Nanotechnology for Materials Physical for Nanotechnology 9 Overview of colloidal nanoparticles: principles, fabrication, characterization, sensors 9 Intermolecular forces ¾ Metallic colloidal nanoparticles: interaction with solvent and substrates, plasmon resonance • Physics of bottom‐up nanofabrication: wetting, dewetting, self‐assembled monolayers, micelles, liquid crystals • Surface microscopy techniques (TEM, SEM, AFM, STM, KPM ) • Molecular modeling and simulation: and Monte Carlo methods and applications

1 Surface forces

2 Interactions in colloids

In colloids and we are interested in the interaction between macroscopic bodies constituted by thousands of , typillicallysphilherical partilicles and flat surfaces. The dominant forces between the constituting atoms are: ‐ dispersion forces (van der Waals, depend on the density of atoms near the surface, ATTRACTIVE ) ‐ ionic forces (due to the surface charges, depend on the number of surface atoms and on the charge density at the surface, REPULSIVE) Assuming addit iv ity of forces, it is possibleto calculate the relevant interaction by integrating on the contribution of each .

3 PART 1 Van der Waals forces in collo ids

4 van der Waals forces between surfaces

5 6 Hamaker constant

7 Hamaker constant

8 Hamaker constant calculation: a microscopic approach For the van der Waals interaction the potential between two atoms or 1 and 2 : where is the interaction constant as defined by London and is specific to the identity of the interacting atoms (C6 ∝ α1*α2. Hamaker then performed the integration of the interaction potential to calculate the total interaction between two macroscopic bodies. This then allows the total force between two arbitrarily shaped bodies to be given by:

And the Hamaker constant is The Hamaker constant has units of energy and is the multiplicative factor of the distance‐dependent functions. Other geometrical constant often appears to make H (or A) transferable between the different possible geometries.

9 Hamaker constant calculation 1) Choose themathemati ca l form of the itintera tom ic/ ii/ionic/ molecular potential, w(r) (e.g. assume an arbitrary power law: w(r)=‐Cr ‐n ) 2) Set up the geometry of the particular interaction being derived 3) Assume "pairwise additivity"; i.e. the net interaction energy of a body is the sum of the individual interatomic/intermolecular interactions of the constituent atoms or molecules which make up that body 4) A solid continuum exists : the summation is replaced by an integration over the volumes of the interacting bodies assuming a number density of atoms/molecules/m3, ρ 5) Constant material properties : ρ and C are constant over the volume of the bodies→ volume integration :

W12(D)= ∫∫∫w12(r) ρ1 ρ2 dV1dV2 10 Hamaker constant calculation: atom‐surface

11 Hamaker constant calculation: atom‐surface

12 Hamaker constant calculation: two planar surfaces

13 Hamaker constant calculation: two planar surfaces

14 Hamaker constant calculation: two spheres

15 Summary of common Hamaker interactions

16 Exercise

U(a)=‐CH r/6 a = ‐CH d/12 a

F(a) = ‐dU/da

17 Strength of VdW forces

2 V=‐AH/12π h per unit area

18 Strength of VdW forces: gecko

19 NATURE |VOL 405 | 8 JUNE 2000 |www.nature.com

Geckos are exceptional in their ability to climb rapidly up smooth vertical surfaces. Microscopy has shown that a gecko's foot has nearly five hundred thousand keratinous hairs or setae. Each 30± 130μmlongsetaisonlyone‐ tenth the diameter of a human hair and contains hundreds of projections terminating in 0.2±0.5μmspatula‐shaped structures2. After nearly a century of anatomical description, here we report the first direct measurements of single setal force by using a two‐dimensional micro‐ electromechanical systems force sensor and a wire as a force gauge. Measurements revealed that a seta is ten times more effective at adhesion than predicted from maximal estimates on whole animals. Adhesive force values support the hypothesis that individual seta operate by van der Waals forces. Suitably orientated setae reduced the forces necessary to peel the toe by simply detaching above a critical angle with the

substratum. 20 21 Gecko spatulae adhesion force

Thesolid line representsasetawith spatltulae projtijecting tdtoward thesurface. The dashed line represents the setal force with spatulae projecting away from the surface 22 The structure similarity between the cross‐section views of the VA‐MWNTs (left) and gecko’s aligned elastic hairs (right). A book of 1480 g in weight suspended from a glass surface with use of vertically aligned‐MWNTs supported on a silicon wafer. The top right squared area shows the VA‐ MWNT array film, 4 mm by 4 mm

23 The effect of the liquid medium The medium between two interacting bodies has the net effect of reducing the attractive interaction with respect to vacuum. The extent depends on the dielectric properties of all themedia ildinvolved.

24 Medium effect: like particles

25 Medium effect: unlike particles

Lorentz‐Berthelot rule

26 Summary of Vdw forces between colloids

• Van der Waals forces in solvents are always attractive for particles of the same material • Approximate macroscopic interaction equations can be derived for different geometries (particle/surface 1 – medium 2‐ particle/surface 3) n • Equations have the form Energy = A123 * f / r A: Hamaker constant, depends on materials 1,2,3 f: geometric factor; n lower than 6 (typically 1 or 3) • The medium (2) has a strong influence on A: higher dielectric constant <‐>lowerHamakerconstant (even negative if 1/=3) • Tabulated Hamaker constants and equations can be useful in understanding the forces acting between your nanoparticles and substrates

27 PART 2 Dielectric permittivity (the medium between the nanopp)articles)

28 Dielectric constant (permittivity)

29 The effect of the liquid medium

VACUUM

VACUUM

30 The effect of the liquid medium

31 The effect of the liquid medium

32 The effect of the liquid medium

Molecular permanent and induced dipoles DIELECTRIC produce local fields in the opposite direction εD > ε0 wrt the applied field

33 Contributions to permittivity: polarization

34 Dipoles in electric field

• Thestrengthofthefieldatagivenpointisdefinedasthe force that would be exerted on a positive test charge of +1 coulblomb pldlaced at tha t poitint; the direc tion of the field is given by the direction of that force. • The definition of dipole orientation is instead from the negative to the positive charge From Coulomb’s law and theabove definitions, it follows that Energy = ‐ E • μ = ‐ E μ cos θ where θ is the angle μ between the two vectors

35 Polarization/2

36 Polarization/3

37 Polarization/4

38 Polarizability Let’s consider a spherical atom. In absence of external field the average charge density has a spherical symmetry centered at the nucleus. Appl yin g a field E the electron cloud will be distorted and attracted to the positive pole of the field, while the nucleus to the negative one, giving rise to the so called induced dipole.

39 Origin of polarizability/1 The nucleus has a point charge +Ze, while the electrons with total charge –Ze are uniformly distributed on the atomic volume,

Electron density:

We try now to evaluate quantitatively the equal and opposite forces that act on electrons and nuclei when they separate of a distance d in presence of an external field.

40 Origin of polarizability/2 Force acting on the nucleus due to the field: At the equilibrium distance d, this force will be balanced by the force exerted by the electrons on the nucleus. As the field due to a hollow charged sphere is 0 (cf Faraday cage/shield), only the electron charge inside the sphere of radius d has to be taken into account:

A spherical charge distribution can be treated as a point charge, hence from Coulomb’ s law:

41 Origin of polarizability/3 Now we can equalize the two forces:

The law obtained is the electrical analogous to Hooke’s law: F=‐kd ‐> F = (Drude oscillator) the distance of equilibrium is: and the induce dipole is μ∗ The total polarization (total dipole per unit of volume) for such a material is therefore ∗ 3 P = Σ μ /V = (Natoms/V)4πε0 R E = (Natoms/V)αE 3 where α= 4πε0 R is the atomic polarizability and E is the LOCAL field

42 Summary of polarization effects DIPOLE POLARIZABILITY Electronic polarization (polarizability)

Orientational polarization (permanent dipoles)

Ionic polarization (ionic solids)

43 The response to electric fields

The electric displacement field,denotedasD, is a vector field that appears in Maxwell's equations. It accounts for the effects of free charges within materials. "D" stands for "displacement," as in the related concept of displacement current in dielectrics.

In a linear, homogeneous, isotropic dielectric with instantaneous response to changes in the electric field, the polarization P depends linearly on the electric field,

P = ε0 (εr‐1) E, D= ε0 εr E As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field.

44 Micro‐macro equations for static permittivity

“Lorentz local field:” Eloc =(εr+2)/3 Eext (takes into account that the field inside the dielectric is different from the external field)

Macro: P = ε0 (εr‐1) Eext Micro: P = (N/V)αEloc = (N/V)α (εr+2)/3 Eext Combining the two equations:

Clausius‐Mossotti: (εr‐1)/(εr+2) = (N/V)αelectronic /(3ε0)

2 Debye: (εr‐1)/(εr+2) = (N/V)(αelectronic +μ permanent/3kT)/ (3ε0)

‐> εr=(2k+1)/(1‐k) with k depending on molecular properties More complex and accurate: Onsager, Kirkwood and Fröhlich

45 Dielectric constants of common materials

46 Dielectric constants of organic solvents

47 Exercise: the effect of polarizability

Calculate the dielectric constant for nitrogen gas (N2), composed by atoms with radius=0.8 Å, having a density = 1.251 g/L, and molllecular weihtight 28 g/l/mol. Do thesamefor liqu id nitrogen (density=0.807 g/mL). Compare the results with the experimental dielectric constants (gas 1.000580, liquid 1.4)

48 Solution

49 Exercise: from macro to micro

50 Solution

51 Exercise

52 Exercise

53 Dielectric constant and refractive index

From Maxwell’s laws it can be derived that the speed of light in a medium m is:

Where ε and μ represent the magnetic and dielectric permeabilities. In vacuum:

Considering anonmagnetic material (μrel=1) if follows that

By definition c0/cm istherefractionindexofthemedium. Therefore we could expect an experimental confirmation of the law

54 Refraction index

55 Dielectric constant and refractive index

2 Actually from measurements often εrel>n e.g. 2 NaCl: εrel=5.9 n =2.25 2 H2O: εrel=81 n =1.75 !!!

Only materials presenting the sole electronic polarization 2 mechanism are in agreement with the prediction εrel=n 2 (diamond: εrel=5.68 n =5.66) The origin of this experimental difference is clear: the refractive index is measured at optical frequencies (1015 Hz) while the dielectric constant at low or zero frequencies

56 The frequency dependence of permittivity

57 Frequency dependent dielectric constant

58 The response to electric fields

At the high‐frequency limit, the complex permittivity is commonly referred to as ε∞. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for altering fields of low frequencies, and as the frequency increases a measurable phase difference δ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength (E0), D and E remain proportional, and

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way: where

ε" is the imaginary part of the permittivity, which is related to the dissipation (or loss) of energy within the medium. ε' is the real part of the permittivity, which is related to the stored energy within the medium.

59 Measuring real and imaginary part

A parallel plate capacitor containing a dielectric material and its equivalent circuit

The dielectric sample placed in the capacitor can be considered

electrically equivalent to a capacitance, Cx, in parallel with a resistance, Rx,whereCx represents the material ability to store charge (ENERGY STORED) and Rx represents its heat related loss (ENERGY LOST)

60 Real and imaginary part of permittivity

These values, Cx and Rx, could be measured using a impedance analyzer and related to the dielectric properties of the sample, knowing the cell capacitance in vacuum C0=Area*ε0 /distance: Real part/ dielectric constant / relative permittivity 2 2 2 Wstored∝ CV = ε’ C0 *E d

Imaginary part 2 2 2 2 2 2 Wlost∝ RI =R(V/R)=Ed /Rx = ε’’ ω C0 E d

Dissipation factor or loss tangent = ε’’/ε’

61 Back to the refractive index

• Taking both aspects into consideration, we can define a complex index of refraction:

• Here, n is the refractive index indicating the phase velocity as above, while κ is called the extinction coefficient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material.

• The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank‐ 2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

62 Example of permittivity spectrum

A dielectric permittivity spectrum over a wide range of frequencies. ε′ and ε″ denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.

63 Water permittivity spectrum

Dielectric ppyermittivity and dielectric loss of water between 0°C and 100°C, the arrows showing the effect of increasing temperature. As the temperature increases, the strength and extent of the dipole-dipole interaction decrease. This (1) lowers both the static and optical dielectric permittivities, (2) lessens the difficulty for the movement dipole and so allows the water to oscillate at higher frequencies, and (3) reduces the drag to the rotation of the water molecules, so reducing the friction and hence the dielectric loss. Note that ε∞ (that is, the dielectric permittivity at short wavelengths) does not change significantly with temperature.Most of the dielectric loss is within the microwave range of electromagnetic. 64 Microwave oven

A microwave oven, or a microwave, is a kitchen appliance that cooks or heats food by dielectric heating. This is accomplished by using microwave radiation to heat water and other polarized molecules within the food. This excitation is fairly uniform, leading to food being adequately heated throughout (except in thick objects), a feature not seen in any other heating technique.

65 Dielectric relaxation models

Debye relaxation is the dielectric relaxation response of an ideal, non interacting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity of a medium as a function of the field's frequency ω:

Where is the permittivity at the high frequency limit and is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium.. BthBoth the Cole‐Cole equation: (good for polymers) and Havrili ak ‐NiNegami reltilaxation are empirical modifications of the Debye relaxation:

66 67 68 Cole‐Cole plot as analysis tool

69 Why Dielectric ?

A material’s dielectric properties are determined by its molecular structure. Other properties of interest can be correlated to die lec tr icproperties. Measurements are: ‐ Fast: Setup in minutes, measure in seconds. ‐ Non‐Destructive: Many materials can be measured “as is”.No sample preparation required ‐ Non‐Contacting: With Free‐Space techniques, your material is not even touched!

70 Example

frequency

71 Example 2

72 73 Capacitance sensor with fringing electric field

wireless soil moisture sensor

74 Retardation

75 Retardation The equation for van der Waals forces (U=K/r6) is valid only at distances less than about 10 nm. At separations greater than this, retardation effects become important and the attractive iiinteraction energy is ilinverselyproportilional to r7. RdiRetardation effects are caused by the fact that the electromagnetic field has to travel farther at greater separations. By the time the field influences the neighbor atom/ molecule, the original atomic/ Hamaker constant vs distance molecular dipole has changed its orientation. This effect causes the interaction to be slightly out of phase. The interaction energy is still attractive but has been reduced. 76 Lifshitz theory of van der Waals Forces Lifshitz (()1956) developed the macroscopic theory (also called the modern or continuum theory) of van der Waals forces between and within continuous materials. He argued that the concept of additivity was unsatisfactory when applied to closely packed atoms in a condensed body. He attributed the non‐additivity to the thermodynamic fluctuations always present in the interior of a material medium. The presence of spontaneous electromagnetic fluctuations in any region will, by Maxwell's equations, change the fluctuation field in any surrounding region. This phenomenon is called screening.

77 Lifshitz theory of van der Waals Forces

78 Lifshitz theory of van der Waals Forces

79 Lifshitz theory of van der Waals Forces

80 Summary

• In dielectrics (liquids), the permittivity depends on the polarizability and on the (squared) dipole moment • As fields take time to travel in a material (molecules take time to move), the dielectric constant depends on the frequency of the E field • The mathematical tool for expressing the frequency dependence is to have a complex dielectric constant • The imaginary part is non zero at frequencies related to the absorption of electric energy by the medium • The plot of real vs imaginary part (Cole‐Cole) is useful for characterizing materials and their absorptions • These absorptions determine retardation effects, i.e. diminish the VdW interaction at long range (>10 nm) • Lifshitz theory takes into account these effects to compute more accurate Hamaker constants

81 PART 3 Ionic Forces in colloids

82 Origin of the surface charge

83 The effect of solvent permittivity

(D=ε r )

84 The diffuse electrical double‐layer

charged surface

“diffuse layer” bulk electrolite solution (charged) (neutral) 85 The diffuse electrical double‐layer

Diffuse Layer: atmosphere of mobile counterions in rapid motion, attractive ionic forces pulling them to surface (l(electrica l migration f)force) causes concentration gradient, gain translational and rotational entropy by moving away from surface (diffusion down the concentration gradient) →theseeffectsarebalanced so their is no net flux of any ionic species Stern or Helmholtz Layer: bound, usually transiently, thickness is a few Å, reflects the size of the charged surface groups and bound counterions → do not completely neutralize the surface charges

86 The starting point: Gauss’s law

87 Potential of a charged sphere

88 Poisson’s equation

89 Approximate solutions for colloids

z or x

90 Derivation of the Poisson‐Boltzmann equation

91 Derivation of the Poisson‐Boltzmann equation

92 Derivation of the Poisson‐Boltzmann equation

93 Derivation of the Poisson‐Boltzmann equation standard value concentration effect electric energy

The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. If a system contains more than one species of particle, there is a separate chemical potential associated with each species(see Atkins pages 105‐109, 141‐).

94 Derivation of the Poisson‐Boltzmann equation

95 Simplifying the Poisson‐Boltzmann equation PROBLEM: there is no general analytical solution for mixed electrolytes, curved surfaces or even spherical particles. There is an asymptotic solution for spherical particles with low charged double layers.

96 Debye‐Hückel approximation

• In the limit of Y<<1 (Zψ(0)q << kBT;ψ(0)< 60meV) sinh Y=Y

≈

• Introducing the Debye length κ‐1 and solving

IONIC STRENGTH

97 Debye‐Hückel potential

98 Counterion concentration

99 Co‐ concentration

100 Physical meaning of the Debye length

101 Physical meaning of the Debye length

Ionic Strength

102 Limits of Debye‐Hückel approximation Comparison of the electrostatic potential profiles calculated with the Gouy‐Chapman PB equation • theory and the corresppgonding Debye‐Hückel approximation for a symmetric electrolyte (Z = 1, C(B) = 1 M) • Note how the Debye‐Hückel approximation overestimates the

pottiltential for ψ(0)q=Φ0e>kBT.

103 Potential between two interacting EDL

(potential at the mid point)

104 Shortcomings of the EDL model

Zeta potential is an abbreviation for electrokinetic potential in colloidal systems. In the colloidal chemistry literature, it is usually denoted using the Greek letter zeta, hence ζ‐ potential. From a theoretical viewpoint, zeta potential is electric potential in the interfacial double layer (DL) at the location of the slipping plane versus a point in the bulk fluid away from the interface. In other words, zeta potentilial isthe potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle.

105 Exercises

(room temperature)

106 Flocculating agents

• Flocculants, or flocculating agents, are chemicals that promote flocculation by causing colloids and other suspended particles in liquids to aggregate, forming a floc. Flocculants are used in water treatment processes to improve the sedimentation or filterability of small particles. • Many flocculants are multivalent cations such as aluminum, iron, calcium or magnesium. These positively charged molecules interact with negatively charged particles to reduce the barriers to aggregation Ionic strength

107 Flocculation Flocculation is, in the field of chemistry, a process where colloids come out of suspension in the form of floc or flakes. The action differs from ppprecipitationinthat, prior to flocculation, colloids are merely suspended in a liquid and not actually dissolved in a solution.

108 Flocculation in simple potentials

109 Interaction between colloids: DLVO theory

Γ0= reduced surface potential 110 DLVO: stable colloid

111 DLVO: flocculation (weak)

112 DLVO: flocculation (strong)

113 DLVO: aggregation (precipitate or cream)

114 Summary of DLVO potential curves

Debye length

115 Limits of DLVO theory

116 Steric stabilization GOOD SOLVENT Steric stabilization (e.g. by polymer grafting) (for the polymer) Hydrophobic molecules chemically or physically attached to the solute surface prevent aggregation of collo ida l partic les. OlOverlap of thesta biliz ing molecules results in a (osmotic) pressure in the overlap region and the stabilized solutes are pushed apart.

BAD SOLVENT

117 Steric stabilization/1

118 Steric stabilization/2

119 Steric stabilization/3

120 Summary “diffuse layer” bulk electrolite solution (neutral) • Electric diffuse double layer: charged surface • Gouy‐Chapman theory leads to Poisson‐Boltzmann equation for flat charged surfaces interacting with point charges: exact but difficult to solve • Debye‐Hückel approximation (electric energy << kT) leads to the simple formula • The Debye length represents the electrostatic screening distance due to the presence of in solution • DbDebye‐Hüc ke l+ Hama ker ‐> DLVO theory for collo ids; a semiquantitative model for the physics of these systems • Steric interaction: a way to stabilize colloidal suspensions

121 Bibliography

• MIT online lectures: http://ocw.mit.edu/OcwWeb/web/home/home/index.htm • Textbook pages: “” by P. W. Atkins , J. de Paula, “Applied Colloid and Surface Chemistry” by Richard Pashley, Marilyn Karaman • Google images • Wikipedia • Other…

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