregates of higher molecular weight. Since each oxygen atom of water has two lone pairs of electrons, it can form two hydrogen bonds in addition to its two covalent bonds. These four bonds are disposed around the oxygen atom towards the corners of a tetrahedron and enable the production of polymeric clusters of water molecules which continually break and reform under thermal agitation with a lifetime of about 10-11 s. This structuring of water confers upon it various anomalous properties such as a high surface tension and dielectric constant and the abnormal effect of temperature and pressure upon its density and viscosity.
Besides producing intermolecular association, the hydrogen bond can form intramolecular bridges. Thus o-nitrophenol forms a bond between the nitro–oxygen and phenol–hydrogen atoms and consequently it remains unassociated and boils at 214K. Because the hydrogen bond is internal the compound is only sparingly soluble in cold water but volatile in steam. The groups of p-nitrophenol, on the other hand, are not close enough to form a hydrogen bond. This compound therefore self associates and boils at the higher temperature of 295K. Since it can form hydrogen bonds with water it is moderately soluble in cold water but it is only slightly volatile in steam. Internal hydrogen bonds play a major role in maintaining the specific configuration of many proteins and their rupture by chemical or thermal means leads to denaturation.
The van der Waals forces are directly involved in the existence of liquid and solid states whereby molecules are held close together and occupy some specific volume depending upon the prevalent temperature and pressure. Because of this a boundary must exist between a condensed phase and a gas or between two condensed phases. Generally speaking, a boundary with a gas is known as a surface, other boundaries being referred to as interfaces. There is however no strict terminology and any boundary may be referred to either as a surface or as an interface.
Liquid–Gas and Liquid–Liquid Interfaces
Surface Tension and Surface Energy
A brief consideration of the kinetic energy of molecules will lead one to realize that the surface of a liquid is in a state of violent agitation with molecules leaving and entering from both the bulk liquid and vapour phases. Calculations show that, for water at any rate, the average residence time of a molecule in the surface is of the order of microseconds. Notwithstanding this, strong cohesion between the molecules causes the position of the surface to be definable to within a few molecular thicknesses (Davies & Rideal, 1961).
Molecules within the bulk of a liquid are subjected to attractive forces equally in all directions whereas those at tile surface have very little attraction from the vapour and so experience a net pull into the liquid. Molecules very near the surface also experience a small net inward attraction and so the surface should be considered more as a region about two or three molecules thick rather than a monomolecular layer. The imbalance of forces results in some molecules being withdrawn into the bulk, thereby slightly increasing the intermolecular distance within the surface giving the latter the character of a stretched elastic membrane, that is the surface is under tension. Such a surface will therefore tend to contract spontaneously and cause droplets or air bubbles to assume a spherical shape.
The surface tension γ is defined as the force, in mN, acting within the surface in a direction normal to any line of length 1m. A force of γ mN.m−1 must therefore be applied tangentially to part that surface.
If the surface area of a given volume of liquid is increased isothermally, work must be done to bring molecules into the surface against the inward pull of the bulk liquid. The work required to expand the surface by 1m2 is γ mJ and therefore the excess free surface energy of a liquid is γ mJ.m−2, which is numerically equal to the surface tension. At constant pressure, the free surface energy per m2 is the excess Gibbs free energy Gs, where the subscript denotes the extra energy of the molecules by virtue of being in the surface. Thus γ = Gs. Although both the surface tension and the excess free surface energy are numerically equal and have the same dimensional units of MLT−2, they are not identical in a qualitative sense since the tension may be considered as the result of the excess energy.
Because surface molecules are not attracted away from the bulk as strongly as they are into it, there is a residual potential attraction energy which constitutes the surface enthalpy Hs mJ.m−2. The molecules in the region of the surface, however, have a greater freedom of movement and so there is an excess entropy in the surface of Ss mJ.m−2.deg−1. Thus if the surface were destroyed, not all of the surface enthalpy would be free to do work and therefore:
Gs is therefore a measure of the thermodynamic instability of the surface and is that part of the surface enthalpy which is capable of doing mechanical work. At very high viscosities, however, the relationship between surface free energy and surface tension no longer holds.
Relationship of Surface Tension and Temperature
The surface tension of almost all liquids decreases with a rise in temperature, an almost linear relationship being found. It follows that the surface tension must vanish at some finite temperature this being the critical temperature Tc above which the cohesive forces are no longer capable of maintaining the integrity of a distinct vapour phase. Since surface tension is a function of the distance between the molecules it is dependent upon the liquid density and Eotvos (1886) suggested the relationship:
where M is the molecular weight ρ is the liquid density and K1 is a constant. van der Waals (1894) proposed a power law of the type:
where B is a constant for a given liquid. B was given a value of 1.5, but this has been shown to vary for different liquids, a value of 1.2 being appropriate for some.
The linear extrapolation of γ(M/ρ)2/3 versus T, however, does not cut the T axis for zero γ at Tc but at a lower temperature (Tc − d) where d is about 6°C. The Eotvos equation was therefore modified by Ramsey and Shields (1893) as:
This correction was shown by Katayama (1916) to be unnecessary if the influence of the vapour upon the surface tension of a liquid is taken into account so that:
where ρl, and ρg are the orthobaric densities of the liquid and vapour, respectively. The constant K4 has a value of about 210 nJ.deg−1 unless there is strong association between molecules when a lower value is found that varies widely between different liquids. Water, for example, gives a value of about 89 nJ.deg−1 at room temperature.
By combining the equations of van der Waals and Katayama and giving B a value of 1.2, Ferguson (1923) deduced that:
where C is a constant for a given liquid. Such an equation was also found by Macleod who confirmed the constancy of C experimentally. If both sides of this last equation are multiplied by the molecular weight and ρg is neglected as being small compared with ρp, the following expression is obtained, which is independent of temperature:
where [P] is known as the parachor (Sugden, 1924b). The parachor may be regarded as the molar volume when the surface tension is unity and affords a useful means of comparing the molar volumes of different liquids since the comparison is made under conditions where the molecular attractions are approximately equal. An important property of the parachor is its additivity since it has been found to be composed of sets of constants which relate both to the atoms composing the molecule and to the way in which the atoms are bonded together. Its importance in elucidating molecular structure has now, however, been completely superseded by other techniques.
Effect of Surface Curvature
The tension forces in a planar surface are balanced within the plane of the surface. Any curvature to the surface produces a resultant normal force which becomes balanced by a pressure change ΔP within the liquid such that
This is the Young–Laplace equation where r1 and r2 are the principal radii of curvature.
Curvature of a liquid surface influences the vapour pressure since it alters the distribution of the attractive forces around the molecules in the surface. A difference in chemical potential can readily be shown to exist between liquid drops of different size since there is a decrease in free surface energy upon mass transfer from a smaller to a larger drop via the vapour phase. Assuming ideal behaviour, the vapour pressure Pr of a spherical drop of radius r compared with that (P) over a plane surface is given by the Kelvin equation:
ρ being the liquid density and M the molecular weight of the vapour Values calculated for water are given in Table 5.2.
Table 5.2 Calculated effect of drop size on vapour pressure of water at 293 K |
|
| Radius of drop (nm) |  |
| 1000 | 1.001 |
| 100 | 1.010 |
| 10 | 1.110 |
| 1 | 2.940 |
It will be noted that the vapour pressure is affected appreciably only at high surface curvature, that is with very small drops. This however is sufficient to render vapour condensation in a dust free atmosphere very difficult unless the atmosphere is highly supersaturated, any condensate nuclei tending to evaporate rather than to grow.
Derivation of the Kelvin equation assumes that the surface tension remains independent of surface curvature. It is to be expected that high curvature, measurable in terms of molecular dimensions, would diminish the surface attraction of molecules thereby resulting in a decrease in the tension. This decrease however is probably small even with droplets consisting of only a few molecules.
Interfacial Tension
When a two component system separates into two liquid phases an interfacial region will exist in which the molecules are subjected to unbalanced forces. This leads to an interfacial tension in just the same way as surface tensions are produced. Incomplete miscibility of the two components arises because of a difference in the attraction forces between the two kinds of molecules and it is this difference which results in the imbalance of attraction forces across the interface.
All spontaneous processes result in a decrease in free energy. It has already been noted that the integrity of the condensed state is due to the van der Waals forces and that these arise due to the loss of energy as molecules come together. Further, it was seen that in some liquids such as water, intermolecular attraction greatly exceeds that due to the London interaction owing to the additional strong Keesom interaction in the form of the hydrogen bond. Interference with hydrogen bond formation by nondipoles would therefore prevent the attainment of the minimum energy state and so the spontaneous process would be a rejection of apolar molecules causing them to occupy some position not interfering with the hydrogen bonds. This means that a separate phase will be formed when water is ‘mixed’ with a hydrocarbon such as octane or liquid paraffin.
Tetrahedral hydrogen bonding which occurs in ice is retained to a lesser extent in liquid water giving rise to ‘flickering clusters’ of structured water (Davies & Litovitz, 1965; Erlander, 1969; Blandamer, 1970). When a solute molecule is dissolved in water it may either (1) substitute for a water molecule in the lattice-like network when there will be competition with solvent molecules for the ‘lattice sites’ or (2) occupy a cavity in the solvent structure. The latter type of ‘interstitial solution’ occurs when nonpolar solutes are dissolved in highly polar solvents, water molecules forming polyhedral structures around the solute molecules. This means that small hydrocarbon molecules can have a slight solubility in water, the solubility being governed by the size of the solute molecule and that of the cavities in the water structure (Abu-Hamdiyyah, 1965). The mean size of the cavities in water is about the size of a C4 or C5 hydrocarbon. However, the microscopic iceberg of structured water which forms around nonpolar molecules (Frank & Evans, 1945) involves a decrease in entropy which overrides the negative enthalpy of dissolution thereby producing an unfavourable positive free energy of dissolution.
If there was no attraction across a water–paraffin oil interface, the interfacial tension γow would be given by the sum of the separate surface tensions since these act together, that is γow = γo + γw. Water and paraffin molecules do, however, attract each other by London interaction, the attraction being given by the geometric mean of the London dispersion force components γd of the surface tension (Fowkes, 1964). Thus for paraffin and water, the attraction across the interface is 
, where 
and 
denoting the polar interaction component. The separate tensions are thereby reduced to:
, where and denoting the polar interaction component. The separate tensions are thereby reduced to:
for the oil and water sides of the interface, respectively, and the interfacial tension is given by their sum:
(5.1)
The superscript in 
denotes that only the dispersion force interaction across the interface is taken into account
denotes that only the dispersion force interaction across the interface is taken into accountConsider now an aqueous solution of a solute whose molecules possess both apolar and polar groups, e.g. a fatty acid or alcohol: the solute and solvent will be miscible in all proportions provided that the apolar group is small as in acetic acid or ethanol (C2). If the hydrocarbon group is larger it will interfere more with hydrogen bond formation and cause a greater structuring of the water around it. Butyric acid and butanol (C4) are therefore only partially miscible with water at room temperature and a mixture where neither component is in large excess will separate into two phases. Longer hydrocarbon chains curl up in order to decrease their contact with water. This effect leads to a decreased entropy of dissolution per CH2 group with increasing chain length.
Since there is polar interaction between the different molecules, the tension of the interface within a two-phase system will be considerably lower than that predicted by Eq. 5.1. Some computed values are given in Table 5.3.
Table 5.3 Effect of interfacial interaction on the tension between water and various organic liquids at 293 K (values in mN.m−1)
Within experimental error, there is seen to be no polar interaction with saturated hydrocarbons. In the case of benzene, a dipole is induced in the π bonds giving a further lowering of 16.5mN.m−1 and a resultant interfacial tension of 35.0mN.m−1. With the last three compounds of Table 5.3, hydrogen bonding involves a much stronger polar interaction with water and consequently leads to low interfacial, tensions. Since an enhanced polar interaction leads to a greater tendency for miscibility of oil with water, the interfacial tension will parallel the mutual insolubility of the two components. An approximate means of calculating interfacial tensions from surface tensions is afforded by Antonoff’s rule which states that the interfacial tension is equal to the difference of the surface tensions of the mutually saturated phases:
While the rule holds for slightly polar oils such as benzene or chloroform against water, it does not hold for more polar oils such as octanol nor for systems with a negative spreading coefficient (see below).
Cohesion, Adhesion and Spreading
If two halves of a column of pure liquid were to be pulled apart, two new liquid–air surfaces would be created. The free energy of the system is therefore increased by 2γ mJ for each m2 cross section of original column, this increase being known as the work of cohesion. For oil and for water the work of cohesion is, respectively:
(5.2)
When an oil and a water surface are brought together, there is a residual energy, γow mJ.m−2, at the interface. To separate these two surfaces, work of adhesion must be supplied for each m2 of interface, that is:
(5.3)
this being known as the Dupre equation.
If a drop of pure liquid paraffin is placed on water, it forms a coherent lens (Fig. 5.2). Let Ao and Aw be the oil–air and water–air surface areas, Aow the oil–water interfacial area and x the horizontal diameter of the lens. Suppose the lens expands over the water surface by an infinitesimal amount dx. The increase in surface energy at the oil–air and oil–water interfaces are respectively γodAow and γowdAow and the increase at the water–air surface is γwdAw. In the latter case dAw is negative since water–air surface is being lost at the expense of the expanding oil lens.
Fig. 5.2 Oil lens–water system parameters
The total surface energy gained by the lens expanding dx is therefore:
It follows that if there is a net loss in surface energy upon expansion of the lens:
and the process of expansion will be spontaneous.
As the diameter x of the lens increases however, the oil–air and oil–water interfacial curvatures decreases so that dAo/dx and dAow/dx increase as x increases. In the case of liquid paraffin on water, a droplet of the oil would expand into a lens until the energy gained by increasing Ao and Aow just balanced that lost by decreasing Aw:
(5.4)
Further expansion of the liquid paraffin lens would increase dAo/dx and dAow/dx still more and involve a net increase in surface energy. The lens therefore acquires an equilibrium diameter with a minimum energy.
Since a lens involves curvature of the oil–air and oil–water surfaces and since also most of the oil floats below the free water–air surface level:
There fore for a lens to occupy a finite equilibrium diameter in accordance with Eq. 5.4, γw −γo −γow <0. For liquid paraffin on water we have 72.8 − 31.8 − 55.6 = −14.6mJ.m−2, clearly a condition for the formation of a stable lens.
Suppose a drop of a polar oil such as octanol is placed on water. For octanol on water at 293K we have: γw −γo − γow =72.8 − 27.5 − 8.5 = +36.8mJ.m−2. Such a lens will continue to expand given enough water surface to do so since the condition for minimum surface energy (Eq. 5.4) cannot be reached. As the lens expands, the oil–air and oil–water surface curvatures approach zero and tend towards the condition –dAw/dx = dAo/dx = dAow/dx. Equilibrium in the form of an extremely thin lens would therefore require the condition (from Eq. 5.4) dA / dx(γw −γo −γow) = 0 then:
(5.5)
Clearly this situation is not achieved with octanol on water and so the oil will spread to form a monolayer, that is, a film one oil molecule thick.
The condition given by Eq. 5.5 therefore forms the dividing line between those systems where immediate spreading towards a monolayer occurs and those whose immediate tendency is to form a lens. The initial spreading coefficient Sinit is defined as:
If Sinit is positive there is an immediate tendency to spread whereas if Sinit is negative as with liquid paraffin on water, spreading does not occur. Table 5.4 lists a few values for the initial spreading coefficient on water.
Table 5.4 Initial spreading coefficients on water at 293 K (values in mJ.m−2)
Combination of Eqs. 5.2, 5.3 and 5.6 gives Sinit = Wow − Wo showing that an oil will spread on water only if it is sufficiently polar to adhere to the water more strongly than it coheres to itself.
The values quoted in Table 5.4 refer only to the initial condition of the system. However the two liquids tend to become mutually saturated, with the result that γw is often found to decrease significantly. Thus although benzene is initially capable of spreading on water, γw becomes reduced to 62.4mJ.m−2 and γo becomes 28.8mJ.m−2. Thus Sfinal = 62.4 − 28.8 − 35.0 = −1.4mJ.m−2 and the benzene-rich phase forms flat lenses with the rest of the water covered by a monolayer of benzene having a surface film pressure π of 10.4mN.m−1 i.e. γw(o) = γw − π = 72.8 − 10.4 = 62.4mN.m−1.
So far only the energetics of the system have been considered but reference to Fig. 5.2 will show that there must also be a balance of forces at the triple point where the three phases, air–oil–water, meet. A condition for equilibrium is:
If it is assumed that the water–air surface is flat and that the oil–air and oil–water interfaces have spherical curvature since this presents the least surface area for a given diameter of the lens, then there will almost certainly be an imbalance of forces as depicted in Fig. 5.3(A). For liquid paraffin on water, θo = 70°, θw = 156° and θa= 134°; therefore, there must be some considerable distortion around the edge of the lens as shown in Fig. 5.3(B).
Fig. 5.3 Liquid paraffin lens floating on water showing necessity for edge distortion: (A) Nonequilibrium; (B) Equilibrium.
It has been seen that a good sample of liquid paraffin cannot spread on water. However, if its trace impurities are oxidized by heating it may spread to a multilayer and show interference colours. This multilayer is known as a duplex film, so called because the oil–water and oil–air interfaces are still sufficiently apart to act independently. Similar behaviour is observed if sufficient ‘spreader’ such as a fatty acid or fatty alcohol is dissolved in liquid paraffin. Thus with 0.5 per cent cetyl alcohol (tetradecanol) the condition for initial spreading is achieved: Sinit = 72 − 32 − 24 = +16mJ.m−2 and given a large water surface, the drop will spread to a duplex film. The high spreading power of fatty alcohols leads to monolayer formation of the alcohol in advance of the spreading oil and if the water surface area is small, the monolayer soon covers the water surface and reduces its tension to 48mN.m−1. This produces a negative spreading coefficient: Sfinal = 48 − 32 − 24 = −8mJ.m−2 which results in the oil contracting into one or more lenses surrounded by a monolayer of the alcohol. The sudden lowering of the surface tension around the spreading lens will cause the forces at the edge to become unbalanced. This produces a sudden distortion of the surface around the edge which may be so violent that the distortion is propagated as a wave throughout the lens causing it to shatter into several hundred small lenses which may subsequently coalesce.
Measurement of Surface and Interfacial Tension
Methods for measuring surface and interfacial tension differ in their suitability for particular purposes. Although the surface of a simple liquid attains an equilibrium tension within a few milliseconds of its formation, the surface of some solutions continues to age for a long time due to slow diffusion of molecules to the surface or molecular reorientation at the surface. It follows that those methods which involve enlargement and breakage of the surface cannot always be used to measure equilibrium tensions. The methods summarized below have been discussed in detail by Adamson (1967).
Capillary Rise
Fig. 5.4 Capillary rise parameters
The difference in pressure ΔP between the bottom of the meniscus O and point A is given by
(5.7)
where Δρ is the density difference of the liquid and the air and b is the radius of curvature at O. Tile term on the right hand side is given by the Young–Laplace equation. At some other point on the meniscus of elevation z above O
(5.8)
R1 and R2 being the principal radii of curvature.
Because ΔP varies with z, the curvature of the meniscus is not uniform and only for very narrow capillaries, when h << ho can γ be calculated from Eq. 5.7 since then b ≈ r. Usually measurement of ho still leaves b unknown and in order to determine γ from Eq. 5.7, a relation between b and r is required. Use is made of the so called capillary constant a2 defined as a2 = bho. For zero contact angles, Sugden (1921) compiled tables of r/b versus r/a and a method of successive approximations may be used to arrive at the value of b.
Although the theory is simple, there are a number of experimental difficulties. Thus it is necessary to ensure that:
A zero contact angle may not be achieved with some solutions due to adsorption of solute on to the glass. With very fine capillaries a further complication due to adsorption is reduction in solute concentration and a consequent change in surface tension.
The necessary condition for zero contact angle may be tested by approaching the equilibrium height ho with a succession of rising and falling menisci. If ho is the same in each case there is perfect wetting. Even so, a rising meniscus may give rise to a nonzero contact angle (see p. 84) and therefore it is best to use data from falling menisci.
In order to eliminate errors due to slight variation in the bore of the capillary, the tube may be immersed in the liquid to a depth sufficient to locate the meniscus at a position where the area of cross section is accurately known. Alternatively, the hydrostatic pressure needed to restore the bottom of the meniscus to the level of the external plane surface may be determined. The method may be adapted for the measurement of interfacial tensions.
Maximum Bubble Pressure
A capillary tube dips into the liquid to an accurately known depth and the minimum pressure is determined at which bubbles of an ‘inert’ gas are just able to grow and detach from the tip. If the bubble formed a surface of spherical curvature, the curvature would be a maximum when a hemispherical bubble had formed and at this point the maximum pressure would be recorded. Eq. 5.7 would therefore be applicable. The pressure difference across the bubble surface however, varies with variation in depth of the meniscus and so the bubble surface will not have spherical curvature. As in the case of capillary rise, Sugden (1922) has published correction factors.
The technique has the advantage that zero contact angle is not necessary. It is however necessary to know whether or not the liquid wets the tube, that is, whether the bubble forms upon the internal or external cross section. Since the method involves an expanding surface, surface ageing cannot be studied, a bubble rate of about 1 S−1 being used. A variation in which two tubes of different diameter are used has also been described by Sugden (1922, 1924a).
Sessile Drop
Fig. 5.5 represents a drop of liquid resting on a nonwetting surface (contact angle greater than 90°). The external fluid may be air or another liquid phase. Alternatively, the figure may be visualized in an inverted position where it represents either a drop of a less dense liquid or a bubble of air.
Fig. 5.5 Sessile drop of water in liquid paraffin.
Given that b is the radius of curvature at the vertex O, and that R1 and r are the radii of curvature at point A in the vertical and equatorial planes respectively, then from Eq. 5.7:
Although neither b nor R1 are known, it has been shown that:
where Δ is a nondimensional correction factor which varies with h and r. The variation of Δ with drop size was determined by Porter (1933) who found that the curve (Fig. 5.6) was fitted by the empirical equation:
whence
The correction term is zero when r = 2h.
Fig. 5.6 Curve of correction factor versus shape factor for the sessile drop method (Porter, 1933).
In order to determine h and r, the drop may be photographed or projected on to a screen. Whilst the correction term only requires determination of the ratio h/r, the absolute value of h is also required. A convenient method is to suspend a rod of known diameter over the vertex of the drop and from the diameter of the image to calculate the enlargement, whence h can be calculated from the size of the drop image (Shotton, 1955).
Significant errors can arise from inaccurate measurement of the drop size parameters but these errors can be minimized using large drops. The main sources of error arise from the difficulty in accurately locating the equatorial plane and from deficiencies in the optical system employed.
Another source of error may be inherent in the liquid being examined. If it is a solution which possesses some sort of structure (exhibits plastic flow), the drop may not flatten to the shape corresponding to the true surface tension. Evidence for this was found by Shotton and Kalyan (1960) when determining the interfacial tensions between benzene and gelatin or sodium alginate solutions. An apparently higher tension was found if a drop of solution was formed in benzene instead of a drop of benzene in solution.
Pendant Drop
Various ways of calculating surface and interfacial tensions from the shape and size of pendant drops, as shown in Fig. 5.7, have been suggested. Eq. 5.8 may be written
(5.9)
b being the radius of curvature at O, the vertex of the drop. β is a dimensionless quantity which depends upon the shape of the drop as does the ratio S = ds/de defined in Fig. 5.7. Therefore S is a function of β. For a given value of S, de is proportional to b and so Eq. 5.9 leads to
where H is a function of S. Measurement of de and ds allows calculation of S from which H may be calculated by the method of Fordham (1948) or read from tables given by the same author and extended by Stauffer (1965).The latter author has shown that greatest accuracy is achieved with S values close to unity.
Fig. 5.7 Pendant drop of water in air.
Small errors in estimating de can lead to considerable errors in ds. This can be minimized by determining the ratio R of the maximum (de) and minimum (dm) diameters which are less dependent on height than is ds. H may then be calculated or read from tables (Winkel, 1965).
Photographic or projection methods are used to determine the shape and size parameters (Douglas, 1950; Campbell, Christian & Eaton, 1955). The pendant drop is more suitable than the sessile drop method for viscous liquids since there is only a small fraction of the drop area in contact with a solid surface. Compared with the sessile drop, viscous drag has less effect in slowing the attainment of hydrodynamic equilibrium. A check for such an equilibrium can be made by measuring the pendant drop diameters at several heights and comparing the drop shape with that predicted by theory (Roe, Bachetta & Wong, 1967).
Drop Volume
If the pendant drop shown in Fig. 5.7 is slowly enlarged, it will grow to a critical length at which it becomes unstable and the waist collapses. Fig. 5.8 shows successive stages in the growth of a drop up to the point of instability, detachment of the drop being illustrated in Fig. 5.9. The extra small drop which is produced arises because of the mechanical instability of the neck. It is evident that not the entire drop becomes detached and as much as 40 per cent may remain attached to the tip.
Fig. 5.8 Girth of a drop of water in liquid paraffin.(Diameter of needle tip is 1.06mm giving γow = 57mNm−1.)
Fig. 5.9 Successive stages in detachment of a drop.
If it is assumed that the surface tension acts vertically, then at the moment preceding detachment mg = vΔρg = 2πry where m and v are the effective mass and the volume of the drop respectively and r is the radius of the tip of the tube. This equation, however, fails to recognize that the volume of the drop is also a function of its shape and that only part of the drop becomes detached.
The shape depends on the ratio between some linear dimension of the tip of the tube such as r and a linear dimension of the drop such as v1/3. Thus:
(5.10)
where f(r/v1/3) is the fraction of the ‘ideal drop’ which actually falls. Harkins and Brown (1919) determined experimentally the correction values plotted in Fig. 5.10. In the case of surface tension measurements, the mean weight of a number of drops may be used to calculate the drop volume v; alternatively, as in the case of interfacial tension measurements, the volume is measured directly by means of a micrometer syringe. Thus r/v1/3 can be calculated and the correction factor read from the graph. Substitution in Eq. 5.10 gives γ. Since the correction factor varies only slightly with change in v near the minimum of the curve (Fig. 5.10), tubes should be selected so that r/v1/3 is as close to the minimum as possible.
Fig. 5.10 Curve of correction factor versus drop shape factor for the drop volume method (Harkins & Brown, 1919).
If the drop is formed too rapidly it becomes too heavy because the waist of the detaching drop gets expanded by the stream of liquid issuing from the tube. The drop must therefore be formed slowly, at least three minutes being required. In practice, when using a micrometer syringe, a rough estimate of the drop volume is obtained rapidly and subsequently about 95 per cent or more of the drop is quickly formed and the process completed slowly over a period of about a minute. In order to know the effective diameter of the tip when determining interfacial tensions, one must make sure that it is completely wetted by one of the liquids. If the tip is wetted by the external liquid, care must be taken to ensure that the interior of the tube is sufficiently wetted by the liquid forming the drop or the point of detachment of the drop may be inside the tube instead of at the tip. Where the drop liquid wets the tip, care must be taken that it does not climb up the outside of the tube and to prevent this, tubes with sharp edges should be used. Whatever the shape of the tip, smoothness is important. Stainless steel hypodermic needles with the ends ground flat are generally suitable. If necessary, they may be rendered hydrophobic by being coated with a film of ferric stearate or silicone.
Du Nouy Ring Tensiometer
This method involves measurement of the force required to detach a platinum wire ring from the liquid surface or from the interface between two liquids. In the latter case the ring must be preferentially wetted by the denser liquid. Careful cleaning of the ring in strong acid or by flaming is essential when it will be preferentially wetted by water. For the determination of γow with an oil as the denser phase, the ring may be rendered hydrophobic with silicone, a pull P exerted on the ring results in some of the liquid being raised (Fig. 5.11), the pull being balanced by the weight of liquid. When the upward pull and the weight of liquid exceed the surface tension force, the ring will become detached. If it is assumed that the surface tension acts vertically on the ring with zero contact angle,
(5.11)
where Pmax is the maximum pull on the ring, m is the mass of liquid raised above the free surface of the liquid and R1 and R2 are the radii of curvature of the inner and outer perimeter of the ring.
Fig. 5.11 du Noüy ring parameters
However, Fig. 5.11 clearly shows that the surface tension does not act vertically on the ring and Harkins and Jordan (1930) showed that Eq. 5.11 gives values of γ which may be in error by up to 45 per cent. They determined experimentally the correction factor F so that:
which reduces to
if r, the radius of the wire, is small so that R1 = R2 = R.
The volume V of liquid upheld by the ring becomes a maximum at the moment of detachment when it will possess a certain shape which is determined by a function of the three variables R, r and V. Determination of F therefore involves the calculation of the two dimensionless quantities R3/V and R/r. Zuidema and Waters (1941) found an empirical relationship which enables the tabulated values of F given by Harkins and Jordan (1930) to be extended for conditions frequently met when determining interfacial tensions. Fox and Chrisman (1952) have also extended the correction tables in the other direction, that is for liquids of low tension but high density where the empirical relationship of Zuidema and Waters was found to be invalid.
In order to obtain accurate data, the following conditions must be met:
1. the ring must lie in a flat plane,
2. the plane of the ring must be horizontal, a tilt of only 1° causing an error of 0.45 per cent,
3. the diameter of the dish must be not less than 8cm since curvature of the meniscus due to the edge of the dish leads to low determined values of γ. When determining interfacial tensions in circumstances where the ring is unavoidably wetted by the upper liquid the force required to push the ring down through the interface may be measured.
Wilhelmy Plate
A thin rectangular plate of glass or mica is suspended vertically from a torsion balance. A dish of liquid is raised under the plate until the liquid surface just contacts the bottom edge of the latter when the surface forces will drag it downwards. The force exerted around the perimeter of the plate may then be measured by the rotation of the torsion wire required to restore the plate to its null position, that is, so that the bottom edge of the plate is coincident with the plane of the free liquid surface (Fig. 5.12).
Fig. 5.12 Plate in null position seen edge on.
Provided that the liquid makes zero angle of contact with the plate
(5.12)
where L and T are respectively the length and thickness of the plate in the horizontal plane and mg is the weight of meniscus liquid. To ensure that L is the measured length of the bottom edge of the plate, the edge must lie in a horizontal plane. This is readily checked when the liquid is raised close to the plate since the edge and its reflection in the liquid surface will then appear to be parallel.
An alternative method, which is particularly useful for following changes in surface tension, is to immerse the plate partially in the liquid and to record the depth of immersion by means of an optical lever. A change in rotation of the torsion wire may then be corrected for the change in buoyancy of the plate to give an accurate measure of the change in the surface force.
A zero contact angle is essential. This may be promoted by careful roughening of the surface of the plate so that the very fine grooves are orientated in all directions (Jordan & Lane, 1964). For airs, a roughened mica plate coated with lampblack is suitable. It is always best to use a receding contact angle (see p. 84) and so, for following decreases in interfacial tension, the lower (denser) phase should be made to wet the plate preferentially.
It is also necessary to use very thin plates since the effective perimeter will tend to differ slightly from 2(L + T) due to the meniscus being continuous around the corners of the plate.For plates not much thicker than 100μm, however, Eq. 5.12 is sufficiently accurate that it may be used without correction.
Surfactants
If a single molecule or ion contains localized regions one of which is lyophibic and the other is lyophobic, the lyophilic part will favour dissolution (affinity for solvent) and the lyophobic part will favour immiscibility (antipathy for solvent). Both of these characters existing in the same molecule or ion red Hartley to coin the term amphipathy.The amphipathic character becomes particularly pronounced if the lyophilic and lyophobic groups are well separated. When considering water as the solvent the groups are termed hydrophilic and hydrophobic. The latter term does not indicate repulsion between the group and water since there is attraction due to dispersion forces, but rather it implies that interference with hydrogen bonding of the water produces a tendency for the hydrophobic group to be expelled to form a separate phase. Since hydrophobic groups tend to be dissolved by nonpolar solvents they are also termed lipophilic. Thus an amphipathic compound contains both hydrophilic and lipophilic groups and therefore may also be called amphiphilic. As a result of the amphipathic character, such compounds will tend to concentrate at the surface of a solution or at the interface between two immiscible liquids or between a liquid and a solid. Such compounds are therefore frequently referred to as surface-active agents or surfactants. Those surface-active agents which are used for removing dirt from solid surfaces are also commonly referred to as detergents. Under suitable conditions of solvent, temperature and concentration, amphipathic compounds spontaneously aggregate to form colloidal structures known as micelles. This spontaneous tendency for association has led to the term association colloid.
Classification of Surfactants
This is made according to the nature of the polar group, which may or may not ionize. There are therefore ionic and nonionic surfactants. Ionic surfactants are further divided into anionic, cationic and ampholytic (or amphoteric) according to the charge carried by the surface-active ion. The charge of an ampholytic ion depends upon the pH, the zwitterion being shown in Table 5.5.
Table 5.5 Examples of surface-active agents
Surface Activity
In dilute aqueous solutions of amphipaths, the hydrophobic group of those solute molecules in the region of the surface will not always be able to re-enter the bulk solution as readily as it entered the surface. Thus, if random motion brings the hydrophobic group into the surface, the amphipathic molecule will stick there (Fig. 5.13) until it is supplied with sufficient suitably directed kinetic energy for it to re-enter the bulk solution. The strength and direction of the force or ‘kick’ needed for desorption will depend on the structure and orientation of the amphipathic molecule and the structure of its immediate environment at that particular instant in time. During the time of residence of the molecules at the surface, other molecules will also arrive there and so there will be an excess of solute molecules at the surface compared with those in a similar volume of bulk solution containing no surface region.
Fig. 5.13 Diagram of surface activity. (Solvent molecules not depicted.)
An increase in size of the hydrophobic group results in increased difficulty of desorption. This means that the average time of residence of solute molecules at the surface/becomes longer and the surface excess will be greater as a result. Indeed, long-chain fatty acids and alcohols (C10 and above), even in extremely dilute systems, reside almost entirely at the surface forming a surface film one molecule thick (monolayer) which can be considerably compressed by a mechanical barrier (see p. 70) before it will collapse into a surface multilayer.
Adsorption must involve a certain degree of orientation within the surface this being opposed by random rotational motion which favours desorption. Perfect orientation will therefore not be achieved. Although the entropy change of adsorption is negative with very short hydrocarbon chains, thus favouring desorption, it becomes increasingly positive with longer chains (Table 5.6). This shows that factors other than orientation of the solute molecules in the surface are important. When surfactant molecules are in bulk solution, interference with hydrogen bonding by hydrocarbon chains increases the structuring of the water and the coiling of the chains. Elimination of these effects upon adsorption of long-chain surfactants results in a net increase of entropy.
Table 5.6 Thermodynamics of adsorption of fatty acids in the water at 293 K (Ward & Tordai, 1946a, b)
Traube’s rule states that the concentrations for equal lowering of the surface tension in dilute solutions decrease by about one third for each additional CH2 group in a homologous series. This concept was extended by Langmuir who showed that if −ΔG° is the work done in transferring one mol of soap molecules from the bulk to the surface, then
(5.13)
where suffix m denotes the number of carbon atoms in the paraffin chain. This arises partly from the increase in molecular orientations available to the chain as each CH2 group is brought above the surface from which one may conclude that the average configuration of the paraffin chains is in the form of the most probable random toil. Traube’s rule is only approximate and this is borne out by comparison of Eq. 5.13 with the fourth column of Table 5.6. Langmuir found a value of 3.4 in Eq. 5.13 whilst Ward found 3.72 by an improved method of calculation giving −3.18 kJ.mol-1 for the methylene group; a value more in accord with Table 5.6 than that from Traube’s rule.
It has been observed that weak attraction of solvent for a substantial part of a solute molecule leads to an accumulation of solute at the surface. On the other hand, if solute–solvent attraction exceeds solvent–solvent attraction, the solute concentration at the surface will be less than in the bulk solution and there will exist a state of negative adsorption leading to an increased surface tension. Such a situation arises with polar organic compounds such as sucrose and with inorganic electrolytes where the effect of the ions follows the lyotropic series (p. 99). The effect of solute concentration upon surface tension is illustrated in Fig. 5.14.
Fig. 5.14 Qualitative effect of various solutes upon the surface tension of water: 1. Inorganic electrolytes and sugars, e.g. NaCl. sucrose; 2. Short-chain acids and alcohols, e.g. tetranoic acid, ethanol; 3. Long-chain ions, e.g dodecanoate, tetradecyltrimethylammonium.
The Gibbs Equation
Fig. 5.15 represents a solution of a surface-active compound whose surface is located somewhere between AA′ and BB’, i.e. AA′ is above the liquid phase and BB′, is below the region of surface irregularity. Region AB therefore contains the whole of the surface whereas region BC consists entirely of bulk solution of ordinary composition.
Fig. 5.15 Hypothetical boundaries defining the surface excess.
Let U joules be the total (internal) energy of region BC. The internal energy is a function of the extensive (quantity) factors, viz. entropy S, volume V and number of mols of solvent (component 1) nl and of solute (component 2) n2 Thus:
Region AB will have a different internal energy from region BC, since the boundaries at A, B and C cannot readily be chosen so as to equate them. The extra energy of region AB is denoted by Us and may be positive or negative. The excess energy Us arises because of:
and 
respectively, where Ss, Vs and ns may be positive or negative. Thus:
Since the system is assumed to be at equilibrium the intensive factors are uniform throughout so that there is no need to add the superscript s to temperature T, pressure P or chemical potential μ. The surface tension γ is restricted to the surface so there is likewise no need for the superscript. The reason for the term γdA being positive is that the tension γ acts to contract the surface unlike the pressure P which tends towards expansion.
so that at constant temperature and pressure:
(5.15)
Considering now each m2 of surface and defining the surface, excess per m2, γ as:
This may be extended for a multicomponent system to:
(5.16)
nS and hence γ may be defined in several ways. For example, C of Fig. 5.15 can be located so that regions BC and AB contain the same volume of liquid (γ(v)), the same total number of mols (γ(N)) or the same total mass (γ(M)). For the purpose considered here however, it is best to define BC and AB as containing the same mass of solvent this being the convention due to Gibbs and denoted by Guggenhein and Adam as (γ(1)). Thus 
.
.For a two component system, i.e. one solvent and one solute, Eq. 5.16 becomes:
whence for dilute solutions, where the activity coefficient is almost unity,
(5.17)
c being the concentration. This is the familiar form of the Gibbs equation and enables the surface excess to be determined from the slope of a plot of surface tension versus the logarithm of the solute concentration (Fig. 5.16).
Fig. 5.16 Effect of amphipath concentration on surface tension.
Fig. 5.16 demonstrates that the slope becomes increasingly negative up to the critical concentration showing that the surface excess increases as the solute concentration increases. It also demonstrates that the Gibbs equation is not valid above the critical micelle concentration.
Application of the Gibbs Equation to Real Aqueous Systems
The form of the Gibbs equation given by Eq. 5.17 is valid only for a two component system and is therefore inapplicable to aqueous solutions since every species must be taken into account. It must be replaced by the general Eq. 5.16 with the adoption of the Gibbs convention, 
. For a single univalent surface-active strong electrolyte (MA) in water:
. For a single univalent surface-active strong electrolyte (MA) in water:
The first requirement is therefore to maintain a constant pH such that:
(5.18)
According to Pethica the surface concentration of the counterion (M+) is much less than that of the amphipathic ion (A−) at low concentrations but it may equal it at higher concentrations when a complete monolayer is formed. The effect of this is seen from the two limits (Eq. 5.19 for very dilute solutions and Eq. 5.20 for less dilute solutions) to Eq. 5.18:
(5.19)
(5.20)
In either case the surface may be considered to be a monolayer of with the diffuse part of the double layer preponderant M+ and H+. The Gibbs equation may thus be written (assuming f± = 1)
(5.21)
where X varies between 1 and 2 according to the concentration of surfactant. Shinoda and Nakayama (Shinoda, 1954) however, have noted that the surface excess of counterions is always less than that of the surface-active ions.
With a surface-active weak electrolyte such as a soap, hydrolysis occurs:
At constant pH:
(5.22)
(5.23)
Since there is a concentration dependence of the effect of the counterions comparison with Eq. 5.21 gives that:
In order to remove the concentration dependence of the effect of the counterions (K+), a constant excess concentration of another electrolyte such as KCl may be added to each soap solution so that:
and the effect of changing the soap concentration will be that:
Under these circumstances
This is nothing more than the simple form of the Gibbs equation (Eq. 5.17) for dilute solutions provided that a pure soap is used and the solutions are all of the same pH and contain a constant amount of counterion, thus:
Because of the relatively high concentration of electrolyte, the electrical double layer is thin and so 
applies to a very thin surface and may be used to calculate surface packing where the area (m2) per mole of a monolayer is given by 
. It has been shown that very little additional electrolyte is in fact required and so an equation of the form of Eq. 5.20 holds only for solutions of extremely pure surface-active electrolytes.
applies to a very thin surface and may be used to calculate surface packing where the area (m2) per mole of a monolayer is given by . It has been shown that very little additional electrolyte is in fact required and so an equation of the form of Eq. 5.20 holds only for solutions of extremely pure surface-active electrolytes.Hydrolysis of soap leads to the formation of free fatty acid which being less soluble than the ion will be more strongly held at the solution surface. If the soap was prepared from impure fatty acid, there will be other anions and free fatty acids present as well, whose area of packing may be different from that of the main constituent. Further, unless very careful purification is accomplished, fatty acids prepared from natural sources will contain traces of fatty alcohols which are very water insoluble.
Hydrolysis of soaps will be enhanced by a drop in pH. This may occur extensively if solutions are exposed to atmospheric carbon dioxide. If the surfactant is an anion, then these will attract countercations some of which will be hydrogen ions. The pH of the surface is therefore lower than that of the bulk and surface hydrolysis is very marked unless the pH is maintained sufficiently high. Conversely, a surface excess of long-chain amines produces a surface with an excess of counteranions and a consequent higher pH than in the bulk solution.
As the concentration of soap in the very dilute solution is increased, the potential for adsorption will also increase because the soap anions, free acids and alcohols will be more strongly competing for a place in the surface. There is therefore a surface pressure π tending to increase the surface area in order to accommodate more solute. This pressure competes against the cause of the surface tension (which is a negative surface pressure) and results in a lower tension than before; consequently:
(5.24)
where γ0 is the surface tension of the solvent alone.
The surface of a soap solution is not filled with soap molecules at low soap concentrations (say at 10−4 molar) but becomes progressively filled as the bulk concentration is increased. Not until the surface is fairly well packed with soap molecules does π increase very much. Then further increase in soap concentration causes a marked reduction in γ with little change in the surface excess (see Fig. 5.16 where the slope of the steep portion remains virtually constant).
When a certain degree of packing at the surface is achieved, the energy required to push more soap into the surface approaches that given up on adsorption. A greater lowering of free energy can now take place by the soap molecules in the bulk aggregating with their paraffin chains together and their polar groups orientated towards the water (Fig. 5.20A). These aggregates are known as spherical micelles and the lowest concentration at which they form in appreciable numbers is called the critical micelle concentration (CMC). The micelles will be formed from all the different amphipathic molecules present and so mixed micelles form in impure systems. Hydrolysis products and other uncharged impurities generally cause formation of micelles to occur at lower concentrations since they are less soluble than soap ions and their lack of a charge lessens the disruptive effect which occurs when like charges are brought together in ionic micelles. Above the CMC there is therefore dynamic equilibrium between three pseudophases; the surface, bulk and micelles. The micellar pseudophase is much larger than the surface pseudophase and will tend to accommodate most of the less soluble components. As a result the surface will become depleted of impurities such as fatty alcohol above the CMC. These impurities will not be replaced in the surface by an equivalent amount of soap because the latter is more soluble and so there will be lowering of π and an increase in γ as the soap concentration is increased a little above the CMC. Further increase in soap concentration will cause increased packing at the surface until the surface tension has decreased again to a constant value.
In theory, the surface is not necessarily one molecule thick, neither does the definition of surface excess require this although studies of the behaviour of dilute surfaces have shown them to be virtually monolayers or at least to act like them. Surface multilayers may occur at concentrations above the CMC. The surface excess will then be underestimated because the Gibbs equation depends on the change in surface tension which resides mainly in the surface molecular layer and amphipathic molecules layered under this will have little influence. Thus dγ/d ln c is not a function of γ for surface excesses more than one molecule thick so that a γ versus ln c plot has zero slope when the monolayer is complete.
Verification of the Gibbs Equation
Some of the first attempts at a direct estimation of the surface excess were made by bubbling air through the solution and collecting the foam. The concentration of solute in the collapsed foam should be greater than that in the solution from which the foam was derived and if the surface area of the bubbles could be accurately estimated then the surface excess could be readily calculated. The results so obtained were unreliable for two reasons; the estimate of the surface area was uncertain and the lifetime of the bubbles was too short for surface equilibrium to be established (Fig. 5.17)
Fig. 5.17 Change of interfacial tension with age.
Successful verification of the Gibbs equation was achieved by McBain and his co-workers who propelled a microtome blade at high speed across the solution surface thereby shaving off a sample to a depth of about 100μm. Analysis of this layer and of the underlying bulk solution enabled the surface excess to be calculated.
They also compressed solution surfaces by a known amount in a special trough and estimated the surface excess after determining the increase in bulk concentration produced by desorption. Similar observations have been made in reverse by calculating surface excess from the decrease in bulk concentration on producing a stable foam of known surface area. Both of these methods have shown good agreement with the Gibbs equation but indicate that factor X of Eq. 5.21 cannot always be ignored.
Radioactive tracer techniques have also been used by a number of workers involving the use of amphipaths tagged with 3H, 14C and 35 S. Low energy radiation is absorbed by water after travelling only a short distance so that a detector placed just above the solution surface will register particles coming only from the surface and a thin layer underlying the surface. In order to correct for the emission received from the thin layer of bulk solution, similar measurements are made using solutions with the same concentration of the isotope contained in nonsurface-active compounds. Coadsorption of counterions may similarly be investigated using unstable metal ions. The accumulation of data appears to support Eq. 5.21.
Surface Ageing
At the instant that a surface is created, the surface will have the same composition as the bulk solution. Attainment of an equilibrium surface excess will therefore exhibit time dependence. Moilliet, Collie and Black (1961) considered two distinct phenomena:
One or both of these may be evident with a given surfactant solution depending upon the solute concentration, presence of electrolyte and temperature. A generalized picture is given in Fig. 5.17.
Diffusion towards the surface is rapid but becomes slower as surfactant ions approach the vicinity of the surface due to electrostatic repulsion from ions already adsorbed there. Thus there is rapid diffusion to the subsurface followed by activated adsorption. Diffusion is also adversely affected if much of the surfactant is aggregated into micelles which, being much larger than single ions or molecules, are relatively slow moving. Surface ageing therefore becomes slowed by substances which encourage aggregation into mixed micelles, such as free acids produced by the hydrolysis of soaps. The rate of diffusion of hydrolysable surfactants is thus increased if the pH is adjusted in the direction which suppresses hydrolysis, e.g. to higher pH with soap solutions.
Diffusion to the surface is also more rapid at higher temperatures and greater surfactant concentrations. Electrolytes also facilitate the diffusion of ionic surfactants but they have little effect upon nonionic surfactants such as methylcellulose. The minimum in the γ−time curve occurs earlier as the concentration is increased, but subsequent ageing effects are little influenced by this factor, leading to the conclusion that only the rapid initial decrease in γ is rate-controlled by diffusion.
The subsequent slower ageing effect appears to be controlled by steric factors such as reorientation and penetration of molecules in the surface. The rise in surface tension is probably associated with a surface phase transition to a condensed state (p. 80) this having been noted with sodium dodecyl sulphate when the area per molecule is reduced to about 0.4 nm2. Electrolytes assist the penetration of the surface by both ionic and nonionic surfactants since expulsion of hydrocarbon chains results in a larger free energy decrease than that resulting from expulsion from a relatively less polar simple aqueous solution (cf. salting out of soaps). Time dependence with surfactants in nonpolar solvents may also arise due to slow breakdown of associated molecules such as dimers
Bulk Properties of Surfactant Solutions
Most physical properties of aqueous solutions of an amphipath change rapidly in magnitude over a limited range of concentration (Figs. 5.18 and 5.19). To account for this, McBain and others postulated the existence of aggregates of amphipathic molecules which, if ionized, are closely associated with a large proportion of the counterions (otherwise known as gegenions). The abrupt nature of the changes in magnitude of the various properties over a limited concentration range suggests that at lower concentrations the molecules are mostly single whereas at higher concentrations they are almost all present as micelles.
Fig. 5.18 Change of physical properties with concentration (Preston, 1948)
Fig. 5.19 Relationship between activity coefficient and concentration for aqueous solutions of sodium dodecylsulphate (from data of Clayfield & Matthews, 1957)
Fig. 5.20 Diagrammatic cross section of spherical micelles in (A) aqueous and (B) nonaqueous solvents.
Hartley suggested that these are essentially spherical: however, at much higher amphipath concentrations than the CMC, other shapes can exist (p. 69). In aqueous solutions micelles have a nonpolar liquid-like interior whereas in nonaqueous solvents inverted micelles are formed, held together by polar interaction of the ‘head’ groups (Fig. 5.20).
Theoretical Treatment of Micelle Formation
When amphipathic molecules associate, they form micelles with a narrow size distribution, the mean number of molecules per micelle being known as the aggregation number. The influence of amphipath concentration c, aggregation number n and charge p of the micelle upon the proportion of aggregated molecules may then be predicted by the law of mass action.
For nonionic molecules, p = 0 and
where c is the concentration in terms of single molecules and x is the unaggregated fraction. Although the activity coefficient does not remain at unity with increasing concentration (Fig. 5.21) yet, assuming its constancy for simplicity, then:
Fig. 5.21 Effect of amphipath concentration upon aggregation into micelles calculated from Eqs. 5.26 and 5.28; (A) Nonionic or completely ionized micelles (for given aggregation numbers); (B) Micelles (aggregation number 50) in different states of ionization (1 = (A50M25)25−, 2 = (A50M43)7−, 3 = A50 and (A50)50−).
The CMC may now be defined less ambiguously than in the previous section in the following terms:
(5.25)
that is, at the CMC:
(5.26)
Eq. 5.26 would also be applicable to ionic molecules provided they were completely dissociated whether present singly or as aggregates. In such a circumstance, however, a large coulomb attraction would exist between the micelles and their counterions and consequently dissociation of the micellar molecules is incomplete. Thus if M+ is a monovalent counterion, p = n − m and
Generally defining (cf. Eq. 5.25) the CMC as
(5.27)
That is, 
(5.28)
from which the concentration relative to the CMC may be computed for all values of x, m and n.
Curves related to Eqs. 5.26 and 5.28 are plotted in Fig. 5.21 showing that formation of micelles with an aggregation number in excess of about ten produces an abrupt change in aggregated fraction around the CMC It will also be observed that partial ionization of micelles enhances this abruptness.
Eqs. 5.25 and 5.27 defining the CMC imply that d2x/dc2 = 0 at the CMC, that is, aggregation is most abrupt at the CMC An analogous method of defining the CMC is to take the point corresponding to the maximum curvature in an ideal solution property/concentration plot as in, that is d3ϕ/dc3 = 0 where ϕ represents the magnitude of a given property such as conductivity.
An alternative approach is to consider micelle formation as a process of phase separation. This theory supposes that micelles begin to form only at the CMC which is then the saturation concentration for single molecules. Consequently the addition of further surfactant produces an equal amount of micellar pseudophase.
Determination of the critical micelle concentration: Reference to Fig. 5.18 will indicate that a plot of any physical property against amphipath concentration gives a rough indication of the CMC of aqueous solutions. Generally the CMC is taken at the point of intersection of the extrapolated straight lines on either side of the break in the curve. Other properties not indicated in the figure which have been applied, include vapour pressure, refractive index, pH and diffusion coefficient. In order to obtain significant data, good temperature control is often essential particularly with refractive index measurements.
The CMC may also be estimated by the addition of traces of a third component whose absorption spectrum depends upon the state of aggregation of the amphipath. Marked spectral changes are frequently observed with ionic amphipaths when a dye of opposite ionic character is used with it. Thus pinacyanol chloride is commonly used for anionic amphipaths and eosin for cationic compounds. Using pinacyanol chloride, the colour change is from blue above to red below the CMC. The technique is to dissolve a small amount of dye in a solution of the amphipath which is above its CMC and to titrate this with an aqueous solution of the dye. The CMC is taken as the concentration of amphipath at which a colour change is first noticed.
Dyes may also be used with nonionic compounds but the colour changes are much less pronounced and instead absorption in the ultraviolet region is followed. Another method which is suitable for use with nonionic compounds is the iodine method. The technique is similar to that using dyes except that a trace of iodine is used and the change in light absorbance with amphipath concentration is followed at a suitable wavelength in the ultraviolet region. A sharp change in the slope of a plot of absorbance versus the logarithm of amphipath concentration indicates the CMC.
Because the interior of micelles in aqueous solution is essentially hydrophobic, water-insoluble compounds are able to dissolve in it. This behaviour is known as solubilization (p. 71). Thus the solubility of the added compound increases markedly at concentrations of amphipath above the CMC In practice an excess of an oil-soluble dye such as Orange at may be used, the amount taken into solution being determined colorimetrically. Alternatively, solutions of varying amphipath concentration may be titrated with the third component when observation of turbidity may be all that is necessary to detect the limit of solubility. Care must however be exercised in order to ensure that equilibrium has been attained.
For further details the reader is referred to the review by Mulley (1964). There is not always perfect agreement between the various methods, particularly when comparing those which do not involve the addition of a third component with those which do. The third component might well be expected to affect micelle formation, the CMC frequently being lowered.
Factors Affecting Micelle Formation
Thermodynamic factors: Hartley suggested that small aggregates begin to form at the CMC and’ that these grow rapidly over a small concentration range to a limiting size which is then independent of further increase in concentration. Mass action theory (Fig. 5.21) shows that slight aggregation is to be expected below the CMC Association between hydrocarbon chains reduces the free energy which results from the disruption of hydrogen bonds in aqueous solution. However, formation of large aggregates in very dilute solutions would be most improbable but dimer formation is known to occur with sodium dodecylsulphate and some soap solutions.
Micelle formation is encouraged by those factors which produce a decrease in the free energy when hydrocarbon–water contact is reduced (p. 49) namely, a decreased water structuring (+ΔS) and a decreased interference with hydrogen bonding (−ΔH). Micellization may be accompanied by an increase in enthalpy but this is more than compensated by a relatively larger entropy increase. Some workers have shown the enthalpy change to decrease as the alkyl chain is lengthened for a homologous series of amphipaths, possibly reflecting an increase in cohesion between alkyl groups in the micelle. Others, however, have reported an increasing enthalpy of micelle formation with increasing alkyl chain length. These differences reflect a different choice of standard state.
Clearly, entropy plays a dominant role in micellization, the overall mechanism being termed ‘hydrophobic bonding’ (Abu-Hamdiyyah, 1965). Ions however, draw water molecules to themselves and reduce their tendency to form hydrogen bonds with neighbouring water molecules. The effect is to reduce the water structure and therefore there tends to be a lower entropy contribution with ionic micelles. This effect would be influenced by the hydration of the counterion and of the polar head group. Various workers have found that counterions higher in the lyotropic series (p. 99) produce higher CMC values and an increase in the size of the polar group of nonionic amphipaths has a similar effect.
Factors other than those due to hydrophobic bonding contribute to the free energy of micelle formation. Poland and Scheraga have considered the influence of aggregation upon (a) the loss of translational and rotational entropy when a molecule enters a micelle and (b) the entropy of the hydrocarbon chain within the micelle. As will be seen from Fig. 5.22, factor (a), curve 1, would favour the molecules remaining in a monomeric form, whereas hydrophobic bonding, curve 2, would favour an aggregate of infinite size. However, when the free energy contribution of factor (b), curve 3, is taken into account, the total free energy curve of the system has a minimum at a finite aggregation number but the free energy in an aggregated form is less than that in the an aggregated form only above the CMC (Fig. 5.23). If the solute is ionic, then formation of micelles will involve bringing like charges together against their mutual repulsion. The general result of this is to cause ionic micelles to have smaller aggregation numbers and larger CMC values than nonionic micelles.
Fig. 5.22 Contribution of various factors to the free energy of an aqueous amphipath solution (not to scale; from Poland & Scheraga, 1965)
Fig. 5.23 Effect of concentration of amphipath showing a finite aggregation number for a minimum free energy. A. Below CMC; B. At CMC; C. Above CMC (from Poland & Scheraga, 1965)
The hydrocarbon group: Increase in the size of the hydrophobic group leads to a greater decrease in the free energy upon micellization, thereby promoting micelle formation. This results in a decreased CMC and an increased aggregation number. The free energy change is however less for branched chains then for straight chains and so the former have larger CMC values for a given number of carbon atoms. For a homologous series, the CMC decreases logarithmically with an increase in the number of carbon atoms. The presence of a double bond increases the CMC three or four times.
The polar group: In contrast to the effect of the hydrophobic group, an increase in length of polar chains (e.g. polyoxyethylene) decreases the free energy change upon micellization thereby increasing the CMC and decreasing the aggregation number. Hydration of the polar group of polyoxyethylene alkyl ethers occurs through hydrogen bonding between the ether oxygens and water. Increasing the length of the oxyethylene chain increases the hydration and solubility, thereby opposing the formation of micelles. When a nonionic molecule is transferred to a micelle, the motion of the oxyethylene chains is restricted to a spiral within the envelope of a truncated cone (Fig. 5.24). This would be expected to reduce the entropy of micelle formation. Since the entropy increase is in fact large, this has been interpreted to mean that micelle formation involves some dehydration of the polar groups. These groups are still however, considerably hydrated in the micelle, Water molecules are probably trapped both within and between the spirals, the proportion of water to number of oxyethylene groups increasing as the chain length increases.
Fig. 5.24 Diagrammatic representation of a nonionic micelle with all but three of the polar groups cut away.
Compared with the large effect that the length of a nonionic polar group has upon the CMC, the type of an ionic group has little effect. The CMC does however increase slightly with increased hydration of the counterions, e.g. Li+ > Na+ > N+(CH3)4, since hydration causes them to be less closely attached to the micelle and therefore to contribute less to its formation. The influence of the ions upon the water structure (see below) also affects the CMC. Compared with Li+ and Na+ which break water structure, tetramethyl ammonium ions enhance water structure thereby producing an appreciably lower CMC.
The position of the polar group in the molecule influences the CMC since moving this group towards the middle of a hydrocarbon chain is equivalent to increasing the branching of the chain which results in an increase in the CMC The presence of a second polar group also increases the CMC particularly if it is ionized; the magnitude of the effect also depends upon the position of the group.
Electrolytes: The addition of salts increases the aggregation number and decreases the CMC of ionic amphipaths. This effect is due mainly to the reduced repulsion between the head groups as a result of the screening action of the counterions but is slightly modified by counterion hydration and the effect on the water structure. The effective degree of dissociation of the enlarged micelles is little influenced. Although ionic micelles appear to be predominantly monodisperse in low salt concentrations, there is evidence of a wider size distribution in the presence of higher salt concentrations.
Salts have a relatively small effect upon nonionic amphipaths, its magnitude probably depending upon the hydration of the salt ions, i.e. a salting out effect, where the more hydrated ions depress the CMC the most. The anions are more effective in this than are the cations. Competition for water of hydration by the salt reduces the hydration of the amphipathic molecule thereby increasing the latter’s tendency to aggregate.
If the added electrolyte is also capable of forming micelles, the CMC of the mixture lies between those of the pure components. There is however preferential aggregation of the longer chain ions so that there is a larger mol fraction of these in the micelle than in the equilibrium unaggregated fraction.
Hydrogen ion concentration: Free fatty acids are less ionized and less soluble than their alkali metal salts, so that a reduction in the pH of a soap solution is accompanied by a reduction in the CMC. As a consequence, the effect of neutral electrolyte upon the CMC is less at a lower pH where the micelles have fewer charges. In contrast to this, the effect of the addition of strong acids to amphipathic strong acids such as dodecylsulphate is comparable to that of neutral electrolytes, the effect of added salts being similar over a wide pH range. Strong acids increase the hydrophilic character of polyoxyethylene groups by oxonium ion formation of the ether oxygens. This effect predominates over the salt effect and results in an increase in the CMC of nonionic compounds.
Nonelectrolytes: The addition of water-soluble compounds such as urea and glycerol generally raises the CMC. The tendency for micelle formation is reduced not only by a decreased structuring of water but also by a decreased dielectric which increases the repulsion between charged head groups, and by an increased solvation of the amphipathic monomer which reduces the enthalpy change of micellization. The overall influence of a particular additive will be governed by a combination of these factors. Such effects are modified if the additive is incorporated into the micelle since this may tend to increase micelle stability· at low additive concentrations. Thus a trace of ethanol is shown (Fig. 5.25) to decrease the CMC only slightly while longer chain alcohols have a much more pronounced effect (Fig. 5.26). Small amounts of ethanol increase the solubility and lower the Krafft-point (see below) but large quantities suppress micelle formation. The hydrocarbon chains of the alcohols penetrate the micelle interior while the polar groups remain on the outside (Fig. 5.34).Long chains in particular enhance the micelle stability since the dispersion force interaction in the micelle interior is increased and the charge density on the micelle surface is probably slightly decreased. The effect of polar compounds which form mixed micelles is both concentration and temperature dependent and as a result the minimum in the CMC–temperature plot with ionic micelles (Fig. 5.27) is eliminated, the CMC decreasing continuously as the temperature falls.
Fig. 5.25 Stability of dodecyltrimethylammonium bromide micelles in aqueous organic solvents at 25°C (from Emerson & Holtzer, 1967). 1. Dioxane; 2. Propan-1:3-diol; 3. Ethylene glycol; 4. Methanol; 5. Ethanol; 6. Propan-2-ol; 7. Propan-1-ol.
Fig. 5.26 Effect of various normal alcohols on the CMC of potassium tetradecanoate at 18°C (from data of Shinoda, 1954)
Fig. 5.27 Variation of CMC with temperature for 1 CH3(CH2)11N(CH3)3Br; 2, CH3(CH2)11SO4Na; 3, CH3(CH2)11(OCH2CH2)7OH (adapted from Schick, 1963)
The solubility of sodium dodecylsulphate is also increased by traces of short (C6–Cs) chain alcohols but large amounts of alcohol, which itself has only a small solubility, cause an alcohol-surfactant gel to separate as a new phase.
Temperature: An increase in temperature influences the stability of micelles by the following means. First, transfer of hydrocarbon groups from an aqueous to a nonpolar environment is normally endothermic and so micelles tend to become more stable by the hydrophobic bond effect. Secondly, the dielectric constant is decreased so that ionic micelles become less stable. Thirdly, thermal agitation increases, resulting in an enhanced disruptive effect. With nonionic compounds the second factor is nonoperative and the first factor outweighs the last. An increase in temperature decreases the hydration of the molecule with a resulting increase in aggregation number and a decrease in the CMC. In the case of ionic compounds, the hydrophobic effect is probably dominant at lower but not at higher temperatures. Such an effect produces a shallow minimum CMC near room temperature. An increase in temperature above that producing the minimum CMC has been shown to decrease the aggregation number and to increase the dissociation of micelles of sodium dodecylsulphate, a sign of the disruptive effect of thermal agitation.
The solubility of an amphipath is influenced by temperature. At low temperatures, solid ionic-type amphipath is in equilibrium with a solution whose concentration is below the CMC. As the temperature is raised the solubility increases until it reaches the CMC at which temperature, called the Krafft-point, the solubility begins to increase very rapidly (Fig. 5.28). If micelles are treated as a separate (pseudo-) phase, then the Krafft-point may be considered as the melting point of the hydrated amphipath above which it becomes dispersed in solution as micelles. The solubility of ionic compounds is limited at higher temperatures by the separation of liquid crystals (see below and Fig. 5.29). Many nonionic compounds do not show a Krafft-point, since the saturated solution is in equilibrium with a liquid crystalline phase (Fig. 5.30). High temperatures however suppress hydrogen bonding thereby reducing the hydration of the polar chains and cause the solution to become turbid, the upper consolute temperature being called the cloud point. The turbidity is caused by the separation of an amphipath-rich phase, which is not surprising since the micelles grow rapidly below the cloud point. The dilute aqueous phase has a concentration roughly that of the CMC and contains few, if any, micelles.
Fig. 5.28 Effect of temperature upon the solubilities of some alkyl sulphonates (after Tartar & Wright, 1939)
Fig. 5.29 Effect of temperature upon aqueous sodium laurate systems (shaped areas indicate the existence of more than one condensed phase) (after McBain et al., 1938)
Fig. 5.30 Effect of temperature upon the system dodecyl-hexaoxyethylene glycol monoether plus water (dashed lines are estimated boundaries; after Balmbra et al., 1962)
Structure of Micelles and Liquid Crystals
Above but close to the CMC, micelles are spherical but as the concentration of amphipath is increased the micelles may grow into cylindrical or rod-shaped structures whose diameter is slightly less than twice the fully extended length of the monomers. The tendency to form rod-shaped micelles depends upon the nature of the hydrocarbon chain, with such factors as length, branching and degree of unsaturation. It is also influenced by the polar group and the counterion and pH of ionic compounds while with the nonionic hexadecylpolyoxyethylene glycol monoether a decrease in polar chain length leads to increased asymmetry of the micelles.
These long micelles often have a random orientation but, as their concentration increases, their freedom of motion becomes restricted producing viscoelastic solutions. Further increase in concentration causes some of them to become closely packed in hexagonal array (Fig. 5.31A) and to separate out as another phase in equilibrium with a more dilute solution containing randomly orientated rods. This new phase which contains the oriented rods is known as the middle phase. Progressive increase in concentration results first in a single middle phase and finally in the separation of solid hydrated amphipath. It should be pointed out however, that the micelles of some compounds, such as cetyltrimethyl ammonium chloride (but not bromide) remain spherical at concentrations below that necessary for the separation of a middle phase. With many compounds a further phase may separate before solid is produced; this is the neat phase which has a lamellar structure (Fig. 5.31C). Both the middle and neat phases form what are known as liquid crystals and are termed mesomorphous (meso = between, morph = form) because they have a structure intermediate between that of true liquids and crystals.
Fig. 5.31 (A) Section of middle phase; (B) Section of invert middle phase; (C) Section of neat phase.
Other mesomorphic states are known. When a water-soluble amphipath, such as sodium octanoate, is dissolved in decanol containing a trace of water, inverted micelles are produced, that is, the nonionized polar groups of the fatty acid salt are orientated inwards around a few water molecules and the hydrocarbon chain extends outwards into the alcohol. If the proportion of decanol is now reduced while maintaining a constant proportion of ‘soap’ and water, an inverted middle phase (Fig. 5.31B) will eventually separate. Continuous decrease of decanol concentration with increase of the ‘soap’ and water produces, in order, invert middle phase, lamellar phase and finally middle phase.
With some systems a face-centred cubic arrangement of spherical aggregates seems to occur. Such structures are generally observed over only a narrow concentration between the lamellar (neat) and the hexagonal (middle) phases. It is probable that with lipids and potassium soaps, the cubic structure is formed from invert spherical aggregates with the hydrocarbon chains filling the gaps between the water ‘cores’. With alkyltrimethyl ammonium bromide the aggregates are probably not spherical.
It will be observed that the structure of the middle and neat phases is anisotropic, that is, it varies with direction. The middle phase is plastic and too stiff to flow under gravity since the close packed regular arrangement of the cylindrical aggregates gives it solid character in two dimensions. In the long direction however, there is liquid-like behaviour since the cylinders probably vary in length and are not regularly arranged. The layered structure of the neat phase gives this a greater fluidity. There is no regular arrangement in the two dimensions parallel to the layers but in the direction normal to the layers there is a crystal like periodic arrangement. The neat phase is therefore solid in one dimension and liquid in two dimensions. It is the ability of the layers to slip over each other that makes the neat phase more mobile than the middle phase. Often, drops of neat phase, which are mobile enough to flow under their own weight, exhibit a stepped surface, each step height being a multiple of the layer thickness.
X-ray diffraction patterns have been used in an attempt to confirm postulated micellar structures. Besides a weak diffraction due to solvent, three bands have been identified in the more concentrated amphipath solutions. The short-spacing (S) band corresponds to the repeat distance of about 0.45 nm for the hydrocarbon chain in the liquid state; the micelle-thickness (M) band has a repeating distance slightly less than twice the length of the hydrocarbon chains and the long-spacing (I) band corresponds to the separation distance of the structures composing the mesomorphous phase (Fig. 5.32).
Fig. 5.32 Interpretation of X-ray diffraction bands.
Liquid crystals are birefringent (have a different refractive index for the two components of the electric vector of light) and may be detected and identified by their effect upon polarized light. Rotation of plane polarized light occurs for light travelling in any direction other than along the optic axis of the uniaxial crystal, so that liquid crystals remain visible when placed between crossed polarizers, whereas light passing through an isotropic phase would be extinguished. Under a polarizing microscope, neat and middle phases may be distinguished by the patterns or ‘textures’ which are produced (Rosevear, 1968).
Mesomorphism as previously described involves the addition of a solvent to the amphipath; this is termed lyotropic mesomorphism. Thermotropic mesomorphism, on the other hand, involves the production of a liquid crystalline phase merely by warming a crystalline, generally organic, solid. This is a transition phase between the solid crystal and isotropic liquid and gives the compound the appearance of having two melting points. Three types of mesomorphous phase have been distinguished, namely smectic (soap-like), nematic (thread-like) and cholesteric (Chapman, 1965).
The neat phase is an example of a lyotropic smectic state. The thermotropic smectic state has a similar layered structure except that the molecules may be arranged as single rather than double layers since one end of the molecules may fit neatly to the opposite end of a molecule in an adjacent layer. Smectic crystals are not orientated in a magnetic field but if formed by cooling the isotropic liquid in such a field, orientated crystals are produced. Examples of compounds exhibiting the smectic state are ethyl p-azoxybenzoate and octyl p-azoxycinnamate.
A nematic structure is occasionally observed with amphipathic solutions which then have a soft, often stringy, consistency. The thermotropic nematic state is produced by heating such compounds as p-azoxyanisole whilst N-(4-methoxybenzylidene)-4-butylanilene is nematic at room temperature. Nematic crystals are less ordered than smectic ones since the elongated molecules are arranged parallel to one another without the formation of a layered structure. Normally they are nor optically active but, if twisted between a microscope slide and cover slip, they exhibit characteristic microscopic textures somewhat different from those given by smectic or lyotropic middle phases. Molecules in the nematic crystal are readily orientated parallel to a weak magnetic field. The viscosity is therefore dependent on the direction of the field in relation to the direction of flow.
Cholesteric phases are produced by a number of cholesteryl esters but not by cholesterol itself. These have a layered structure quite different from the smectic type. The elongated molecules lie parallel to each other within the plane of a layer, their orientation in adjacent layers being slightly displaced. This displacement arises because the shape of the molecules influences their fit in adjacent layers and so these latter trace out a helical path which causes rotation of polarized light. With nonpolarized light, circular dichromism is observed giving rise to a multicoloured effect.
Some compounds can exhibit more than one type of mesophase. Ethyl anisal-p-aminocinnamate, for example, forms smectic crystals which change to nematic at a higher temperature. A number of cholesteryl fatty acid esters also form smectic crystals but these become cholesteric on further warming.
Stability of Micelles
The equilibrium between n1icelles and monomers is a dynamic one. When suddenly diluted to below the CMC, some ionic micelles have been shown to have a half-life of a centisecond or even less. Micelle stability is slightly increased by an increase in alkyl chain length and is affected to a small extent by the nature of the counterion. Traces of hydrocarbon dissolved in the micelle interior appear to reduce the rate of micelle breakdown but the formation of mixed micelles by the addition of long-chain alcohol may not have an appreciable effect. Micelles formed from mixed anionic and cationic amphipaths however do appear to breakdown less easily. Micelle stability increases with increasing concentration of amphipath above the CMC but decreases with increasing temperature.
Fig. 5.33 Types of thermotropic liquid crystals.
Fig. 5.34 Diagrammatic representation of various sites of solubilization in aqueous solution. Open figures are solubilizer; closed figures are solubilizate. Left: ionic half micelle section; right: nonionic half micelle section
Solubilization
The interior of a micelle has a different dielectric from the surrounding solvent. Solutions of amphipaths therefore have the ability of dissolving substances which are insoluble or at best only sparingly soluble in the solvent alone. This is known as micellar solubilization, the solubilized substance being termed the solubilizate.
Solubilizates are incorporated into the micelle in different ways according to their chemical structure, in particular the presence and disposition of polar and nonpolar groups. Saturated hydrocarbons dissolve in the nonpolar part of the micelle which in aqueous solutions forms the micellar interior (Fig. 5.34a). Since the M-spacing and thus the volume of the micelle is increased, more amphipath molecules are incorporated into the micellar surface to maintain a suitable packing density. There are therefore larger but fewer micelles and so the I-spacing also increases. Amphiphilic solubilizates, such as n-alcohols, form mixed micelles; that is, they are incorporated into the palisade layer of the micelle (b), where the molecules are orientated to juxtapose the polar groups. Weakly polar or polarizable groups, such as double bonds, enable the solubilizate to be drawn deeper into the palisade layer (c). Incorporation of polar solubilizates tends to result in increased micellar asymmetry with only slight changes in the M and I-spacing. Some water-soluble polar solubilizates are thought to be adsorbed on to the micelle surface (d), but the validity of this has been questioned (Osipow, 1962). It appears, however, that complex formation by hydrogen bonding occurs between polyoxyethylene groups and aromatic carboxylic acids such as phydroxybenzoic acid or phenolic compounds such as chloroxylenol (e).
Micellar solubilization can only occur in the presence of micelles. Often the amphipath forms micelles in the absence of the solubilizate but occasionally aggregation takes place only upon addition of the latter. Thus the addition of an amphiphilic solubilizate to amphipath concentrations below the CMC may reduce the CMC sufficiently for mixed micelles to form: a process known as comicellization. Incorporation of a fourth component to the amphipath–water solubilizate system may increase the solubility of the solubilizate; for example, octanol forms mixed micelles with sodium 3-hendecanesulphate in water thereby increasing the solubility of a hydrocarbon.
Enhanced solubility of organic compounds may sometimes be achieved by the addition of nonmicelle forming compounds. Thus sodium acetate or salicylate greatly increases the solubility of the bromine. While such an effect is gerterally due to complex formation or to decreased hydrophobic bonding, various workers have shown that there is a continuous gradation in solubilizing behaviour between high molecular weight micelle-forming amphipaths and low molecular weight nonmicelle-forming amphipaths. The effect of nonmicelle-forming amphipaths is known as hydrotropy and is due to a ‘cosolvent’ action, the distinction between hydrotropy and solubilization being rather vague. Even below their CMC, soaps are known to increase the solubility of phenolic compounds.
Factors Affecting Solubilization
Concentration of amphipath: Marked increase in solubility occurs above the CMC, the solubility often being a linear function of the amphipath concentration for dilute solutions (Fig. 5.35). If the amphipath is more concentrated however, the solubility may increase more rapidly, reflecting the changing size and shape of the micelles.
Fig. 5.35 Solubilization of Cresol BP in potassium oleate solution at pH 10.6 and 23°C (soap prepared from Oleic Acid BP)
Nature of the amphipath: For a homologous series of amphipaths, solubilization of hydrocarbons is increased by an increase in alkyl chain length but is decreased by chain branching. The effect of unsaturation in the chain probably depends upon the solubilizate. Enlargement of polyoxyethylene chains of nonionic compounds appears slightly to reduce their solubilizing power even when calculated on the basis of weight of solubilizate incorporated per unit weight of micelle interior. The type of ionic group also has an effect which is slightly modified by the nature of the counterion. However, although McBain and Richards found dodecylamine HCl to be a much better solubilizer than K laurate (C12), data collected by Mulley (1964) suggest that superior solubilizing power is not inherent in any particular class of amphipath. The influence of alkyl chain length upon solubilization of polar compounds is complicated since these solubilizates are incorporated in the palisade layer. Instead of excess solubilizate separating as another phase virtually devoid of amphipath and water, as occurs with hydrocarbons, an excess of alcohols, aldehydes and fatty acids causes separation into two phases each containing a considerable proportion of amphipath. Each of these phases may be isotropic, one containing ordinary micelles and excess water (L1 phase) and the other containing inverted micelles and excess amphiphile (L2 phase). Alternatively, the two phases may be L1 and liquid crystalline (LC) phases. In either case, a turbid emulsion results due to the different refractive index of the phases. L1 and L2 systems separate with amphipaths containing the shorter alkyl chains when increasing chain length results in increasing amount of amphiphile which can be solubilized before separation of an L2 phase. However, solubilization with the longer alkyl chain compounds is limited by the separation of an LC phase at a lower amphiphile concentration than when an L2 phase separates, and this concentration decreases as the amphipath alkyl chain is lengthened.
Nature of the solubilizate: Elworthy et al. (1968) have discussed the effect of the nature of a solubilizate upon the extent of its incorporation into micelles and conclude that generalizations apply only to the simplest compounds. Polarity and polarizability are important, besides molecular geometry. For a homologous series, whether polar or nonpolar, an increase in the size of a hydrocarbon group decreases the solubility in a solution of an amphipath of given concentration. Branching of alkyl chains has only a small effect but unsaturation increases the solubility. Cyclization also tends towards enhanced solubility but this remains low for the more rigid polycyclic compounds. Polar compounds are usually more soluble than nonpolar ones although octanol is less soluble than octane in a decinormal solution of dodecylamine HCl at 25°C, presumably for reaSl1llS discussed in the preceding paragraph. Octanol is, however, more soluble than octane in several other amphipath solutions. The effect of the type of polar group and alkyl chain length upon the system sodium caprylate–water–solubilizate has been illustrated by Mandell et al. (1967a, b).
Effect of electrolytes: Salts have been shown to enhance the solubility of hydrocarbons in amphipath solutions. Besides promoting the formation of micelles as evidenced by a lowering of the CMC, salts cause an increase in micelle size with a consequent increase in the volume–surface ratio and presumably therefore an increased solubilizing capacity. The solubility of some polar compounds is reduced in the presence of salt and is probably associated with a decreased solubility causing separation of a liquid crystalline phase.
Effect of nonelectrolytes: Mention has already been made (p.66) of the effect of water-soluble substances on the CMC. Similar effects have been noted with regard to the solubilizing power of amphipaths towards hydrocarbons. For example, ethanol suppresses both micelle formation and the heptane and benzene-solubilizing power of cetrimide. Solubilization behaviour is however influenced by the solvent effects of the additive and there will exist a partition of both the hydrocarbon and the polar additive between the micelles and the extramicellar solution. Sometimes the addition of water-soluble compounds enables a given amount of solubilizate to be dissolved using less amphipath: glycerol, sorbitol and sucrose, for example, reduce the amount of nonionic amphipath required to solubilize vitamin A. The addition of hydrocarbons may also increase the solubility of sparingly soluble compounds. This effect, known as cosolubilization, is thought to be due to the increased volume of the micelles upon solubilization of the hydrocarbon. The cosolubilizing action of long-chain alcohols and hydrocarbons has already been mentioned (p. 72). Other polar compounds are also effective cosolubilizers such as long-chain amines or mercaptans.
Effect of temperature: There is little effect upon the solubility of liquid hydrocarbons in dilute solutions of ionic amp hi paths although an increase in temperature can markedly increase the solubility of solid solubilizates since the crystal lattice becomes less stable. The solubility of polar solubilizates, however, may increase considerably with temperature particularly when solubility is limited by the separation of a liquid crystalline phase. Increase in’ temperature may also increase the solubilizing power of nonionic amphipath solutions but many solubilizates, particularly amphiphilic ones, lower the cloud point so that the solubility decreases again as higher temperatures are reached (Fig. 5.36).
Fig. 5.36 Solubilization and cloud point curves of 1 per cent C10H21(OCH2CH2)10OCH3 solution in the presence of (A) Decane; (B) Decanol (after Nakagawa & Tori, 1960)
Effect of pH: Although pH has little effect on the CMC of such compounds as alkyl sulphates, the CMC of fatty acid soaps decreases with a reduction in pH. With the latter compounds a small reduction in pH results in enhanced solubilizing power. When the solubilizate is a weak electrolyte, its degree of ionization is influenced by the prevailing pH. Since the extent of solubilization may frequently be considered in terms of the distribution between the lipophilic interior or palisade layer of the micelle and the extramicellar aqueous solution (Evans & Dunbar, 1965), a shift in pH in the direction which suppresses ionization and makes the solubilizates less hydrophilic will increase the proportion solubilized. Some compounds such as benzoic acid also appear to associate with the polar chains of nonionic amphipaths by hydrogen bonding. A decrease in pH decreases the ionization of the acid and would increase such binding to the micelle surface. Regardless of the position in the micelle occupied by the benzoic acid, it follows that a decrease in pH results in a decreased extramicellar concentration for a given total benzoic acid concentration.
Ternary Phase Diagrams
Various physicochemical characteristics of three component systems at constant temperature and pressure can be mapped out using triangular coordinates. The diagrams are constructed using equilateral triangles which have the unique property that the lengths of the three lines radiating from any point within the triangle, parallel to the sides and disposed at an angle of 60° to each other,” add up to the length of one side. Similarly the composition fractions of the three components add up to the whole system. Thus point A of Fig. 5.37 represents 50 per cent X, 20 per cent Y and 30 per cent Z; point B represents 30 per cent X and 70 per cent Z and point C represents pure component Z.
Fig. 5.37 Triangular diagram.
Line CF contains every possible mixture with components X and Y in the ratio 5:2. Dilution of composition F with component Z causes the resultant composition to pass through A and to reach the corner at C at infinite dilution. Line DE contains every possible mixture with 20 per cent Y, the proportion of X and Y varying according to the position on the line. (See also p.9.)
Triangular diagrams are constructed as contour maps each contour joining those compositions having some property in common. Thus a phase diagram has contours which circumscribe those compositions exhibiting the same phase type.
A four component system might be represented as a solid figure in the form of a regular tetrahedron or an equilateral triangular prism. Alternatively, the vertical scale of the prism may be used to represent the influence of temperature or of amphiphile chain length upon a three component system.
Fig. 5.38 illustrates the triangular phase diagram. Compositions within the regions denoted by L1 (high proportion of water) and L2 (high proportion of decanol) are isotropic containing micelles and inverted micelles respectively except right in the corners where the soap concentration is below the CMC. The smectic liquid crystalline region is delineated near the centre of the triangle and the two hexagonal mesophases are also shown. Regions between these single phases are composed of a mixture of the relevant phases. Thus at low soap concentrations the mixture separates into two phases (L1 and L2) except when water or decanol is in extreme excess. Mixtures on the tie-line MN separate into the two phases whose compositions are M and N. The proportion of phases of composition M and N depends on the position of the mixture on the tie-line. Other mixtures separate into L1 and neat phase, L1 and middle phase or L2 and invert middle phase. Thus mixtures lying on the tie-line OP separate into two phases of composition O and P. Between the single liquid crystalline phases are two phase regions, each phase consisting of a different type of liquid crystalline arrangement, for example neat and middle. It will also be noted that there are a number of triangular regions within which the mixture separates into three phases. According to the phase rule these form invariant systems, that is, any mixture within a triangle will separate always into the phases whose compositions are denoted by the corners of that triangle. For a given temperature and pressure, only the relative proportion of the three phases will vary with change in composition of the mixture within the triangle. Thus mixtures within ABC separate into L1 + L2 + neat; mixtures within. DEF separate into L1 + middle + neat and mixtures within GHI separate into middle + neat + hydrated solid.
Fig. 5.38 Phase equilibria for the system sodium octanoatedecanol–water at 20°C (slightly modified from Mandell et al., 1967)
Pharmaceutical Application of Solubilization
Drugs of limited aqueous solubility have been solubilized for both internal and external use. These include oil soluble vitamins, steroid hormones and antimicrobial agents, the resulting systems being thermodynamically stable.
Solubilization of orally administered drugs results in an· improved appearance and may improve an unpleasant taste. It also enables both oil- and water-soluble compounds to be combined in a single phase system as with multivitamin preparations. The amphipath must be nontoxic in the quantities used, have an agreeable taste and odour and be compatible with other ingredients. It should also possess good solubilizing power and stability. Cationic amphipaths are the most toxic but nonionic compounds generally have a low toxicity and therefore are to be preferred. Because many nonionic compounds are bitter, careful formulation is required to keep their concentration to a minimum. The monoesters of polyoxyethylene sorbitan or of sucrose are commonly used, often with cosolubilizing agents which may also help to improve the taste.
Solubilization may lead to enhanced absorption and an increased biological activity. It improves the intestinal absorption of vitamin A and the percutaneous absorption of oestrone. Drug absorption from ointment bases and suppositories may also be increased by the presence of an amphipath.
Clear solutions of corticosteroids for ophthalmic use are achieved using nonionic agents. In order to minimize any irritant effect, low concentrations of amphipath are required, a good solubilizing power being essential.
Amphipaths are more toxic when given pilrenterally. Ionic compounds readily produce haemolysis and therefore only nonionic compounds are suitable, such as a polyethylene glycol ether which is used to disperse phytomenadione.
Aqueous concentrates of volatile oils have also been prepared. Careful formulation is necessary in order to achieve a concentrate which remains clear upon dilution. Soaps are used to solubilize phenolic compounds for use as disinfectants, familiar examples being Lysol and Roxenol. The soap enables the phenol to become readily dispersed throughout systems, such as drains, to which it is added and assists penetration into greasy or coagulated material. While only the phenol in the extramicellar solution is bactericidal active, the micelle acts as a reservoir tending to maintain a constant extramicellar concentration as phenol is used up due to reaction with protein or removal by dirt. The partial phase diagram for a Lysol is given in Fig. 5.39. Line AF represents all compositions containing 51 per cent w/w (50 per cent v/v) Cresol. Compositions B, C and D are clear solutions (L2 phase only) whereas E is cloudy (L1 + L2). If composition B is diluted with water it will remain clear although L1 is eventually produced. It is also apparent that composition C contains the minimum of soap that will enable a Lysol to remain clear on dilution. Composition D, although clear at first, will pass through the L1 + L2 region upon dilution but will eventually become clear as the single L1 region is reached. This illustrates the use of the ternary diagram which enables the formulator to decide upon a suitable composition. Different soaps and grades of cresol can have a profound effect upon the phase diagram for Lysol. Burt (1965) showed that p-cresol with sodium oleate produced a three phase region (L1 + L2 + LC) so that dilution of such a Lysol always made it pass through a cloudy region.
Fig. 5.39 Equilibrium phase diagram for the system potassium oleate– cresol–water at pH 10.7 and 20°C
Disinfectants are also prepared by solubilizing iodine with nonionic amphipaths. Such a product is called an iodophore (meaning iodine carrier) and probably involves a certain degree of complexing between the iodine and the polar groups. The use of an amphipath obviates the necessity of solvents of high glycol content and enables dilution without precipitation of the iodine. Solubilized iodine volatilizes less readily and is less liable to stain fabric or corrode metal. It has however good bactericidal activity and most of the iodine is readily released to the bacterial cell.
Liquid preparations generally require preservation against microbial spoilage. The preservative exerts its effect by dissolution in the aqueous phase. Absorption into micelles reduces its effective concentration and so larger amounts of preservative need to be added to maintain a suitable extramicellar concentration.
The stability of a drug may be enhanced by solubilization. Vitamin A is more resistant to autoxidation in aqueous nonionic amphipath solutions than in oil. Similarly the oxidation of aliphatic and aromatic aldehydes is less when they are solubilized rather than emulsified. It is probable that the reduced rate of oxidation within micelles is due to dilution of the solubilizate by the amphipath making free-radical propagation more difficult. The catalytic effect of metal ions is also reduced when an aldehyde is solubilized.
Solubilization can also reduce the rate of hydrolysis of ’drugs such as benzocaine and methantheline bromide. Anionic amphipaths are better than cationic ones in protecting against alkaline hydrolysis since hydroxyl ions are repelled by micelles of the former but attracted by those of the latter. On the other hand, acid hydrolysis may be promoted by anionic compounds which attract hydrogen ions, but some measure of protection may be afforded by the addition of salts whose cations compete with the hydrogen ions at the micelle surface. Nonionic amphipaths are also capable of reducing hydrolysis rates, but if solubilization occurs among the polar groups rather than in the micelle, interior protection from hydroxyl ions is not so marked. The site of the hydrolytic process is in the extramicellar solution. Increasing the amount of amphipath increases the partition of drug into the micelle thereby reducing the fraction of drug in the aqueous environment with a consequent reduction in the rate of hydrolysis. However, solubilized systems possess a large micellar surface area enabling rapid transfer of drug in to the hydrolysis-promoting environment. Emulsified oils with a much smaller oil–water interfacial area therefore tend to hydrolyse more slowly.
Further details may be obtained from the reviews by Mulley (1964), Swarbrick (1965) and Elworthy et al. (1968), which also contain good bibliographies.
Insoluble Monolayers
If the nonpolar groups of an amphipathic compound are sufficiently large to render it almost completely insoluble, then traces of the compound will usually spread over a water surface to form a monomolecular layer. In such a case the whole of the ‘solute’ may be said to form the Gibbs excess. As usual, such molecules will have a preferred orientation with the polar groups directed towards the water. Such monolayers may spread not only from liquid amphipaths, or from their solutions in nonaqueous solvents, but also from the solid. Spreading from the solid involves a positive entropy change whereas from the liquid it is negative showing that the monolayer is disorientated relative to the solid but more orientated than in the bulk liquid. Such monolayers behave somewhat like two-dimensional matter in possessing two-dimensional physical states and exhibiting a surface pressure, a surface viscosity and a surface potential. They can form at water–air or organic liquid–air surfaces and also at liquid–liquid interfaces.
Formation of a Monolayer
The liquid or liquids forming the interface must be very carefully purified by redistillation in scrupulously clean apparatus and the trough in which the monolayer is to be spread must also be freed from minute traces of surface-active contaminants. When the water–air or water–oil interface has been formed, residual contamination can be sucked away from the interface through a capillary.
Spreading of the monolayer is generally achieved by preparing a dilute solution of the surfactant using a pure solvent which itself has a positive spreading coefficient. For fatty acids and alcohols at the water–air surface, petroleum spirit (boiling point 60–80°C) is suitable. An accurately known volume (0.01–0.05cm3) is injected near the surface using a micrometer syringe. The solution floats to the surface, spreads over it and the solvent completely disappears by evaporation and by dissolution in the ‘subphase’ after a few seconds. Solvent retention at the surface can occur and this may influence the properties of a monolayer. Such retention is minimized by employing the smallest amount of solvent consistent with complete spreading. If a hydrocarbon is unsuitable as a solvent, a water-soluble alcohol, ethanol or propanol, may be used. Alcohol–water mixtures are used to spread polypeptides, electrolytes being included for protein solutions. The solvent soon disappears from the interface by dissolution in the bulk phase in which it becomes extremely dilute. To ensure that the solute is carried to the surface, the solvent must have a suitable density, or in the case of an oil–water interface, the solution must be injected on a suitable side of it.
Surface Pressure and its Measurement
Since the molecules of the monolayer are confined to an interface, their kinetic motion in the interfacial plane will produce a two-dimensional pressure (n) measured in terms of such units as mN.m−1. This pressure is augmented by repulsion of ionized groups and opposes the surface tension of the substrate liquid, or the interfacial tension between two immiscible liquids, giving rise to a lower tension (Eq. 5.24). The surface pressure may therefore be conveniently determined by measuring the reduction in the surface or interfacial tension using a Wilhelmy plate (p. 56).
Just as a change in volume produces a change in three-dimensional pressure, so a change in the area occupied by a monolayer results in a different surface pressure. Measurement of these changes enables π−A plots to be constructed, analogous to P–V plots, A being the average area occupied per molecule or per milligram in the case of proteins.
Knowing, the area of the interface and the amount of surfactant injected into it, the area available to each molecule can readily be calculated provided that the molecular weight is known. The surfactant can be injected in small increments and the resulting tension measured each time. A disadvantage of this technique, however, is the necessity to wait many times for the disappearance of solvent from the interface.
Studies at a water–air surface may be facilitated by using a rectangular trough made of Teflon, or of glass rendered hydrophobic by being spread with a Teflon aerosol or coated with paraffin wax. The water level may then be made to stand a little above the level of the wall of the trough so that a hydrophobic bar placed across the trough isolates the liquid surfaces on either side of the bar. The Wilhelmy plate dips into the liquid at one end of the trough and the monolayer is spread on the same ·side of the bar. By moving the bar towards the plate, the area available to each molecule in the monolayer is decreased (Fig. 5.40). The production of nonzero contact angles, particularly with rigid films at high surface pressures, is a source of error.
Fig. 5.40 Diagrammatic representation of the trough and Wilhelmy plate (A) Side view, (B) Plan view. (The surfactant molecules are not drawn in proportion.)
Direct measurement of the surface pressure can be made using a film balance otherwise known as a Langmuir–Adam trough (Fig. 5.41). The Wilhelmy plate is replaced by a ‘float’ or ‘boom’, a thin strip of mica rendered hydrophobic in the same manner as the trough. The float lies flat on the water surface and is suspended from a horizontal torsion wire ·by means of a stiff wire or rigid plastic plate. It extends almost the whole width of the trough, the narrow gaps at the ends being sealed using thin gold or platinum foil or silk treated with petroleum jelly or plastic threads to prevent leakage of the monolayer past the float. Motion of the float is monitored by means of an optical lever. The forte exerted on the float by the monolayer can then be determined by the displacement of a reflected light spot, or by the angle of twist to the torsion wire needed to restore the light spot to its null position.
Fig. 5.41 Diagrammatic representation of a film balance: (A) Side view, (B) Plan view. (The surfactant molecules are not drawn in proportion.)
Preliminary cleaning of the water surface is facilitated by the use of movable bars since these can be used to sweep adsorbed contaminants out of the area to be covered by the monolayer. A clean surface can then be demonstrated by reducing the confined surface area when no reduction in surface tension or build up of surface pressure should be evident. With the film balance it is important to clean the surface on both sides of the float.
When the spread monolayer is highly insoluble, the float of the Langmuir–Adam trough acts as a perfect two-dimensional semipermeable membrane by allowing free passage only of the substrate liquid. Direct measurements of the surface pressure of soluble monolayers can, however, be made using a PLAWM trough (PLAWM = Pockels–Langmuir–Adam–Wilson–McB ain). This is similar to the Langmuir–Adam trough, but it is divided into two compartments by a thin highly convoluted rubber membrane under the float. This membrane prevents the seepage of solution to the surface behind the float.
An alternative method of studying soluble monolayers is to of how the changes in tension by means of a strain gauge as a film of solution is stretched between two platinum rings.
Various modifications are necessary for studies at the liquid–liquid interface. For example, the trough and barriers may need to be made of a less hydrophobic material than Teflon, such as nylon, while it may be more convenient to· use a taut plastic band rather than bars to confine the interfacial film.
Surface Potential and Its Measurement
Adsorption of a monolayer of surfactant molecules at a water–air or water–nonpolar oil interface changes the potential across the interface due to the separation of charges in or around the polar groups. This potential change (ΔV) is known as the surface or interfacial potential and may be measured by the air electrode or vibrating plate methods (Alexander & Johnson, 1950; Davies & Rideal, 1961).
Surface potential measurements can give information relating to the structure of the monolayer. Fluctuation so the potential between different parts of the surface indicate a variation in the packing of the molecules while the actual value of the potential can give information regarding their orientation. The surface potential is analogous to the potential difference between the plates of a parallel-plate capacitor giving the equation:
(5.29)
where N is the number of charges of magnitude Q coulombs per square metre, d is the distance between the plates, ε is the dielectric constant (relative permittivity) and εo is the permittivity of free space (= 4π × 10−7 c2, where c is the velocity of light in free space in ms−1). For a molecule, the product of the charge Q and the separation distance is the dipole moment μ. The distance d is however the separation distance resolved normal to the plane of the monolayer so that if the dipoles make an angle θ to the normal then Qd = μcosθ. Eq. 5.29 then becomes
with N equal to the number of monolayer molecules in each square metre of surface and ε generally being taken as unity. Caution must however be exercised in interpreting surface potentials solely in terms of 1he monolayer structure since the presence of a monolayer can affect the structure of an aqueous substrate adjacent to the monolayer.
Surface Rheology and Its Measurement
Two-dimensional distortion and flow of a monolayer is analogous to the behaviour of a bulk phase in three dimensions. A wide range of surface mechanical behaviour can be demonstrated such as viscosity and elasticity in both dilatation and shear. Thus the surface elastic modulus Ks is given by:
and its reciprocal is the surface compressional compliance which, for a perfectly elastic film, is also the surface compressibility Cs given by
(5.30)
so that 
which is known as the surface compressional modulus. The compressibility is readily calculated from π–A plots and its value depends upon the state of the film, it being small for closely packed monolayers having the character of a two-dimensional condensed state. The compressibility of a film and its relaxation time are important parameters governing the stability of foams and perhaps some emulsions. The various surface elastic moduli and their interrelationship have been described by Tschoegl (1958).
which is known as the surface compressional modulus. The compressibility is readily calculated from π–A plots and its value depends upon the state of the film, it being small for closely packed monolayers having the character of a two-dimensional condensed state. The compressibility of a film and its relaxation time are important parameters governing the stability of foams and perhaps some emulsions. The various surface elastic moduli and their interrelationship have been described by Tschoegl (1958).Resistance to a shear stress in the plane of a monolayer is measured in terms of a surface viscosity (ηs), the unit being a surface Poise (sP = 1mN.s.m−1). For a surface,tangential force per metre = ηs × shear rate
so that ηs has the dimensions MT−1 and is related to the three-dimensional shear viscosity (η) by:
(5.31)
d being the film thickness which is of the order of 2 nm for monolayers. Thus a monolayer having a surface viscosity of 10−4 sP has a viscosity of about 500 P (= 50 N.s.m−2) over its thickness. A liquid with such a high viscosity would be termed ‘very thick’ so that monolayers having surface viscosities of the order of 1 sP or more would generally be regarded as solid.
Surface viscosities can be determined in a number of ways, two-dimensional analogues of tube and Couette viscometers often being employed. The canal method (Fig. 5.42) employs a trough with movable barriers to apply a surface pressure to the film and the pressure difference (Δπ) between the ends of the canal can be measured using a film balance.
Fig. 5.42 Canal method for measuring the surface viscosity of an insoluble film.
By analogy with the Poiseuille equation
(5.32)
where W and L are the width and length of the canal, respectively. Since the flowing film drags some of the underlying water, a correction must be applied so that Eq. 5.32 becomes
where ηo is the ‘ordinary’ viscosity of the substrate liquid and a depends upon the width and depth of the canal. This method can be used only for insoluble mono layers since partially soluble ories would tend to desorb from the region of high surface pressure and be readsorbed on the other side of the canal.
The drag of the substrate liquid is employed in the so-called viscous traction method. In one variant of this, a Petri dish containing the spread film is slowly rotated and a canal between two concentric wire rings suspended at the surface retards the motion of the film. The rings are so fixed that the meniscus between them is slightly concave upwards and talc particles sprinkled on it enable the rate of rotation of the midline to be determined. The retardation of the rate of rotation between the rings is a function of the interfacial viscosity, the latter being determined by calibration of the instrument with films of known interfacial viscosity. This method is suitable for both water–air and water–oil interfacial films.
Another method, suitable for liquid–air surfaces, also uses two concentric rings but the inner one is rotated while the liquid is kept stationary. Liquid motion imparted by the rotating ring is kept to a minimum by placing a sheet of mica about 1mm below the surface. Other techniques that have been used include those which determine the damping of the oscillations of a torsion pendulum.
Creep tests can also be used to study the viscoelastic behaviour of films. This is best achieved by applying a constant torsional stress to a ring or biconical disc suspended in the interface from a torsion wire and observing the displacement with time. The ring or disc must be concentric with the circular dish.
Large increases in surface or interfacial viscosity can result from the formation of mixed monolayers between oil-soluble and water-soluble surfactants, such as between lauryl alcohol and sodium lauryl sulphate or cetyl alcohol and cetrimide. Such monolayers are capable of stabilizing emulsions and foams. The increased surface viscosity resulting from the addition of a little fatty alcohol to an ionic surfactant solution is probably the prime cause of the enhanced foam stability which is obtained (Davies & Rideal, 1961).
Structure and State of Monolayers
Just as the bulk state is affected by the three dimensional pressure (P) so changes in surface pressure (π) can alter the state of a monolayer. By the equation analogous to Eq. 5.31:
it follows that if the monolayer thickness is 2nm, then a surface pressure of 2mN.m−1 is equivalent to a bulk pressure of 1MN.m−2 or about ten times atmospheric pressure. Since surface pressures well in excess of 20mN.m−1 are frequently encountered, it will be appreciated that such monolayers have a strong tendency to maintain their integrity without collapsing and in so doing they exhibit a surface compressibility equivalent to the bulk compressibility of solids or liquids.
The surface pressure of a monolayer is the resultant of three components: that due to kinetic motion (πk), that due to electrostatic repulsion (πr), and the opposing effect due to cohesional forces (π) which is always negative. Thus:
The effect of these pressure components can be seen in Fig. 5.43. Unionized molecules of straight chain fatty acids or alcohols can pack very closely to each other in vertical orientation and cohesion due to van der Waals forces is large. If a large area is available, these form large clusters up to several millimetres in diameter and exert only a very small surface pressure. These clusters behave rather like a two-dimensional liquid with a few ‘gaseous’ molecules in between: Only when the available area is decreased so that the groups begin to pack tightly does π increase appreciably. Further compression of the monolayers results in large values of π with only a small decrease in A when the compressibility becomes even smaller due to the attainment of a two-dimensional solid state. A little more compression then results in collapse of the film.
Fig. 5.43 π−A curves illustrating the effect of: (A) A straight hydrocarbon chain, octadecanoic acid (stearic); (B) A hydrocarbon chain bent by a double bond in the middle, cis-9-octadecanoic acid (oleic); (C) An ionized polar group, octadecyltrimethylammonium ions on M/2 NaCl
A double bond in the hydrocarbon chain particularly with a cis configuration prevents close packing of the molecules and reduces interchain cohesion. The monolayer therefore is much more expanded than with saturated-chain acids. Similar behaviour is also observed with other polarizable groups in the hydrocarbon chain such as with hydroxy acids. At large areas both polar groups become anchored to the aqueous substrate so that the molecule occupies a much larger area than when vertically orientated. Appreciable pressures are required to detach the second polar group and to force the molecules into a more or less vertical Orientation. This monolayer then has the character of an expanded liquid with the hydrocarbon chains in a more random state than with straight saturated chains and consequently a higher compressibility. Further compression will force the molecules into a closer packing with a smaller compressibility having the character of a condensed film before collapse occurs.
An ionized monolayer, spread on a salt solution in order to reduce its solubility, has an appreciable surface pressure even at very large available areas. Repulsion between the polar groups overcomes the cohesional forces causing the molecules to fill the surface in a manner analogous to a gas filling an available space. These monolayers are therefore termed gaseous.
Interchain cohesion increases with length of the hydrocarbon chains. Figs. 5.44 and 5.45 show the effect of chain length, and temperature upon the π−A curves for fatty acids. Fig. 5.44 demonstrates a remarkable similarity to Andrew’s P–V curves for a gas above and below the critical temperature suggesting that an increase in interchain cohesion results in a higher critical temperature. The shallow curves on the right hand side of the figure represent highly compressible gaseous films while the steep portions on the left are produced by liquid-like films of low compressibility. For fatty acids below the critical temperature (in Fig. 5.44 at about 14° with more than 12c atoms) a horizontal portion represents a first order transition, that is, dA/dπ becomes infinite. Monolayers having a mean area per molecule lying within this transition are composed of discontinuous regions some/with coherent molecules, the rest being gaseous. Such a condition leads to surface potentials which fluctuate from one part of the surface to another.
Fig. 5.44 π−A curves for a series of straight chain saturated fatty acids between 12 and 16°C (from Adam, 1930)
Fig. 5.45 π−A curves for tetradecanoic acid at various temperatures (from Adam, 1930)
The extreme left hand portion of Fig. 5.44 for the C14 acid has been enlarged in Fig. 5.45, which shows the effect of temperature upon the liquid-like state. The steep curves on the right hand side are those for the liquid-expanded state whilst on the left hand side the condensed state is found. Between these two states is another transition, this time for a second order where dA/dπ is finite. It is known as a liquid-intermediate state and has a higher compressibility than either the liquid-condensed or liquid-expanded states. This transition probably involves the loss of rotational freedom about the long axis of the molecules thereby creating further area for close packing. The sequence of phase changes is summarized in Fig. 5.46. Not all states are given by every compound or at all temperatures. For example, at low temperatures a liquid condensed film may change directly into a gaseous film. Some workers have also distinguished other states such as the superliquid, which is rheologically liquid but has a compressibility like that of a solid film; cetyl alcohol behaves in this manner.
Fig. 5.46 Diagrammatic sequence of monolayer states (values on abscissa represent only the usual order of magnitude)
By analogy with a gas, the simple equation of state for a gaseous film is
(5.33)
where k is the Boltzmann constant, that is, the gas constant per molecule (R/N). This assumes that the film pressure has no contribution from cohesional or electrostatic repulsive forces and is due only to kinetic motion so that π = πk Thus Eq. 5.33 is a limiting law and can be represented by a rectangular hyperbola having a value of about 4 × 10−21 J at room temperature (see also Fig. 5.44). Gaseous films tend to approximate to such a curve only at areas per molecule approaching 100 nm2 as shown in Fig. 5.47.
Fig. 5.47 πA−π and π−A−1 plots for ethyl hexadecanoate at 16−17°C
This ideal gaseous film equation (Eq. 5.33) suggests that A becomes vanishingly small as π becomes very large. A better equation of state is obtained by taking into account the effective area (Ao) of the molecule when
(5.34)
Films likely to obey Eq. 5.34 are those which are uncharged (πr = 0) and which have no interchain cohesion (πo = 0). Interchain cohesion is virtually eliminated at an oil–water interface since the oil molecules penetrate between the hydrocarbon chains and satisfy their London attraction forces (Fig. 5.48). This leads to greater surface pressures for a given area per molecule at the oil–water interface. The surface pressures of films of quaternary ammonium ions do however have an electrostatic repulsion component and the π(A − Ao) hyperbola is therefore not coincident with the curve for the oil–water interface. Approximate coincidence with the curve for the air–water surface implies that the cohesive forces and the electrostatic repulsive forces are opposite and of nearly equal magnitude.
Fig. 5.48 π−A curves for hexadecyltrimethylammonium bromide on salt solution at 20°
Interchain cohesion becomes appreciable in gaseous films under high surface pressure or in liquid-expanded films. For an uncharged film π=πk+π0 whence πk=π−π0 and Eq. 5.34 becomes
This equation of state is analogous to the van der Waals equation for real gases and in the case of liquid-expanded films π0 is nearly independent of A.
Since condensed films have nearly linear π−A plots, their equation of state can be written
where a and b are constant for a given compound and set of physicochemical conditions. Comparison with Eq. 5.30 shows that aA is tilt: surface compressional modulus 
.
.Extrapolation of a linear plot to zero pressure gives information concerning the packing and orientation of the film molecules. Saturated straight long-chain fatty acids at room temperature show both the liquid and solid condensed states with limiting areas (A0) per molecules of about 0.24–0.26 nm2 and 0.203–0.205 nm2, respectively. These areas are independent of chain length at a water air surface (although not at a water–oil–interface) and indicate a vertically orientated monolayer. That of the solid condensed state is about 0.02nm2 greater than the area of cross section (0.185 nm2) in the three-dimensional crystal.
At high surface pressures, many straight-chain compounds have limiting areas of about 0.205nm2 Besides the fatty acids, this is found with alcohols, amides and esters where staggering of the polar groups allows the area to be limited only by the interlocked hydrocarbon chains. Since the cross section of the polar groups is larger than that of the chains, a larger limiting area is often found at low surface pressures; for example, alcohols and esters pack to about 0.22 nm2 and nitriles to about 0.27 nm2. However, association between the polar groups by hydrogen bonding, with amides for example, may ensure a low limiting area given at low surface pressures.
For a given surface pressure, the area per molecule of a liquid-condensed film decreases slightly with an increase in chain length due to the increased interchain cohesion. Thus at 5mN.m−1 and 25° a hexadecanoic acid molecule occupies 0.24 nm2 whereas an eicosanoic acid molecule occupies only 0.23 nm2. Under these conditions, tetradecanoic acid would form only an expanded film with an area per molecule of 0.37 nm2.
The effect of an unsaturated bond depends upon the position and configuration. A double bond near the middle of the chain with a cis configuration produces too much bending of the chain for a condensed film to form whereas the chain of a trans compound is straighter and can produce such a film. Condensed films also form if the double bond is next to a terminal carboxyl group, the liquid condensed phase then having a limiting area of 0.287 nm2.
Sterichindrance to close packing also results from chain branching. Thus 16-methylheptadecanoic acid forms a much more expanded film than octadecanoic acid although both are saturated C18 acids. If the methyl group is near the middle of the chain, the film is even more expanded for a given surface pressure.
Sterols such as cholesterol with a hydroxyl group at the 3 position also form condensed films with a limiting area between 0.37 and 0.42 nm2 according to the stereochemical configuration of the A and B rings. Such molecules tend to stand nearly upright at a surface, packing closely to form a film of low compressibility. The tilt of the sterol nucleus is affected by the type and position of the polar group. Sometimes the molecules are caused to lie nearly flat at low surface pressures and consequently they possess large limiting areas. Such films have a much greater compressibility than cholesterol.
A liquid condensed film of a straight-chain fatty acid has been shown to have an appreciable compressibility, the area per molecule decreasing from about 0.25 to about 0.205 nm2 due to a rearrangement of the carboxyl groups. The polar groups of L-monoglycerides are also closely packed with a limiting area of about 0.263 nm2 but are not rearranged by compression. Their low compressibility suggests that the molecules are held apart bi structure formation between the polar groups. The head groups of p-alkyl-phenols are also closely packed with a limiting area of 0.24 nm2. Low compressibility in this case suggests that the benzene rings are vertically orientated in a closely packed layer about 0.6 nm thick and that these take up the whole of the compression.
The areas quoted above refer to unionized films and for fatty acids apply to films spread on dilute HCl or on fresh very pure water. On water, trace contaminants such as polyvalent cations may eliminate the liquid-condensed state causing the solid state to persist down to low film pressures. Alkaline substrates cause ionization and expansion of fatty acid monolayers. Although electrostatic repulsion between ionized carboxyl groups would be expected to have some effect, association of the carboxyl groups with the hydrated cations appears to be the prime cause of the expansion. Liquid-expanded films of octadecanoic acid are produced at pH 13 on solutions of LiOH, NaOH and KOH and it is probable that the size of the hydrated cation in the surface increases in the order Li, Na and K, producing the same order of effect upon expansion of the monolayer. Traces of Ca ions, however, markedly reduce the expansion effected by monovalent cations. At pH 8.5, using bicarbonates, compression of the now incompletely ionized monolayer causes monovalent cations to be ejected. These changes in the condition of the monolayer are reflected in tile surface potential which becomes considerably reduced by dipolar opposition of the counterions.
The conclusion, therefore, is that the state of a given monolayer and its transition from one state to another is influenced by three independent variables. These are the temperature, surface pressure and the extent to which the polar groups interact with the liquid substrate. In the latter case this is affected by the composition of the substrate such as the presence of electrolytes and the prevailing pH. Additional to these factors the configuration and size of the hydrophobic groups influence the cohesive forces and extent of packing of different monolayers.
Mixed Monolayers
While monolayers of water-insoluble amphipaths such as octadecanoic acid tend to be coherent, those composed of less hydrophobic substances tend to be fully solvated, that is, they are penetrated by water to form two-dimensional aqueous solutions. Compression of such monolayers causes the water molecules to be squeezed out at some critical pressure which is less than 0.1mN.m-1 in the case of octadecanoic acid but between 28 and 30mN.m−1 for sodium dodecylsulphate.
Water-soluble surfactants can also penetrate into an insoluble monolayer, the free energy of the penetrant molecules being decreased by dilution in the monolayer. The excess free energy of mixing monolayers of long-chain alcohols with some ionic surfactants has been shown to be a minimum when the monolayer contains equimolecular proportions of the two components. Interchain cohesion and screening of the ionic groups by the intervening alcohol groups are both probably involved in maintaining the integrity of the mixed monolayer.
Molecular association between two components requires a strong interaction between the polar groups but this is accompanied by monolayer penetration only if the soluble molecules have large enough hydrocarbon groups whereby they can form a strong interchain cohesion with the insoluble molecule. Sometimes the association in the· monolayer is strong enough for stoichiometric complexes to be distinguished. Thus a cholesterol film on NaCl solution has a limiting area per molecule of about 0.4 nm2. If the film is expanded and sodium hexadecylsulphate is injected underneath, a very large rise in surface pressure occurs upon recompression with collapse at about 0.6 nm2 per cholesterol molecule. This indicates that some hexadecylsulphate molecules become squeezed out of the monolayer upon compression but that type remaining molecules strongly adhere in a 1:1 ratio. The surface complex is less stable in the absence of salt. Some hexadecylsulphate also appears to be adsorbed underneath the monolayer and this may enhance the stability of the film.
Similar 1:1 complexes have been found to form between other amphipaths such as between cholesterol and the steroidal saponin digitonin while 1:2 or 1:3 complexes have been detected with some other substances. Many other mixed monolayers also have collapse pressures and areas larger than those of the pure insoluble component but not all form stable ‘complexes’. Thus molecular association occurs between long-chain alcohols and sodium hexadecylsulphate but a stoichiometric relationship is not found. In such cases the association is strongest when the alcohol possesses a straight hydrocarbon chain.
Surface association is paralleled in bulk solution by the formation of mixed micelles or liquid crystals. A thermolabile complex of cholesterol and digitonin can crystallize from alcohol. Complexing may also affect certain properties in the bulk phase. Thus sodium hexadecylsulphate loses its haemolytic power and is not precipitated by silver nitrate when its solutions contain an equimolar proportion of cholesterol. The extent of molecular association at a water–oil interface is also implicated in the stability of emulsions and may involve multilayer formation.
Monolayer penetration may be accompanied by a change in the physical state of the film. Thus a solid film of ergosterol becomes liquid on injection of saponin or sodium hexadecylsulphate into the substrate whilst a liquid film of cholesterol becomes solid op injection of saponin but remains unchanged with sodium hexadecylsulphate.
If cholesterol is added to a liquid expanded film of a long-chain fatty acid in the molecular proportion of 1:4, the bulky cholesterol molecule restricts the oscillatory motion of the acid molecules causing them to occupy the smaller area of a condensed film. A mixed monolayer of cholesterol and lecithin also occupies a smaller area for a given surface pressure than the sum of the areas of the separate components. Although it has been suggested that the components of the film form molecular complexes, it would seem that the mean molecular area simply reflects the best arrangement of the molecules in accordance with their mutual attractive forces.
Macromolecular Films
Like monolayers of simple molecules, macromolecular films can exist in various physical states ranging from gaseous to solid, compressed films frequently being viscoelastic. They are more expanded at the water–oil interface than at the water air surface because of reduced intermolecular cohesion due to the penetration of oil molecules between the hydrophobic groups.
Proteins are able to form monomolecular films on water having a thickness of 1 nm or more. This involves the unfolding of the molecule and may result in film formation being irreversible because of the diminution in solubility which accompanies the exposure of hydrophobic side chains. At low surface pressures, protein films occupy an area around 1m2mg-1 with the unfolded molecules lying flat on the surface. Higher pressures cause the hydrophobic side chains to become orientated away from the water and at collapse pressures an insoluble coagulum is produced. Interfacial tensions are a minimum when solutions of proteins such as gelatin are at the isoelectric point because the molecules then have a relatively large intra and intermolecular cohesion and so are likely to pack better at the surface. This tends to be accompanied by a maximum interfacial viscosity. Greater adsorption on to solids has been noted by a number of workers at or near the isoelectric point.
Many hydrocolloids are adsorbed at solid–water and oil–water interfaces whereby they can stabilize dispersions. The interface between benzene and acacia, sodium alginate or gelatin solutions takes several hours to achieve an equilibrium tension presumably due to multilayer formation. Highly branched stiff-coiled molecules such as acacia form rigid films whereas unbranched chains such as sodium alginate tend to produce liquid films. The properties of these films have an important bearing upon the stability of emulsions. No denaturation of acacia occurs upon adsorption although the first layer appears to be irreversibly adsorbed with the film becoming noticeably rigid within about 20 seconds. The thickness of an acacia multilayer varies with the type of oil. Strong films of potassium arabate about 100 nm thick appear to be formed with aliphatic hydrocarbons whereas weaker thin films form with benzene.
Various statistical mechanical models have been proposed for the conformation of adsorbed macromolecules (Hoeve et al., 1965). It is suggested that the molecule may become anchored at the interface by only a fraction of the groups along its length, the rest of the molecule forming loops into the water. In such an event it is likely that the ends of the molecule are also anchored at equilibrium. The extent of looping into the solvent and the resulting thickness of the adsorbed layer can depend on the solvent. Polymer coils tend to extend in good solvents and would produce thick layers of low density. The density of multilayers probably decreases, that is solvation increases, with distance from the interface.
Biological Membranes
Multimolecular films form the boundary around all living cells and organelles such as mitochondria and chloroplasts. They are composed of lipids and protein molecules held together by hydrophobic and electrostatic interactions to form lipoprotein.
The lipids consist of nonionic amphipaths such as cholesterol and amphoteric amphipaths called compound lipids. Compound lipids are long-chain fatty acid esters of alcohols containing other polar groups; for example, a glycerophospholipid is based upon glycerol which contains a phosphoric acid residue and a nitrogenous base this being choline in the case of lecithins (p. 57). Another phospholipid is based upon inositol while the sphingolipids are derived from the unsaturated amino alcohol, sphingosine. Hydrolysis of some compound lipids also yields sugars such as galactose from cerebrosides and sialic acids from gangliosides. Phospholipids predominate in animal cells.
Membranes differ considerably in their lipid and protein content according to their source and function. Thus myelin membranes contain a dry weight fraction of over 0.7 of lipid whereas rat liver mitochondria membranes contain less than 0.3, proteins making up most of the remaining weight. The proportion of cholesterol in the lipid also varies considerably, constituting about 0.4 of myelin lipid but only 0.06–0.14 of mitochondrial lipid and being absent from chloroplast lipid, while bacterial membranes contain little or no cholesterol.
Unlike films formed at an air–water or an oil–water interface, biological membranes are bounded on either side by an essentially aqueous environment. This requires that a multilayer be formed with the hydrophobic groups orientated towards each other inside the membrane, with the more polar groups or molecules exposed on the surface. If the lipids are extracted from erythrocyte membranes and spread as a monolayer upon water they occupy twice the area of the original erythrocyte surface demonstrating that the lipid portion of the membrane is two molecules thick.
Danielli and Davson proposed that cell membranes consist of a biomolecular lipid leaflet bounded on either side with a layer of protein molecules (Fig. 5.49). Electron microscopic evidence for such a structure in membranes forming the myelin sheath of nerves has been obtained by Robertson (1967) where the bimolecular lipoprotein forms a repeating unit about 7.5 nm thick. However the protein does not appear to be held on the outside of the lipid bilayer merely by coulomb interaction but some portions of the protein molecules must penetrate to the interior where they are held by hydrophobic interactions.
Fig. 5.49 The Danielli–Davson concept of a cell membrane as modified by Robertson (1967)
Myelin lipids contain a high proportion of saturated and long-chain fatty acids which favours the formation of a condensed bilayer. Although containing a considerable proportion of polyunsaturated fatty acid esters, human erythrocyte lipids can also form a condensed bilayer within most of the membrane due to the condensing effect of the high proportion (0.42) of cholesterol. Such tightly organized membranes function as permeability barriers. On the other hand, membranes such as those of mitochondria and chloroplasts, which are associated with the metabolic activity of the cell, contain lipids which are rich in polyunsaturated or branched-chain fatty acid esters and contain little or no cholesterol. These are unable to form condensed bilayers and are associated with a high proportion of protein. Other, more loosely organized, structures have been proposed for organelle membranes which involve two dimensional arrays of protein interposed by lipid (O’Brien, 1967). One such model (Fig. 5.50) consists of a bimolecular array of globular protein having predominantly nonpolar amino acids towards the centre of the layer, the cavities between the protein molecules containing the lipid bilayer. It would also appear that some protein molecules extend right across a single membrane and may even extend across two membranes when these are stacked together.
Fig. 5.50 Protein crystal model for membranes showing a double layer of protein molecules (large circles) with liquid bilayer regions filling the pores (after Vanderkooi & Gree, 1970, 1971).
Membranes are not only essential for maintaining the integrity of cells and organelles, but play other vital roles. They are selective to the passage of substances into and out of the cell, ranging from electrolytes to large molecules. ‘Active transport’ mechanisms are frequently encountered which can be biochemically modified. The difference in permeability of ions together with the Donnan equilibrium gives rise to a resting electrical potential which if affected by increasing the permeability of Na+ ‘during stimulation’ gives rise to an action potential. Many enzymes are bound to membranes which therefore become the sites for certain metabolic pathways such as protein synthesis and respiration. Immunological reactions are also based at the membrane.
The structure and functions of biological membranes have been reviewed by Gross (1971). Their study has been facilitated by the use of artificial membranes. The formation and properties of the latter have been reviewed by Castleden (1969). Chapman (1968) and Sutton (1969) have also written short articles.
Liquid–Solid Interface
Interfacial Energy, Contact Angle and Adhesion
A surface free energy γs exists at solid surfaces just as it does at liquid surfaces. If a liquid surface rests at equilibrium on a solid then the surface free energies are related by the equation:
(5.35)
where γsl is the interfacial free energy, γ1, is the liquid surface energy (numerically the surface tension) and θ, measured within the liquid, is known as the contact angle (Fig. 5.51). The surface energies may be considered as surface tensions but surface tensions cannot be directly measured in the case of solids. Strictly, Eq. 5.35 should be corrected for the equilibrium film pressure πe of adsorbed vapour on the solid–air surface when:
Fig. 5.51 Equilibrium contact angle for a liquid on a solid
The work of adhesion Wsl between the solid and liquid is given by the Dupre equation (see p. 51):
(5.36)
Strong adhesion between the solid and liquid produces low contact angles and if the adhesion is equal to, or greater than, the cohesion of the liquid (2πl) a zero contact angle is obtained, This is really only a statement of the obvious since if the molecules of liquid attract those of the solid more strongly than they do each other, then the liquid will spread and completely cover the solid surface.
Since van der Waals forces operate across all solid–liquid interfaces there can never be a contact angle of 180°C, that is, Wsl is never zero. Water on a smooth surface of a pure solid paraffin wax gives an angle of about 110°C corresponding to Wsl=48mJ.m−2, a figure close to the interfacial tension between paraffins and water (see Table 5.4). Polyethylene and polytetrafluoroethylene give contact angles with water of 94 and 108°C, respectively at room temperature. If a block of stearic acid is cut, the contact angle will be anywhere between 50 and 105°C according to the direction of the cut with respect to the bimolecular layered structure of the crystal.
If the surface is composed of small patches of two different kinds of surface, then the effective contact angle θ is given by the relationship
(5.37)
where fl is the fraction of surface having a contact angle of θ1.
Surface roughness influences the contact angle. If θ on a smooth surface exceeds 90°C, roughness increases the contact angle, whereas if θ is less than 90°C, a decrease is found. Thus roughened paraffin wax with water can exhibit an apparent contact angle in excess of 130° whilst mica is roughened for use as a Wilhelmy plate to ensure an effective zero angle of contact (see p. 56).
Contact angles are generally affected by motion of the liquid edge on the solid. When the liquid advances over a dry solid surface, the advancing contact angle increases while the receding angle decreases (Fig. 5.52). The reason is that the receding edge is in contact with a more lyophilic surface due either to liquid penetration into the surface or to removal of lyophobic contaminants.
Fig. 5.52 Dynamic contact angles
Adsorption on to Solids
Many compounds are adsorbed on to solids from solution. Clay minerals such as kaolin, bentonite and attapulgite and antacid powders such as aluminium or magnesium hydroxides or silicates have been shown to adsorb such drugs as alkaloids, phenothiazine derivatives and the B vitamins. In such cases the rate and even extent of adsorption of drugs from the gastrointestinal tract may be considerably reduced.
Adsorption is often physical due to van der Waals forces of interaction between the adsorbate and the substrate in which case it is largely reversible by changes in pH or electrolyte concentration, but it may be chemical which involves a chemical reaction at the substrate surface when desorption may be far less easily accomplished. The extent of adsorption is influenced by the solvent since this too may be adsorbed, the polar/nonpolar natures of the solute, solvent and substrate all being involved. The nature of composite and individual isotherms has been discussed by Gregg (1961) and Kipling (1964) while Giles et al. (1960) have classified composite isotherms relating their shape to orientation of the adsorbate and penetration of the adsorbed layer by solvent molecules.
Rowland et al. (1965) have discussed the adsorption of macromolecules at the liquid–solid interface. Natural and synthetic hydrocolloids such as acacia and the cellulose derivatives and amphipaths such as the saponin glycosides of quillaia are adsorbed on to the surface of suspended particles which are not normally wetted by water. This produces an increase in the hydrophilic character of the solid surface thereby promoting wetting.
Promotion of Wetting
If a nonzero angle of contact exists between a solid and a liquid, a suspension of powdered solid in the liquid will contain a considerable proportion of the ‘powder floating at the surface attached to air bubbles thereby producing an unsightly scum or froth. Homogeneity of the suspension may be difficult or impossible to achieve and shaking may make matters worse. Such a state of affairs may be observed if aspirin or precipitated sulphur is dispersed in water. Wetting of these two solids is accomplished using acacia and the saponins in quillaia extract respectively. There are many wetting agents available, initial selection being determined by the use of the suspension. Gums, cellulose derivatives and nonionic amphipaths such as the polysorbates are suitable for internal use. For external applications, ionic amphipaths can be used but some gums are precluded on account of their sticky nature. The addition of a wetting agent is essential for the adequate dispersion in water of a number of powders apart from those already quoted, cortisone and the sulphonamides being examples. The foaming and resuspensiori properties will influence the final choice of wetting agent. A wetting and deflocculating action may frequently go hand in hand since the wetting agent increases the attraction for the liquid dispersion medium and, if ionic, confers repulsive charges on the solid particles.
Young’s equation (p. 84) shows that increasing the work of adhesion between the solid and the liquid decreases the contact angle. A similar reduction in contact angle can be achieved without adsorption on the solid if an amphipath is merely adsorbed at the liquid–air surface since the surface tension is thereby reduced. Frequently poor wettability of a powder arises from the porous nature of the particle surface which entraps air: penetration of the pores by the liquid is then required. If r is the effective mean pore radius, then the capillary pressure ΔP is given by the Young–Laplace equation (p. 49) as:
This shows that for good pore penetration γl should be as large as possible. The better wetting agents therefore act mainly by a specific adsorption on to the solid.
Wetting agents are necessary for fungicide and insecticide sprays since these have to spread over a waxy cuticle or penetrate hairy surfaces of leaves and insects (see below). Some of the best compounds have a high CMC because of their irregular molecular shape and so high monomer concentrations are obtainable giving their solutions exceptionally low surface tensions. One of the better known of these wetting agents is sodium dioctylsulphosuccinate. A good wetting action is also necessary for detergent solutions since intimate contact with the cleaned surface is a prerequisite for the removal of dirt.
Detergency
This is the removal of solid particles and greasy films from solid surfaces by means of solutions containing amphipaths. Adsorption of amphipath occurs at both the solid–liquid and oil–liquid interface thereby reducing γsw and γow. Eq. 5.38, analogous to Eq. 5.35, shows that θ must decrease as a result:
(5.38)
thereby causing a greasy dirt to ‘roll up’ and to become more readily detached from the solid surface by gentle agitation (Fig. 5.53). These globules then form an emulsion or are solubilized within the micelles. Solubilization probably occurs to a considerable extent with polar dirt. Lawrence (1961) showed that soap is concentrated at the surface of amphiphilic dirt and slowly penetrates it, forming a liquid crystalline phase. This phase becomes sufficiently mobile to break away upon agitation, when it becomes diluted to form an L1 isotropic phase with or without globules of L2 phase dispersed in it (see p. 72).
Fig. 5.53 Illustration of the ‘rolling-up’ process: A. Oil film in the absence of soap; B. Globules produced in the presence of soap
Emulsification or solubilization of liquid dirt, or adequate wetting and deflocculation of solid particles is necessary in order to prevent the dirt from becoming redeposited. Emulsification is assisted by a scouring action and by removal of the dirt into the foam while further adhesion is prevented by the ‘adsorption of amphipath on to the cleansed surface. Rinsing involves dilution with water and may enable some deposition of L2 phase. In such cases a preliminary rinse in dilute detergent solution should be undertaken.
Commercial detergent formulations contain a number of additives or ‘builders’. A high proportion of a sequestering (chelating) agent such as pyrophosphate is incorporated. This removes polyvalent ions such as calcium present in hard water which might otherwise cause the detergent–dirt mixture to produce an insoluble scum and become redeposited. Much of the solid dirt is also held on to the substrate by adsorbed polyvalent ions. The inclusion of sodium carboxymethylcellulose in the detergent also reduces the risk of redeposition of dirt by forming an hydrated adsorbed layer over the substrate. Liquid detergents also include a hydrotrope (p. 72) in order to increase the surfactant solubility. Since rapid solubilization of fatty dirt requires this to melt, simple surfactant solutions may have an efficient detergent action only at high temperatures. The inclusion of an organic solvent such as xylene, in a solubilized form, enables the detergent to function at lower temperatures.
Dry cleaning may involve the use of detergent solutions containing predominantly an organic solvent and water solubilized within inverted micelles. The water-soluble dirt is solubilized in this case.
Water Repellenyc
Cationic amphipaths are not good wetting agents for many purposes. A dilute solution of cetrimide, for example, can render a clean glass surface hydrophobic by exchange of the cations so that the hydrocarbon chains of the adsorbed monolayer extend outward. A high contact angle can also be obtained by treating such surfaces as glass or steel with a silicone or a methylchlorosilane. Even the single layer of methyl groups in a monolayer of these compounds greatly decreases the adhesion of the surface to water.
Water repellency may be desirable for a number of reasons. For example, rendering the surface of steam condensers hydrophobic prevents the formation of a continuous film of condensed water which would impede the removal of heat by conduction. A water-repellent surface inside glass vials may also be desirable so that solutions or suspensions can be drained completely from them. Dimethicones are suitable for this purpose.
In mineral flotation the wanted ore is rendered hydrophobic while impurities remain wetted. Small amounts of ‘collectors’ such as zanthates are added to the slurry. These coat the surface of the ore particles with the alkyl groups on the exterior (provided only a monolayer is adsorbed) so that the air bubbles become attached and float the ore. The ξ potential (p. 100) of the ore particle should be reduced to zero in order that the work of adhesion we should be as low as possible since for hydrophobicity Ws1<2γl
Woven fabrics may be rendered hydrophobic by coating the fibres with silicone polymers or with cationic amphipaths. The regular small mesh structure of fabrics leads to especially large effective contact angles. If f1 is the solid fraction of the surface, Eq. 5.37 reduces to:
since θ2 = 180°.Thus even if θ1 < 90°, the average angl 
can be considerably greater than 90°C. This is essential for tents and raincoats where the receding contact angle must be greater than 90°C for raindrops to ‘pearl’ off the fabric.
can be considerably greater than 90°C. This is essential for tents and raincoats where the receding contact angle must be greater than 90°C for raindrops to ‘pearl’ off the fabric.To prevent water penetration through a porous fabric however, θ1 must exceed 90°C (Fig. 5.54). The curvature of the meniscus between the fibres produces a negative capillary pressure which can become very large if the gaps between the fibres are very small. Some microporous plastics are also water-repellent and water-proof wound dressings made from these are required to withstand a hydrostatic head of 50cm. Such dressings prevent liquid water from penetrating to the wound while allowing the free passage of air and water vapour.
Fig. 5.54 A contact angle greater than 90° makes a fabric water-repellent
Numerous examples of water repellency assisted by a broken surface are to be found in nature. The regular structure of the feathers of aquatic birds is one such example where it is essential that the barbules are kept hydrophobic by the secretion from a special oil gland. Some aquatic insects are covered with fine hydrophobic hairs which maintain a thin but permanent film of air around the body. Gaseous exchange occurs across this air film rendering it unnecessary for the insect to replenish its air supply. Many plants also possess highly water-repellent surfaces, due either to a rough waxy cuticle or to a covering of hydrophobic trichomes.
References
Abu-Humdiyyah M. J. Phys. Chem., Ithaca. 1965;69:2720.
Adam NK. The Physics and Chemistry of Surfaces. London: Oxford University Press; 1930.
Adamson AW. Physical Chemistry of Surfaces. 2nd ed. New York: Interscience; 1967.
Alexander AE, Johnson P. Colloid Science. London: Oxford University Press; 1950.
Balmbra RR, Clunie JS, Corkill JM, Goodman JF. Trans. Faraday Soc. 1962;58:1661.
Blandamer MJ. Q. Rev. Chem. Soc. 1970;24:169.
Burt BW. J. Soc. Cosmet. Chem. Br. Edn. 1965;16:465.
Campbell JA, Christian JE, Eaton JR. J. Am. Pharm. Ass. sci. Edn. 1955;44:501.
Castleden JA. J. Pharm. Sci. 1969;58:149.
Chapman D. Sci, J., Lond. 1965;1(8):32.
Chapman D. Set. J., Lond. 1968;4(3):55.
Clayfield EJ, Matthews JB. In: Schulman JW, ed.
Proc. 2nd Intern. Congr. Surface Activity.1. London: Butterworths; 1957:172–187.
Davies CM, Litovitz TA. J. Chem. Phys. 1965;42:2563.
Davies JT, Rideal EK. Interfacial Phenomena. New York: Academic Press; 1961.
Douglas HW. J. Sci. Instrum. 1950;27:67.
Elworthy PH, Florence AT, Macfarlane CB. Solubilisation by Surface-Active Agents. London: Chapman & Hall; 1968.
Emerson MF, Holtzer A. J. Phys. Chem., Ithaca. 1967;71:3320.
Erlander SR. Sri. J., Land. 1969;5A(5):60.
Evans WP, Dunbar SF, Monograph SCI. Surface Activity and the Microbial Cell. 1965;19:169.
Ferguson A. Trans. Faraday Sue. 1923;19:407.
Fordham S. Proc. R. Sac. 1948;A194,:1.
Fowkes FM. Ind. Engng. Chem. Int. Edn. 1964;56(12):40.
Fox HW, Chrisman CH. J. Phys. Chem., Ithaca. 1952;56:284.
Frank HS, Evans MW. J. Chem. Phys. 1945;13:507.
Giles CH, MacEwan TH, Nakhwa SN, Smith D. J. Chem. Soc. 1960.3973.
Gross W. Angew. Chem. Int. Edn. 1971;10:388.
Harkins WD, Brown FE. J. Am. Chem. Soc. 1919;41:499.
Harkins WD, Jordan HF. J. Am. Chem. Soc. 1930;52:1751.
Hartley GS. Solutions of Paraffin Chain Salts: A Study in Micelle Formation. Paris: Hermann et cie; 1936.
Hoeve CAJ, DiMarzio EA, Peyser P. J. Chem. Phys. 1965;42:2558.
Jordan DO, Lane JE. Aust. J. Chem. 1964;17:7.
Kipling JJ. Chemy Ind. 1964.1007.
Lawrence ASC. Chemv Ind. 1961.1764.
Mandell L, Ekwall P. In: Overbeek J Th G, ed. Proc. 4th Intern. Congr. Surface Active Substances (Brussels, 1964). New York: Gordon & Breach; 1967:659–671.
Mandell L, Fontell K, Ekwall P, Washingto DC. Advances in Chemistry. 1967.89–124. American Chemical Society Series No. 63
McBain JW, Brock GC, Vold RD, Vold MJ. J. Am. Chem. Soc. 1938;60:1870.
Moilliet JL, Collie B, Black W. Surface Activity. 2nd ed. London: Spon; 1961.
Mulley BA. In: Bean HS, Beckett AH, Carless JE, eds.
Advances in Pharmaceutical Sciences.1. London: Academic Press; 1964:86–194.
Nakagawa T, Tori K. Kolloid-Zeitschrift. 1960;168:132.
O’Brien JS. J. Theor. Biol. 1967;15:307.
Osipow. LI. Surface Chemistry. New York: Reinhold; 1962.
Poland DC, Scheraga HA. J. Pins. Chem., Ithaca. 1965;69:2431.
Porter AW. Phil. Mag. 1933;15:163.
Preston J. J. Phys. Coll. Chem. 1948;52:85.
Robertson JD. Protoplasma. 1967;63:218.
Roe RJ, Bachetta VL, Wong PMG. J. Phys. Chem., Ithaca. 1967;71:4190.
Rosevear FB. J. Soc. Cosmet. Chem. 1968;19:581.
Rowland F, Bulas P, Rothstein E, Eirich FR. Ind. Engng.Chem. Int. Edn. 1965;57(9):46.
Schick MJ. J. Phys. Chem., Ithaca. 1963;67:1796.
Shinoda K. J. Phys. Chem., Ithaca. 1954;58:1136.
Shotton E. J. Pharm. Pharmac. 1955;7:990.
Shotton E, Kalyan K. J. Pharm. Pharmac. 1960;12:116.
Stauffer CE. J. Phys. Chem., Ithaca. 1965;69:1933.
Sugden S. J. Chem. Soc. 1921;119:1483.
Sugden S. J. Chem. Soc. 1922;121:858.
Sugden S. J. Chem. Soc. 1924;125:27.
Sugden S. J. Chem. Soc. 1924;125:32.
Sutton A. New Scient. 1969;42:125.
Swarbrick J. J. Pharm. Sri. 1965;54:1229.
Tartar HV, Wright KA. J. Am. Chem. Soc. 1939;61:539.
Tschoegl NW. J. Colloid Sri. 1958;13:500.
Vanderkooi G, Green DE. Proc. Nat. Acad. Sci U.S. 1970;66:615.
Vanderkooi G, Gree D. New Scient. 1971;50:22.
Ward AFH, Tordai L. Nature, Lond. 1946;158:416.
Ward AFH, Tordal L. Trans. Faraday Soc. 1946;42:413.
Winkel D. J. Phys. Chem., Ithaca. 1965;69:348.
Zuidema HH, Waters GW. Ind. Engng. Chem. Ananlyt. Edn. 1941;13:312.