Adsorption of Surfactants and Contact Interactions

The data presented in the previous chapter clearly indicate the universal importance of investigating particle cohesion under various conditions for establishing the scientific basis for explaining (and controlling) the mechanical properties of disperse systems in various natural and industrial processes� An important aspect of such investigations is the use of surface-active substances (surfactants), which at a low bulk concentration accumulate at the interfaces and radically change their properties� Before addressing specific results pertinent to the studies of contacts between particles of various natures in various surfactant solutions, let us briefly summarize the concepts of the adsorption of surfactants, primarily of the thermodynamics of adsorption�

At constant temperature, T, and volume, V, the condition of stable equilibrium corresponds to the minimum of free energy� Due to the uncompensated interactions, there is an excess of free energy associated with the surface (or interface)� In the absence of external forces, such as gravity, the minimum of free energy at constant T and V is reached at the lowest value of the surface free energy, σS, where S is the surface area of a body� At a constant value of the specific surface energy, σ, the minimum in energy for liquid surfaces is reached at the minimum surface area, which explains the spherical shape of small droplets, bubbles, or the meniscus in a capillary� For crystalline solids, this corresponds to the minimum of the sum of the product of the surface area of faces and their specific surface free energy (Curie-Wulff theorem)�

In the case when one or both of the neighboring phases contain a component capable of lowering the interfacial energy σ, the minimum in the surface energy is reached when this component is concentrated within the interfacial layer, that is, when adsorption takes place� It must be emphasized here that the concentration of surface-active component at the interface and the lowering of the free surface energy are mutually related: the spontaneous concentration of a surfactant takes place because it can cause a lowering of free energy, and conversely, the surfactant lowers the free energy because it is concentrating at the interface�

Adsorption as a physical-chemical parameter can be defined in a number of ways� We will focus here on the same principle as we used in defining the surface energy, that is, the principle of surface excess, originally introduced by Gibbs [1-4]�

Figure 2�1 shows a column that consists of two neighboring volumes of two-component phases, for example, an aqueous solution of n-hexanol, C6H13OH (phase δ) in equilibrium with its own vapor phase (phase δ′)� The square cross section represents the unit of surface area� The horizontal axis shows the concentrations in these two phases: of water and water vapor, c1(z), and of the additiven-hexanol and its vapors, c2(z)�

Outside of a particular layer of a finite thickness (the physical discontinuity surface), the concentrations of water and hexanol in the corresponding phases are constant� These concentrations are, respectively, c1¢ and c2¢ in an aqueous solution (liquid) and c1² and c2² in vapor� Within the discontinuity surface, there is a transition between these concentrations: either a smooth one or a positive “tongue-like” one, as shown in Figure 2�1b� In order to determine the excess (or the “difference in the composition” to be more precise) of a given component in a surface layer, we need to use Gibbs’s method and extrapolate these four constant values to the dividing surface (the geometric discontinuity surface located within the physical discontinuity surface)� The excess (adsorption) of

component I (1-water, 2-hexane) in moles per unit surface area within the discontinuity surface is given by the following integral:

G - - - - -¥

i i i i i i i ic c z c c z c c z c= ¢ + ¢¢ » ¢ + ¢ò ò ò ò + ²0

( ) ( ) ( ) (d d d ’

¢c zi )d

As in the case of the definition of σ, the values of these integrals may depend on the choice of dividing surface position� Without getting into details (see [1,4,5]), we will just emphasize that in the case of a large area of the “tongue” on the graph of c2(z), there is no need to focus on the actual position of the dividing surface, while in the case of a smooth transition in the plot of c1(z), the value of integral Γ1 may be both positive and negative (Figure 2�1)� For this reason, it is convenient for one to choose a position of dividing surface that yields Γ1 = 0�

At constant temperature and volume, the system is characterized by a single parameter, which in our case is the concentration of additive, (of n-hexanol in a liquid phase), c2¢ = c� The adsorption of this additive, Γ1 = Γ, is a function of the concentration, c: Γ = Γ(c)� Establishing this dependence for different interfaces is in fact the main subject of the theoretical physical chemistry of surface phenomena� Both rigorous solutions, originating in the fundamental Gibbs equation, as well as the simpler approximations, leading to the thermodynamic dependence Γ = Γ(c), have been reported in the literature [1-8]� These approaches are based on the fact that the chemical potentials of each component in the phases coexisting in equilibrium are equal� The latter can be illustrated as follows� In the case of an equilibrium distribution of an additive between phases, the chemical potential of that additive, μ, should be the same everywhere: in a liquid phase (solution with the concentration c [moles per volume]), in a rarified gaseous phase, and in the transition layer represented by the physical discontinuity surface with the surface excess Γ (moles per surface area)� The value of σ (J/m2) is the result of mechanical work spent on the creation of a unit surface area, while the Γμ product (mol/m2 × J/mol = J/m2) is the chemical component of that work� At equilibrium, the variation of the sum σ + Γμ should be equal to zero, that is, δσ + Γδμ + μδΓ = 0� By restricting ourselves to the dependence between the concentration of the additive c and the surface tension at the equilibrium value of Γ (i�e�, δΓ = 0), we establish that δσ + Γδμ = 0� Since there is only one independent variable in the system, it is possible to use the full derivative, that is,

G - s

m = d

d (2�1)

Equation 2�1 is the well-known Gibbs adsorption equation describing the adsorption of a single component� The corresponding thermodynamic consideration produces a general equation for the case of multicomponent systems [1,3,4]� Within the limits of the low concentration of the additive in the bulk liquid phase, c2¢ = c, the chemical potential of the additive is given by μ(c) = RT ln c + const, and dμ = RT dc/c�These relationships together with Equation 2�1 readily yield Gibbs law in its most common form

G - s= æ

è ç

ö ø ÷

c

T cR d d

(2�2)

The meaning of Gibbs law (Equation 2�2) can be formally stated as follows: the surface tension of a liquid phase (or in the general case the interfacial free energy), σ(c), decreases with the increase in surfactant concentration c of the adsorbing component, and the rate of this (relative) decrease at constant temperature is described by the value of adsorption, Γ�

Gibbs equation contains three parameters� Being a typical thermodynamic equation, it does not produce solutions for the surface tension isotherm, σ = σ(c), and for the adsorption isotherm, Γ = Γ(c)� Here, the term “isotherm” emphasizes constant temperature� Consequently, in order to integrate Equation 2�2, one needs an additional independent equation relating the same parameters� This equation can be both theoretical (utilizing some kind of a model, most commonly a molecular one) or experimental� For a liquid interface, the experimental surface tension isotherm, σ = σ(c), can serve as the second relationship and can allow one to obtain the adsorption isotherm Γ(c) from the Gibbs equation�Conversely, for the solid-gas and solid-liquid (S/L) interfaces, experimental measurements yield the adsorption isotherm, Γ(c), which together with the Gibbs equation produces the surface free energy isotherm, σ(c)�

The common universal approximation frequently utilized in thermodynamics is that of ideal solutions, that is, a restriction to the limit of very low concentrations� When the parameters of linear thermodynamic equations have very small values, they are usually proportional to each other�This proportionality provides the necessary additional universal relationship�

Let us demonstrate the application of this approach to the analysis of adsorption from an aqueous solution at the air-liquid interface� The experimentally determined change in the surface tension of solutions, σ, as a function of the additive concentration, c, typically shows linear dependence at low values of c (we will further present an explanation of what specifically can be used as a criterion to judge whether c is low enough):

lim [( / )/( / ])c

c ®

æ è ç

ö ø ÷ = =0

2 3-sd d

const mJ m mol dmG (2�3)

The constant G characterizes the surface activity of the additive� Consequently, at low values of c, the Gibbs equation has the form –dσ = (RTΓ/c)dc, where RTΓ/c = G = const (and Γ/c = const as well)� Integration with the initial condition of σ = σ0 at c = 0 yields σ – σ0 = Δσ = –Gc, or

-Ds s - s G( )c T= =0 R (2�4)

Equation 2�4 can be utilized to present the concept of surface activity or inactivity (Figure 2�2)� The adsorption Γ, viewed as the excess of a component in the interfacial layer and given by the

“tongue” area (see Figure 2�1b), may be presented as the averaged difference between volume-based concentrations of the additive in the surface layer, cs, and the concentration in the bulk, c, multiplied by the volume (thickness) of the interfacial layer, namely,

G - d= ( )c cs (2�5)

which yields

- = =d

d R R ss G d -

c

T c

T c c c

( ) (2�6)

When the adsorption is positive, that is, when there is a real excess of a component in the surface layer, cs > c, or cs ≫ c, and Equation 2�6 becomes

- =d

d R ss

c

Tc c

(2�7)

In the aforementioned expression, the ratio cs/c can be very large, up to several thousands� This is due to the surface activity, which is characterized by the decaying linear portion of the isotherm 1 in Figure 2�2� This type of strong surface activity is the general feature of all surface-active substances to which we will devote a lot of attention throughout this book� Conversely, under the conditions when the adsorption layer is deficient in a given component, the absolute value of cs-c cannot exceed the value of c, and hence the value of (c-cs)/c does not exceed 1� In Figure 2�2, this situation is reflected in the small positive slope of isotherm 2, which corresponds to a surface-inactive substance� One example of such a system is a NaCl solution: 100% negative adsorption simply corresponds to the absence of salt in the surface layer�

Equation 2�4 was experimentally verified by Irving Langmuir and eventually allowed him to propose an explanation of the adsorption layer structure� To further discuss adsorption, we will summarize the essence of Langmuir’s method and introduce the concept of two-dimensional (or surface) pressure, πs (Figure 2�3) [5]�

Langmuir and Pockels [5-8] have offered an experimental method for studying surfactant adsorption layers at the air/water interface� Their technique, focused on the use of insoluble or sparingly soluble surfactants, has had a significant impact on the development of colloid science� A shallow rectangular cuvette with hydrophobic walls (paraffin-coated glass or Teflon) is filled with water, which carries an adsorption layer of an insoluble surfactant on its surface� Although in these systems the concentration c2¢ is very low, it does not prevent the formation of an equilibrium adsorption layer, corresponding to a very high value of the surface activity, G� A moving barrier separates the surface of liquid in the cuvette into two portions, with surfactant being present only on one side of the barrier� Since the surfactant is insoluble, there is no diffusion to the other side of the barrier via the liquid phase�

The experiment is conducted by first placing on the one side of the barrier a drop of very dilute surfactant solution in a volatile organic solvent (for instance, a drop of 10 mol/dm3 solution of cetyl alcohol in cryoscopically pure benzene) and allowing the solvent to evaporate� Since the surfactant solution concentration in benzene is known, the mass of the surfactant in the adsorption layer is also known-it is typically of the order of fractions of a milligram� When the molecular weight of a surfactant is known, one can estimate the average area occupied by a single molecule, sm� Repositioning the moving boundary allows one to change the area occupied by the adsorption layer and hence the value of sm� A sensitive dynamometer connected to the barrier allows one to measure the surface tension gradient existing across the barrier� It is obvious and can indeed be confirmed experimentally that the barrier should slide in the direction of a surfactant-free surface� That is, the barrier slides toward the side containing pure solvent with the surface tension of σ0 (for a very pure double-distilled water, σ0 = 72�8 mJ/m2), which is greater than the surface tension on the other side of the barrier� The difference in the surface tension can be written as

Ds s - s G= =0 RT

From this description, it is clear that the adsorption layer exerts “pressure” on the barrier, referred to as the two-dimensional (or surface) pressure per unit length of the barrier:

p G s - ss R= =T 0 (2�8)

Dividing both sides of Equation 2�8 by the value of Γ and by Avogadro’s number, NA = 6�02 × 1023 mol−1, yields

p G

AN k= T

(2�9)

where the value 1/Γ = Sm is the area occupied in the adsorption layer by a mole of surfactant, while 1/ NAΓ is the corresponding area per molecule� Consequently, one can write that

p ps m s mR and kS T s T= =, (2�10)

This remarkable Langmuir’s relationship is, in fact, the equation of the state of a rarified (ideal) twodimensional “gas” consisting of adsorbed molecules� This equation is analogous to a well-known gas law describing a “conventional” three-dimensional ideal gas�

This theoretically (thermodynamically) derived equation has been verified experimentally for the dilute adsorption layers (where the concentrations are low, not only in the bulk phase but also in the

adsorption layer) formed by numerous common organic surfactants, such as fatty acids CnH2n+1OH, fatty amines CnH2n+1NH2, and many other water-insoluble higher members of homologous series� These particular experiments further extended to include the case of dense adsorption layers (high cs and Γ values), allowed Langmuir to achieve an understanding of the structure of the adsorption layers at the molecular level, and hence explain the nature of the surface activity�

A typical two-dimensional isotherm of a (surface) pressure, πs = πs(sm), as a function of area per molecule is shown in Figure 2�4� The right-hand side of the isotherm expression describes rarified adsorption layers in which sm ~ 103 Å2� This portion of the isotherm is similar to the isotherm of a three-dimensional ideal gas� When sm is lowered, the adsorption layer becomes denser, and the value of πs increases� When a certain value of πs = πcis reached, the so-called two-dimensional condensation takes place� The latter corresponds to the formation of liquid-or solid-like isles in equilibrium with the two-dimensional gaseous phase� The portion of the isotherm associated with the two-dimensional condensation is associated with a constant value of πs = πc� When a certain small area per molecule, sm(min) = s1, has been reached, a sharp rise in πs is observed� For long-chain fatty acids, the value of s1 is around 21 Å2� The onset of this rise corresponds to the critical concentration, cs and adsorption Γ� This region of the isotherm reflects a compression of the two-dimensional condensed phase down to the point at which it loses elastic stability and develops local fractures (“hummocks”) at πs = πfr, which is on the order of several tens of dyn/cm� Langmuir has shown that the critical value of area per molecule, s1, corresponds to the cross-sectional area of a surfactant molecule in a dense adsorption layer� In such an arrangement, hydrophilic polar groups of surfactant molecules stay immersed in the polar liquid, while hydrophobic hydrocarbon chains are expelled from the aqueous phase into the “most nonpolar medium of all,” that is, into the air� This situation is schematically illustrated in Figure 2�5� Similar values of s1 = 20-21 Å2 obtained for many members of homologous series of fatty acids, fatty amines, and fatty alcohols indicate that this characteristic value corresponds to the cross-sectional area of a hydrocarbon chain and not the polar group� It has also been demonstrated that a similar arrangement takes place at the interface between aqueous and hydrocarbon phases (i�e�, at the oil/water interface): polar groups are contained in water, and hydrocarbon chains stay in the nonpolar phase�

The value of sm(min) = s1 numerically yields the value of the limiting adsorption, Γmax, which further yields the thickness of the adsorption layer, δ, that is, the length of the hydrocarbon chain, δ = l�

The estimation of the distance between neighboring –CH2 groups as projected on the axis of a molecule allows one to compare the data obtained from adsorption measurements to the same data independently obtained from the x-ray diffraction analysis [9]� Both methods yield similar results: 1�3 Å in the former case and 1�2 Å in the latter� The value of the valence angle CH2-CH2-CH2 is in good agreement with the known value of 109°� Table 2�1 contains the corresponding sm(min) and l data for stearic and oleic acid, as well as for their triglycerides� These data clearly indicate that the unsaturated oleic acid molecule is “twice-folded” as compared to the stearic acid molecule�

This specific structure of adsorption layers allows one to explain the phenomenon of surface activity� The molecules of organic surfactants containing both polar and nonpolar features within the same molecule form a bridge between neighboring polar and nonpolar phases� This bridge evens out the transition between two antagonistic phases by compensating otherwise uncompensated interactions across the interface� This leads to a lowering of the free energy excess at the interface, σ, which manifests itself as the observed lowering of surface or interfacial tension� Rehbinder has shown [9,10] that one can formulate the polarity equalization rule by stating that the adsorption layer equalizes the polarities of the neighboring phases, decreases the extent of their antagonism toward each other, and as a result lyophilizes the system�

The same considerations regarding the structure of the adsorption layer also hold true for S/L interfaces� At the interface between a polar solid phase (such as mineral salts, oxides, hydroxides, and glasses) and a nonpolar organic phase (e�g�, lubricating oils, organic monomers, and oligomers), the polar groups face the surface of a solid phase, while hydrophobic hydrocarbon chains float freely in nonpolar liquids� Conversely, in the case of a hydrophobic nonpolar solid surface (e�g�, crude oil paraffins, solidified fats and waxes, plant leaves, animal fur, bird feathers, human skin, and soot particles) and a polar liquid medium (e�g�, water), the hydrophobic chains of surfactant molecules face the nonpolar surface, while the hydrophilic groups are immersed into an aqueous phase�

The aforementioned is true for both water-soluble and water-insoluble surfactants� In the first case, one measures the surface tension of the corresponding aqueous solutions, while in the second case,

TABLE 2.1 Values of sm(min) and l for Stearic and Oleic Acid, and of Their Triglycerides

the primary experimental method consists of measuring the two-dimensional pressure, πs(sm), with a Langmuir balance� Some common examples of water-soluble surfactants include soaps formed with monovalent cations, Na+, K+, NH4+, while water-insoluble surfactants are soaps formed with polyvalent cations, Ca2+, Al3+, etc�, and containing a well-developed hydrophobic portion� The waterinsoluble surfactants form a class of oil-soluble additives, which adsorb on surfaces from a nonpolar liquid phase� In all cases, including those in which adsorption from both polar and nonpolar liquid phases takes place (as, e�g�, in emulsions), the polarity equalization rule determines the adsorption layer structure�

Our understanding of a tendency to minimize surface free energy as the general cause of adsorption can be enhanced by emphasizing the role of the chemical potential in the transition of surfactant molecules to the interface� It is the gradient of chemical potential that always determines the direction of mass transfer resulting in the equalization of the chemical potential in all of the phases in contact�

It is worth pointing out that despite the similar arrangement of the hydrophilic and hydrophobic segments of surfactant molecules at the water/hydrocarbon interface, there is a principal difference in the driving force of the adsorption from the aqueous phase and from the liquid hydrocarbon phase (Figure 2�6)�

The obvious reason for an “immersion” of the polar surfactant group into the aqueous phase is the work of hydration, that is, a strong nondispersion interaction between the polar head and the molecules of water� The driving force for a transition of hydrocarbon chain CnH2n+1 into a nonpolar medium is different� While a detailed discussion of this rather complex issue is beyond the scope of this book, we will provide the following qualitative explanation� The introduction of a hydrocarbon chain into an aqueous phase disrupts the icelike structure of the water in which dipoles are present in a partially ordered arrangement with a tetrahedral coordination� The reorganization of the water molecules around the hydrocarbon chain in an attempt to restore tetrahedral coordination (i�e�, to restore ordering) is related to a decrease in entropy� An increase in entropy upon the “expulsion” of the hydrocarbon chain from an aqueous phase into a nonpolar phase is the driving force for adsorption in this case� This applies to the adsorption at the aqueous solution-air interface, aqueous solution-liquid hydrocarbon interface, and to some extent to the adsorption at the interface between an aqueous solution and a solid hydrophobic surface� Similar considerations are also applicable to the formation of surfactant micelles in aqueous media� Micelles are spherical nanoparticles formed by the association of the surfactant molecules�In these nanostructures, surfactant hydrocarbon chains are associated together into a core, while polar groups form an interface with the aqueous solution� Micellar cores are capable of dissolving (solubilizing) nonpolar phases� At high surfactant concentrations, spherical micelles undergo transformation into cylindrical micelles, form liquid crystalline phases and sponge phases� It is also worth emphasizing here the exceptional role of bilayers formed in aqueous media by phospholipids� In these bilayers, the hydrocarbon tails of phospholipid molecules are associated with

each other, while the polar heads are oriented toward the aqueous solution� These bilayer membranes are the main building blocks of all living organisms� The specifics of the water structure and the related entropic nature of adsorption constitute one of the main factors of life�

Direct methods of measuring adsorption are essentially restricted to the measurements of adsorption at the solid surface from a gas phase� At sufficiently high specific surface area, S1 (m2/g), the adsorption can be determined directly as the increase in mass� High-surface-area materials include highly disperse adsorbents with fine pores, such as activated charcoal, zeolites, and various catalysts for which the surface area is on the order of tens and hundreds of square meters per gram� The adsorption on such surfaces from a gas phase can also be determined by measuring a decrease in the gas (vapor) pressure, p, in a closed vessel� The multilayer adsorption of noncorrosive gasses is a commonly used method to determine the surface area of adsorbents on the basis of the BrunauerEmmett-Teller (BET) theory included in all classic texts on physical chemistry�

The adsorption at the S/L interface between finely dispersed powders and aqueous solutions is measured by the same principle as described earlier, that is, by a decrease in the concentration c in solution, which can be most simply measured by the increase in the surface tension at the airliquid interface� However, this method lacks accuracy, once the surfactant concentration exceeds the critical micellization concentration� For more accurate determination of surfactant concentrations, spectroscopic or chromatographic-mass spectrometric methods can be used� A review of the methods used to determine the concentration of different surfactants can be found in [5]�

It is important to recognize that the measured change in the surfactant concentration due to the adsorption from solutions at the solid surfaces yields the adsorbed mass, Γ*� In order to determine a specific adsorption (per unit area), Γ, for a given fine disperse adsorbent, the surface area, s, needs to be determined from the independent measurements from the gas phase (e�g�, by the BET method), that is, Γ = Γ*/s�

For both solid-gas and S/L interfaces, the isotherm of interfacial free energy can be determined from the combination of Gibbs equation and adsorption isotherm�

At the same time, the change in the σSL value during wetting can be estimated from the work of wetting, Ww = σLcosθ, if the adsorption on the hydrophobic surface is negligible, which together with Gibbs equation allows one to estimate the adsorption�

The adsorption at the S/L interface is different from the adsorption at the solid-gas interface in numerous aspects� Yet, the thermodynamics used to describe adsorption at the solid-gas interface are applicable to the S/L interface, and the polarity equalization rule remains valid�

The specifics of adsorption at S/L interfaces are utilized in many different applications, including the removal of toxic substances from wastewaters; liquid chromatography; using the surfactants to direct, alter, and control wetting, which plays a critical role in crude oil recovery; and using polar adsorbents, such and clays and zeolites, to purify a nonpolar medium from oil-soluble surfactants� The mosaic structure of the solid surfaces, and specifically the role of nonuniformities, plays a significant role in adsorption and chemisorption phenomena� There are pronounced differences between these two phenomena� As will be shown later in this book, chemisorption plays a significant role in modifying the strength of contacts between the particles and in influencing the mechanical properties of various materials� Surfactant chemisorption plays a critical role in the hydrophobization of solid surfaces in the aqueous medium, which is employed in mineral flotation�

A peculiar adsorption behavior is observed in the case of adsorption at the solid surface from a mixture of two mutually miscible liquids, A and B (see Figure 2�7)� The isotherms may have significantly different shapes depending on the extent of the surface activity (e�g�, of liquid B)� Case I, represented by a diagonal, corresponds to zero surface activity, which is the case when two liquids have a similar nature: the increase in the surface concentration is the same as the increase in the bulk concentration, which corresponds to Γ = 0� Case II represents the strong surface activity of liquid B: surface concentration of B and surface excess initially grow rapidly as the mole fraction of liquid B, xB, is initially increased until a maximum in adsorption is reached� A further increase in xB results in a lesser accumulation of liquid B in the surface layer; the concentration of B in the bulk increases to a larger extent

than the surface concentration and a decrease in adsorption is observed� Case III corresponds to a weak surface activity of liquid B: an initial increase in xB results in an increase in the surface excess of B, but to a much smaller extent than in case II: similar behavior is now observed at much lower values of xB� At some point, the surface concentration of B becomes equal to the bulk concentration, and hence Γ = 0� A further increase in xB results in a higher bulk concentration than the surface concentration and thus yields Γ < 0� The understanding of this behavior plays an essential role in the analysis of the effectiveness of the decontamination of liquids by adsorption, for example, by charcoal adsorbents�

Let us provide an example of how one can estimate the change of the surface free energy from the σ(pv) isotherm� This method will be utilized again further in the book� Preweighed samples of microporous magnesium oxide with high surface area are kept in the atmosphere of vapors of the adsorbing component (water vapor at different pressures, pv) until the equilibrium has been reached� From the gain in weight of the specimen, one can then determine the adsorption Γ(pv) (mol/m2)� It is assumed here that the surface area of the specimen is known, for example, from low-temperature nitrogen adsorption (BET) measurements� The parameter pv plays here the same role as the concentration, c, in Equation 2�2� According to the Gibbs equation, we can write that

G - s s -G( ) , ( )p p

T p p T

p pv

vR d d

or d R d= æ è ç

ö ø ÷ =

æ

è ç

ö

ø ÷ (2�11)

By integrating without the low pv limitation, one gets

-Ds s - s G( ) ( ) ( )p p T p p p

R d= = ò0 0

(2�12)

Further, with some additional data, one can also estimate the value of σ0�

Depending on the physical-chemical nature of the adsorbent solid surface and on the nature of the adsorbing molecules, the Γ(p), Γ(c), and the interfacial free energy isotherms may have significantly different shapes� For example, the Γ(p) isotherm may contain a region corresponding to the adsorption of vapors in narrow capillaries�

The adsorption at the liquid/gas interface can be determined by using a microtome or molecular tracers; however, the most precise and universal methods of determining Γ(c) at mobile interfaces, liquid-gas and liquid-liquid, are indirect methods based on the simultaneous use of the Gibbs equation and the surface (or interfacial) tension isotherm, σ(c) (Figure 2�8)�

A series of σ(c) isotherms of aqueous solutions of fatty acids CnH2n+1COOH is used as a common example� Formic and acetic acids, which do not contain a long-enough hydrocarbon chain, do not

reveal any surface activity� Higher acid homologues are water-insoluble� The data for intermediate members of the acid series, that is, C3-C6, shown in Figure 2�8, were originally obtained by Szyszkowski, who has generalized them empirically by introducing the following expression, now known as Szyszkowski isotherm: σ0 – σ(c) = B ln(1 + Ac)� While parameter B is a constant for the entire homologous series, the parameter A increases by a factor of 3-3�5 upon the transition from one homologue to the next one in order� This means that the concentration needed to reach equal lowering of the surface tension is decreased by a factor of 3-3�5 upon the transition to the next member of the homologous series (the Ducleaux-Traube rule)� At low concentrations c ≪ 1/A, the isotherm σ(c) displays a linear decay, while at high concentrations c ≫ 1/A, the isotherm decays logarithmically (within the solubility limits)� Substitution of Szyszkowski’s isotherm into the Gibbs equation yields

G - s G( ) ( ) ( )maxc

c

T c c

T BA

Ac Ac Ac= æ

è ç

ö ø ÷ =

æ è ç

ö ø ÷ +

= + R

d d R 1

1 (2�13)

Equation 2�13 is the Langmuir adsorption isotherm, originally derived by Irving Langmuir� As seen in Figure 2�8, the initial portion of the Γ(c) Langmuir isotherm is linear, and at high concentrations, the value of Γ(c) approaches the limiting value, Γmax = B/RT� This limiting value can be viewed as representing the dense packing of surfactant molecules in the adsorption layer, thus 1/Γmax = RT/B is the area occupied by a mole of surfactant molecules� Furthermore, the value of 1/NAΓmax yields the area s1 occupied by a single molecule in the close packed layer, that is, it is the surfactant molecule crosssectional area� Indeed, such calculations for water-soluble surfactants coincide with the values of the cross-sectional area obtained from measurements with a Langmuir balance using insoluble surfactants�

Figure 2�9 briefly summarizes the main steps in the analysis of the experimental adsorption data using the Langmuir isotherm� It also schematically shows the structure of the solid surface assumed by Langmuir and the summary of the derivation of the isotherm equations from the kinetics of adsorption

and desorption� In Langmuir’s approach, the surface is modeled as a chessboard, each site of which can host only one adsorbing molecule (Figure 2�9a)� The treatment is restricted to the case of localized adsorption, that is, the possibility of migration of molecules from one site to another is not considered� The adsorption and desorption rates, va and vd, are proportional only to the fractional surface coverage, θ, with the proportionality rate constants ka and kd, respectively, and an equilibrium between adsorption and desorption, that is, va = vd, readily yields Langmuir’s isotherm (Figure 2�9b)� The ratio of the rate constants, ka/kd yields A = 1/α, which is the constant of the Langmuir isotherm� The plot of experimental data in the c/Γ – c coordinates yields a straight line when the adsorption follows Langmuir’s isotherm and allows one to obtain the values of Γmax and sm(min) by linear regression (Figure 2�9c)�

The adsorption phenomena discussed in the previous section utilized amphiphilic molecules of organic surfactants (predominantly synthetic ones)� Here, it would be worthwhile to provide a brief description and classification of the most common types of organic surfactants [11-24]�

Synthetic surfactants represent important commodity products, with an annual production estimated at tens of millions of tons� While most of commercially available surfactants have been in production for years, new surfactant molecules, frequently targeting specialized applications, continue to appear�

According to their physical-chemical nature, the surfactants can be subdivided into ionic (about 70% of all surfactants manufactured) and nonionic� In nonionic surfactants, the polar part mainly consists of multiple ethylene oxide units, –(CH2CH2O)n-, while ionic surfactants can be subdivided into three large classes based on the charge of their polar group, that is, anionic, cationic, and zwitterionic (amphoteric) surfactants�

In anionic surfactants, the “carrier of surface activity,” that is, the hydrocarbon chain, is a part of an anion having a negative charge� A well-known example of such surfactants are soaps, which are the alkali or ammonium salts of carbonic acids, for example, CnH2n+1COO-Na+� The hydrocarbon chains of such soaps typically contain 12-18 carbon atoms�

The excellent environmental properties of natural soaps come with a serious drawback: weakly ionizable carboxylic groups do not show good solubility in hard water and in acidic medium, while the main commercial source for manufacturing these surfactants is livestock fat�

The most common and inexpensive anionic surfactants are the aryl benzenesulfonates, R-C6H4SO3− Na+, containing a strong sulfonic group� Such surfactants are used as wetting agents, dispersion stabilizers, dispersants, and foamers� The drawback of these surfactants is their poor biodegradability� Nonbranched alkylsulfates have better biodegradability characteristics� The example of a common surfactant of such a type is sodium dodecyl sulfate (SDS), C12H25-OSO3−Na+� Other common anionic surfactants are alkenesulfonates, containing double bonds�

In cationic surfactants, the carriers of the surface activity are positively charged cations� Along with weakly dissociating alkylamines (e�g�, octadecylamine C18H37NH2), these include quaternary ammonium salts, such as cetyltrimethylammonium bromide (CTAB), C16H33N+(CH3)3Br –,and cetylpyridinium chloride C16H33N+C5H5 Cl−� The strong ionic group ensures high surface activity and sufficient solubility over a broad pH range and in hard water� Cationic surfactants have numerous industrial applications, such as in corrosion inhibitor formulations and in specialty compositions used to stimulate oil and gas wells�

The main representatives of the amphoteric surfactants class are amino acids, including α-amino acids, RCH(NH2)COOH� These molecules are surface active if they contain a substantially developed hydrocarbon chain� In the acidic medium, such molecules can act as bases by accepting a proton, thus forming NH3+ ions, while in the basic medium, they dissociate as acids forming COO− anions� Amino acids play a significant role in nature� For example, α-aminoacetic acid (glycine) is 1 of the 20 amino acids that take part in the protein synthesis occurring in the human body� Other synthetic amphoteric surfactants may simultaneously contain anions and cations on the same chain� Such surfactants are referred to as zwitterionic-they include betaines and hydroxysultanes, often used as foamers�

In common nonionic surfactants, the polar group consists of chains of polyethylene oxide, –(CH2CH2O)n-� These surfactants reveal surface activity at all pH values and in hard water� Some of the oldest and cheapest surfactants of this kind are octyl and nonyl phenol ethoxylates, C8H17C6H4(OC2H4)nOH, and C9H19C6H4(OC2H4)nOH, respectively� In these surfactants, the values of n are typically in the range of 4-20� Low n values typically correspond to oil-soluble surfactants, while high n values correspond to water-soluble surfactants� The largest drawback of these surfactants is their poor biodegradability and their hazard to the environment� In many countries, these surfactants have already been banned for use in some applications (e�g�, in hydraulic fracturing)� For these reasons, these materials are being displaced by polyethoxylated alcohols and ethers containing no aromatic rings� Along with alkylsulfates, these surfactants are the main components of synthetic detergents�

An important class of nonionic surfactants is the so-called poloxamers or Pluronics� They represent block copolymers of ethylene oxide (hydrophilic portion of the molecule) and propylene oxide (hydrophobic portion of the molecule): H(OC2H4)m(OC3H6)p(OC2H4)nOH, which are commonly used in oil production and in other areas�

The surfactant market is highly versatile, with numerous brands and compositions available to the end user� Among the major consumers of surfactants are such industries as mining, metalworking, petroleum (demulsifiers, wetting modifiers, foamers, etc�), construction (additives to cement and asphalt), transportation (lubricating oils and greases, lubricating cooling liquids, etc�), pharmaceutical, food, cosmetic, paper, and water treatment� The use of surfactants in household and industrial detergents still remains a major area, consuming significant volumes of manufactured surfactants�

Along with the classification by chemical nature, one can also classify surfactants with respect to the mechanism of their behavior at interfaces� Two main mechanisms of surfactant action at S/L interfaces are of importance in their relevance to contact interactions between particles that we will address broadly throughout this book� These are (1) the control of the wetting of a solid surface by a liquid (see Figures 2�10 and 2�11) and (2) the dispersion action, which facilitates the fracture of solids, which we will discuss in detail in the second part of this book�

The main characteristics and principal concepts of wetting were described in Section 1�1� Here, we will address some specific examples of how surfactants can be used to control wetting in different processes encountered in various areas of technology�

1� Improving surface wetting by water, that is, the hydrophilization of a hydrophobic surface, can be achieved by the adsorption of water-soluble surfactants from an aqueous phase� The molecules of such surfactants typically contain a strong polar group promoting good water solubility and a developed hydrophobic chain, which serves as a carrier of surface activity� In accordance with the polarity equalization rule, hydrophobic tails in the adsorption layer are oriented toward the solid surface, while the polar groups are “floating” (hydrated) in the aqueous phase� The most common example of such action is detergency, in which the first step is the act of the wetting of soot or fat particles, fabric fibers, or even the surface of a skin in the course of handwashing� Hydrophilization of plant leaves, chitin shells of insects, and animal furs is necessary for successful application of pesticides, herbicides, and other agricultural chemicals� Surfactants promote the wetting of coal dust in mines and of surfaces in fire extinguishing, etc� In all these applications, which involve making hydrophobic surfaces hydrophilic, the adsorption mechanism is “nonspecific�” This means that at the interface between nonpolar surfaces in contact with the nonpolar segments of the surfactant molecules, the compensation of nondispersion interactions takes place, while the interaction between relatively distant polar groups and the surface is of no significance (Section 1�3)� As evident from the examples that follow, this is true for both ionic and nonionic surfactants�

2� The hydrophobization of hydrophilic solid surface involves principally different mechanism� In this case, the adsorption layer of oil-soluble surfactants with a strongly developed nonpolar part is formed at the S/L interface� Polar group may not necessarily be strongly ionizable but should have a specific affinity to a given solid surface� That is, anionic surfactants with a negatively charged polar group are most effective in the hydrophobization of surfaces bearing a positive charge, such as carbonates and other basic minerals, while cationic surfactants are effective at surfaces bearing a negative charge, such as acidic minerals, clays, oxides and hydroxides, quartz, and glass� Nonionic surfactants can be effective in both of these cases�

It is worth noting that in nature, the surfaces of living organisms are mostly hydrophobic, while those of inanimate matter are mostly hydrophilic�

Traditionally, one of the first and still predominate areas in which hydrophobization has been sought is the textile industry� Natural fibers, such as cotton, flax, silk, or synthetic fibers require “oiling” so that their mutual cohesion can be minimized (see Section 1�3 on the interactions between nonpolar and polar particles), and weaving and spinning processes are facilitated� Another technological example is the modification of polymer filler particles, such as chalk or clay, so that they can be introduced into liquid nonpolar media without undergoing aggregation and caking�

Similar issues are encountered in road construction, where gravel surfaces must be modified for an effective contact with hydrophobic asphalt, or in printing, where selective wetting by a

hydrophobic ink is desired on particular sections of the printing plates� All of these examples represent direct applications of physical-chemical mechanics�

A standalone and particularly important example of rendering particles hydrophobic is encountered in ore enrichment� A crushed ore containing a mixture of useful minerals and barren rock undergoes the process of froth flotation, in which it is exposed to an aqueous surfactant solution� The surfactants used in this process are capable of selectively “oiling” only one of these fractions (typically desired mineral particles)� Upon bubbling the suspension with air, hydrophobized particles, such as particles of metal sulfides, are attached to the hydrophobic air bubbles and are carried to the surface with the froth, while the particles of barren rock (such as sulfates or quartz) undergo settling at the bottom�

In froth flotation, the adsorption takes place from the polar aqueous phase, but the hydrophobic portions of the surfactant molecules in the adsorbed layer must be facing the polar phase, which appears to contradict the thermodynamically based polarity equalization rule� This is possible only if surfactant molecules are chemisorbed at the mineral surface� Chemisorption involves the formation of a very strong “chemisorption bond” between the polar groups and the particle surface� The energy of such a bond needs to well exceed 50 mJ/m2 in order to compensate for the free energy increase associated with the formation of the hydrocarbon/water interface (Figure 2�11c)�

Therefore, the use of surfactants for the modification of interfaces is very versatile, both with respect to the nature of the interfaces (between solid and liquid, polar and nonpolar), as well with respect to the assortment of the available surfactants� Up to this point, we have been talking about amphiphilic synthetic organic surfactants� However, the adsorption phenomenon is universal in nature and industry and takes place at all interfaces without any exceptions� It is worth emphasizing one more time that the general reason for the accumulation of surface-active substances at interfaces is the lowering of free energy as a result of the partial compensation of the disrupted bonds between interfacial atoms�

It is noteworthy and rather remarkable that we can distinguish water as a surface-active substance (Figure 2�12)� Such a distinction was first made by Rehbinder in his classic work [9,25]� While a molecule of water is not amphiphilic, that is, it does not contain a hydrocarbon tail like a surfactant, it does have a large dipole moment and is capable of forming hydrogen bonding� These two features stipulate the capability of water to form strong bonding with the surface of various ionic substances, for example, salts, oxides, and hydroxides� In this situation, the lowering of surface energy of various ionic substances from hundreds of mJ/m2 to the values close to that of the surface tension of water takes place� One can talk about the universal nature of the adsorption of substances with a low work of cohesion, low melting point, and low value of σ at the surfaces characterized by a high work of cohesion and high values of σ� An example would be the adsorption of hydrocarbons, alcohols, ethers, and volatile substances on the surfaces of substances with ionic or covalent bonding and metals� This is especially well seen at low temperatures, as illustrated by the well-known example of using nitrogen adsorption for the determination of the surface free area of various disperse materials (BET isotherm measurements)�

Good wetting of solid metals by metallic melts of certain fusible metals with a low value of σ (such as wetting of iron by zinc, or of copper by bismuth) indicates high values of the work of adhesion, Wa, and the possibility of effective adsorption from either vapor or melt in another low-meltingpoint metal� This will be the subject of a more detailed discussion on the interactions between solid and liquid metals later in this book�

Adsorption phenomena may also take place at the interfaces between solid phases, such as different phases in multiphase compositions, as well as at grain boundaries in single-phase systems (granulated materials, metals, salts, etc�)� All grain boundaries carry uncompensated interactions between the surface atoms and thus have a certain excess of free surface energy associated with them� One typical example of adsorption in such systems is that of sulfur and phosphorus at the grain boundaries in steel� Such adsorption results in a significant change in the mechanical properties of steel and in particular in the lowering of strength� Conversely, the adsorption of carbon or boron results in a significant improvement in the steel quality�

It is worth pointing out here some limitations on the use of the polarity equalization rule� This rule can be directly applied to systems in which one can distinguish the presence and significant role of the dispersion component of interactions in comparison with nondispersion interactions, which are mainly molecular� Obviously, in the case of a contact between a solid and a liquid metal or of a contact between an ionic or covalent body with a fusible salt, or in the case of the adsorption of a third component at such interfaces from a liquid phase, the polarity equalization concept has a more general meaning� Indeed, the rule emphasizes the physical-chemical similarity or dissimilarity between contacting phases� In this sense, the adsorption of water at the surface of ionic compounds “equalizes” the interface with air by replacing a surface having a surface energy on the order of hundreds of mJ/m2 with the one that has a surface energy on the order of the surface tension of water (~72 mJ/m2)

These basic concepts of surfactant adsorption that we have reviewed should be sufficient for the discussion of the influence of surfactant adsorption on the interactions between disperse phase particles, which we will address in the next section�

In this section, we will continue the discussion started in previous chapter and address the role of surfactant adsorption in contact interactions� In Section 1�3, we described the concept and experimental method for measuring contact adhesive forces between molecularly smooth particles of different nature in different media� In detail, we explored two limiting cases: that of very hydrophilic glass surfaces and that of hydrophobic surfaces of methylated glass� Along with the air, which is the “most nonpolar” medium, we also compared interactions in liquid media, which can be subdivided into three characteristic groups: (1) adhesive interactions in water and hydrocarbon; (2) adhesive interactions in a series of liquids, including alcohols of different polarities; and (3) interactions in liquids comprising a “continuous spectrum” of polarities, that is, in aqueous solutions of alcohols� Calculations based on Derjaguin’s theorem have shown that the free energy of interaction covers a very broad range of four orders of magnitude�

Now, we will turn our attention to experimental studies on contact interactions in the presence of surfactants� Here, we will also devote special attention to the studies involving molecularly smooth surfaces, which allow one to quantitatively estimate the free energy of interaction as the main invariant parameter characterizing adhesion� We will also look at polar and nonpolar particles in various aqueous and nonaqueous media but focus on the role of the presence of small amounts of different surfactants�

Figures 2�13 and 2�14 show the isotherms of the free energy of contact interaction (½ F) between methylated glass spheres as a function of surfactant concentration in aqueous solution for two model ionic surfactants: SDS and CTAB, respectively�

In the solutions of both the anionic SDS surfactant (Figure 2�13) and the cationic CTAB surfactant (Figure 2�14), the systems are in equilibrium and display full reversibility� To some extent, this is also true for a nonionic surfactant, oxyethylated ether (Figure 2�15), but in this case, equilibrium is not reached instantaneously, as the value of p1 shows strong time dependence (Figure 2�16)� In the

latter case, equilibrium is achieved only after the adsorption layer of oxyethylated ether has already formed� Time dependence indicates that for nonionic surfactant, this process is much slower than for a cationic or anionic surfactant�

Figure 2�15 shows a comparison between the interaction of two nonpolar particles and the interaction of a nonpolar particle and a partially lyophilized particle (hydrophilized with acetyl cellulose)� In the first case, ½ F reaches a limiting value corresponding to low but constant attraction, while in the second case, ½ F drops below zero, which is the real, experimentally observed change in the sign of the disjoining pressure, that is, a transition from attraction to repulsion�

All of the illustrated examples correspond to equilibrium and reversible conditions� The response of the adsorption layer formed at the hydrophobic methylated surfaces to the applied compression is schematically illustrated in Figure 2�17� The layer is displaced from the contact zone as the particles are compressed against each other and returns when the particles are pulled apart� The situation is principally different in the case of polar particles and adsorption from nonpolar media, such as in the case of amines on glass spheres� The chemisorption that takes place leads to the formation of an adsorbed layer that has its own mechanical strength, in which case a critical compressive force, fc, needs to be applied for the adsorption layer to rupture and to be displaced from the contact zone (Figures 2�17b and 2�18)�

In retrospectively looking at the data presented in this chapter and the data reported earlier in Section 1�3, we again need to emphasize here that the spectrum of the values of the free energy of

interaction is very broad and covers four orders of magnitude� If we were to consider particles with sizes ranging from tens of nanometers to millimeters, the spectrum of the adhesive contact forces, p1, would cover nine orders of magnitude, from 10−8 dynes to tens of dynes, and the corresponding spectrum of the interaction energy values would range from 10−15 to 10−6 ergs� This can be directly related to the subject of colloid stability, which will be addressed in detail in Chapter 4� A characteristic critical value w ~ 10kT falls in the middle of this range, and thus lyophilicity and lyophobicity are manifested as properties of the system�

The described experimental method for measuring the contact interactions between solid particles influenced by surfactant adsorption from various electrolyte solutions allows one to observe transition from lyophilicity to lyophobicity and to study the role of the electrolyte in this transition�

While we will continue to address the subject of using surfactants to control contact interactions in Chapters 3 and 4, we would like to devote the remainder of this chapter to a discussion of the cohesive forces between anisometric particles-specifically, between the cellulose fibers� This study conducted by the authors and their colleagues is of importance in papermaking applications�

The papermaking process is based on the dewatering of a suspension of cellulosic fibers mixed with mineral fillers (e�g�, calcium carbonate, clay, titanium dioxide) on a moving wire� In the papermaking process, a suspension of cellulosic fibers undergoes a significant change in rheology dictated by the fiber concentration and fiber-fiber or fiber-filler interactions at different stages in the papermaking process� First, a rather thick and concentrated pulp stock with a consistency of around 4% is diluted with water and pumped at a high rate into the head box of the paper machine� Various papermaking chemicals, such as starches, polymeric coagulants, polymeric flocculants, and colloid particulate suspensions (the so-called “microparticles,” most commonly colloidal silica sols) are added to the flowing pulp suspension� From the head box, the pulp is delivered to a moving wire on which water is removed by gravity drainage or vacuum drainage� The paper

web is then transferred to the pressing section and the drying section to make a finished product� Contact interactions play an essential role at various stages of this process: upon the dilution of the pulp, they are responsible for controlling rheology; on the moving wire, they control the rate of dewatering and eventually are responsible for good sheet formation, which determines the end quality paper; and in the drying stage, contact interactions determine the strength of the paper web, which makes it possible to operate modern paper machines at high speeds [32]� All of these complex phenomena, occurring at various stages of the papermaking process, provide one with a good reason to investigate fiber-fiber contact interactions and to study the impact of common papermaking chemicals on them�

Certain peculiarities of fiber systems are determined, on the one hand, by the inhomogeneous, mosaic-like nature of the fiber surface, and, on the other hand, by their anisometric shape, which allows for the formation of a number of local bonds� The rheological properties of the fiber structures control the resistance of interfiber bonds, both to their rupture under the effect of normal stresses and to shear under tangential ones� This means that friction forces play a significant role in the contact interactions of fibers�

At the same time, the normal loads on the fibers may be small, or absent at the stage of the homogenization of the paper pulp, but significant during pressing� In the case of pulp homogenization, the friction force is caused by the molecular attraction between the fibers and depends essentially on their surface properties� Changing the surface properties of the fibers in a desired way can be achieved with the use of surfactants or polymers� This allows one to control the rheological behavior of the pulp and obtain effective adhesion between the paper fibers and the particles of filler, and also to achieve optimal conditions for textile fiber spinning, etc� [33,34]�

A small radius of curvature in the fiber-fiber contact areas results in cohesive forces of a very small magnitude� This necessitates the use of highly precise methods and devices for measuring these small forces in contacts between individual fibers, under both normal rupture and shear� Such measurements provide one with the means for the evaluation of a contribution of molecular attraction to friction forces and allow one to obtain a thermodynamic value of the free energy of interaction� The latter is a general characteristic of interactions in a given medium, invariant with respect to the specimen geometry�

The technique and suitable instrumentation for determining the cohesive forces under rupture caused by normal force was reviewed in Section 1�3, and the suitable experimental device was shown in Figure 1�28� The measurement of the cohesion under shear also requires a specialized experimental setup, developed for this purpose, and the scheme of such an apparatus is presented in Figure 2�19� This technique was originally proposed by Yakhnin [35] and subsequently modified and used extensively at the Chemistry Department of Moscow State University [36-40]�

The main element of the apparatus for measuring the friction forces between individual fibers is the tilting Π-shaped frame (1)� The frame is connected to an electric motor (2) via a reducing gear box (3) to which a rigid holder (4) is attached� The tilt angle of the frame, α, is determined by the position of the pointer (5) on the scale (6)� Two threadlike samples (i�e�, fibers) (7) and (8) are attached to the frame at end points A and B, as well as C and D, respectively� The fibers contact each other at point O at a right angle� The free end D is connected to the weight P, which exerts a compressive force, N, between the fibers at the point of contact� The crossing fibers can be immersed into a cuvette (10) for measurements in a liquid medium� In the original prototype of this measuring device, the fibers were attached to the frame with glue, which is undesirable since the glue alters the surface properties of the fibers� To alleviate this problem, the fibers were tied with small knots to the extending thread lines, which were mounted in a device with specially designed microclamps� The thread connected to the ends of fiber CD is connected to the frame at point C� The thread (11) connected to the end D is placed over the pulley (12) with a very small friction torque, and a weight (13) is connected to its free end� For more precise measurements, a torsion scale in place of a weight can be used, which allows one to extend the range of the loads P, down to ~10 μN� All of the parts of this setup are mounted on a stand (14)�

At the initial, horizontal position of the frame, N = P� Tilting the frame by an angle α exerts a shear force F = P sin α; the corresponding normal force equals N = P cos α� The critical value of the angle α corresponding to a tilt at which fiber CD starts to move along fiber AB is recorded in the course of the measurement with the help of a microscope (9)�

The described method was used to measure cohesive force under applied shear between individual cotton fibers in aqueous solution and in solutions of polyethyleneimine (PEI), which is a common papermaking coagulant with a molecular weight of around 60,000 Da� The fiber thickness as determined by microscopy was ~10 μm, and the compression load was varied between 0�057 and 1 g, with a minimum of 30 measurements conducted at each load�

The resulting measurements of the friction force, F, between cotton fibers in water as a function of the applied load, N, is shown in Figure 2�20� As expected, the values of angle α obtained with the same pair of fibers at similar loads, P, show a rather broad scatter� Correspondingly, there is a large amount of scatter in the values F = P sin α and N = P cos α� Each set of normal and tangential force data is represented by a histogram showing the distribution of these values� The maxima of these distributions are shown as circles, while the distributions themselves are represented by inclined lines for each load P�

Figure 2�20 shows that overall the F(N) data fall on a straight line passing through the origin� Similar trends were also observed for the measurements in air and in PEI solutions� The slope of this line reveals the friction coefficient, μ, which is an invariant characteristic describing fiber-fiber interaction� The described measurements allow one to measure μ with a precision of 5%–10%� Friction coefficients measured between cotton fibers are rather high, μ ~ 0�4-0�5, which may be attributed to the surface roughness of the fibers� This is also supported by experimental observations, which revealed that the highest values of μ are typically obtained in the first few measurements carried out with a given fiber pair� Values of the friction coefficient tend to decrease with consecutive measurements, perhaps due to some “smoothening” of the fiber surface in repeated shearing�

An interesting finding is that similar values of the friction coefficient were obtained in the course of the measurements carried out in air and in water, μ = 0�41 ± 0�03 and μ = 0�47 ± 0�03, respectively� One possible reason for this similarity may be related to the hygroscopic nature of cotton fibers, that is, due to the uptake of moisture in air�

The friction coefficients between cotton fibers measured in solutions of PEI of different concentrations are shown in Figure 2�21� The concentration dependence of the friction coefficient reveals a maximum at low concentrations, essentially constant μ in the range of concentrations between 4 × 10−7 and 4 × 10−5 mol/dm3, and a sharp drop at high concentrations of the polyelectrolyte� These findings agree qualitatively with the known mechanism by which PEI flocculates cellulosic pulp� It is well known that the adsorbed PEI forms positively charged patches on the negatively charged fiber surface, and the flocculation is driven by the attraction of the “positive patch” of one fiber to the bare surface of the other fiber [32,41,42]� Optimum flocculation efficiency is typically observed at about or less than 50% surface coverage and does not require complete surface charge neutralization� In the case of two fibers, this “optimum flocculation” condition logically corresponds to the observed maximum in the friction coefficient� Further increases in PEI concentration would result in increased coverage of the fiber surface by PEI and eventually into a complete charge reversal resulting in electrostatic repulsion between two similarly charged fibers� The trends in the friction

coefficient observed in Figure 2�21 appear to support this argument� These results also agree qualitatively with the trends in the electrokinetic data for cellulosic fibers in the presence of PEI with a lower molecular weight of 30,000 Da [42]� With the addition of PEI zeta potential changes from the original negative values to positive ones, passing through the isoelectric point at c ~ 10−8 M of PEI, reaching maximum of +40 mV at c ~ 10−6 M and then decaying to 15-18 mV at c ≥ 10−4 M� At concentrations c ~ 4 × 10−7 to 4 × 10−4 M, the values of the friction coefficient are comparable to the values determined in water without any PEI added� This value probably corresponds to the value of μ measured under conditions corresponding to the absence of electrostatic attraction� One has to keep in mind that despite a general similarity in the trends for the electrokinetic and friction coefficient measurements, it does not seem feasible to completely transfer the results of one set of studies to the other, as the properties of PEI used in the friction coefficient measurements and electrokinetic measurements may be substantially different�

The results presented in Figure 2�21 correspond to the measurements of friction coefficients conducted after the fibers were in contact with PEI for several hours� Data shown in Figure 2�21 also indicate that after ageing the fibers left in PEI solutions for 1-2 days, significantly higher values of the friction coefficient were observed at the corresponding PEI concentrations� Possible reasons for this may include fiber swelling leading to an increase in surface roughness, as well as possible adjustment in the conformation of the adsorbed polymer with time [42,43]�

In agreement with Derjaguin’s molecular theory of friction [44,45], the friction force is defined as F = μ(N + N0), where N is the normal component of the applied load and N0 is the sum of the molecular attractive forces� In the absence of external loads, the friction force results solely from the action of the forces of molecular attraction, that is, N0: Fm = μN0� Under realistic conditions, that is, in the course of pulp mixing in the chests, the normal loads N are not high, and the contributions of N0 to the net friction force between the fibers can be quite substantial� However, in the experiments involving the measurement of friction coefficients where there is a small curvature radii of the fibers, the values of N0 are much smaller than the applied normal loads N� This does not allow one to isolate the portion of the friction force responsible solely to the molecular attraction, that is, F = μN0, by a simple extrapolation of F = F(N) data to yield the

N = 0 value� This is an unfortunate drawback of the experimental technique: the estimate for the N0 is essential for an understanding of the mechanisms of the influence of surfactants and polymers on the friction between fibers and on the pulp rheology� This estimate can be obtained from direct measurements of cohesive forces in contacts between crossed fibers� The molecular component of the cohesion, p, in the contact between two fibers can be obtained as the force necessary to rupture the fiber-fiber contact� The shear friction force can then be determined as the product of the previously determined friction coefficient, μ, and the normal force N0� In the absence of an external load, the value of N0 is solely the result of the molecular attraction forces, that is, F = μN0 = μp�

The cohesive forces between 10 μm fibers are very small but can be measured using the setup described in Section 1�3 and shown in Figure 1�28� The individual fibers can be mounted on the L-shaped holders by using the specialized clamp illustrated in Figure 2�22� The fiber 1 is placed across the end of a hollow polystyrene cylinder 2 and is fixed with a tight-fitting ring 3� In order to achieve better fixation of the fiber, small grooves were made on the outer edge of the cylinder�

The results of the contact force measurements, p, between cellulosic fibers in the presence of PEI are shown in Figure 2�23� The observed trends are similar to those shown in Figure 2�21 for the friction coefficient measurements and are also in good agreement with the “patch model” of flocculation by the PEI, and the previously mentioned electrokinetic studies, as reflected by the maximum in the cohesive force at low concentrations of PEI� As expected, after reaching the maximum, the cohesive forces decrease with a further increase in the PEI concentration� This corresponds to the decrease in the fraction of available negatively charged patches for interaction with PEI and the increase in the electrostatic repulsion due to a continued adsorption of PEI�

The component of the shear friction force in contact, F, due solely to molecular attraction, can be obtained from the measured values of the cohesive forces, p, using the values of the friction coefficients, μ, shown in Figure 2�21� These data are illustrated in Figure 2�24, which shows the friction force, F = μp as a function of the PEI concentration� In the concentration interval c ~ 10−9-10−7 M, the addition of PEI results in an increase in the friction force up to a maximal value observed in the concentration interval between c ~ 10−9-10−8 M� At PEI concentrations greater than ~10−6 M, the friction force decreases and reaches values nearly two times less than in water�

The obtained values of the friction force are related to a single “point-like” contact between two fibers crossed at a 90° angle� In the pulp suspension of papermaking consistency, the fibers form numerous such contacts� The summed molecular attractive forces are responsible for the friction forces that determine the rheological properties of the pulp at different stages of the papermaking

process� The interactions between fibers in the presence of added papermaking chemicals influence the dewatering of the pulp on the wire and the end sheet formation� The control of the rheological properties is thus dependent on the ability to influence the molecular component of the friction force�

The cohesive force between cellulosic fibers from Kraft bleached pulp was also measured in solutions of a cationic surfactant, tetrabutylammonium iodide (TBAI)� The cohesive force, p, as a function of the concentration of TBAI is shown in Figure 2�25� As seen in this figure, the cohesive force undergoes a sharp decrease as the TBAI concentration increases and then passes through a

minimum at c ~ 10−3 M� Further increases in the TBAI concentration result in an increase in the cohesive force up to a maximum value at c = 3�2 × 10−3 M� Measurements of the ζ-potential in the same system revealed that, at concentrations of TBAI of less than 10−3 M, the ζ potential value decreases from –2 to –10 mV, most likely due to the specific adsorption of I− ions� Such an increase in the negative ζ potential results in an increase in electrostatic repulsion, which in turn results in a weakening of the cohesive force� However, the observed decrease in the cohesive force is too large to be explained solely by the modest change in the ζ potential� Most likely, the main contribution to the contact strength decrease is caused by the formation of a hydrophilic adsorption layer� Independent adsorption measurements indicated that at TBAI concentration, c = 3�2 × 10−3 M, the adsorption is 3�7 × 10−4 mol/cm2, which corresponds to the formation of a polymolecular adsorption layer�

Further increases in the TBAI concentration result in an increase in the ζ-potential due to the preferential adsorption of the surfactant cation� The isoelectric point is observed at c ~ 3�2 × 10−3 M, and the ζ-potential further increases to +10 mV at c ~ 2�5 × 10−2 M� Around the isoelectric point, the maximum in the cohesive force has been observed� Further increases in the TBAI concentration result in an increase in electrostatic repulsion, which is the likely reason for the cohesive force increase�

Another interesting study on the cohesive force between bleached pulp fibers was carried out in a solution of papermaking high-molecular-weight cationic polyacrylamide (CPAM) flocculant [46]� In contrast to PEI, the molecular weight of this polyacrylamide flocculant was on the order of tens of millions of daltons� The results of this study are shown in Figure 2�26 in the form of a histogram of the repeated force measurements between two fibers immersed in a 0�001% solution of CPAM� Initially, the cohesive force between the fibers was so high that the contact could not be ruptured by the applied current and had to be ruptured manually� When the fibers were brought together again, the cohesive force was not nearly as high: the contact could be ruptured with a current of sufficient magnitude, and the measured cohesive force, p, was about 4�5 μN� The cohesive force upon the subsequent contact between the same fibers gradually decreased until it reached the minimum value of ~0�5 to 0�5 μN� These observations agree very well with the bridging mechanism of pulp using high-molecular-weight flocculants [32]� When the flocculant is added to the flowing pulp slurry,

very large and strong flocs are initially formed by means of polymer “bridges” between neighboring fibers� These flocs are then broken down by shear stresses, for example, in paper machine screens� Once the flocculated pulp passes the shear point and the relief of shear stress occurs, a reflocculation takes place, but the newly formed flocs are smaller and weaker� The rupture of flocs under papermaking conditions results in the physical rupture of the flocculant molecules with the subsequent reconformation of the residual flocculant on the fiber surface� The degradation in the polymer molecular weight and the reduction in the number of “loops” and “tails” extending into the pulp slurry leads to the formation of smaller and weaker flocs, that is, to the degradation of flocculation effectiveness� This process was to some extent simulated by the repeated rupture of cohesive contacts between the fibers�