Chemical Physics lecture notes

what physics is in chemical engineering and what does chemical physics involve pdf free download
MarthaKelly Profile Pic
MarthaKelly,Mexico,Researcher
Published Date:12-07-2017
Your Website URL(Optional)
Comment
Birdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 197 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems 7 and Interfaces Peter A. Kralchevsky, Krassimir D. Danov, and Nikolai D. Denkov CONTENTS 7.1 Introduction................................................................................................................................................................... 199 7.2 Surface Tension of Surfactant Solutions....................................................................................................................... 200 7.2.1 Static Surface Tension....................................................................................................................................... 200 7.2.1.1 Nonionic Surfactants........................................................................................................................... 200 7.2.1.2 Ionic Surfactants.................................................................................................................................. 206 7.2.2 Dynamic Surface Tension................................................................................................................................. 213 7.2.2.1 Adsorption under Diffusion Control................................................................................................... 213 7.2.2.2 Small Initial Perturbation..................................................................................................................... 214 7.2.2.3 Large Initial Perturbation..................................................................................................................... 215 7.2.2.4 Generalization for Ionic Surfactants.................................................................................................... 217 7.2.2.5 Adsorption under Barrier Control....................................................................................................... 218 7.2.2.6 Dynamics of Adsorption from Micellar Surfactant Solutions............................................................ 220 7.3 Capillary Hydrostatics and Thermodynamics............................................................................................................... 225 7.3.1 Shapes of Fluid Interfaces................................................................................................................................. 225 7.3.1.1 Laplace and Young Equations............................................................................................................ 225 7.3.1.2 Solutions of Laplace Equations for Menisci of Different Geometry.................................................. 227 7.3.1.3 Gibbs–Thomson Equation................................................................................................................... 230 7.3.1.4 Kinetics of Ostwald Ripening in Emulsions....................................................................................... 231 7.3.2 Thin Liquid Films and PBs............................................................................................................................... 233 7.3.2.1 Membrane and Detailed Models of a Thin Liquid Film..................................................................... 233 7.3.2.2 Thermodynamics of Thin Liquid Films.............................................................................................. 234 7.3.2.3 Transition Zone between Thin Film and PB....................................................................................... 236 7.3.2.4 Methods for Measuring Thin Film Contact Angles............................................................................ 239 7.3.3 Lateral Capillary Forces between Particles Attached to Interfaces................................................................... 239 7.3.3.1 Particle–Particle Interactions............................................................................................................... 239 7.3.3.2 Particle–Wall Interactions.................................................................................................................... 243 7.3.3.3 Electrically Charged Particles at Liquid Interfaces............................................................................. 244 7.4 Surface Forces............................................................................................................................................................... 248 7.4.1 Derjaguin Approximation.................................................................................................................................. 248 7.4.2 van der Waals Surface Forces........................................................................................................................... 249 7.4.3 Electrostatic Surface Forces............................................................................................................................... 251 7.4.3.1 Two Identically Charged Planes.......................................................................................................... 251 7.4.3.2 Two Nonidentically Charged Planes................................................................................................... 253 7.4.3.3 Two Charged Spheres......................................................................................................................... 254 7.4.4 Derjaguin–Landau–Verwey–Overbeek (DLVO) Theory.................................................................................. 255 7.4.5 Non-DLVO Surface Forces............................................................................................................................... 255 7.4.5.1 Ion Correlation Forces......................................................................................................................... 255 7.4.5.2 Steric Interaction.................................................................................................................................. 256 7.4.5.3 Oscillatory Structural Forces............................................................................................................... 259 7.4.5.4 Repulsive Hydration and Attractive Hydrophobic Forces.................................................................. 263 7.4.5.5 Fluctuation Wave Forces..................................................................................................................... 267 197 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 198 14.10.2008 10:28am Compositor Name: DeShanthi 198 Handbook of Surface and Colloid Chemistry 7.5 Hydrodynamic Interactions in Dispersions................................................................................................................... 268 7.5.1 Basic Equations and Lubrication Approximation............................................................................................. 268 7.5.2 Interaction between Particles of Tangentially Immobile Surfaces.................................................................... 271 7.5.2.1 Taylor and Reynolds Equations, and Influence of the Particle Shape................................................ 271 7.5.2.2 Interactions among Nondeformable Particles at Large Distances....................................................... 272 7.5.2.3 Stages of Thinning of a Liquid Film................................................................................................... 275 7.5.2.4 Dependence of Emulsion Stability on the Droplet Size..................................................................... 278 7.5.3 Effect of Surface Mobility................................................................................................................................. 280 7.5.3.1 Diffusive and Convective Fluxes at an Interface—Marangoni Effect................................................ 281 7.5.3.2 Fluid Particles and Films of Tangentially Mobile Surfaces................................................................ 282 7.5.3.3 Bancroft Rule for Emulsions............................................................................................................... 285 7.5.3.4 Demulsification.................................................................................................................................... 287 7.5.4 Interactions in Nonpreequilibrated Emulsions.................................................................................................. 288 7.5.4.1 Surfactant Transfer from Continuous to Disperse Phase (Cyclic Dimpling)...................................... 288 7.5.4.2 Surfactant Transfer from Disperse to Continuous Phase (Osmotic Swelling).................................... 289 7.5.4.3 Equilibration of Two Droplets across a Thin Film............................................................................. 290 7.5.5 Hydrodynamic Interaction of a Particle with an Interface................................................................................ 291 7.5.5.1 Particle of Immobile Surface Interacting with a Solid Wall............................................................... 291 7.5.5.2 Fluid Particles of Mobile Surfaces...................................................................................................... 293 7.5.6 Bulk Rheology of Dispersions.......................................................................................................................... 295 7.6 Kinetics of Coagulation................................................................................................................................................. 299 7.6.1 Irreversible Coagulation..................................................................................................................................... 299 7.6.2 Reversible Coagulation...................................................................................................................................... 302 7.6.3 Kinetics of Simultaneous Flocculation and Coalescence in Emulsions............................................................ 303 7.7 Mechanisms of Antifoaming......................................................................................................................................... 305 7.7.1 Location of Antifoam Action—Fast and Slow Antifoams................................................................................ 306 7.7.2 Bridging–Stretching Mechanism....................................................................................................................... 307 7.7.3 Role of the Entry Barrier................................................................................................................................... 308 7.7.3.1 Film Trapping Technique (FTT)......................................................................................................... 309 7.7.3.2 Critical Entry Pressure for Foam Film Rupture.................................................................................. 309 7.7.3.3 Optimal Hydrophobicity of Solid Particles......................................................................................... 310 7.7.3.4 Role of the Prespread Oil Layer.......................................................................................................... 311 7.7.4 Mechanisms of Compound Exhaustion and Reactivation................................................................................. 312 7.8 Electrokinetic Phenomena in Colloids.......................................................................................................................... 314 7.8.1 Potential Distribution at a Planar Interface and around a Sphere..................................................................... 315 7.8.2 Electroosmosis................................................................................................................................................... 317 7.8.3 Streaming Potential............................................................................................................................................ 319 7.8.4 Electrophoresis................................................................................................................................................... 319 7.8.5 Sedimentation Potential..................................................................................................................................... 322 7.8.6 Electrokinetic Phenomena and Onzager Reciprocal Relations......................................................................... 323 7.8.7 Electric Conductivity and Dielectric Response of Dispersions......................................................................... 324 7.8.7.1 Electric Conductivity........................................................................................................................... 324 7.8.7.2 Dispersions in Alternating Electrical Field......................................................................................... 325 7.8.8 Anomalous Surface Conductance and Data Interpretation............................................................................... 328 7.8.9 Electrokinetic Properties of Air–Water and Oil–Water Interfaces.................................................................... 329 7.9 Optical Properties of Dispersions and Micellar Solutions............................................................................................ 330 7.9.1 Static Light Scattering....................................................................................................................................... 330 7.9.1.1 Rayleigh Scattering.............................................................................................................................. 330 7.9.1.2 Rayleigh–Debye–Gans (RDG) Theory............................................................................................... 332 ................. 335 7.9.1.3 Theory of Mie..................................................................................................................... 7.9.1.4 Interacting Particles............................................................................................................................. 335 7.9.1.5 Depolarization of Scattered Light....................................................................................................... 338 7.9.1.6 Polydisperse Samples.......................................................................................................................... 339 7.9.1.7 Turbidimetry........................................................................................................................................ 339 7.9.2 DLS.................................................................................................................................................................... 340 7.9.2.1 DLS by Monodisperse, Noninteracting Spherical Particles................................................................ 340 7.9.2.2 DLS by Polydisperse, Noninteracting Spherical Particles.................................................................. 342 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 199 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 199 7.9.2.3 DLS by Nonspherical Particles........................................................................................................... 344 7.9.2.4 Effect of the Particle Interactions........................................................................................................ 345 7.9.2.5 Concentrated Dispersions: Photon Cross-Correlation Techniques, Fiber Optics DLS, and Diffusing Wave Spectroscopy (DWS)......................................................................................... 349 7.9.3 Application of Light Scattering Methods to Colloidal Systems....................................................................... 350 7.9.3.1 Surfactant Solutions............................................................................................................................. 350 7.9.3.2 Dispersions.......................................................................................................................................... 352 7.9.4 Recent Developments in Light Scattering Techniques..................................................................................... 353 7.9.4.1 Opaque Systems.................................................................................................................................. 353 7.9.4.2 Small Angle Light Scattering.............................................................................................................. 353 7.9.4.3 Multispeckle DLS................................................................................................................................ 354 Acknowledgment..................................................................................................................................................................... 354 References................................................................................................................................................................................ 355 7.1 INTRODUCTION Acolloidalsystemrepresentsamultiphase(heterogeneous)system,inwhichatleastoneofthephasesexistsintheformofvery small particles: typically smaller than 1mm but still much larger than the molecules. Such particles are related to phenomena like Brownian motion, diffusion, and osmosis. The terms microheterogeneous system and disperse system (dispersion) are more general because they include also bicontinuous systems (in which none of the phases is split into separate particles) and systems containing larger, non-Brownian, particles. The term dispersion is often used as a synonym of colloidal system. Classification of the colloids with respect to the state of aggregation of the disperse and the continuous phases is shown in Table 7.1. Some examples are following. 1. Examples for gas-in-liquid dispersions are the foams or the boiling liquids. Gas-in-solid dispersions are the various porous media like filtration membranes, sorbents, catalysts and isolation materials. 2. Examplesforliquid-in-gasdispersionsarethemist,theclouds,andotheraerosols.Liquid-in-liquiddispersionsarethe emulsions. At room temperature there are only four types of mutually immiscible liquids: water, hydrocarbon oils, fluorocarbon oils, and liquid metals (Hg and Ga). Many raw materials and products in food and petroleum industries exist in the form of oil-in-water (O=W) or water-in-oil (W=O) emulsions. The soil and some biological tissues can be considered as liquid-in-solid dispersions. 3. Smoke, dust, and some other aerosols are examples for solid-in-gas dispersions. The solid-in-liquid dispersions are termed as suspensions or sols. The pastes and some glues are highly concentrated suspensions. The gels represent bicontinuous structures of solid and liquid. Solid-in-solid dispersions are some metal alloys, many kinds of rocks, some colored glasses, etc. Below we will consider mostly liquid dispersions, i.e., dispersions with liquid continuous phase like foams, emulsions, and suspensions. Sometimes these are called complex fluids. 3 In general, the area of the interface between the disperse and continuous phases is rather large. For instance, 1 cm of 2 dispersion with particles of radius 100 nm and volume fraction 30% contains interface of area about 10 m . This is the reason why the interfacial properties are of crucial importance for the properties and stability of colloids. The stabilizing factors for dispersions are the repulsive surface forces, the particle thermal motion, the hydrodynamic resistance of the medium, and the high surface elasticity of fluid particles and films. TABLE 7.1 Types of Disperse Systems Continuous Phase Disperse Phase Gas Liquid Solid Gas — Gas in liquid Gas in solid Liquid Liquid in gas L in L Liquid in solid 1 2 Solid Solid in gas Solid in liquid S in S 1 2 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 200 14.10.2008 10:28am Compositor Name: DeShanthi 200 Handbook of Surface and Colloid Chemistry On the contrary, the factors destabilizing dispersions are the attractive surface forces, the factors suppressing the repulsive surface forces, and the low surface elasticity, gravity and other external forces tending to separate the phases. Sections7.2and7.3considereffectsrelatedtothesurfacetensionofsurfactantsolutionandcapillarity.Section7.4presents a review on the surface forces due to the intermolecular interactions. Section 7.5 describes the hydrodynamic interparticle forcesoriginatingfromtheeffectsofbulkandsurfaceviscosityandrelatedtosurfactantdiffusion.Section7.6isdevotedtothe kinetics of coagulation in dispersions. Section 7.7 discusses foams containing oil drops and solid particulates in relation to the antifoaming mechanisms and the exhaustion of antifoams. Finally, Sections 7.8 and 7.9 address the electrokinetic and optical properties of dispersions. 7.2 SURFACE TENSION OF SURFACTANT SOLUTIONS 7.2.1 STATIC SURFACE TENSION As a rule the fluid dispersions (emulsions, foams) are stabilized by adsorption layers of amphiphile molecules. These can be ionic1,2 and nonionic 3 surfactants, lipids, proteins, etc. All ofthem have the property to lower the value of the surface (or interfacial) tension, s, in accordance with the Gibbs adsorption equation 4–6 X ds¼ Gdm (7:1) i i i where G is the surface concentration (adsorption) of the ith component i m is its chemical potential i The summation in Equation 7.1 is carried out over all components. Usually an equimolecular dividing surface with respect to the solvent is introduced for which the adsorption of the solvent is set zero by definition 4,5. Then the summation is carried out over all other components. Note that G is an excess surface concentration with respect to the bulk; G is positive for i i surfactants, which decrease s in accordance with Equation 7.1. On the contrary, G is negative for aqueous solutions of i electrolytes, whose ions are repelled from the surface by the electrostatic image forces 5; consequently, the addition of electrolytes increases the surface tension of water 6. For surfactant concentrations above the critical micellization concentra- tion (CMC) m is equal to constant and, consequently, s is also equal to constant (see Equation 7.1). i 7.2.1.1 Nonionic Surfactants 7.2.1.1.1 Types of Adsorption Isotherms Consider the boundary between an aqueous solution of a nonionic surfactant and a hydrophobic phase, air or oil. The dividing surface is usually chosen to be the equimolecular surface with respect to water, that is G ¼0. Then Equation 7.1 reduces to w ds¼G dm , where the subscript 1 denotes the surfactant. Because the bulk surfactant concentration is usually not too high, 1 1 we can use the expression for the chemical potential of a solute in an ideal solution: (0) m ¼m þkT ln c , 1 1 1 where k is the Boltzmann constant T is the absolute temperature c is the concentration of nonionic surfactant 1 (0) m is its standard chemical potential, which is independent of c 1 1 Thus the Gibbs adsorption equation acquires the form ds¼kTG dln c (7:2) 1 1 The surfactant adsorption isotherms, expressing the connection between G and c , are usually obtained by means of some 1 1 molecular model of adsorption. Table 7.2 contains the six most popular surfactant adsorption isotherms, those of Henry, Freundlich 7, Langmuir 8, Volmer 9, Frumkin 10, and van der Waals 11. For c 0 all isotherms (except that of 1 Freundlich)reducetotheHenryisotherm:G =G ¼Kc .ThephysicaldifferencebetweentheLangmuirandVolmerisotherms 1 1 1 is that the Langmuir isotherm corresponds to a model of localized adsorption, while the Volmer corresponds to nonlocalized © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 201 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 201 TABLE 7.2 Types of Adsorption and Surface-Tension Isotherms Type of Isotherm Surfactant Adsorption Isotherms (for Nonionic Surfactants:a  c ) 1s 1 G 1 Henry Ka ¼ 1s G 1  1=m G 1 Freundlich Ka ¼ 1s G 1 G 1 Langmuir Ka ¼ 1s G G 1 1  G G 1 1 Volmer Ka ¼ exp 1s G G G G 1 1 1 1  G 2bG 1 1 Frumkin Ka ¼ exp  1s G G kT 1 1  G G 2bG 1 1 1 van der Waals Ka ¼ exp  1s G G G G kT 1 1 1 1 Surface-Tension Isotherms¼s kTJþs (for Nonionic Surfactants:s  0) 0 d d Henry J¼G 1 G 1 Freundlich J ¼ m G 1 Langmuir J¼G ln(1 ) 1 G 1 G G 1 1 Volmer J ¼ G G 1 1  2 G bG 1 1 Frumkin J¼G ln 1  1 G kT 1 2 G G bG 1 1 1 Van der Waals J ¼  G G kT 1 1 Note: Surfactant adsorption isotherm and surface-tension isotherm, which are combined to fit experimental data, obligatorily must be of the same type. adsorption.TheFrumkinandvanderWallsisothermsgeneralize,respectively,theLangmuirandVolmerisothermsforcase,in which the interaction between neighboring adsorbed molecules is not negligible. (If the interaction parameter b is set as zero, the Frumkin and van der Walls isotherms reduce to the Langmuir and Volmer isotherms, correspondingly.) The comparison between theory and experiment shows that for air–water interfaces b0, whereas for oil–water interfaces we can set b¼0 12,13. The latter facts lead to the conclusion that for air–water interfaces b takes into account the van der Waals attraction betweenthehydrocarbontailsoftheadsorbedsurfactantmoleculesacrossair;suchattractionismissingwhenthehydrophobic phase is oil. The adsorption parameter K in Table 7.2 characterizes the surface activity of the surfactant: the greater the K, the 0 0 higher the surface activity. K is related to the standard free energy of adsorption,Df ¼m m , which is the energy gain for 1 1s bringing a molecule from the bulk of the aqueous phase to a diluted adsorption layer 14,15: (0) (0) d m m 1 1 1s K ¼ exp (7:3) G kT 1 where d characterizesthethicknessoftheadsorptionlayer;d canbeset(approximately)equaltothelengthoftheamphiphilic 1 1 molecule G represents the maximum possible value of the adsorption 1 In the case of localized adsorption (Langmuir and Frumkin isotherms) 1=G is the area per adsorption site. In the case of 1 nonlocalized adsorption (Volmer and van der Waals isotherms) 1=G is the excluded area per molecule. 1 As mentioned earlier, the Freundlich adsorption isotherm, unlike the others in Table 7.2, does not become linear at low concentrations, but remains convex to the concentration axis. Moreover, it does not show saturation or limiting value. Hence, fortheFreundlichadsorptionisotherminTable7.2G isaparameterscalingtheadsorption(ratherthansaturationadsorption). 1 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 202 14.10.2008 10:28am Compositor Name: DeShanthi 202 Handbook of Surface and Colloid Chemistry This isotherm can be derived assuming that the solid surface is heterogeneous 16,17. Consequently, if the data fits the Freundlich equation, this is an indication, but not a proof, that the surface is heterogeneous 6. The adsorption isotherms in Table 7.2 can be applied to both fluid and solid interfaces. The surface-tension isotherms in Table 7.2, which relate s and G , are usually applied to fluid interfaces, although they could also be used for solid–liquid 1 interfaces if s is identified with the Gibbs 4 superficial tension. (The latter is defined as the force per unit length which opposes every increase of the wet area without any deformation of the solid.) Thesurface-tensionisothermsinTable7.2arededucedfromtherespectiveadsorptionisothermsinthefollowingway.The integration of Equation 7.2 yields s¼s kTJ (7:4) 0 where s is the interfacial tension of the pure solvent and 0 c G 1 1 ð ð dc dlnc 1 1 J  G ¼ G dG (7:5) 1 1 1 c dG 1 1 0 0 The derivative dlnc =dG is calculated for each adsorption isotherm, and then the integration in Equation 7.5 is carried out 1 1 analytically. The obtained expressions for J are listed in Table 7.2. Each surface-tension isotherm, s(G ), has the meaning 1 of a two-dimensional equation of state of the adsorption monolayer, which can be applied to both soluble and insoluble surfactants 6,18. An important thermodynamic property of a surfactant adsorption monolayer is its Gibbs (surface) elasticity  s E G (7:6) G 1 G 1 T Expressions for E , corresponding to various adsorption isotherms, are shown in Table 7.3. The Gibbs elasticity characterizes G the lateral fluidity of the surfactant adsorption monolayer. At high values of the Gibbs elasticity the adsorption monolayer behavesastangentiallyimmobile.Insuchacase,iftwoemulsiondropletsapproacheachother,thehydrodynamicflowpattern, and the hydrodynamic interaction as well, is almost the same as if the droplets were solid. For lower values of the surfactant adsorption the so-called Marangoni effect appears, which is equivalent to the appearance of gradients of surface tension due to gradients of surfactant adsorption:r s¼(E =G )r G wherer denotes surface gradient operator. The Marangoni effect s G 1 s 1, s : can considerably affect the hydrodynamic interactions of fluid particles (drops and bubbles) (see Section 7.5). 7.2.1.1.2 Derivation from First Principles Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tensionandsurfacechemicalpotential,canbederivedfromanexpression forthesurfacefreeenergy, F ,whichcorrespondsto s agivenphysicalmodel.Thisderivationhelpsusobtain(oridentify)theself-consistentsystemofequations,referringtoagiven TABLE 7.3 Elasticity of Adsorption Monolayers at a Fluid Interface Type of Isotherm (cf. Table 7.2) Gibbs Elasticity E G Henry E ¼ kTG G 1 G 1 Freundlich E ¼ kT G m G 1 Langmuir E ¼ kTG G 1 G G 1 1 2 G 1 Volmer E ¼ kTG G 1 2 (G G ) 1 1  G 2bG 1 1 Frumkin E ¼ kTG  G 1 G G kT 1 1  2 G 2bG 1 1 van der Waals E ¼ kTG  G 1 2 (G G ) kT 1 1 Note: Above expressions are valid for both nonionic and ionic surfactants. © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 203 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 203 TABLE 7.4 Free Energy and Chemical Potential for Surfactant Adsorption Layers Type of Isotherm Surface Free EnergyF (T, A, N)(M¼G A) s 1 1 (0) Henry F ¼ N m þkTN ln(N =M)N s 1 1 1 1 1s kT (0) Freundlich F ¼ N m þ N ln(N =M)N s 1 1 1 1 1s m (0) Langmuir F ¼ N m þkTN lnN þ(MN )ln(MN )MlnM s 1 1 1 1 1 1s (0) Volmer F ¼ N m þkTN lnN N N ln(MN ) s 1 1 1 1 1 1 1s 2 bG N 1 (0) 1 Frumkin F ¼ N m þkTN lnN þ(MN )ln(MN )MlnMþ s 1 1 1 1 1 1s 2M 2 bG N 1 (0) 1 van der Waals F ¼ N m þkTN lnN N N ln(MN )þ s 1 1 1 1 1 1 1s 2M Surface Chemical Potentialm (uG =G ) 1s 1 1 (0) Henry m ¼m þkTlnu 1s 1s kT (0) Freundlich m ¼m þ lnu 1s 1s m u (0) Langmuir m ¼m þkTln 1s 1s 1u u u (0) Volmer m ¼m þkTð þln Þ 1s 1s 1u 1u u (0) Frumkin m ¼m þkTln 2bG 1 1s 1s 1u  u u (0) van der Waals m ¼m þkT þln 2bG 1 1s 1s 1u 1u model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say Langmuir adsorption isotherm with Frumkin surface-tension isotherm) is incorrect and must be avoided. The general scheme for derivation of the adsorption isotherms is the following: 1. With thehelpof statistical mechanicsanexpressionisobtained,say, forthecanonicalensemblepartitionfunction, Q, from which the surface free energy F is determined 11: s F (T,A,N )¼ kTlnQ(T,A,N)(7:7) s 1 1 where A is the interfacial area N is the number of adsorbed surfactant molecules (see Table 7.4) 1 2. Differentiating the expression for F , we derive expressions for the surface pressure, p , and the surface chemical s s potential of the adsorbed surfactant molecules, m 11: 1s   F F s s p s s¼ , m ¼ (7:8) s 0 1s A N 1 T,N T,A 1 Combiningtheobtainedexpressionsforp andm ,wecandeducetherespectiveformoftheButlerequation19(see s 1s Equation 7.16). 3. The surfactant adsorption isotherm (Table 7.2) can be derived by setting the obtained expression for the surface chemical potential m equal to the bulk chemical potential of the surfactant molecules in the subsurface layer (i.e., 1s equilibrium between surface and subsurface is assumed) 11: (0) m ¼m þkTln(a d =G)(7:9) 1s 1 1 1s 1 Wherea istheactivityofthesurfactantmoleculeinthesubsurfacelayer;a isscaledwiththevolumepermoleculeinadense 1s 1s (saturated) adsorption layer, v ¼d =G , where d is interpreted as the thickness of the adsorption layer, or the length of an 1 1 1 1 adsorbed molecule. In terms of the subsurface activity, a , Equation 7.9 can be applied to ionic surfactants and to dynamic 1s © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 204 14.10.2008 10:28am Compositor Name: DeShanthi 204 Handbook of Surface and Colloid Chemistry processes. In the simplest case of nonionic surfactants and equilibrium processes we have a c , where c is the bulk 1s 1 1 surfactant concentration. First, let us apply the above general scheme to derive the Frumkin isotherm, which corresponds to localized adsorption of interacting molecules.(Expressions corresponding to the Langmuir isotherm can be obtained bysettingb¼0 in the respective expressionsfortheFrumkinisotherm.)Letusconsidertheinterfaceasatwo-dimensionallatticehavingMadsorptionsites.The corresponding partition function is 11  2 M n wN c 1 Q(T,M,N )¼ q(T)N exp  (7:10) 1 1 N (MN ) 2kTM 1 1 Thefirstmultiplierintheright-handsideofEquation7.10expressesthenumberofwaysN indistinguishablemoleculescanbe 1 distributed among M labeled sites; the partition function for a single adsorbed molecule is q¼q q q , where q , q , and q are x y z x y z one-dimensional harmonic-oscillator partition functions. The exponent in Equation 7.10 accounts for the interaction between adsorbed molecules in the framework of the Bragg–Williams approximation 11. w is the nearest-neighbor interaction energy of two molecules and n is the number of nearest-neighbor sites to a given site (e.g., n ¼4 for a square lattice). Then, we c c substitute Equation 7.10 into Equation 7.7 and using the known Stirling approximation, ln M¼M lnMM, we get the expression for the surface free energy corresponding to the Frumkin model: 2 n wN c 1 F ¼ kTN lnN þ(MN )ln(MN )MlnMN lnq(T)þ (7:11) s 1 1 1 1 1 2M Note that M ¼G A, N ¼G A (7:12) 1 1 1 1 whereG is the area per one adsorption site in the lattice. Differentiating Equation 7.11 in accordance with Equation 7.8, we 1 deduce expressions for the surface pressure and chemical potential 11: 2 p ¼G kTln(1u)bG (7:13) s 1 1 u (0) m ¼m þkTln 2bG (7:14) 1 1s 1s 1u where we have introduced the notation G n w 1 c (0) u¼ , b¼ , m ¼kTlnq(T)(7:15) 1s G 2G 1 1 WecancheckthatEquation7.13isequivalenttotheFrumkin’ssurface-tensionisotherminTable7.2foranonionicsurfactant. Furthermore, eliminating ln(1u)betweenEquations7.13 and7.14, weobtainthe Butler equationin thefollowing form 19 (0) 1 m ¼m þG p þkTln(g u) (Butler equation) (7:16) s 1s 1s 1 1s where we have introduced the surface activity coefficient  bG u(2u) 1 g ¼ exp  (for Frumkin isotherm) (7:17) 1s kT (In the special case of Langmuir isotherm we have b¼0, and then g ¼1.) The Butler equation is used by many authors 1s : 12,20–22 as a starting point for the development of thermodynamic adsorption models. It should be kept in mind that the specific form of the expressions for p and g , which are to be substituted in Equation 7.16, is not arbitrary, but must s 1s correspondtothesamethermodynamicmodel(tothesameexpressionfor F—inourcaseEquation7.11).Finally,substituting s Equation 7.16 into Equation 7.9, we derivethe Frumkin adsorption isotherm in Table 7.2, where K is defined by Equation 7.3. Now, let us apply the same general scheme, but this time to the derivation of the van der Waals isotherm, which corresponds to nonlocalized adsorption of interacting molecules. (Expressions corresponding to the Volmer isotherm can be © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 205 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 205 obtained by setting b¼0 in the respective expressions for the van der Waals isotherm.) Now the adsorbed N molecules are 1 considered as a two-dimensional gas. The corresponding expression for the canonical ensemble partition function is  2 1 n wN c N 1 1 Q(T,M,N )¼ q exp  (7:18) 1 N 2kTM 1 where the exponent accounts for the interaction between adsorbed molecules, again in the framework of the Bragg–Williams approximation. The partition function for a single adsorbed molecule is q¼q q , where q is one-dimensional (normal to the xy z z interface) harmonic-oscillator partition function. On the other hand, the adsorbed molecules have free translational motion in the xy-plane (the interface); therefore we have 11 2pmkT e q ¼ A (7:19) xy 2 h p where me is the molecular mass h is the Planck constant p 1 1 A¼ AN G is the area accessible to the moving molecules; the parameter G is the excluded area per molecule, 1 1 1 which accounts for the molecular size Having in mind that M G A, we can bring Equation 7.18 into the form 1  2 1 n wN c N 1 N 1 1 Q(T,M,N )¼ q (MN ) exp  (7:20) 1 1 0 N 2kTM 1 where 2pmkT e q (T) q (T)(7:21) 0 z 2 h G 1 p Further, we substitute Equation 7.20 into Equation 7.7, using the Stirling approximation, we determine the surface free energy corresponding to the van der Waals model 11,18,23: 2 n wN c 1 F ¼ kTN ln N N N lnq (T)N ln(MN )þ (7:22) s 1 1 1 1 0 1 1 2M Again, having in mind that MG A, we differentiate Equation 7.22 in accordance with Equation 7.8 to deduce expressions 1 for the surface pressure and chemical potential: u 2 p ¼G kT bG (7:23) s 1 1 1u  u u (0) m ¼m þkT þln 2bG (7:24) 1 1s 1s 1u 1u (0) wherem ¼ kTlnq (T)andbisdefinedbyEquation7.15.WecancheckthatEquation7.23isequivalenttothevanderWaals 0 1s surface-tensionisotherminTable7.2foranonionicsurfactant.Furthermore,combiningEquations7.23and7.24,weobtainthe Butler equation (Equation 7.16), but this time with another expression for the surface activity coefficient  1 bG u(2u) 1 g ¼ exp  (for van der Waals isotherm) (7:25) 1s 1u kT (In the special case of Volmer isotherm we have b¼0, and then g ¼1=(1u).) Finally, substituting Equation 7.24 into 1s Equation 7.9, we derive the van der Waals adsorption isotherm in Table 7.2, with K defined by Equation 7.3. In Table 7.4 we summarize the expressions for the surface free energy, F , and chemical potential m , for several s 1s thermodynamic models of adsorption. We recall that the parameter G is defined in different ways for the different models. 1 On the other hand, the parameter K is defined in the same way for all models, viz. by Equation 7.3. The expressions in Tables 7.2 through 7.4 can be generalized for multicomponent adsorption layers 18,27. © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 206 14.10.2008 10:28am Compositor Name: DeShanthi 206 Handbook of Surface and Colloid Chemistry Attheendofthissection,letusconsiderageneralexpression,whichallowsustoobtainthesurfaceactivitycoefficientg 1s directly from the surface pressure isotherm p (u). From the Gibbs adsorption isotherm, dp ¼G dm , it follows that s s 1 1s   m 1 p s 1s ¼ (7:26) G G G 1 1 1 T T Bysubstitutingm fromtheButler’sEquation7.16intoEquation7.26,andintegratingwecanderivethesoughtforexpression: 1s u ð (1u) p du s lng ¼ 1 (7:27) 1s G kT u u 1 0 Wecancheckthatsubstitutionofp fromEquations7.13and7.23intoEquation7.27yields,respectively,theFrumkinandvan s der Waals expressions for g , viz. Equations 7.17 and 7.25. 1s 7.2.1.2 Ionic Surfactants 7.2.1.2.1 Gouy Equation The thermodynamics of adsorption of ionic surfactants 13,24–28 is more complicated (in comparison with that of nonionics) becauseofthepresenceoflong-rangeelectrostaticinteractionsand,inparticular,electricdoublelayer(EDL)inthesystem(see Figure 7.1). The electrochemical potential of the ionic species can be expressed in the form 29 (0) m ¼m þkTlna þZec (7:28) i i i i where e is the elementary electric charge c is the electric potential Z is the valence of the ionic component i i a is its activity i In the EDL (Figure 7.1) the electric potential and the activities of the ions are dependent on the distance z from the phase boundary: c¼c(z), a ¼a(z). At equilibrium the electrochemical potential, m, is uniform throughout the solution, including i i i the EDL (otherwise diffusion fluxes would appear) 29. In the bulk of solution (z1) the electric potential tends to a constant value, which is usually set equal to zero, that isc 0 andc=z 0 for z1. If the expression form at z1 i andthatform atsomefinitezaresetequal,fromEquation7.28weobtainaBoltzmann-typedistributionfortheactivityacross i the EDL 29:  Zec(z) i a(z)¼ a exp  (7:29) i i1 kT where a denotes the value of the activity of ion i in the bulk of solution. If the activity in the bulk, a , is known, then i1 i1 Equation 7.29determinestheactivity a(z)ineach point oftheEDL. Agoodagreementbetween theory andexperiment canbe i achieved 12,13,27 using the following expression for a : i1 a ¼g c (7:30) i1 i1  where c is the bulk concentration of the respective ion i1 g is the activity coefficient calculated from the known formula 30  pffiffi AjZ Z j I þ  pffiffi logg ¼ þbI (7:31)  1þBd I i which originates from the Debye–Hückel theory; I denotes the ionic strength of the solution: X 1 2 I  Z c (7:32) i1 i 2 i © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 207 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 207 Aqueous phase Coions Counterions Diffuse layer Surfactant Stern layer adsorption layer of adsorbed counterions Counterions C ∝ Coions 0 z FIGURE 7.1 Electric double layer in the vicinity of an adsorption layer of ionic surfactant. (a) Diffuse layer contains free ions involved in Brownian motion, while Stern layer consists of adsorbed (bound) counterions. (b) Near the charged surface there is an accumulation of counterions and a depletion of coions. where the summation is carried out over all ionic species in the solution. When the solution contains a mixture of several electrolytes, then Equation 7.31 definesg for each separate electrolyte, with Z and Z being the valences of the cations and  þ  anions of this electrolyte, but with I being the total ionic strength of the solution, accounting for all dissolved electrolytes 30. TheloginEquation7.31isdecimal,d istheionicdiameter,A,B,andbareparameters,whosevaluescanbefoundinRef.30. i 1=2 1=2 Forexample,ifIisgiveninmolesperliter(M),theparametersvaluesareA¼0.5115M ,Bd ¼1.316M ,andb¼0.055 i 1 M for solutions of NaCl at 258C. The theory of EDL provides a connection between surface charge and surface potential (known as the Gouy equation 31,32 of Graham equation 33,34), which can be presented in the form 27,35 () 1=2 N N X X 2 ¼ zG a exp(zF )1 (Gouy equation) (7:33) i i i1 i s k c i¼1 i¼1 whereG (i¼1,...,N)aretheadsorptionsoftheionicspecies,z ¼Z=Z ,andtheindexi¼1correspondstothesurfactantions i i i 1 2 2 2Z e Z ec 1 2 1 s k  , F  (7:34) s c « «kT kT 0 « is the dielectric permittivity of the medium (water), c ¼c(z¼0) is the surface potential. Note that the Debye parameter is s 2 2 k ¼k I. c For example, let us consider a solution of an ionic surfactant, which is a symmetric 1:1 electrolyte, in the presence of a symmetric, 1:1, inorganic electrolyte (salt). We assume that the counterions due to the surfactant and salt are identical. For example, this can beasolution of sodiumdodecylsulfate(SDS) in the presence ofNaCl.We denote by c , c ,and c the 11 21 31 bulkconcentrationsofthesurface-activeions,counterions,andcoions,respectively(Figure7.1).ForthespecialsystemofSDS © 2009 by Taylor & Francis Group, LLC Nonaqueous phase Nonionic concentrationBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 208 14.10.2008 10:28am Compositor Name: DeShanthi 208 Handbook of Surface and Colloid Chemistry  þ  with NaCl c , c , and c are the bulk concentration of the DS ,Na , and Cl ions, respectively. The requirement for the 11 21 31 bulk solution to be electroneutral implies c ¼c þc . The multiplication of the last equation by g yields 21 11 31  a ¼ a þa (7:35) 21 11 31 Theadsorptionofthecoionsofthenonamphiphilicsaltisexpectedtobeequaltozero,G ¼0,becausetheyarerepelledbythe 3 similarly charged interface 27,36–38. However, the adsorption of surfactant at the interface,G , and the binding of counter- 1 ions in the Stern layer, G , are different from zero (Figure 7.1). For this system the Gouy equation (Equation 7.33) acquires 2 the form  4 pffiffiffiffiffiffiffiffi F s G G ¼ a sinh (Z :Z electrolyte) (7:36) 1 2 21 1 1 k 2 c 7.2.1.2.2 Contributions from the Adsorption and Diffuse Layers e Ingeneral,thetotaladsorptionG ofanionicspeciesincludecontributionsfromboththeadsorptionlayer(surfactantadsorption i layer and adsorbed counterions in the Stern layer), G, and the diffuse layer, L 13,24,26,27: i i e G ¼G þL, i i i where 1 ð L  a(z)a dz (7:37) i i i1 0 e G represents a surface excess of component i with respect to the uniform bulk solution. Because the solution is electroneutral, i P P N N e we have zG ¼ 0. Note, however, that zG 6¼ 0, see the Gouy equation (Equation 7.33). Expressions forL can be i i i i i i¼1 i¼1 obtained by using the theory of EDL. For example, because of the electroneutrality of the solution, the right-hand side of Equation 7.36 is equal toL L L , where 2 1 3 1 1 L ¼ 2a k exp(F =2)1; L ¼ 2a k exp(F =2)1, j¼ 1,3: (7:38) 2 21 s j j1 s 2 2 (k ¼k I; Z :Z electrolyte). In analogy with Equation 7.37, the interfacial tension of the solution, s, can be expressed as a 1 1 c sum of contributions from the adsorption and diffuse layers 24,27,32: s¼s þs (7:39) a d where 1 ð 2 dc s ¼s kTJ and s ¼« « dz (7:40) a o d 0 dz 0 Expressions for J are given in Table 7.2 for various types of isotherms. Note that Equations 7.39 and 7.40 are validunder both equilibrium and dynamic conditions. In the special case of SDSþNaCl solution (see above), at equilibrium, we can use the theory of EDL to express dc=dz; then from Equation 7.40 we can derive 24,27,32   pffiffiffiffiffiffiffiffi 8kT F s s ¼ a cosh 1 (Z :Z electrolyte, at equilibrium) (7:41) d 21 1 1 k 2 c Analytical expressions for s for the cases of 2:1, 1:2, and 2:2 electrolytes can be found in Refs. 27,35. d In the case of ionic surfactant Equation 7.1 can be presented in two alternative, but equivalent forms 27,35 N X e ds¼kT Gdlna (T ¼ constant) (7:42) i i1 i¼1 N X ds ¼kT Gdlna (T ¼ constant) (7:43) a i is i¼1 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 209 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 209 where a ¼a(z¼0) is the subsurface value of activity a. From Equations 7.29 and 7.34, we obtain is i i a ¼ a exp(zF)(7:44) is i1 i s ThecomparisonbetweenEquations7.42and7.43showsthattheGibbsadsorptionequationcanbeexpressedeitherintermsof e s,G, and a , or in terms of s ,G, and a . Note that Equations 7.42 and 7.44 are valid under equilibrium conditions, while i i1 a i is Equation 7.43 can also be used for the description of dynamic surface tension (Section 7.2.2) in the case of surfactant adsorption under diffusion control, assuming local equilibrium between adsorptions G and subsurface concentrations of the i respective species. The expression s ¼s kTJ, with J given in Table 7.2, can be used for description of both static and dynamic surface a 0 tension of ionic and nonionic surfactant solutions. The surfactant adsorption isotherms in this table can be used for both ionic and nonionic surfactants, with the only difference that in the case of ionic surfactant the adsorption constant K depends on the subsurface concentration of the inorganic counterions 27 (see Equation 7.48). 7.2.1.2.3 Effect of Counterion Binding As an example, let us consider again the special case of SDSþNaCl solution. In this case, the Gibbs adsorption Equation 7.1, takes the form ds ¼kT(G dlna þG dlna)(7:45) a 1 1s 2 2s  þ where, as before, the indices 1 and 2 refer to the DS and Na ions, respectively. The differentials in the right-hand side of Equation 7.45 are independent (we can vary independently the concentrations of surfactant and salt), and moreover, ds is an a exact (total) differential. Then, according to the Euler condition, the cross derivatives must be equal 27: G G 1 2 ¼ (7:46) lna lna 2s 1s Asurfactantadsorptionisotherm,G ¼G (a ,a ),andacounterionadsorptionisotherm,G ¼G (a ,a ),arethermodynam- 1 1 1s 2s 2 2 1s 2s ically compatible only if they satisfy Equation 7.46. The counterion adsorption isotherm is usually taken in the form G K a 2 2 2s ¼ (Stern isotherm) (7:47) G 1þK a 1 2 2s where K is a constant parameter. The latter equation, termed the Stern isotherm 39, describes Langmuirian adsorption 2 (binding) of counterions in the Stern layer. It can be proven that a sufficient condition G form Equation 7.47 to satisfy the 2 Euler’s condition (Equation 7.46), together with one of the surfactant adsorption isotherms for G in Table 7.2, is 27 1 K ¼ K (1þK a)(7:48) 1 2 2s where K isanotherconstantparameter. In other words, if KisexpressedbyEquation7.48,the Sternisotherm (Equation 7.47) 1 is thermodynamically compatible with all the surfactant adsorption isotherms in Table 7.2. In analogy with Equation 7.3, the (0) parametersK andK arerelatedtotherespectivestandardfreeenergiesofadsorptionofsurfactantionsandcounterionsDm : 1 2 i (0) d Dm i i K ¼ exp (i¼ 1,2)(7:49) i G kT 1 where d stands for the thickness of the respective adsorption layer. i 7.2.1.2.4 Dependence of Adsorption Parameter K on Salt Concentration The physical meaning of Equation 7.48 can be revealed by chemical-reaction considerations. For simplicity, let us consider Langmuir-typeadsorption,i.e.,wetreattheinterfaceasatwo-dimensionallattice.Wewillusethenotationu forthefractionof 0  the free sites in the lattice, u for the fraction of sites containing adsorbed surfactant ion S , and u for the fraction of sites 1 2 containing the complex of an adsorbed surfactant ion and a bound counterion. Obviously, we can write u þu þu ¼1. The 0 1 2 adsorptions of surfactant ions and counterions can be expressed in the form: G =G ¼u þu ; G =G ¼u (7:50) 1 1 1 2 2 1 2 © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 210 14.10.2008 10:28am Compositor Name: DeShanthi 210 Handbook of Surface and Colloid Chemistry  Following Kalinin and Radke 119, we consider the reaction of adsorption of S ions:   A þS ¼ A S (7:51) 0 0 whereA symbolizesanemptyadsorptionsite.Inaccordancewiththerulesofthechemicalkinetics,wecanexpresstheratesof 0 adsorption and desorption in the form: r ¼ K u c , r ¼ K u (7:52) 1,ads 1,ads 0 1s 1,des 1,des 1 where c is the subsurface concentration of surfactant 1s K and K are the rate constants of adsorption and desorption 1,ads 1,des InviewofEquation7.50,wecanwriteu ¼ (G G )=G andu ¼ (G G )=G .Thus,withthehelpofEquation7.52we 0 1 1 1 1 1 2 1 obtain the net adsorption flux of surfactant: Q  r r ¼ K c (G G )=G K (G G )=G (7:53) 1 1,ads 1,des 1,ads 1s 1 1 1 1,des 1 2 1 Next, let us consider the reaction of counterion binding:  þ A S þM ¼ A SM (7:54) 0 0 The rates of the direct and reverse reactions are, respectively, r ¼ K u c , r ¼ K u (7:55) 2,ads 2,ads 1 2s 2,des 2,des 2 where K and K are the respective rate constants 2,ads 2,des c is the subsurface concentration of counterions 2s Having in mind that u ¼(G G )=G and u ¼G =G , with the help of Equation 7.55 we deduce an expression for the 1 1 2 1 2 2 1 adsorption flux of counterions: Q  r r ¼ K c (G G )=G K G =G (7:56) 2 2,ads 2,des 2,ads 2s 1 2 1 2,des 2 1 If we can assume that the reaction of counterion binding is much faster than the surfactant adsorption, then we can set Q  0, 2 and Equation 7.56 reduces to the Stern isotherm (Equation 7.47) with K  K =K . Next, a substitution of G from 2 2,ads 2,des 2 Equation 7.47 into Equation 7.53 yields 35 1 Q  r r ¼ K c (G G )=G K (1þK c ) G =G (7:57) 1 1,ads 1,des 1,ads 1s 1 1 1 1,des 2 2s 1 1 Equation 7.57 shows that the adsorption flux of surfactant is influenced by the subsurface concentration of counterions, c . 2s At last, if there is equilibrium between surface and subsurface, we have to set Q  0 in Equation 7.57, and thus obtain the 1 Langmuir isotherm for an ionic surfactant: Kc ¼G =(G G ), with K  (K =K )(1þK c)(7:58) 1s 1 1 1 1,ads 1,des 2 2s Note that K  K =K . This result demonstrates that the linear dependence of K on c (Equation 7.48) can be deduced 1 1,ads 1,des 2s from the reactions of surfactant adsorption and counterion binding (Equations 7.51 and 7.54). (For I0.1 M we have g 1  and then activities and concentrations of the ionic species coincide.) 7.2.1.2.5 Comparison of Theory and Experiment As illustration, we consider the interpretation of experimental isotherms by Tajima et al. 38,40,41 for the surface tension s versus SDS concentrations at 11 fixed concentrations of NaCl (see Figure 7.2). Processing the set of data for the interfacial  þ tensions¼s(c , c ) as a function of the bulk concentrations of surfactant (DS ) ions and Na counterions, c and c , 11 21 11 21 we can determine the surfactant adsorption, G (c , c ), the counterion adsorption, G (c , c ), the surface potential, 1 11 21 2 11 21 c (c , c ), and the Gibbs elasticity E (c , c ) for every desirable surfactant and salt concentrations. s 11 21 G 11 21 Thetheoreticaldependences¼s(c ,c )isdeterminedbythefollowingfullsetofequations:Equation7.44fori¼1,2; 11 21 the Gouy equation (Equation 7.36), Equation 7.39 (with s expressed by Equation 7.41 and J from Table 7.2), the Stern d © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 211 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 211 80 NaCl concentration 70 0 mM 0.5 mM 60 0.8 mM 1 mM 50 2.5 mM 4 mM 5 mM 40 8 mM 10 mM 30 20 mM 115 mM 20 0.1 1 10 SDS concentration (mM) FIGURE 7.2 Plot of the surface tension s versus the concentration of SDS, c , for 11 fixed NaCl concentrations. The symbols are 11 experimentaldatabyTajimaetal.38,40,41.Thelinesrepresentthebestfit42withthefullsetofequationsspecifiedinthetext,involving the van der Waals isotherms of adsorption and surface tension (Table 7.2). isotherm 7.47, and one surfactant adsorption isotherm from Table 7.2, say the van der Waals one. Thus we get a set of six equations for determining six unknown variables: s, F , a , a , G , and G . (For I0.1 M the activities of the ions can be s 1s 2s 1 2 replaced by the respective concentrations.) The principles of the numerical procedure are described in Ref. 27. Thetheoretical model contains fourparameters,b,G ,K ,andK ,whosevalues aretobeobtained from thebestfitofthe 1 1 2 experimentaldata.Notethatall11curvesinFigure7.2arefittedsimultaneously42.Inotherwords,theparametersb,G ,K , 1 1 and K are the same for all curves. The value of G , obtained from the best fit of the data in Figure 7.2, corresponds to 2 1 2 3 1 1=G ¼29.8 Å . The respective value of K is 99.2 m mol , which in view of Equation 7.49 gives a standard free energy of 1 1 (0)  1 4 3 surfactant adsorptionDm ¼ 12:53 kT perDS ion,that is30.6 kJ mol .Thedetermined value ofK is6.510 m =mol, 2 1 (0) þ whichaftersubstitutioninEquation7.49yieldsastandardfreeenergyofcounterionbindingDm ¼ 1:64 kT perNa ion(i.e., 2 1 4.1 kJ mol ). The value of the parameter b is positive, 2bG =kT¼þ2.73, which indicates the attraction between the 1 hydrocarbon tails of the adsorbed surfactant molecules. However, this attraction is too weak to cause two-dimensional phase transition. The van der Waals isotherm predicts such transition for 2bG =kT6.75. 1 Figure 7.3 shows calculated curves for the adsorptions of surfactant, G (the full lines), and counterions, G (the dotted 1 2 lines),versustheSDSconcentration,c .TheselinesrepresentthevariationofG andG alongtheexperimentalcurves,which 11 1 2 correspondtothelowestandhighestNaClconcentrationsinFigure7.2(viz.c ¼0and115mM).WeseethatbothG andG 31 1 2 1.0 – DS adsorption + 0.8 Na adsorption 115 mM NaCI 0.6 No salt 0.4 0.2 0.0 0.01 0.1 1 10 SDS concentration (mM)  þ FIGURE7.3 PlotsofthedimensionlessadsorptionsofsurfactantionsG =G (DS ,solidlines),andcounterionsG =G (Na ,dottedlines), 1 1 2 1 versus the surfactant (SDS) concentration, c . The lines are calculated 42 for NaCl concentrations 0 and 115 mM using parameter values 11 determined from the best fit of experimental data (Figure 7.2). © 2009 by Taylor & Francis Group, LLC Surface tension, s(mN/m) Dimensionless adsorptionBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 212 14.10.2008 10:28am Compositor Name: DeShanthi 212 Handbook of Surface and Colloid Chemistry 6 are markedly greater when NaCl ispresent in the solution.The highest values ofG for the curves in Figure 7.3 are 4.210 1 6 2 and 4.010 mol m for the solutions with and without NaCl, respectively. The latter two values compare well with the saturation adsorptions measured by Tajima et al. 40,41 for the same system by means of the radiotracer method, 6 2 6 2 viz. G ¼4.310 mol m and 3.210 mol m for the solutions with and without NaCl. 1 Forthe solution without NaCl the occupancyof theStern layer,G =G rises from 0.15 to0.73 and then exhibits atendency 2 1 to level off. The latter value is consonant with the data of other authors 43–45, who have obtained values of G =G up to 2 1 0.700.90 for various ionic surfactants; pronounced evidences for counterion binding have also been obtained in experiments with solutions containing surfactant micelles 46–50. As it could be expected, bothG andG are higher for the solution with 1 2 NaCl. These results imply that the counterion adsorption (binding) should be always taken into account. The fit of the data in Figure 7.2 gives also the values of the surface electric potential, c . For the solutions with 115 mM s NaClthemodelpredictssurfacepotentialsvaryingintherangejcj¼5595mVwithintheexperimentalintervalofsurfactant s concentrations,whereasforthesolutionwithoutsaltthecalculatedsurfacepotentialishigher:jc j¼150–180mV(forSDSc s s has a negative sign). Thus it turns out that measurements of surface tension, interpreted by means of an appropriate theoretical model, provide a method for determining the surface potential c in a broad range of surfactant and salt concentrations. The s describedapproachcouldbealsoappliedtosolvetheinverseproblem,viz.toprocessdataforthesurfacepotential.Inthisway, the adsorption of surfactant on solid particles can be determined from the measured zeta-potential 51. It is remarkable that the minimal (excluded) area per adsorbed surfactant molecule, a 1=G , obtained from the best fit 1 of surface-tension data by the van der Waals isotherm practically coincides with the value of a estimated by molecular- size considerations (i.e., from the maximal cross-sectional area of an amphiphilic molecule in a dense adsorption layer) (see Figure 7.1inRef. 34).This isillustrated inTable7.5,which containsdata foralkanols,alkanoicacids,SDS,sodiumdodecyl benzene sulfonate (DDBS), cocamidopropyl betaine (CAPB), and C -trimethyl ammonium bromides (n¼12, 14, and 16). n The second column of Table 7.5 gives the group whose cross-sectional area is used to calculate a. For molecules of circular 2 cross section, we can calculate the cross-sectional area from the expression a¼pr , where r is the respective radius. 2 2 2 For example 52, the radius of the SO ion is r¼3.09 Å, which yields a¼pr ¼30.0 Å . In the fits of surface-tension 4 2 data by the van der Waals isotherm, a was treated as an adjustable parameter, and the value a¼30 Å was obtained from the bestfit. As seen in Table 7.5, excellent agreement between the values ofa obtained from molecular size and from surface- tension fits is obtained also for many other amphiphilic molecules 52–59. It should be noted the above result holds only for the van der Waals (or Volmer) isotherm. Instead, if the Frumkin (orLangmuir)isotherm isused,thevalueofaobtained fromthesurface-tensionfitsiswithabout33%greaterthanaobtained from molecular size 42.Apossibleexplanation ofthis differencecouldbethefactthat the Frumkin (andLangmuir) isotherm is statistically derived for localized adsorption and are more appropriate do describe adsorption at solid interfaces. In contrast, the van der Waals (and Volmer) isotherm is derived for nonlocalized adsorption, and they provide a more adequate theoretical description of the surfactant adsorption at liquid–fluid interfaces. This conclusion refers also to the calculation of surface (Gibbs) elasticity by means of the two types of isotherms 42. The fact that a determined from molecular size coincides with a obtained from surface-tension fits (Table 7.5) is very useful for applications. Thus, when fitting experimental data, we can use the value of a from molecular size, and thus to decrease the number of adjustable parameters. This fact is especially helpful when interpreting theoretical data for the surface tension of surfactant mixtures, such as SDSþdodecanol 52, SDSþCAPB 57, and fluorinatedþnonionic surfactant 59. (0) An additional way to decrease the number of adjustable parameters is to employ the Traube rule, which states that Dm 1 increases with 1.025 kT when a CH group is added to the paraffin chain (for details see Refs. 52,53,58. 2 TABLE 7.5 Excluded Area per Molecule,a, Determined in Two Different Ways a from Molecular a from Surface-Tension 2 a 2 Amphiphile Group Size (Å ) Fits (Å ) References Alkanols Paraffin chain 21.0 20.9 52  Alkanoic acids COO 22–24 22.6 53,54 2 SDS SO 30.0 30 42,55 4 DDBS Benzene ring 35.3 35.6 56 þ CAPB CH–N –CH 27.8 27.8 57 3 3 þ C TAB (n¼12, 14, 16) N(CH ) 37.8 36.5–39.5 55,58 n 3 4 a Fit by means of the van der Waals isotherm. © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 213 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 213 7.2.2 DYNAMIC SURFACE TENSION If the surface of an equilibrium surfactant solution is disturbed (expanded, compressed, renewed, etc.), the system will try to restore the equilibrium by exchange of surfactant between the surface and the subsurface layer (adsorption–desorption). The change of the surfactant concentration in the subsurface layer triggers a diffusion flux in the solution. In other words, the process of equilibration (relaxation) of an expanded adsorption monolayer involves two consecutive stages: 1. Diffusion of surfactant molecules from the bulk solution to the subsurface layer 2. Transfer of surfactant molecules from the subsurface to the adsorption layer; the rate of transfer is determined by the height of the kinetic barrier to adsorption (Inthecaseofdesorptiontheprocesseshavetheoppositedirection.)Suchinterfacialexpansionsaretypicalforfoamgeneration and emulsification. The rate of adsorption relaxation determines whether the formed bubbles=drops will coalesce upon collision, and in final reckoning—how large will be the foam volume and the emulsion drop-size 60,61. Below we focus our attention on the relaxation time of surface tension, t , which characterizes the interfacial dynamics. s The overall rate of surfactant adsorption is controlled by the slowest stage. If it is stage (i), we deal with diffusion control, while if stage (ii) is slower, the adsorption occurs under barrier (kinetic) control. Sections 7.2.2.1 through 7.2.2.4 are dedicated to processes under diffusion control (which are the most frequently observed), whereas in Section 7.2.2.5 we consider adsorption under barrier control. Finally, Section 7.2.2.6 is devoted to the dynamics of adsorption from micellar surfactant solutions. Variousexperimentalmethodsfordynamicsurface-tensionmeasurementsareavailable.Theiroperationaltimescalescover different time intervals 62,63. Methods with a shorter characteristic operational time are the oscillating jet method 64–66, the oscillating bubble method 67–70, the fast-formed drop technique 71,72, the surface wave techniques 73–76, and the maximum bubble pressure method (MBPM) 77–82. Methods of longer characteristic operational time are the inclined plate method 83, the drop-weight=volume techniques 84–88, the funnel 89 and overflowing cylinder 58,90 methods, and the axisymmetric drop shape analysis 91,92 (see Refs. 62,63,93 for a more detailed review). Inthissection,devotedtodynamicsurfacetension,weconsidermostlynonionicsurfactantsolutions.InSection7.2.2.4,we address the more complicated case of ionic surfactants. We will restrict our considerations to the simplest case of relaxation of an initial uniform interfacial dilatation. The more complex case of simultaneous adsorption and dilatation is considered elsewhere 62,78,82,90,93. 7.2.2.1 Adsorption under Diffusion Control Here we consider a solution of a nonionic surfactant, whose concentration, c ¼c (z,t), depends on the position and time 1 1 because of the diffusion process. (As before, z denotes the distance to the interface, which is situated in the plane z¼0.) Correspondingly, the surface tension, surfactant adsorption, and the subsurface concentration of surfactant vary with time: s¼s(t), G ¼G (t), c ¼ c (t). The surfactant concentration obeys the equation of diffusion: 1 1 1s 1s 2 c c 1 1 ¼ D (z 0, t 0)(7:59) 1 2 t z where D is the diffusion coefficient of the surfactant molecules. The exchange of surfactant between the solution and its 1 interface is described by the boundary conditions dG c 1 1 c (0,t)¼ c (t), ¼ D , (z¼ 0, t 0)(7:60) 1 1s 1 dt z The latter equation states that the rate of increase of the adsorptionG is equal to the diffusioninflux of surfactant per unit area 1 of the interface. Integrating Equation 7.59, along with 7.60, we can derive the equation of Ward and Tordai 94: 2 3 rffiffiffiffiffiffi t ð pffiffi D c (t) 1 1s 4 5 G (t)¼G (0)þ 2c t pffiffiffiffiffiffiffiffiffiffidt (7:61) 1 1 11 p tt 0 SolvingEquation7.61togetherwithsomeoftheadsorptionisothermsG ¼G (c )inTable7.2,wecaninprincipledeterminethe 1 1 1s twounknownfunctionsG (t)andc (t).BecausetherelationG (c )isnonlinear(exceptfortheHenryisotherm),thisproblem, 1 1s 1 1s oritsequivalentformulations,canbesolvedeithernumerically95,orbyemployingappropriateapproximations 78,96. © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 214 14.10.2008 10:28am Compositor Name: DeShanthi 214 Handbook of Surface and Colloid Chemistry In many cases it is convenient to use asymptotic expressions for the functionsG (t), c (t) ands(t) for short times (t 0) 1 1s and long times (t1). A general asymptotic expression for the short times can be derived from Equation 7.61 substituting c c (0)¼constant: 1s 1s pffiffiffiffiffiffiffiffiffiffiffiffi pffiffi G (t)¼G (0)þ2 D =p c c (0) t (t 0)(7:62) 1 1 1 11 1s Analogousasymptoticexpressioncanbeobtainedalsoforthelongtimes,althoughthederivationisnotsosimple.Hansen97 derived a useful asymptotics for the subsurface concentration: G G(0) 1e c (t)¼ c  pffiffiffiffiffiffiffiffiffiffiffi (t1)(7:63) 1s 11 pD t 1 where G is the equilibrium value of the surfactant adsorption. The validity of Hansen’s Equation 7.63 was confirmed in 1e subsequent studies by other authors 98,99. Below we continue our review of the asymptotic expressions considering separately the cases of small and large initial perturbations. 7.2.2.2 Small Initial Perturbation When the deviation from equilibrium is small, then the adsorption isotherm can be linearized:  G 1 G (t)G  c (t)c(7:64) 1 1,e 1s e c 1 e Hereafter the subscript e means that the respective quantity refers to the equilibrium state. The set of linear Equations 7.59, 7.60, and 7.64, has been solved by Sutherland 100. The result, which describes the relaxation of a small initial interfacial dilatation, reads ffiffiffiffiffi   r s(t)s G (t)G t t e 1 1,e ¼ ¼ exp erfc (7:65) s(0)s G (0)G t t e 1 1,e s s where  2 1 G 1 t  (7:66) s D c 1 1 e is the characteristic relaxation time of surface tension and adsorption, and 1 ð 2 2 erfc(x)pffiffiffiffi exp(x )dx (7:67) p x istheso-calledcomplementaryerrorfunction101,102.Theasymptoticsofthelatterfunctionforsmallandlargevaluesofthe argument are 101,102:   2 x 2 e 1 3 erfc(x)¼ 1pffiffiffiffixþO(x ) for x 1; erfc(x)¼pffiffiffiffi 1þO for x 1 (7:68) 2 p px x Combining Equations 7.65 and 7.68, we obtain the short-time and long-time asymptotics of the surface-tension relaxation: " rffiffiffiffiffi  3=2 s(t)s G (t)G 2 t t e 1 1,e ¼ ¼ 1pffiffiffiffi þO (tt)(7:69) s s(0)s G (0)G p t t e 1 1,e s s rffiffiffiffiffi  3=2 s(t)s G (t)G t t e 1 1,e s s ¼ ¼ þO (tt)(7:70) s s(0)s G (0)G pt t e 1 1,e © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 215 14.10.2008 10:28am Compositor Name: DeShanthi Chemical Physics of Colloid Systems and Interfaces 215 Equation7.70isoftenusedasatesttoverifywhethertheadsorptionprocessisunderdiffusioncontrol:datafors(t)areplotted pffiffi versus1= t anditischeckediftheplotcomplieswithastraightline;moreover,theinterceptofthelinegivess .Werecallthat e Equations 7.69 and 7.70 are valid in the case of a small initial perturbation; alternative asymptotic expressions for the case of large initial perturbation are considered in the next Section 7.2.2.3. With the help of the thermodynamic Equations 7.2 and 7.6, we derive 2 G G s G kT 1 1 1 ¼ ¼ (7:71) c s c c E 1 1 1 G Thus Equation 7.66 can be expressed in an alternative form 35:  2 2 1 G kT 1 t ¼ (7:72) s D c E 1 1 G e Substituting E from Table 7.3 into Equation 7.72, we can obtain expressions for t corresponding to various adsorption G s isotherms. In the special case of Langmuir adsorption isotherm, we can present Equation 7.72 in the form 35 2 2 1 (KG ) 1 (KG ) 1 1 t ¼ ¼ (for Langmuir isotherm) (7:73) s 4 4 D D (1þKc ) (1þE =(G kT)) 1 1 1 G 1 Equation7.73 visualizes the very strong dependence of the relaxation timet onthe surfactant concentration c ; in general,t s 1 s canvarywithmanyordersofmagnitudeasafunctionofc .Equation7.73showsalsothathighGibbselasticitycorrespondsto 1 short relaxation time, and vice versa. 3 1 2 6 2 1 Asaquantitativeexampleletustaketypicalparametervalues:K ¼15m mol ,1=G ¼40Å ,D ¼5.510 cm s , 1 1 1 6 and T¼298 K. Then with c ¼ 6.510 M, from Table 7.3 (Langmuir isotherm) and Equation 7.73 we calculate 1 1 4 4 E 1.0 mN m and t 5 s. In the same way, for c ¼6.510 M we calculate E 100 mN=m and t 510 s. G s 1 G s To directly measure the Gibbs elasticity E , or to precisely investigate the dynamics of surface tension, we need an G experimental method, whose characteristic time is smaller compared to t . Equation 7.73 and the latter numerical example s show that when the surfactant concentration is higher, the experimental method should be faster. 7.2.2.3 Large Initial Perturbation By definition, we have large initial perturbation when at the initial moment the interface is clean of surfactant: G (0)¼ 0, c (0)¼ 0 (7:74) 1 1s In such case, the Hansen Equation 7.63 reduces to G 1,e pffiffiffiffiffiffiffiffiffiffiffi c (t)¼ c  (t1)(7:75) 1s 11 pD t 1 By substituting c (t) for c in the Gibbs adsorption Equation 7.2, and integrating, we obtain the long-time asymptotics of the 1s 1 surface tension of a nonionic surfactant solution after a large initial perturbation:  1=2 2 G kT 1 1 s(t)s ¼ (large initial perturbation) (7:76) e c pD t 1 1 e with the help of Equation 7.72, we can bring Equation 7.76 into another form:  1=2 t s s(t)s ¼ E (large initial perturbation) (7:77) e G pt where E is given in Table 7.3. It is interesting to note that Equation 7.77 is applicable to both nonionic and ionic surfactants G with the only difference that for nonionics t is given by Equation 7.66, whereas for ionic surfactants the expression for t is s s somewhat longer 35,103. © 2009 by Taylor & Francis Group, LLCBirdi/Handbook of Surface and Colloid Chemistry 7327_C007 Final Proof page 216 14.10.2008 10:28am Compositor Name: DeShanthi 216 Handbook of Surface and Colloid Chemistry Equations 7.70 and 7.77 show that in the case of adsorption under diffusion control the long-lime asymptotics can be expressed in the form 1=2 s¼s þSt (7:78) e 1=2 In view of Equations 7.70 and 7.77, the slope S of the dependence s versus t is given by the expressions 103  1=2 t s S ¼ s(0)s (small perturbation) (7:79) s e p  1=2 t s S ¼ E (large perturbation) (7:80) l G p Asknown,thesurfactantadsorptionG monotonicallyincreaseswiththeriseofthesurfactantconcentration,c .Incontrast,the 1 1 slopeS isanonmonotonicfunctionofc :S exhibitsamaximumatacertainconcentration.Todemonstratethatwewillusethe l 1 l expression 2 G kT l,e S ¼ pffiffiffiffiffiffiffiffiffi (7:81) l c pD 1 1 which follows from Equations 7.76 and 7.78. In Equation 7.81, we substitute the expressions for c stemming from the 1 Langmuir and Volmer adsorption isotherms (Table 7.2 with c ¼a ); the result reads 1 1s e S ¼u(1u) (for Langmuir isotherm) (7:82) l  u e S ¼u(1u)exp  (for Volmer isotherm) (7:83) l 1u e where u and S are the dimensionless adsorption and slope coefficient: l pffiffiffiffiffiffiffiffiffi G S pD 1,e l 1 e u¼ and S ¼ (7:84) l 2 G 1 kT KG 1 e Figure 7.4 compares the dependencies S (u) given by Equations 7.82 and 7.83: we see that the former is symmetric and has a l maximum at u¼0.5, whereas the latter is asymmetric with a maximum at u0.29. We recall that the Langmuir and Volmer isothermscorrespondtolocalizedandnonlocalizedadsorption,respectively(seeSection7.2.1.1.2).ThenFigure7.4showsthat e the symmetry=asymmetry of the plot S versus u provides a test for verifying whether the adsorption is localized or l nonlocalized. (The practice shows that the fits of equilibrium surface-tension isotherms do not provide such a test: theoretical isotherms corresponding to localized and nonlocalized adsorption are found to fit equally well surface-tension data) 0.25 Langmuir 0.20 0.15 0.10 Volmer 0.05 0.5 0.29 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless adsorption, Γ /Γ 1e ∞ e FIGURE 7.4 Plot of the dimensionless slope, S, versus the dimensionless equilibrium surfactant adsorption, u¼G =G , in accordance l 1e 1 with Equations 7.82 and 7.83, corresponding to the cases of localized and nonlocalized adsorption. © 2009 by Taylor & Francis Group, LLC Dimensionless slope, S l

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.