Dispersing Pigments and Fillers E-Book

Jochen Winkler

• Herausgeber: Vincentz Network
• Kategorie: Wissenschaft und neue Technologien
• Serie: European Coatings TECH FILES
• Sprache: Englisch

Seitenzahl: 318

Veröffentlichungsjahr: 2014

Beschreibung

This eBook describes the most important and complex step during paint manufacturing: The dispersion of pigments and fillers. Wetting, mechanical destruction and stabilization occur simultaneously and each of these steps must be faultless in order to get an optimal dispersion. The eBook deals with physical and chemical basics of dispersing. Your benefit: A detailed access to dispersion technology and basics.

Leseprobe

Jochen Winkler

Dispersing Pigments and Fillers

Cover: Evonik Degussa GmbH, Essen/Germany

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Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliographie; detaillierte bibliographische Daten sind im Internet über http://dnb.ddb.de abrufbar.

Jochen Winkler

Dispersing Pigments and Fillers

Hanover: Vincentz Network, 2012

EUROPEAN COATINGS TECH FILES

ISBN 3-86630-811-6

ISBN 978-3-86630-811-4

© 2012 Vincentz Network GmbH & Co. KG, Hanover

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ISBN 3-86630-811-6

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EUROPEAN COATINGS TECH FILES

Jochen Winkler

Dispersing Pigments and Fillers

Preface

For the last 100 years or so, our everyday life has been predominated by the use polymeric materials. Tires for automobiles, for example, grant us the mobility that is a paramount precondition for our modern civilization to function as it does. More and more, polymeric materials are substituting metals or, as in the case of coatings, are delivering important contributions to the protection of valuable resources.

It is true for almost all polymeric materials that they gain their full potential of use only by the incorporation of pigments and fillers (extenders). Usually, one finds that the pigments only develop their beneficial action when they are evenly distributed within the polymeric matrices. Most pigments and fillers are produced and marketed as dry powders that “agglomerate” due to mutual attraction and therefore present themselves as larger, fairly spherical entities. The destruction of these structures by mechanical forces in polymer melts or polymer solutions, yielding a homogenous distribution of the single pigment particles is called “dispersing”. In that sense, dispersing is the elemental step in the production of any composite materials, especially in the case of coatings.

In spite of the huge importance of dispersion processes in the production of composite materials, dispersing itself is still often looked upon as being more of an art rather than a fundamental, scientifically underlain technical process. The reason for this may lie in the fact that a number of separate steps take place simultaneously during dispersions. These steps are the wetting of pigment surfaces, the mechanical disruption of agglomerates and the stabilization of the single “primary” pigment particles obtained against renewed agglomeration, which is called “flocculation”. The interrelations appear complicated and confusing to some. None of the single steps can be studied completely isolated from one another in a quantitative manner. Yet, there is profound knowledge concerning the physical and chemical background to these three steps. And, although quantitative prognoses are difficult, there are a number of perceptions and theories that lead to a sound understanding of the influencing parameters. With this knowledge, it is possible to find the reasons for possible failure on a case to case basis in a structured manner and to then find solutions to the problems quickly. In doing so, mathematical formulas are of great assistance since they enable an operator to find the influencing parameters at a glance. In the simplest case, a formula depicts which parameters facilitate a process, which impede it and which don’t play a role at all.

In order to satisfy the pretension of this book, namely to convey fundamental knowledge of dispersion processes, the basic interactions between atoms and/ or simple molecules is treated first. In that way, the wordings that are used to describe colloidal interactions are filled with meaning.

Following this are chapters in which the elemental steps of dispersions are discussed. These are explicitly:

•the wetting of pigment (filler or extender) surfaces by liquid components of a mill base

•the mechanical breakage of pigment (filler or extender) agglomerates

•the stabilization of the dispersed pigments against flocculation.

The pretension of this book is formulated in the title: “Dispersing Pigments and Fillers”. It is meant to serve the experienced practitioner as well as the novice as a source for information for their everyday work.

Krefeld, January 2012

Jochen Winkler

Contents

1Physical interactions of atoms and molecules

1.1Periodic table of the elements

1.1.1Covalent and ionic bonds

1.1.2Electronegativity of elements

1.1.3Ionic contribution to a chemical bond

1.2Physical interactions

1.2.1Dielectric substances in a capacitor

1.2.2Electron polarization

1.2.3Orientation and molar polarization

1.3Energies and forces of attraction

1.3.1Dipole-dipole interaction

1.3.2Induced dipole interactions

1.3.3London-van der Waals interaction

1.3.4Born interaction

1.3.5Total interaction energy

1.3.6Lennard-Jones potential

1.4Hydrogen bonds

1.5Range of physical interaction energies

1.6Interactions at interfaces

1.7Literature

2Properties of pigments and fillers

2.1Dispersing and milling

2.2Particle size determination

2.3Interactions between pigment particles

2.4Van der Waals attraction between particles

2.5Surface treatment of pigments

2.6Organic and inorganic pigments

2.7Literature

3Wetting of pigment surfaces

3.1Relevancy of wetting for the dispersion process

3.2Surface tension

3.3Young equation

3.3.1Critical surface tension according to Zisman

3.3.2Approach of Good und Girifalco

3.3.3Approach of Fowkes

3.3.4Approach of Owens und Wendt

3.3.5Approach of Wu

3.3.6Interfacial tension at complete wetting

3.4Wetting of pigments

3.4.1Measuring the free surface energy of pigments

3.4.2Kinetics and thermodynamics of pigment wetting

3.4.3Wetting volume

3.5Mill base rheology and mill base optimization

3.5.1Mill base rheology

3.5.2Mill base optimization for bead mills; determination of binder demand

3.6Literature

4Dispersing equipment

4.1High speed impellers

4.2Roller mills (three roll mill)

4.3Kneaders and extruders

4.4Bead mills (high speed attritors)

4.4.1Milling beads

4.5Determining dispersion time

4.6Literature

5Mechanical breakage of agglomerates

5.1Measuring dispersion success

5.2Principle of mechanical breakage; hammer–walnut experiment

5.3Dispersion equation

5.3.1Stress probability

5.3.2Breaking probability

5.3.3Total probability

5.3.4Determination of the energy density

5.3.5Colour strength development function

5.3.6Experimental results; using the dispersion equation

5.3.6.1Dispersion experiment ; variation of the bead filling degree

5.3.6.2Dispersion experiment; variation of dispersing time at different mechanical powers

5.3.6.3Dispersing of nano-particles

5.4Mechanical power and dispersion results in bead mills

5.5Literature

6Stabilization against flocculation

6.1Flocculation kinetics

6.1.1Spontaneous flocculation kinetics

6.1.2Measurement of flocculation rates

6.1.3Delayed flocculation

6.2Sedimentation

6.3Potential curves

6.4Electrostatic stabilization

6.4.1Electrostatic charging of pigment particles

6.4.2Fundamentals of electrostatics

6.4.3Potential distribution surrounding an electrostatically charged particle

6.4.4Zeta potential

6.4.5Electrostatic repulsion energy

6.5Steric stabilization

6.5.1Macromolecules in solution

6.5.2Macromolecules on pigment surfaces

6.6Solubility parameters

6.7Adsorption of polymers on pigment surfaces

6.8Let-down

6.9Flocculation stabilization by rheology control

6.10Literature

Appendix

Author

Index

Buyers’ guide

1Physical interactions of atoms and molecules

The periodic table arranges the elements in a clear way and, amongst other things, provides the basis for understanding chemical bonds. Covalent and ionic chemical bonds are distinguished. Depending upon the extent of the ionic portion of a chemical bond, the molecules exhibit different dipole moments and, depending upon their size, their electron shells are more or less easily polarizable. This chapter deals with the methods for determining dipole moments and polarizabilities of dielectric substances and how these two properties can be used to estimate the physical interaction energies and forces between atoms or molecules. In order to get a feeling for the relative significance of the different interaction principles, it will be demonstrated how the boiling points (as a measure for these interactions) rely on the dipole moments and polarizabilities. This chapter forms the fundament for later chapters in which the physical interactions in colloidal systems are discussed.

1.1Periodic table of the elements

The chemical elements are assorted in the so called “periodic table of the elements”, shown in Figure 1.1. Every element consists of an atomic nucleus with positively charged protons and a varying number of uncharged (“electroneutral”) neutrons. The nuclei, which comprise almost the whole mass of the elements, are surrounded by the negatively charged electrons. Since the number of protons and electrons in an element are equal in number, the positive and negative charges compensate each other and the whole atom (= atomic nucleus plus electrons) carries no externally measurable net charge.

The electrons are located in the so called “orbitals”. Orbitals differ in shape and size from one another. Every orbital has room for two electrons only. According to the Heisenberg uncertainty principle, it is not possible to denote both the location and the impuls (= mass · velocity) of an electron simultaneously1. For that reason, orbitals are to be understood as probable areas in which electrons may be encountered. Illustrations of orbitals represent exactly that, as shown in Figure 1.2.

Figure 1.1: Periodic table of the elements with atomic numbers and electronegativities listed

The periodic table consist of rows and columns. Starting from the first element, hydrogen (symbol H), and when going from left to right from one element to the next, every element is distinguished from its neighbour to the left in that it has both one additional proton and one further electron. Apart from that, a number of neutrons may accrue. When reaching the right end of a row, the next element with one additional proton and electron is the element standing at the left side of the following row. The number of electrons and protons that an atom has can therefore be taken from its “atomic number”, a simple, continuous numbering of the elements in the periodic table.

A further feature of chemical elements is that they have different tendencies to attract the binding electrons in chemical bonds. The greater the electron drawing effect of an element is, the more “electronegative” it is. The electronegativity increases from left to right within a row and declines from the top to the bottom of a column. The highest electronegativities belong to the elements in the upper right corner of the periodic table. In the last column, next to the “halogens” fluorine (F), chlorine (Cl), bromine (Br), iodine (J) and astatine (As), are the “noble gases” helium (He), neon (Ne), argon (Ar), krypton (Kr) and radon (Rn). The name “noble gas” expresses both the gaseous nature of the elements and that they usually do not undergo chemical reactions (they are “noble”). This is due to the fact that the electrons are in a “low energy state”, the so called “noble gas electron configuration” . This constellation is energetically favourable so that the other elements also exhibit the tendency to reach this state. They do this either by sharing electrons amongst each other, by rendering electrons to a chemical reaction partner, or by gathering electrons from them. Hence, the driving force for the generation of chemical bonds is explained by the reaction partners either sharing electrons, gaining electrons or releasing electrons to reach the noble gas electron configuration, i.e. having eight electrons in their outer electron shell (“octet rule”). For the elements in the first (second) column of the periodic table, it is more favourable to reach this state by rendering one (two) electron(s) to its partner in a chemical reaction, whereas an element in the seventh (sixth) column preferably takes up on (two) electron(s). The electrons participating in a chemical bond are called “binding electrons”.

Figure 1.2: Drawings of 1s, 2s, 2p, 2p and 2p orbitals as spaces with certain probabilities for the electrons to be located in.

Source: Wikipedia; Search word “Orbital”; public domain

1.1.1Covalent and ionic bonds

When chemical elements (= atoms) react with one another, molecules are formed. Depending upon the differences in electronegativities between the reaction partners, the binding electrons are distributed more or less equally in the chemical bond. Therefore, covalent and ionic chemical bonds are distinguished. Covalent bonds are formed when partners with the same electron drawing properties react with each other. This is especially the case when two atoms of the same sort, such as two chlorine atoms, react. Covalent bonds are formed when two orbitals with one single electron each overlap, so that a binding orbital is formed which is filled with two electrons. In that case, one may say that the two reaction partner mutually “share” the electrons.

The second possibility is that one reaction partner delivers one or more electrons to the other reaction partner. In this case, ionic bonds are formed in which one of the reactants obtains a positive charge (cation) and the other a negative charge (anion). The two oppositely charged ions attract each other electrostatically, so that the atoms become chemically connected. An example for this is the reaction of lithium (Li) with fluorine (F) to form the “salt” lithium fluoride (LiF). By releasing one electron, lithium obtains the electron structure of the noble gas helium (He) and fluorine that of neon (Ne). When adding lithium fluoride to water, the salt dissolves to form lithium cations Li+ and fluoride anions F–.

The two borderline cases are distinguished by the following reaction equations in which a dot represents the potential binding electron of an element whereas a line between two atoms symbolizes a binding orbital filled with two binding electrons.

and

Normally, chemical bonds in molecules are not completely of ionic nature but are best described as covalent bonds with larger or smaller ionic contributions. The extent of ionic binding depends upon the differences in electronegativities of the binding partners.

1.1.2Electronegativity of elements

The electronegativity describes the tendency of an element to attract the binding electrons in a chemical bond. In a molecule, the binding electrons on a time average are closer to the more electronegative reaction partner than the less electronegative (more electropositive) atoms. Thus, the chemical bonds obtain an ionic portion and the molecules have a permanent dipole moment.

According to a semi-empirical concept of Pauling2, the electronegativities X of elements can be ascertained and the ionic fraction of a chemical bond A-B can be estimated. For that, the geometric mean of binding energies of the molecules A-A and B-B is calculated and subtracted from the binding energy determined experimentally for that molecule3. The idea behind this rationing is that the binding energies DA-A and DB-B of the chemical compounds A-A and B-B have only covalent binding energy, whereas DA-B is put together from covalent as well as ionic parts. Binding energies may be determined from electron spectra in the visible and UV section of the spectrum and are listed in reference books such as the “Handbook of Chemistry and Physics” (CRC Press). The difference Δ is described in Equation 1.1.

Pauling took the square root of Δ as a measure for the electronegativity difference between the atoms A and B.

The electronegativities of other elements are found accordingly. In theory, given that the binding energies are determined accurately enough, the electronegativities found are independent of the calculation path. In practice, however, published binding energies differ from source to source. This is also attributed to the fact that in molecules containing more than two atoms, the strength of chemical bonds will depend upon the molecules themselves. For example, the binding energy of the O-H bond in H-O-H (water) is 497 kJ/mole, whereas in CH3-O-H (methanol) its value is only 440 kJ/mole. The electronegativities of the elements are listed in the periodic table in Figure 1.1.

1.1.3Ionic contribution to a chemical bond

Molecules or sections of molecules with unsymmetrical distributions of electrons possess a dipole moment. The dipole moment is an aligned parameter, that is to say, it may be characterized as a vector. In the case of linear molecules, the dipole moment is positioned along the axis of the chemical bond. The dipole moment5 is the product between the electrical charges e and their separation distance .

Table 1.1: Binding energies of some molecules from identical atoms

Binding energy

Bond

kJ/mol

eV

H-H

435

4.51

N-N

945

9.79

F-F

159

1.65

Cl-Cl

245

2.54

Br-Br

194

2.01

J-J

153

1.59

O-O

498

5.16

S-S

425

4.40

C-C

618

6.40

Table 1.2: Calculation of electronegativities of halogen atoms

In case of angulated molecules, the separate dipole moments of the chemical bonds add in a vectorial manner as shown in Figure 1.3.

In practice, the distribution of charges in a molecule is initially unknown. It is however possible to draw conclusions about the distribution of charges from measured dipole moments and also to estimate the share of ionic bonding in a chemical bond. For a molecule consisting of two atoms, this is done by multiplying the formal charges of the bond with the separation distance between the nuclei of the atoms in that chemical compound. In doing so, the dipole moment is calculated for the case that the bond is one hundred per cent ionic in nature. The measured dipole moment may be expressed as a percentage of the calculated dipole moment, thus revealing the ionic contribution.

Example: Hydrochloric acid, HCl:

Figure 1.3: Vectorial addition of dipole moments along two different bonds in a molecule to a resulting dipole moment

The measured dipole moment is 3.44 · 10-30Cm. This corresponds to an ionic contribution to the bond of about 17%.

Knowing the charge distribution in molecules is helpful when trying to understand or to foresee interactions between molecules6. In colloidal chemistry, this knowledge is beneficial for example when making assumptions about what section of a molecule will have the tendency to adsorb onto solid surfaces.

Figure 1.4: Capacitor without (left) and with (right) a dielectric substance

1.2Physical interactions

Physical interactions between molecules arise either by virtue of their having a dipole moment (only molecules) or them being polarizable, i.e. that the electrons may be shifted relative to the atomic nuclei, thus making them dipolar (molecules and atoms). Substances that are either dipolar in nature or, otherwise, become dipolar when brought into an electric field, yet are not conductive, are called “dielectric”. The dielectric properties of substances are determined by measuring the capacitance of a capacitor with (C) and without (C0) the substance.

1.2.1Dielectric substances in a capacitor

In the simplest case, a capacitor (see Figure 1.4) is comprised by two metal plates that face each other without touching. The two plates can be oppositely charged by applying a voltage U. The capacitance of the capacitor tells how many charges Q appear on one of the two plates. The capacitance C is larger, the more charges are generated at a given voltage.

An electric field with a field force (or field strength) E then fills the gap between the two plates. The electric field force is defined as the potential difference U between the plates separated by the distance d.

In the case that vacuum is in the gap between the plates, the electric field force E is directly proportional to the number of charges Q per unit area A of one capacitor plate. The proportionality constant ε0 is called the “permittivity of vacuum”.

ε0 has the value 8.854 · 10-12C · V-1 · m-1. When, instead of vacuum, a dielectric substance fills the gap between the capacitor plates, the number of charges rise by a factor which is characteristic for every individual substance. This is taken into account by introducing a relative dielectric constant εr as an expansion factor in Equation 1.6.

By substituting the potential U in Equation 1.4 by E · d (Equation 1.5) and further substituting E by Q/A εr ε0 (Equation 1.7), a new expression for the capacitance of a capacitor filled with a dielectric is found.

Since, by definition, εr in vacuum is unity, εr may consequently be determined from the ratio of the capacitance of a capacitor with (C) and without (C0) the presence of the dielectric. “Without a dielectric” means either in vacuum or, adequately enough, in air, since air has a low density and since the molecules in the atmosphere are barely polarizable.

According to Equation 1.8, the introduction of a dielectric always leads to an increase in capacitance or, respectively, to a drop of the electric field force in the capacitor (Equation 1.7).

The reason for this lies in the “polarization” P of the dielectric. Polarization depicts the formation of an electric field within the capacitor opposite to the applied outer electric field, thereby weakening it. The total effect is composed of a contribution coming from the displacement of the electrons relative to the nuclei of the dielectrics (induced displacement or electron polarization Pi) and a second component stemming from the orientation of dipolar molecules (orientation polarization P0).

Both types of polarization are schematically presented in Figure 1.5. When the polarization is standardized to one mole of substance, it is called “molar polarization” PM.

Figure 1.5: Emergence of electron polarization in a capacitor by displacement of the electrons relative to the nuclei (top) and orientation polarization by the orientation of dipolar molecules in the electric field of the capacitor (bottom).

The electron polarization happens very rapidly and is independent of the temperature since it relies only upon the mobility of the electrons in a molecule or atom. Electrons follow any changes of the electric field instantaneously because of their small mass. Typically, electrons are excited by electromagnetic waves with a frequency of 1014 Hertz. This corresponds to the region of visible light (approximately 4 - 7 · 1014 Hz).

This is different in the case of orientation polarization. Orientation happens more slowly and is temperature dependent. As opposed to electrons, molecules are heavy so that they possess inertia, letting them follow changes in the electric field only with a certain delay time. So, when an alternating electric field is applied to a capacitor, it will depend upon the frequency of the field change if dipolar molecules have enough time to adjust to the field or not. Low molecular weight materials such as common solvents need about 10-12 seconds time to follow the field. That’s why electric field frequencies of 105 Hz are usually applied when the contribution of orientation polarization to the relative dielectric constant is to be accessed. The temperature dependency of the orientation polarization is due to the thermal fluctuation of molecules, which is known as the “Brownian motion”. The thermal movement counteracts the orientation of the molecules.

Both the electron polarization and the orientation polarization create an electric field with a magnitude of P/(3 · ε0) that opposes the applied electric field, whereby P has the physical dimension of a charge density (C · m-2) or a volume based dipole moment (C · m · m-3), respectively.

1.2.2Electron polarization

The electron polarization is described by the “Clausius-Mosotti” equation (Equation 1.16). The starting point for its derivation[1] is to consider the dielectric substance as a single dipole consisting of many dipoles, each having an induced dipole moment of μi.

The induced dipole moment is proportional to the effective field strength F. The proportionality constant is the polarizability α.

α has the dimension of a reciprocal volume. The effective field strength within the dielectric substance is the vector sum of the externally applied electric field with the field strength E and the opposed field.

From Equations 1.12 and 1.13 follows for the induced dipole moment

The polarization is expressed by

Solving Equation 1.11 for μi and Equation 1.15 for E and inserting these expressions for μi and E into Equation 1.14, followed by rearrangement leads to the Clausius-Mosotti equation7.

The Clausius-Mosotti equation enables the determination of the polarizabilities of dielectric substances by measuring their relative dielectric constants. As stated in the introduction to this chapter, dispersive interactions between molecules can be calculated from their polarizabilities.

Alternatively to capacity measurements in a capacitor, the portion of the dielectric constant coming from electron polarization can be calculated from the refractive indexes n of dielectric substances. According to Maxwell’s theory of electromagnetic radiation:

1.2.3Orientation and molar polarization

The Clausius-Mosotti equation regards only the contribution of the displacement of the electrons to the molar polarization PM. Since the total dipole moment may also have a contribution resulting from the orientation polarization, this part must be added to the induced polarization of the electrons. As already stated, the orientation polarization decreases with increasing temperature. Using the Boltzmann equation and applying a number of simplifying assumptions, an expression for the molar orientation polarization is found [1].

Equation 1.10 shows that the total polarization is put together by parts coming from induction and from orientation. Hence, Equation 1.16 and 1.18 are combined to the so called Debye equation:

Two methods are used to ascertain polarizabilities α and dipole moments μ. Either the relative dielectric constants are measured at different temperatures and PM is plotted against the reciprocal temperature 1/T. Then, a straight line is found with the slope of μ2/3k and an intercept with the y-axis at Lα/3 ε0. The dipole moment can then be calculated from the slope, whereas the polarizability is found from the intercept. Alternatively, εr may be calculated according to Equation 1.17 from the refractive index and then the polarizability α can be found from Equation 1.16. Subsequently, the dipole moment can be calculated with Equation 1.19 from the dielectric constant εr measured with the capacitor and from the α-value acquired via the refractive index.

Table 1.3 gives an overview of the dielectric constants of a number of solvents widely used in paint formulations. The dielectric constants of liquids are also a good measure for their ability to stabilize ions by solvation. The higher they are, the greater the ability of the liquids is to dissolve salts. The relative dielectric constant of water, for example, lies at about 80.

Table 1.3: Relative dielectric constants of some organic solvents

Compound

rel. DK

Cyloaliphatic carbohydrates

Dekaline

2.20

Cyclohexane

2.00

Aromatic carbohydrates

ortho-Xylene

2.57

meta-Xylene

2.37

para-Xylene

2.30

Ethylbenzene

2.30

Alcohols

Isopropanol

26.00

Ethanol

22.40

n-Propanol

22.20

Isobutanol

18.40

n-Butanol

18.20

Cyclohexanol

15.00

Tert.-Butanol

10.90

Isooktylalcohol

10.00

Ethers und glykole ethers

Ethylglykole

13.70

Butyldiglykole

11.00

Butylglykole

9.20

Tetrahydrofurane

7.60

Esters

Butyldiglykol acetate

7.00

Ethylacetate

5.60

n-Butylacetat

4.50

Ketones

Cyclohexanon

18.30

Methyl-n-propylketone

15.40

Methyl isobutyl ketone

13.11

Glykoles

Ethylenglykole

41.20

Diethylenglykole

32.00

Triethylenglykole

24.00

1.3Energies and forces of attraction

In case the polarizabilities and the dipole moments of dielectric substances are known, the energies of attraction between congeneric or between disparate molecules can be calculated using rather simple equations.

The following interactions are distinguished

•Interactions between two dipoles (dipole-dipole or “Keesom” interaction)

•Interactions between a dipole and a polarizable substance (induced dipole or “Debye” interaction)

•Interactions between two polarizable substances (dispersive or “London-van der Waals” interaction)

All of these interactions are extremely distance-dependent. In fact, the energies of interaction are proportional to the reciprocal value of the distance between the molecules (or atoms) to the power of six. In general terms, energy is the ability to perform work. Energy and work are two expressions for one and the same thing. When two molecules (or atoms) are a distance of r to each other and have an interaction energy Er, then Er is the work that must be applied in order to separate them to a distance where they no longer attract each other. Theoretically, this is the case at an infinite distance. In practice, taking the high distance dependency into account, interactions decay to small values at only moderate separations already. Since E is proportional to 1/r6, by doubling the separation distance of the two molecules, the interaction energies diminish to 1.5 % of the original value. Attractive interactions get a negative sign, whereas repulsive interactions are depicted by their positive sign.

An even more descriptive quantity than “energy” is “force”. Energy is defined as the product between force and distance. Therefore, by mathematical differentiation of energy with respect to the distance, the attractive (or repulsive) force is obtained.

1.3.1Dipole-dipole interaction

The dipole-dipole (Keesom) interactions come into place when two permanent dipoles align in such a way that the negatively polarized part of the first molecule shows towards the positively polarized part of the second molecule, so that an attractive force is generated along their dipole axes. Dipole-dipole interactions are only possible between molecules and not between atoms since the latter have no permanent dipole moment.

The dipole-dipole interaction energy is, according to Equation 1.20, proportional to the square of the dipole moments of both molecules A and B. When dipole-dipole interaction takes place between alike molecules, the interaction energy is proportional to the power of four of the dipole moment, thus proportional to μ4. The inverse dependency towards the separation distance of the molecules to the power of six was already mentioned. The factor 1/(4π ε0)2 arises in the derivation of Equation 1.20 due to geometric considerations of the interactions of point charges in a three-dimensional space. The factor 1/kT results from considering the thermal movement of the molecules which counteracts their alignment.

1.3.2Induced dipole interactions

The induced dipole (Debye) interaction comes into effect when a dipolar molecule changes the electron density distribution of a neighbouring molecule or atom relative to its nucleus. Debye interactions therefore take place either between molecules or between molecules and atoms, yet never between atoms alone. The interaction energies in this case are proportional to the sums of the products of the polarizabilities and the square of the dipole moments of the molecules A and B.

In the specific case that the interactions take place between equal molecules, the term in brackets of Equation 1.21 is substituted by 2 · μ2α. The remarks to the other factors standing in front of the brackets that were made in the preceding section apply here also.

1.3.3London-van der Waals interaction

In the case of identical molecules or atoms, αAαB is replaced by α2.

The characteristic frequency v0 may be determined with the help of Cauchy’s dispersion formula by measuring the refractive index n of the chemical at different light wave frequencies v:

Equation 1.23 represents a straight line with a slope of Const. and an intercept with the ordinate at Const. · v02. By plotting the left hand side of the equation (n2 + 2)/(n2 – 1) against the square of the frequency of the light, the characteristic frequency v0 may be assessed. Since the frequency dependency of the refractive index or any other physical property of matter is termed “dispersion”, the Londenvan der Waals interaction is also called “dispersive interaction”.

If dispersive interactions between different atoms or molecules were to be calculated, one must theoretically consider that they might have different characteristic frequencies. Some authors propose to use 1 MJ/mole as a general value for h v0. In that case, the characteristic frequency would be assumed to lie in the ultraviolet region of the spectrum at a frequency of 2.5 · 1015 Hertz, corresponding to a fixed wavelength of 120 nm. Provided the characteristic frequencies are known, calculations could be carried out by using a mean value of the characteristic frequencies. A third possibility is expressing the term h v0 by the ionization potentials IA and IB of the molecule kinds A and B:

Ionization energies are the energies necessary to remove an electron from the most outer orbital of a molecule or an atom and bring it to an infinite distance. For electrons in organic substances, they lie in the order of magnitude of about 6 · 105 J/mole, or 10-18 J for a single electron, respectively.

1.3.4Born interaction

Up to now, only attractive forces between atoms or molecules have been considered. There are, however, also repulsive forces. If molecules could approach each other unhindered, their electron shells would overlap so that the orbitals would be populated with more than two electrons. In that way, there would be electrons with identical quantum numbers in some of the orbitals. This contradicts the “Pauli exclusion principle” which explains why one substance cannot penetrate another. One way to account for this is to assume that, at a certain minimal distance, an infinitely large repulsive energy comes into place. According to Born, the repulsive energy EB may also be expressed as a high power function of the distance r.

In Equation 1.25, m is in the order of magnitude between 9 and 12. A value of 12 is commonly used for m, whereas B is a constant.

1.3.5Total interaction energy

The total interaction energy Etot at a certain separation distance between molecules or atoms is given by the sum of all attractive and repulsive energies at that distance. Equations 1.20 to 1.22, 1.24 and 1.25 therefore yield:

Accordingly, by adopting the complete expressions of the individual interaction energy contributions:

As mentioned earlier, “energy” is the product of “force” times “distance”. The resulting attractive force Ftot is therefore found by differentiating Equation 1.27 with respect to the distance r.

Equation 1.28 in comparison to Equation 1.27 illustrates that attractive forces between molecules and/or atoms are even more distance dependent than the corresponding energies. Due to the mathematical operation of differentiation, the attractive forces obtain a positive sign, which accounts for the fact that negative forces do not exist 8.

1.3.6Lennard-Jones potential

Unfortunately, Equation 1.27 is however not applicable without further ado, since B is unknown. Lennard-Jones[2] expressed Equation 1.27 in the general form:

whereby A and B both depend upon σ, the distance of the centres of species A and B at which the attractive energy is zero, so that they neither attract nor repel each other, and ε, the highest attractive energy.

By substituting Equation 1.30 and 1.31 into 1.29 and differentiating with respect to the distance r and then fixing Etot to zero, the distance rm is found, at which the highest attractive energy is in place.

Figure 1.6 shows the principle dependency of the total interaction energy as a function of the distance between two atoms or molecules. The figure therefore also represents the total interaction energy between atoms or molecules according to Equation 1.27. At large separation distances, the atoms or molecules undergo little or no interaction. However, when approaching each other, the attractive interaction energy obtains progressively larger, negative values. At a distance rm, the interaction energy runs through a minimum, so that the attractive energy is highest at that point. As the distance becomes even shorter, the attraction becomes less and finally turns into a repulsive interaction at further approach. The consequence is that the two atoms or molecules that are associated vibrate around a mean separation distance rm.

Figure 1.6: Lennard-Jones potential between two atoms

In the further course of this book it will be shown that colloidal particles exhibit similar interaction energy curves like atoms and molecules as long as both attractive and repulsive forces are in place.

1.4Hydrogen bonds

A further physical interaction type that, however, may only occur between certain molecules, is the hydrogen bond. Hydrogen bonds are characterized by one molecule acting as a proton donor and another as a proton acceptor. Within the proton donor molecule, a hydrogen atom must form a chemical bond with an atom that has a high electronegativity. The proton acceptor, on the other hand, should have lone electron pairs. As an example, Figure 1.7 shows how two water molecules are polarized while forming a hydrogen bond. The molecules arrange in a way that the overlap between the protons of the proton donor and the free electron pair of the proton acceptor maximizes. In liquid water, clusters of approximately nine individual molecules exist. The high conductivity of water is, for example, attributed to the switching of hydrogen bonds to chemical bonds, whereby positive charges are transferred without mass transport being involved.

Figure 1.7: Polarization of electrons in a hydrogen bond

Strong hydrogen bonds of the type X-H----Y occur mainly when the atom X is either an oxygen, a nitrogen or a halogen atom (fluorine, chlorine, bromine or iodine) whereas the atom Y may be oxygen, nitrogen, sulfur or a halogen atom. Hydrogen bonds play an important role in many processes taking place in the animated and non-animated nature. In colloidal chemistry, hydrogen bonds are often determining factors as well because of their strength and their special alignment.

There is as yet no coherent method for calculating interaction energies from hydrogen bonds between molecules. They usually lie in the order of about 10 to 50 KJ/mole. The F-H------F bond is the most powerful hydrogen bond. A bonding energy of 160 to 170 KJ/mole is attributed to it.

1.5Range of physical interaction energies

Polarizabilities, dipole moments and values of h · v0 for a number of molecules as well as the contributions of attraction energies coming from the different mechanisms are presented in Table 1.4. It shows that dispersive forces always play a major role in intermolecular interactions. In contrast, appreciable dipole-dipole interactions only come into being when dipole moments exceed 1.3 Debye (= 4.3 · 10-30 C · m). Induced dipole interactions are very weak in all cases.