Surface Interactions in Aqueous and Nonaqueous Solutions

Thesis

Reference

Surface Interactions in Aqueous and Nonaqueous Solutions

STOJIMIROVIC, Biljana

Abstract

The main focus of this thesis is interaction forces between colloidal particles in liquids of different polarities. Atomic force microscopy (AFM) colloidal probe was the main technique used for measurements, besides dynamic light scattering (DLS) and electrophoresis in the case of nonpolar solvents. Since the surfaces are often electrostatically stabilized, to obtain a stable dispersion it is important to know how the surfaces are charged, or what the characteristics of the solvent are, in order to predict the amount of charges present in the system. The thesis addresses interactions between charged silica particles in the presence of multivalent coions and experimental evidence of algebraically decaying double layer interactions, both in aqueous solutions. The aggregation behavior of three kinds of particles in nonpolar solvent (decane) in the presence of surfactant dioctyl sodium sulfosuccinate (AOT) is investigated. Finally, interactions in presence of 1:1 electrolytes are measured in solvent with intermediate polarity (isopropanol).

STOJIMIROVIC, Biljana. Surface Interactions in Aqueous and Nonaqueous Solutions . Thèse de doctorat : Univ. Genève, 2020, no. Sc. 5453

DOI : 10.13097/archive-ouverte/unige:138144 URN : urn:nbn:ch:unige-1381442

Available at:

http://archive-ouverte.unige.ch/unige:138144

Disclaimer: layout of this document may differ from the published version.

1 / 1

Département de chimie minérale et analytique

Dr Gregor Trefalt

Surface Interactions in Aqueous and Nonaqueous Solutions

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès science, mention chimie

par

Biljana STOJIMIROVIĆ née UZELAC de

Banatsko Karadjordjevo (Serbie)

Thèse N° 5453

GENÈVE Atelier ReproMail

2020

FACULTE DES SCIENCES

DOCTORAT ES SCIENCES, TUENTION CHITMIE

Thdse de Madame Biljana STOJ|M|ROVIC

intitulee

<<Surface lnteractions _in Aqueous and Nonaqueous Solutions)

La Faculte des sciences, sur le pr6avis de fi/onsieur [/. BORKOVEC, professeur ordinaire

et directeur de thdse (Departement de chimie min6rale et analytique), Monsieur G. TREFALT, docteur et codirecteur de thdse (Departement de chimie min6rale

et analytique), Monsieur R. D. MILTON, professeur assistant (D6partement de chimie min6rale et analytique) et [Monsieur R. TUINIER, professeur (Chemical Engineering and Chemistry, Eindhoven University of Technology, Eindhoven, The Netherlands), autorise l'impression de la pr6sente thdse, sans exprimer d,opinion sur les propositions qui y _sont 6nonc6es.

Gendve,

le

17 juin

2020

Thdse ^- 5453

-

Le Doyen

La these doit porter la d6claration pr6c6dente et remplir les conditions enum6r6es dans les ,'lnformations relatives aux thdses de doctorat d l'Universit6 de Gendve,'.

N.B. -

Table of Contents

Abstract 1

Résumé 3

List of publications 5

1. Introduction 7

2. Surface forces 9

2.1 DLVO theory 9

2.1.1 Van der Waals Forces 9

Dipole-dipole interaction (Keesom) 10

Dipole-induced dipole interaction (Debye) 10

Dispersion interaction (London) 11

Van der Waals force between Macroscopic bodies 12

Derjaguin approximation 14

Lifshitz theory 16

Combining relations 17

2.1.2 Double layer forces 17

Diffuse layer overlapping 22

2.2 Non DLVO forces 23

3. Materials 29

3.1 Colloidal silica 29

Charging of the surface 29

3.2 Solvents 31

Charge formation 32

4. Methods 37

4.1 AFM colloidal probe 37

4.2 DLS 42

4.3 Electrophresis 45

5. Interactions between silica particles in the presence of multivalent coions 51 6. Experimental evidence for algebraic double layer forces 61

7. Surfactant mediated particles in nonpolar solvents 73

8. Forces between silica particles in isopropanol solutions of 1:1 electrolytes 85

9. Conclusion 95

1

Abstract

The main focus of this thesis is interaction forces between colloidal particles in liquids of different polarities. Atomic force microscopy (AFM) colloidal probe was the main technique used for measurements, besides dynamic light scattering (DLS) and electrophoresis in the case of nonpolar solvents.

The first two chapters contain theoretical background of the field. Theory of surface forces expected and measured and colloidal behaviour are described, in particular DLVO (Derjaguin, Landau, Verwey, and Overbeek) and non-DLVO forces. Chapter three deals with materials used in the experiments, particularly silica particles, and main characteristics of solvents. Attention is given to the formation of charge on the surface and in the solvent.

Chapter four explains the instruments and principles of measurements for AFM, DLS and electrophoresis.

Interactions between charged silica particles in solutions in the presence of multivalent ions with the same charge as particle surface (coions) are measured and analysed in the fifth chapter. The double layer behaviour with the change of concentration of 1:z potassium salts (where z as the valence of the ion goes from 1-4) is studied. All the forces are registered with the help of AFM. The force profiles can be nicely described with DLVO theory. Several parameters are extracted from the force curves, such as diffuse layer potentials and charge regulation parameters. Double layer interactions exhibit a sigmoidal shape. They are exponential only at large separation distances, and algebraic at small distances. This results from the expulsion of the multivalent coions from the space between the two surfaces approaching each other. With the increasing coion valence, a plot of diffuse layer potential as a function of concentration of salt shows that the potential shifts to lower concentrations.

The same happens with the regulation parameter. In case when these quantities are plotted as a function of the ions charged oppositely than the particles (counterions), the same profiles now collapse to a single curve.

Chapter six presents experimental evidence of algebraically decaying double layer interactions. Again, AFM is used for measurement of the forces between colloidal silica particles in aqueous solutions of different concentrations of monovalent salt (KCl) on various pH values. Normally, exponential decay with separation distance is expected for these forces. This behaviour is reported both in the counterion-only regime, as for the strongly overlapping double layers. For small separation distances, the disjoining pressure decays inversely with the separation distance. At larger separation distances, the pressure profile follows inverse square distance dependence.

Chapter seven changes focus to nonpolar solvents. The aggregation behaviour of particles in

decane in the presence of surfactant dioctyl sodium sulfosuccinate (with the commercial

2

name Aerosol-OT, or AOT) is investigated using different methods. Three different types of particles are used: silica and sulphate latex particles that are negatively charged, and amidine latex which are positively charged. Aggregation kinetics was studied by means of DLS, by measuring the particle suspension stability ratios. To obtain Hamaker constant and determine van der Waals attraction of the system, the AFM colloidal probe technique was employed. Charging of the particles is evaluated with the help of electrophoretic mobility measurements. Particles are weakly charged at low surfactant concentrations, therefore making the suspensions unstable, as in the pure solvent. With increasing AOT concentration, all three types of particles become more charged, and at very high surfactant concentrations, they become neutralized and aggregate fast. DLVO theory can explain the suspension behaviour.

Alcohols, as the solvent with intermediate polarity are studied in the chapter eight. Forces between micron-sized silica particles were measured with AFM colloidal probe method. The influence of 1:1 electrolytes on forces between surfaces in isopropanol solutions is investigated, and a quite rich behaviour is observed. When the salt concentrations are low, the double layer repulsive forces are dominant, and at high concentrations, van der Waals forces take over. The decay of double layer forces is much larger than predicted from the Debye length calculations from the nominal salt concentrations. The cause of this effect is the ion-pairing in the system examined. Besides two mentioned types of forces, there are other forces present in these isopropanol solutions, such as attractive non-DLVO forces and repulsive solvation. Below 2 nm of surface separation, layering and oscillatory forces are observed.

Chapter nine concludes and summarizes all of the main topics from the thesis.

3

Résumé

Cette thèse porte principalement sur les forces d'interaction entre les particules colloïdales dans les liquides de différentes polarités. La sonde colloïdale de la microscopie à force atomique (AFM) a été la principale technique utilisée pour les mesures, outre la diffusion dynamique de la lumière (DLS) et l'électrophorèse dans le cas de solvants non polaires.

Les deux premiers chapitres contiennent des informations théoriques sur ce domaine. La théorie des forces de surface attendues et mesurées et le comportement colloïdal sont décrits, en particulier les forces DLVO (Derjaguin, Landau, Verwey et Overbeek) et les forces non DLVO. Le chapitre trois traite des matériaux utilisés dans les expériences, en particulier les particules de silice, et des principales caractéristiques des solvants. L'attention est portée sur la formation de la charge à la surface et dans le solvant. Le chapitre quatre explique les instruments et les principes de mesure pour l'AFM, le DLS et l'électrophorèse.

Les interactions entre les particules de silice chargées dans les solutions en présence d'ions multivalents ayant la même charge que la surface des particules (bobines) sont mesurées et analysées dans le cinquième chapitre. Le comportement de la double couche avec le changement de concentration des sels de potassium 1:z (où z comme la valence de l'ion passe de 1 à 4) est étudié. Toutes les forces sont enregistrées avec l'aide de l'AFM. Les profils de force peuvent être décrits avec précision grâce à la théorie DLVO. Plusieurs paramètres sont extraits des courbes de force, tels que le potentiel de la couche diffuse et le paramètre de régulation de la charge. Les interactions de la double couche présentent une forme sigmoïdale. Elles ne sont exponentielles qu'à de grandes distances de séparation, et algébriques à de petites distances. Cela résulte de l'expulsion des bobines multivalentes de l'espace entre les deux surfaces qui se rapprochent l'une de l'autre. Avec l'augmentation de la valence des spires, un graphique du potentiel de la couche diffuse en fonction de la concentration de sel montre les changements de potentiel vers des concentrations plus faibles. La même chose se produit avec le paramètre de régulation. Lorsque ces quantités sont tracées en fonction des ions chargés de manière opposée aux particules (contre-ions), les mêmes profils s'effondrent en une seule courbe.

Le chapitre six présente des preuves expérimentales d'interactions à double couche en

décroissance algébrique. Là encore, l'AFM est utilisée pour mesurer les forces entre les

particules de silice colloïdale dans des solutions aqueuses de différentes concentrations de

sel monovalent (KCl) à des valeurs de pH variées. Normalement, on s'attend à une

décroissance exponentielle avec la distance de séparation pour ces forces. Ce

comportement est signalé aussi bien en régime de contre-ion seulement, que pour les

forces qui se chevauchent fortement. Pour les petites distances de séparation, la pression

de disjonction diminue de manière inversement proportionnelle à la distance de séparation.

4

Pour des distances de séparation plus importantes, le profil de pression suit la dépendance de la distance au carré inverse.

Le chapitre sept se concentre sur les solvants non polaires. Le comportement d'agrégation des particules dans le décane en présence de l'agent tensioactif dioctyle sodium sulfosuccinate (avec le nom commercial Aerosol-OT, ou AOT) est étudié à l'aide de différentes méthodes. Trois types différents de particules sont utilisés : les particules de latex de silice et de sulfate qui sont chargées négativement, et le latex d'amidine qui est chargé positivement. La cinétique d'agrégation a été étudiée au moyen de la diffusion dynamique de la lumière, en mesurant les rapports de stabilité des suspensions de particules. Pour obtenir la constante de Hamaker et déterminer l'attraction de van der Waals du système, la technique de la sonde colloïdale AFM a été utilisée. La charge des particules est évaluée à l'aide de mesures de mobilité électrophorétique. Les particules sont faiblement chargées à de faibles concentrations de tensioactifs, ce qui rend les suspensions instables, comme dans le solvant pur. Avec l'augmentation de la concentration en AOT, les trois types de particules deviennent plus chargés, et à des concentrations très élevées de tensioactifs, elles sont neutralisées et s'agrégent rapidement. La théorie DLVO peut expliquer le comportement des suspensions.

Les alcools, en tant que solvant de polarité intermédiaire, sont étudiés dans le chapitre huit.

Les forces entre les particules de silice de taille micronique ont été mesurées avec la méthode de la sonde colloïdale AFM. L'influence des électrolytes 1:1 sur les forces entre les surfaces dans les solutions d'isopropanol est étudiée, et un comportement assez riche est observé. Lorsque les concentrations de sel sont faibles, les forces de répulsion de la double couche sont dominantes, et à des concentrations élevées, les forces de van der Waals prennent le dessus. La décroissance des forces de la double couche est beaucoup plus importante que ce qui avait été prévu dans les calculs de longueur de Debye à partir des concentrations nominales de sel. La cause de cet effet est l'appariement des ions dans le système examiné. Outre les deux types de forces mentionnés, d'autres forces sont présentes dans ces solutions d'isopropanol, telles que la force d'attraction non DLVO et la solvatation répulsive. En dessous de 2 nm de séparation de surface, des forces de stratification et oscillatoires sont observées.

Le chapitre neuf conclut et résume tous les principaux sujets de la thèse.

5

List of Publications

 B. Uzelac, V. Valmacco, G. Trefalt, Interactions between Silica Particles in the Presence of Multivalent Coions, Soft Matter, 2017, 13, 5741

 B. Stojimirović, M. Vis, R. Tuinier, A.P. Philipse, G. Trefalt, Experimental Evidence for Algebraic Double-Layer Forces, Langmuir, 2020, 36, 47−54

 M. Farrokhbin, B. Stojimirović, M. Galli, M.K. Aminian, Y. Hallez, G. Trefalt , Surfactant Mediated Particle Aggregation in Nonpolar Solvents, Phys.Chem.Chem.Phys., 2019, 21, 18866

 B. Stojimirović, M. Galli, G. Trefalt , Forces between Silica Particles in Isopropanol

Solutions of 1:1 Electrolytes, Physical Review Research, 2020, 2, 023315

6

7

1. Introduction

The term colloid describes a mixture composed of at least two parts, where one phase with size that ranges from 1 nm to 1 μm is suspended in another one.

^1-3

The lower size regime distinguishes them from molecular systems, whereas the upper size domain is determined by the fact that colloids are systems for which thermal motion (or thermal energy) dominates gravitational energy. The area which separates two phases from each other is called an interface, no matter their state of matter. Suspensions of particles are usually called colloidal suspensions. These particles are not soluble, and they are dispersed in another medium. However, the colloidal characteristics are sometimes present in systems with particles larger or smaller than those defined above. Therefore, the system is referred to as a colloidal if it “looks and acts like one”.

⁴

The surface to volume ratio in colloids is very big, so the properties and forces acting at the interfaces determine their behavior. The stability of dispersions is dependent on the balance between the attractive and repulsive forces. Even the slightest changes in the composition, such as addition of salt, can affect the phase behavior.

The properties of the bulk usually deviate from the ones on the nano or micro level.

Surfaces are often electrostatically stabilized. Therefore, to obtain a stable dispersion it is important to know how the surfaces are charged, or what the characteristics of the solvent are, in order to predict the amount of charges present in the system. One of the useful bulk properties in this case can be the dielectric permittivity of the liquid media.

Colloids are used a lot, both throughout the history and today. From stained chapel windows and paints, to mayonnaise or wastewater treatment, it is crucial to know the behavior of particles with variation of different parameters for all of these systems.

Therefore, main techniques involved in the investigation include measurement of forces

between constituents of such systems, or their aggregation rates. These techniques are

often developed based on explained phenomena that originated from colloids. The irregular

movement of pollen in water observed by Brown and blue eye color contributed to Tyndall

scattering are used in dynamic light scattering technique.

¹

Einstein later established a

relationship between Brownian motion and diffusion, and Perrin relied on it to determine

the precise value of Avogadro’s number. In general, even though colloids were used since

the ancient times, most of the theoretical work and understanding was initiated only

halfway the 20th century. A major step in gaining insights into the stability of charged

colloids was made through DLVO theory.

^5-7

8 References

1. Evans, D. F.; Wennerstrom, H., The Colloidal Domain. John Wiley: New York, 1999.

2. Butt, H.-J.; Graf, K.; Kappl, M., Physics and Chemistry of Interfaces. Wiley-VCH: 2003.

3. Dhont, J. K. G., An Introduction to Dynamics of Colloids. Elsevier: Amsterdam, 1996.

4. Kontogeorgis, G. M.; Kiil, S., Introduction to Applied Colloid and Surface Chemistry. Wiley:

Denmark, 2016.

5. Derjaguin, B., A theory of interaction of particles in presence of electric double-layers and the stability of lyophobe colloids and disperse systems. Acta Phys. Chim. 1939, 10 (3), 333-346.

6. Derjaguin, B.; Landau, L. D., Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes. Acta Phys. Chim.

1941, 14 (6), 633-662.

7. Verwey, E. J. W.; Overbeek, J. T. G., Theory of Stability of Lyophobic Colloids. Elsevier:

Amsterdam, 1948.

9

2. Surface Forces

2.1 DLVO theory

Derjaguin-Landau-Vervey-Overbeek (DLVO)

¹

theory describes the forces acting between two surfaces in liquid as a sum of two different contributions. One of those is the van der Waals interaction F

vdW

, and the other electrical double layer force F

dl

, resulting from overlapping of the electrical double layers:

dl vdW

F  F  F _{. (2.1)}

If we consider two identical particles, double layer force will always be repulsive, and the van der Waals part always attractive, which is illustrated in the Figure 2.1.

Figure 2.1 Schematic illustration of DLVO theory, separately shown force profiles for double layer force, van der Waals, and sum of both.

The theory itself is named after its developers: two Russian scientists, Boris Derjaguin

²

and Lev Landau

³

, and two from Netherlands Evert Verwey

⁴

and Theodoor Overbeek. It was established in the 1940s, and still widely used for describing interactions between colloidal particles in liquids.

2.1.1 Van der Waals Forces

These interactions are named after Dutch scientist Johannes Diderik van der Waals.

Three different types of interactions are included in the name van der Waals (vdW) forces:

dipole-dipole or Keesom, dipole-induced dipole or Debye, and dispersion, also known as

London interaction. The first two are in a sense related to the coulombic force, since they

10

have electrostatic background, but do not include charged species. The London dispersion force is quantum-mechanical in origin.

^5-7

Dipole-dipole interaction (Keesom)

If two molecules with dipole moments u

1

and u

2

are close to each other in a vacuum, with a distance h between them, as on the Figure 2.2, a dipole-dipole interaction is formed. The energy of those dipoles relation is:

).

cos sin sin cos

cos 2 4 (

) , , ,

(

_{3 1 2 1 2}

0 2 1 2

1

    

 



  

h u h u

U (2.2)

Here θ

1

and θ

2

are angles of dipoles at which they are oriented to the line joining them, φ is the angle which corresponds to rotation, and ε

0

is vacuum permittivity.

Figure 2.2 Dipole-dipole interaction.

When the separations between dipoles or dielectric constant of the medium are large, system thermal energy will make the dipoles rotate and move around. Therefore, the dipole-dipole interaction can be averaged over the angles, and the resulting potential will be:

6 2 0

2 1 2

1

, , ) 3 ( 4 )

,

( k Th

u h u

U



B





  . (2.3)

Here k

B

T is the thermal energy of the system, with T being the temperature, and k

B

the Boltzmann constant.

Dipole- induced dipole interaction (Debye)

When a polar molecule is in the proximity of a nonpolar polarizable one, the former’s dipole

electric field can induce a dipole in the latter. The field can be written as:

11



²



^1/2

3 0

1 3cos 4 E u

h





   , (2.4)

where u is the dipole moment of a polar molecule, and θ is the orientation angle between the two molecules. The energy of this interaction will be:

 

2 2

2 0

0 2 6

0

1 3cos ( , ) 1

2 2(4 )

U h E u

h

 

 



     , (2.5)

where α

0

is polarizability. When averaging over the orientation angles, cos

²

θ average will give 1/3, so the net energy is:

2 0

2 6 0

( ) (4 )

U h u

h



   , (2.6)

Dispersion interaction (London)

London force may be the most important of the three parts of the vdW force. It is universal, and not tied to the nature of the species involved. These forces are nonadditive, can be attractive or repulsive, and are relatively long ranged.

The rapid movement of the electrons in the outermost occupied valence shell of the atom/molecule results in fast fluctuating dipoles. The interaction energy for two dissimilar atoms/molecules is given by:

 

01 02 1 2

2 6

0 1 2

( ) 3

2 (4 )

h

P

U h h

   

  

   , (2.7)

In the equation (2.7), h

P

is Planck constant and ν

1

and ν

2

ionization frequencies.

For two identical atoms/molecules the expression (2.7) turns into:

2 0

2 6 0

( ) 3

4 (4 )

h

P

U h h

 

   , (2.8)

12

All three interactions involve inverse sixth power of the distance between the two interacting centers.

Total van der Waals potential is then given by:

6

. h

U

_{vd W}

  C

^{vd W}

(2.9)

where C

vdW

is van der Waals constant that includes all three above mentioned interactions:

orient

.

ind disp

vdW

C C C

C    (2.10)

The full equation for two different molecules is:

   

 4  ^.

3 2

3

2 6 0

2 2 2 1 01 2 2 02 2 1 2

1

2 1 02 01

6

h

T k

u u u

h u

h C C

U C

^B

P orient

ind disp

vdW





 









  

 



   

 

 

 

 (2.11)

And for two same polar molecules (2.11) takes the form:

 ⁴  ^.

2 3 4

3

2 6 0

4 0 2 1 2 0

6

h

T k u u

h

h C C

U C

^B

P orient

ind disp

vdW



 



 





 



  



 

 

 (2.12)

There are two approaches to calculate van der Waals forces between surfaces; microscopic and macroscopic. The first scientist to calculate the total interaction energy between colloidal particles was Hamaker.

⁸

He used a microscopic approach and summed up all of the interaction energies that act between all of the molecules of which the two bodies consist.

After him, Lifshitz

⁹

developed a macroscopic theory in which he uses the properties of the bodies for the calculation, precisely their electro-optical features.

Both of the theories as a result present the force as a product of an energetic term in the form of Hamaker constant and geometric part.

Van der Waals force between Macroscopic bodies – (Microscopic) Hamaker approach

Following the work of Boer

¹⁰

and Bradley

¹¹

, Hamaker developed a concept based on

pairwise additivity of the dipole interactions between two bodies. This assumption does not

include the effect of the surroundings. Therefore, any other atom or molecule in the

proximity is neglected.

13

Figure 2.3 shows the molecule A and an infinite plane which is made from molecules B. First we assume the pair potential U

AB

between them is

^{7, 12}

:

 

₆

. h h C

U

_AB

 

^AB

(2.13)

The potential energy of the total interaction between A and all the molecules in B shown in Figure 2.3 can be acquired by integrating the molecular density ρ over the whole body B volume:

Figure 2.3 Van der Waals force between macroscopic body B and a molecule A.

 

 ²

^{2 2}



³

^.

6 2

/

^{ }  ^{ }  _{ }

r x h

rdrdx C

h dV C

W

_{mol plane}



^B

 

(2.14)

Substituting 2rdr = d(r

²

), and solving the integral, the resulting expression is :

6

³

.

/

h

W

_{mol plane}

C 

^B



 (2.15)

If molecule A is replaced by another macroscopic body, then the resulting interaction energy between the A and B will be:

12 h

²

. W C 

^A



^B



 (2.16)

The van der Waals interaction in (2.16) is inversely proportional to second power of distance between the macroscopic bodies, whereas the one between molecules decays as h

^-6

.

Taking in consideration the Hamaker constant definition, as a coefficient that is providing

quantitative information of intermolecular forces:

14

B

C

A

H  

²

  , (2.17)

And by substituting (2.17) in (2.16), the resulting expression for energy per unit area for two plates is:

12 h

²

. W H

 

 (2.18)

Van der Waals energy between macroscopic solids with different geometries is also possible, although sometimes difficult. An important example for colloidal chemistry is the interaction between two spherical particles with radii R

1

and R

2

with the distance h between them is:

6

_{1 2}

.

2 1

R R

R R h U

_vdW

H

 

 (2.19)

The force is the derivative of the energy and therefore:

6

_{1 2}

.

2 1

2

R R

R R h F

_vdW

H

 

 (2.20)

Derjaguin approximation

Even though the interaction energies are more often used for molecular relations, forces between particles or macroscopic bodies are easier to measure. Boris Vladimirovich Derjaguin, a Russian physicochemist, who also worked with Hamaker, was able to relate the interaction energy between two planar surfaces F(h) to the force acting between bodies of any geometry F(h)

^{12, 13}

:

).

( 2

)

( h R W h

F

_sphere

 

_{eff planes}

(2.21)

Figure 2.4 shows two spheres with radii R

1

and R

2

at a distance h. If the condition R>>h is fulfilled for both of them, then the force acting between the spheres can be acquired by integrating the force between locally flat small circular regions of area 2πxdx on the surfaces and at the distance z=h+z

1

+z

2

. The force is then:

. ) ( 2 ) ( 

^





^Z

h Z

z xdxf h

F  (2.22)

15

Figure2.4 Derivation of Derjaguin approximation.

Here, f(z) is the normal force per unit area between two flat surfaces. Using geometry, and applying x

²

=2R

1

z

1

=2R

2

z

2

:

1 . 1

2

_{1 2}

2 2

1

 

 



 









 R R

h x z z h

z (2.23)

Differentiating the (2.23), the following is obtained:

1 . 1

2 1

R xdx dz R  

 



 

 (2.24)

Therefore, the Derjaguin approximation is:

).

( 2

) ( 2

) (

2 1

2 1 2

1 2

1

W h

R R

R dz R

z R f R

R h R

F

h

 

 





 

 

 





 

^

^  ^{ (2.25)}

where

.

2 1

2 1

R R

R R

_eff

R

  (2.27)

This approximation is valid for any geometry, not just spheres, and is used when the decay

length of the surface force is much smaller than the curvature of the surfaces. In case of a

sphere-plane interaction, the R

2

→∞, so the effective radius is R

eff

= R

1

.

16

Lifshitz theory (Macroscopic calculation; continuum approach)

The influence of the neighbouring atoms is completely disregarded in the Hamaker approach. Lifshitz theory treats macroscopic bodies as continuous media. It takes into account induced dipoles or the fact that the effective polarizability changes depending of the surrounding. The interaction energies equations earlier mentioned are still applicable, but the Hamaker constant is obtained in a different way.

The Hamaker constant is calculated using the properties of the material such as refractive indices and dielectric constants. The nonretarded Hamaker for molecules 1 and 2 interaction across medium 3 is given by:

   

       

    ^.

4 3 4

3

3 2

3 2

3 1

3 1

3 2

3 2 3 1

3 1

1

i dv i

i i

i i

i i

T h k

H

_B ^P

 

 







 



 







 

 

 







 



 







 _ ^  _ ^ _ ^ _ ^ _ 

^

_ ^ _ ^ _ ^{ } _{ } ^ _ ^ _ ^ _ ^



(2.28)

Here ε

1,

ε

2

, and ε

2

are dielectric constants of each media, ε(iν) are the values of ε at imaginary frequencies. The first term in this equation represents Keesom and Debye contributions, and results in zero frequency energy of the system. London contribution is given by the second term. The (2.28) shows only the first two terms in an infinite series, so even though not exact, it still works quite well, given the fact that the rest of the terms usually don’t contribute to more than 5% of the total value. ε(iν) varies with frequency, and can be represented with the following:

. 1 1 1 ) (

2 2 2

e

i n



 





 

 (2.29)

After inserting (2.29) into (2.28) for each of three media, the following approximation of the Hamaker constant is:

  

          ^.

2 8 3 4

3

2 3 2 2 2

3 2 1 2 3 2 2 2 3 2 1

2 3 2 2 2 3 2 1 3

2 3 2 3 1

3 1

n n n

n n n n n

n n n n T h

k

H

_B ^{P e}













 

 

 







 



 







  















 (2.30)

In the case where two same phases are interacting across medium 3, the expression simplifies to:

 

  ^.

2 16 3 4

3

2 / 2 3 3 2 1

2 2 3 2 1 2

3 1

3 1

n n

n n T h

k

H

_B ^{P e}



 

 

 







  







 (2.31)

Van der Waals force between the same particles in a liquid is always attractive.

17 Combining (mixed) relations

There are several useful relations of acquiring values of unknown Hamaker constants using the values of known ones.

¹⁴

For example, if H

132

is Hamaker constant for media 1 and 2 which interact across medium 3, it can be related to constants of symmetric situations in which two media 1 or 2 interact across medium 3:

232 131

132

H H

H 

. (2.32)

If 1 and 2 are interacting across vacuum, H

12

can be given in terms of symmetric cases of two of each of media are interacting across vacuum:

2 2 1 1

1 2

H H

H 

. (2.33)

Another interesting relation is:



11 33



²

13 33

11 313

131

H H H 2 H H H

H      

, (2.34)

which when combined with the (2.32) gives the following:



11 33



22 33



132

H H H H

H   

. (2.35)

In case when media 1 and 2 interact across medium 3, and H

11

< H

33

< H

22

, the resulting Hamaker constant will be negative, making van der Waals interaction repulsive.

2.1.2 Double layer forces

When a colloidal particle is immersed in water or a liquid with high dielectric constant it will usually develop a surface charge through one of two main mechanisms

¹²

:

- Ionization or dissociation of surface groups

- Adsorption of ions that were already in the solution to the neutral surface

Acquired surface charge causes an electric field. This field will therefore start to attract ions

with opposite charge from the surface – counterions. At this point, the particle will be

surrounded by a so called electric double layer. The first layer is consisted of the surface

charge, and the second one is made up by the counterions. The first and most simple

18

representation of this layer would be when the oppositely charged ions bind to the surface and neutralize the existing charge and can be compared to a plate capacitor. This is called the Helmholtz layer, named after the Ludwig Helmholtz who worked on electric capacitors.

This model could not explain the capacitance that could be measured, and it is only as thick as a molecular layer.

Figure 2.5 From left to right: Helmholtz model, Gouy-Chapman (diffuse) layer model, and Stern double layer model. Surface potential ψ

0

is the potential at the surface that is in contact with ions. Diffuse layer potential ψ

D

is the potential at the diffuse part of the double layer. Electrokinetic potential ζ is the difference in potential between the medium and the liquid layer that is surrounding the particle; it is similar, but always smaller than ψ

D

.

Taking into consideration the thermal motion of the ions, it is obvious that the charge will be driven away from the surface over a certain distance. This is the diffuse layer, and it extends further away and gradually dies out reaching the bulk phase. This representation is called the Gouy-Chapman model.

^{15, 16}

The diffuse layer is described with a decaying profile with a thickness of 1-100 nm. The charge in the proximity of the surface, up to 1 nm distance, neutralizes the charge in the diffuse layer.

¹⁷

Due to all of this, an electrical potential profile arises across the layers. The diffuse layer potential is the potential value at the plane of origin (Figure 2.5).

Gouy-Chapman theory is applicable to many systems, but still the biggest flaws are the facts that it does not take in consideration the molecular nature of the liquid, and the ions are treated as a continuous charge distribution, disregarding their finite dimensions. This theory can provide a general model for the interactions, but still does not explain all of its aspects.

The modification of the Gouy-Chapman layer model led to creating a new one – Stern

model.

¹⁸

Stern basically combined previously mentioned versions, the Gouy-Chapman and

the Helmholtz layer. Therefore, there is a layer of ions close to the surface, and the diffuse

19

layer of mobile ions. The potential at the border of these two layers is called electrokinetic or zeta potential, by the Greek letter used to denote it (ζ) (Figure 2.5). The capacitance of the Stern layer is given as

^{7, 19}

:

,

0 D

C

S







  (2.36)

with ς being the surface charge density. The potential at the surface is ψ

0

and ψ

D

is the potential at the Stern plane. The total capacitance of the double layer depends on both Stern layer and the diffuse layer capacitance:

1 . 1 1

D

S

C

C

C   (2.37)

Electric double layer forces are usually described with Poisson-Boltzmann theory (PB further in the text). This model describes ions as charged spheres, but still describes the surrounding liquid as a continuous phase, only within its dielectric characteristics

^{5, 13}

.

The electric potential caused by the charge density distribution is described with the Poisson equation:

.

0 2

2 2 2 2 2 2









   



 



 



 

 x y z (2.38)

The Boltzmann distribution gives the local cation and anion concentrations in the solution c

+

and c

-

:

.

;

^{k T}

e B T

k e B

B

B

c c e

e c c









^ _



(2.39)

Here c

B

is the concentration in the bulk. The work needed for bringing one ion with charge q from the bulk (infinite distance) close to the surface to a place with a potential ψ is given by qψ, assuming that only electric work is necessary. The next assumption is that there is only 1:1 salt in the liquid. Also, the concentration of this salt is considered to be much greater than that of the ions dissociated from the surface.

The local charge density is given in the form of:

, )

(  





 



 







_{ } ^{ k T}

e T k

e B

B

B

e

e qc c c q





 (2.40)

20

where ψ is the function of all the coordinates ψ=f(x,y,z). After inserting (2.38) into Poisson equation (2.36), the resulting relation is called Poisson-Boltzmann equation:

.

0

2

 





 



 





^{k T}

e T k

e

B _{B B}

e qc e

^{ }

  (2.41)

Considering the simple case of the planar surface which is infinitely extended, the one dimensional case is used to describe the system. Since the potential close to the charged plate located in x,y plane does not change in x and y coordinates, the PB equation can be written just for the third dimension, the z direction:

.

0 2 2

 





 



 

 

^k T

e T k

e

B _{B B}

e ec e

dz







 (2.42)

For low potentials, when qψ<<k

B

T, the exponential function in the brackets can be expanded into series:

! ...

3

! 1 2

3

2

 







 x x

x

e

^x

(2.43)

If only the first linear term is taken into account and the rest are disregarded, the PB takes form:

2 . ...

1 1

0 2

0 2

2













T k

c e T

k e T

k ec e

dz

_B

B B

B

B

  

 



   

 

(2.44) The equation (2.42) is referred to as the linear PB equation, or the Debye-Hückel (DH) approximation. By solving this equation, the general solution obtained is:

z

z

C e

e C

z

^{ }

 ( ) 

₁



₂ ^

_{, (2.45)}

where κ is the inverse Debye length:

T k

e c

B B



 

0

2

2



.

(2.46)

21

Constants C

1

and C

2

from (2.43) are determined by applying boundary conditions. The potential at the surface is ψ(x = 0) = ψ

0

, and it gradually decreases with distance from the surface with the decay length κ

^-1

until it reaches zero ψ(x →∞) = 0. So, C

1

= ψ

0

and C

2

=0, making the (2.43):

e

z

z 

^

 ( ) 

₀ ^

. (2.47)

Grahame derived an equation based on Gouy-Chapman theory, which gives a link between the surface charge density ς and surface potential:

2 . sinh

8

₀ ⁰

 

 



 

T k T e

k c

B B

B

 

 (2.48)

It then makes it possible to calculate the potential or the capacitance of the double layer if the ς is known.

Similar to the PB equation, here sinh can be expanded into a series:

! ...

5

! sinh 3

5

3

 



 x x

x

x (2.49)

Again, for low potentials, the series can be approximated by taking only the first term:







 

_{0 0}

. (2.50)

The capacitance of the double layer as a plate capacitor is given by:

1

.

0 0

0







 

 

 



C

D

(2.51)

The double layer behavior can be compared to that of the plate capacitor, and the Debye

length represents the distance between the plates. Since the Debye length decreases with

increasing salt concentration, the ability to store charge also increases.

22 Diffuse layer overlapping

The electrostatic double layer force appears when two surfaces approach each other, and their double layers start to overlap. The disjoining pressure is created since there is now a difference between the pressure in between the surfaces, and the one in the bulk:

2 . 1 cosh

2

0





 



 

 

 



 



 dz

d T

k Tc e

k

B B

B





 (2.52)

where coshx=(e

^x

+e

^-x

)/2. When the potential profile is known, the disjoining pressure is easily calculated. For very low potentials, the DH approximation with the series expansion can be applied:

2 .

2 2

0 2

 





 



 



 



 



 dz

d 



 

(2.53)

Integration of the pressure gives the interaction energy:

'.

) ' ( ) ( ^ 

^

^

h

d l

h h dh

W (2.54)

When the distance between the surfaces is sufficient enough, the superposition approximation states that each of the potential profiles is not affected by the other one. The disjoining pressure is therefore:

. 2 

₀



²



_D²

e

^h



 (2.55)

The double layer force decays exponentially with the Debye length as the decay length.

While the surfaces approach each other, their charges may vary. In case the charge of the inner layer remains constant, this situation is referred to as the constant charge (CC) condition. Another case is when the diffuse layer potential does not change at the approach, so this is called the constant potential (CP) conditions. These two are considered special cases, and usually, the most realistic situation is when both surface charge and potential are regulated upon approach, which is called charge regulation (CR).

The parameter which can describe how the charge and the potential actually vary in this

case is called the regulation parameter (p)

²⁰

:

23 .

D S

D

C C p C

  (2.56)

where C

S

and C

D

are capacitances of the Stern and diffuse layer. The parameter takes values from 0 to 1. The Figure 2.6 shows an example of the force vs. distance curve profile with extreme cases of charge regulation. When p=1, the surface charge does not change when the surfaces approach (CC), and when p=0, the potential is the one that remains constant (CP). What usually happens is that the p value lies between those two, as presented in the figure.

Figure 2.6 Charge regulation of surfaces: p=1 (constant charge, CC), 0<p<1 (charge regulation, CR), p=0 (constant potential, CP).

2.2 Non-DLVO Forces

DLVO theory can usually very well describe the force acting between two solid surfaces in liquid at large separations. When the distance between them becomes comparable to the molecular size, other factors start contributing to the force, such as size and shape of the solvent, or the very nature of it. These characteristics are not considered in the DLVO theory. The additional forces that arise due to these other effects are called non-DLVO forces. They can be either repulsive or attractive, or even oscillatory.

^{7, 19}

Also, at small separations, non-DLVO can dominate over DLVO forces. These forces can be quite important, for example, to determine the coagulation in biological systems.

²¹

Derjaguin and Kussakov were the first to suggest that when surfaces are introduced to

liquid, the molecular structure in the bulk will differ from the one on the interface. They

called the force that arises when the surfaces are in close proximity – solvation force. Quite

often, they are also referred to as structural forces, since the solvent forms particular

structured layer on the surfaces, which is disturbed upon their approach.

24

Solvation forces are not only dependent on the solvent characteristics, but also the surface roughness, hydrophobicity, or shape. All of these can affect the structure of liquid between the surfaces.

The pressure in space between two parallel plates filled with fluid is related to the local number density of molecules in proximity to walls, which is distance dependent ρ

0

(h). When the walls start approaching each other, the density changes. The contact theorem describes the force acting in this situation:



₀

( ) 

₀

(  ) 

 k T  h 

f

_B

. (2.57)

where ρ

0

(∞) is the number density at infinite distance, which is the same as the one for isolated wall. When the density of the liquid between the walls increases, the force will be repulsive, and on the other hand, when the density decreases the surfaces will attract each other. During the approach, each surface forms the layered structure of the solvent at the other one, and so the local number density of molecules changes. These fluctuations of density are influencing the force that varies periodically. For spherical molecules of solvent between two hard walls, this force is usually an oscillatory function that decays with distance. This is shown in Figure 2.7. The solvation force per unit area between two interfaces on a distance h and with layer thickness d

0

is:

2 . cos

0



^h

o

e

d f h

f  

^

 



  (2.58)

By applying Derjaguin approximation and integrating (2.58), the force between a sphere and a plane is given by:

2 . cos )

(

0

0

 

 



 



^_

 

d e h

F h F

h

(2.59)

In this equation, φ is introduced as the phase shift.

There is no need for the existence of any attractive forces between solvent molecules, or

solvent and the walls for oscillatory forces to appear. If solvent molecules are asymmetric,

or their interaction potentials are not pairwise additive, the solvation force can have a

monotonically attractive or repulsive component.

25

Figure 2.7 Schematic illustration of oscillatory force as a result of the change in layer structures depending on the distance.

If two hydrophilic surfaces in aqueous solution are in contact, a repulsive force at around 1

nm appears. This force is credited to the energy that is needed to remove hydration water

from the surface, or from the species adsorbed on the surface. The energy of wetting this

kind of surface in water is low. This force is actually a solvation force in water, and therefore

named hydration force.

^{22, 23}