Surface sensitive techniques for studying ultrathin layers adsorbed on solid substrates

Thesis

Reference

Surface sensitive techniques for studying ultrathin layers adsorbed on solid substrates

PORUS, Mariya Vladimirovna

Abstract

This thesis is focused on studying the adsorption of thin layers on solid substrates.

PORUS, Mariya Vladimirovna. Surface sensitive techniques for studying ultrathin layers adsorbed on solid substrates. Thèse de doctorat : Univ. Genève, 2012, no. Sc. 4454

URN : urn:nbn:ch:unige-223975

DOI : 10.13097/archive-ouverte/unige:22397

Available at:

http://archive-ouverte.unige.ch/unige:22397

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

1 / 1

1 UNIVERSITÉ DE GENÈVE

Section de chimie et biochimie Département de chimie minérale et analytique

FACULTÉ DE SCIENCES Professeur Michal Borkovec

Surface Sensitive Techniques for Studying Ultrathin Layers Adsorbed on Solid Substrates

THÈSE

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

par

Mariya Vladimirovna Porus de

Moscou (Russie)

Thèse N°4454 GENEVE Atelier ReproMail

2012

2 Content

1. Introduction 7

1.1 Adsorption of ions 7

1.2 Adsorption of polyelectrolyte monolayers 11

1.3 Multilayer systems formed by layer-by-layer (LBL) technology. 13

1.4 Surface sensitive techniques 14

1.4.1 Quartz Crystal Microbalance (QCM) 14

1.4.2 Optical surface sensitive techniques 19

1.4.2.1 Ellipsometry 30

1.4.2.2 Reflectometry 34

1.4.2.3 Surface Plasmon Resonance (SPR) 37

1.5 Outline of the thesis 39

2. Highly-sensitive reflectometry setup capable of probing the electrical double layer on silica

43 3. Adsorption of monovalent and divalent cations on planar water-silica

interfaces studied by optical reflectivity and Monte Carlo simulations

44 4. Zipper and layer-by-layer assemblies of artificial photosystems analyzed by

combining optical and piezoelectric surface techniques

45 5. Structure of adsorbed polyelectrolyte monolayers investigated by combining

optical reflectometry and piezoelectric techniques

46 6. Probing conformational changes of polyamidoamine (PAMAM) dendrimers

adsorbed on silica substrates

47 7. Ion-specific responsiveness of polyamidoamine (PAMAM) dendrimers

adsorbed on silica substrates

48

8. Conclusions 49

9. Acknowledgement 51

References 52

3 Abstract

This thesis is focused on studying the adsorption of thin layers on solid substrates. The first chapter describes the theoretical background of adsorption of ions and polyelectrolytes to planar surfaces of opposite charge. Besides that, the principle of several surface sensitive techniques used for probing the adsorption of thin layers on planar substrates is discussed.

The chapter provides detailed discussion of mathematical interpretation of experimental data obtained by optical techniques.

The second chapter presents the study of electrical double layer formation on planar silica surface. The study was performed by home-made reflectometry setup supplied with thermally stabilized laser and lock-in amplifier detection scheme. It was demonstrated that experimentally measured amount of adsorbed cations on silica can be successfully interpreted in the frames of the 1-pK basic Stern model.

The third chapter extends this reflectometric study of double layer formation. The adsorption of different monovalent and divalent cations on silica was monitored. The experimental data were compared to grand canonical Monte Carlo titration simulations at the primitive model level. Theoretical calculation predicted much higher surface charge excess in case of divalent cations compared to monovalent cations, which was confirmed experimentally. Ion specificity at silica/water interface was interpreted qualitatively. Simulations together with experimental studies of pH dependence of silica charging strongly suggest the existence of two types of silanol groups.

The fourth chapter presents the study of formation of multilayered films of photochemically active organic macromolecules on gold surface. Multilayers were formed by applying layer- by-layer technology, namely by sequential adsorption of oppositely charged macromolecules on the surface. Surface plasmon resonance (SPR) and quartz crystal microbalance (QCM) were used to monitor the growth of adsorbed mass of the layers. It was demonstrated that combination of data obtained from both techniques enables to characterize the properties of multilayered films and the supramolecular architecture of the layers.

4 Development of a method of complete characterization of nanometer thick adsorbed layers is presented in the fifth chapter. Monolayers of linear polyelectrolytes adsorbed on silica substrates were probed by reflectometry and QCM in parallel. Thickness and water content of the layers were calculated. Systematic study of adsorption of polyelectrolytes at different salt conditions revealed that at high ionic strength polyelectrolytes tend to adsorb in coiled conformation while at low ionic strength polyelectrolytes adsorb in a flat conformation and form thin and rigid layers. Study of pH dependence of polyelectrolyte adsorption illustrated conformational transitions of polylysine.

The sixth chapter reports the study of stimuli-responsive behavior of dendrimers adsorbed on silica surface. AFM imaging and QCM measurements demonstrated that changes in chemical composition of surrounding solution affect the conformational state of dendrimer molecules.

At conditions of low pH and high salt dendrimers obtain swollen conformation, while the opposite conditions force dendrimers to collapse. This phenomenon was interpreted in terms of electrostatic interactions between the dendrimers and the surface.

The seventh chapter presents an extended study of dendrimer swelling on silica substrates.

The variation of adsorbed mass of dendrimer layers under different environmental conditions was monitored by reflectometry and QCM in parallel. pH and salt concentration dependencies of dendrimer swelling were obtained. It was demonstrated that the extent of dendrimer swelling appears to be ion specific and depends on the type of the cation of the background electrolyte.

The eighth chapter contains the basic conclusions of the thesis.

5 Résumé

Cette thèse est centrée sur l’étude de l’absorption des couches fines sur des substrats solides.

Le premier chapitre décrit les bases théoriques de l’absorption des ions et des polyélectrolytes par les surfaces planes de charge opposée. Parallèlement à cela, une investigation est menée sur les différentes techniques basées sur des surfaces sensibles permettant l’absorption de couches fines sur des substrats plats. Ce chapitre fournit une discussion détaillée sur l’interprétation mathématique des données expérimentales obtenues par des techniques optiques.

Le deuxième chapitre présente l’étude de la formation de double couche électrique sur des surfaces de silice planes. L’étude a été réalisée à partir d’un dispositif de réflectométrie disposant d’un laser stabilisé thermiquement et d’un système de détection synchronisée. Les mesures expérimentales ont démontré que la quantité de cations absorbés sur la silice peut être interprétée selon le modèle 1-pK de Stern.

Le troisième chapitre étend l’étude de la réflectométrie de la formation double couche.

L’absorption de différents cations monovalents et bivalents a été étudiée. Les données expérimentales ont été comparées aux simulations de titrage Monte Carlo grand canonique au niveau d’un modèle primitive. Les calculs théoriques prédisaient un excès de charge en surface bien plus élevé dans le cas de cations bivalents en comparaison aux cations monovalents, ce qui a été confirmé expérimentalement. La spécificité des ions à l’interface silice/eau a été interprétée qualitativement. Des simulations appuyées par des études expérimentales sur la dépendance du pH de la silice fortement chargée ont suggéré l’existence de deux types de groupes silanol.

Le quatrième chapitre présente l’étude de la formation de films multicouches composés de macromolécules organiques actives d’une manière photochimique sur une surface d’or. Les multicouches ont été formées en appliquant une technologie couche-par-couche, c’est-à-dire par l’absorption séquentielle sur la surface de macromolécules de charges opposées. La résonance plasmon de surface et la microbalance à quartz de crystal ont été utilisées pour étudier la croissance de la masse absorbée couche-par-couche. Il a été démontré que la combinaison des données obtenues par les deux techniques permet de caractériser les

6 propriétés des films multicouches et de prédire l’architecture supramoléculaire de ces dernières.

Le développement d’une méthode de caractérisation complète des couches nanométriques absorbées est présenté dans le cinquième chapitre. L’absorption des monocouches de polyélectrolytes sur des substrats de silice a été étudiée en parallèle par la réflectométrie et la microbalance de quartz. L’épaisseur de ces couches et la quantité incorporée d’eau ont été calculées. Une étude systématique de l’absorption des polyélectrolytes dans différentes conditions salines a démontré que dans les solutions d’une force ionique élevée les polyélectrolytes s’adsorbent dans une structure de pelote alors dans les solutions d’une force ionique faible ils absorbent dans une structure aplatie. L’étude de la dépendance du pH de l’absorption de la polylysine a illustré le rôle importante les transitions de conformation.

Le chapitre six relate l’étude du comportement de denrimères absorbés sur une surface de silice. L’imagerie par la microscopie à force atomique et les mesures par la microbalance à quartz ont démontré que les modifications de la composition de la solution modifient l’état de conformation des denrimères. Dans des conditions de faible pH et haute salinité, les denrimères ont une conformation gonflée, alors que dans des conditions opposées les dendrimères adoptent une conformation aplatie. Ce phénomène a été interprété par les interactions électrostatiques entre les denrimères et la surface.

Le septième chapitre présente l’étude approfondie du changement de conformation des denrimères sur des substrats de silice. La variation de la masse absorbée des couches de denrimères dans les solutions de composition différentes a été étudiée en parallèle par la réflectométrie et la microbalance de quartz. Les changements de conformation des denrimères en fonction du pH et de la concentration de sel ont été mis en évidence. Il a été démontré que le gonflement des dendrimers est spécifique aux ions présents et dépend de la nature du cation contenu dans le milieu électrolyte.

Le huitième chapitre contient les conclusions de base sur l’ensemble de la thèse.

7

CHAPTER 1

INTRODUCTION

1.1. Adsorption of ions

Adsorption of ions takes place on charged surfaces immersed in electrolyte solutions. Surface charges cause an electrical field, which attract the ions with the opposite charge (counter ions). Localization of the counter ions in the vicinity of a charged surface results in the formation of electrical double layer (EDL). In the simplest model, the counter ions bind strongly to the surface and neutralize the surface charge just like in a plate capacitor. The layer of adsorbed ions is called Helmholtz layer in honor of Ludwig Helmholtz and his work on electric capacitors.

The physical model of EDL was further improved by Gouy and Chapman, who took into account the thermal motion of the ions, which lead to the concept of the diffuse layer. In their model the ions are considered as point charges distributed in the continuum medium with the dielectric constant ɛ. The surface is homogeneous with a uniform charge density σ. The surface charge generates the surface potential ψ . Surface charge density and potential are related by the Poisson equation:

8

2 2 2

2

2 2 2

0 e

x y z

ρ

ψ ψ ψ

ψ εε

∂ ∂ ∂

∇ = + + = −

∂ ∂ ∂ (1)

where ρ_e is the local charge density, ε is dielectric constant, and ε₀ is the dielectric permittivity in vacuum . The spatial distribution of ions c_i is described by Boltzmann equation:

/ 0

i B

W k T

ci =c e⁻ (2)

where c₀ is the bulk concentration of the monovalent electrolyte, k_Bis Boltzmann constant, T is a temperature, and W_iis the work required to bring an ion from the infinite distance to a certain position towards the charged surface, which is approximated by the electrostatic energy according to:

W⁺ =eψ (3)

W⁻ = −eψ (4)

where W⁺ is the electric work required to bring the cation to a place with potential ψ , W⁻ is the corresponding value for anion, eis the elementary charge.

Local charge density ρ_e can be equally expressed in the following way:

( ) 0 ( ^{B B} )

e e

k T k T

e e c c c e e e

ψ ψ

ρ = ⁺− ⁻ = ⁻ − (5)

Thus, the Poisson equation can be rewritten as:

2 0

0

( ^{B B} )

e e

k T k T

c e e e

ψ ψ

ψ εε

∇ = − − (6)

which is the so called Poisson-Boltzmann equation. For low potential (≤25 mV) one can expand the exponential functions into a series and neglect all but the first term:

9

2

2 0

0

2

B

c e ψ k Tψ

∇ =εε (7)

Further we will consider only planar substrates, meaning that potential changes only in one direction. The general solution of eq. 7 is:

1 2

( )x C e ^κx C e ^κx

ψ = ⁻ + ⁻ (8)

2 0 0

2

B

c e κ k T

= εε (9)

κ⁻1 is a so-called Debye length that defines the thickness of the diffuse layer. The constants C₁ and C₂ are obtained from the boundary conditions:

(x 0) 0, (x ) 0

ψ = =ψ ψ → ∞ = (10)

Thus, the electric potential ψ profile is obtained as:

0

e ^κx

ψ ψ= ⁻ (11)

The surface charge density σis related to the electric charge density ρ_e according to the electroneutrality conditions:

2

0 2 0

0 0

( 0)

e

d d

dx dx x

dx dx

ψ ψ

σ = −^∞

∫

∫

The famous Grahame equation which describes charge/potential relations:

0 0

2 sinh( )

2

B

B

e k T

e k T

σ = εε κ ψ (13)

Gouy-Chapman model of EDL was further modified by Stern, who divided double layer into two parts: the inner part, the Stern layer, and an outer part, the diffuse layer. Stern layer is a layer of counter ions that are directly adsorbed on the surface and are immobile. At the same time, diffuse layer consists of mobile counterions. Introduction of the immobile layer of

10 adsorbed counter ions leads to the additional parameter of EDL, capacitance C_S, which is defined as

0 S

d

C σ

=ψ ψ

− (14)

where ψ_d is the diffuse layer potential.

The amounts of adsorbed ions Γ⁺ and Γ⁻ depend on the ψ_d according to:

2 0

( 1)

d B

e A k T

N c e

ψ

κ

+ −

Γ = − (15)

2 0

( 1)

d B

e A k T

N c e

ψ

κ

Γ =− − (16)

where N_A is Avogadro number.

Particular attention in the present work was given to the silica/water interface. Charging behavior of silica can be interpreted in terms of 1-pK basic Stern model. This model takes into account the charging of silica according to the following reaction of deprotonation of silanol groups:

K - +

SiOH←→SiO +H (17)

The mass action law can be written as:

0

- +

SiO H

SiOH

B

e

a k T

Ke Γ ψ

Γ = (18)

where Γ_SiO−and Γ_SiOHare the site densities of deprotonated and protonated form of silanol groups, respectively, a_H+is the activity of protons and Kis the dissociation constant. The total site density is defined according to:

0 SiO- SiOH

Γ = Γ + Γ (19)

11 The surface charge density σis related to the density of deprotonated silanol groups:

SiO-

σ = − Γe (20)

Eq. 13-16 and 18-20 form a system of nonlinear equations that can be solved numerically, giving an accurate description of charging behavior of silica/water interface.

The Stern model is valid only for monovalent ions, low surface charges and low electrolyte concentrations. For strongly coupled systems, namely the systems containing multivalent ions, high salt level and high surface charge densities, this model fails to precisely describe the formation of EDL. Thus, Stern model is unable to explain the phenomenon of overcharging, caused by strong accumulation of counter ions close to the charged surfaces, which was observed experimentally in many different systems^1,2. This is due to the fact that described theory does not take into account ion-ion correlations, which become essential for multivalent ions and concentrated electrolyte solutions. For the exact description of EDL one should use more accurate theory³ or Monte-Carlo simulations⁴.

1.2. Adsorption of polyelectrolyte monolayers.

Polyelectrolytes (PEs) are polymer chains that contain ionizable functional groups.

Depending on the sign of their charge one refers to polyanions, polycations and polyampholytes. Another classification divides PE on weak and strong. Strong PE, such as poly(styrene sulphonate) (PSS) or poly(diallyldimethylammonium chloride) (PDADMAC), are fully charged independently of pH. On the contrary, ionization of weak PE, such as polylysine (PLL) or poly(allylamin hydrochloride) PAH, is pH dependent and can be described by a dissociation constant.

PE chains are flexible and attain different conformations in solution. In poor solvents with low affinity to PE segments, PE chains tend to be in a compact conformation in order to minimize the area of contact between the chains and the solvent. In good solvents, on the contrary, PEs tend to assume the conformation of a swollen coil, as the contact with the solvent becomes energetically favorable. Ionic strength of the solution also affects the

12 conformational state of PEs. At low ionic strength the charges of PEs are weakly screened, forcing the chains to stretch in order to maximize the distance between the charges. High salt conditions lead to the screening of the PE charges, thus, PEs form more compact structures similar to neutral polymers in good solvents.

Adsorption of PEs on solid substrates plays a crucial role in numerous industrial applications, including papermaking⁵, bioengineering, ceramics and water treatment⁶. For this reason adsorption has been extensively studied during the past four decades^7-12.

Some of the theoretical models of PE adsorption are based on a self-consistent-field theory of Scheutjens and Fleer ¹⁰. In this theory, the shape of the concentration profile near a surface is is found by minimization of free energy. In this way the electrostatic contributions to the free energies directly affect the concentration profile⁸. The approach was introduced by Van der Schee and Lyklema¹³. They showed that strong electrostatic repulsion between charged monomers is responsible for the formation of thin PE layers. On the contrary, when this electrostatic repulsion is screened, PEs form thicker layers.

In general, adsorption of PEs is determined by several factors. First of all, the adsorption is dictated by electrostatic and non-electrostatic interactions between the PE chains and the surface. Consequently, the parameters, which tune these interactions, affect the amount of the adsorbed PE. An important factor is also a gain in entropy, as adsorption of PE chains is followed by the release of the solvent molecules. Finally, one should not forget about the interaction between the PEs and the solvent. Poor solvent favors the adsorption of PEs, while a good solvent opposes the adsorption.

The structure of the adsorbed layer depends on the conformational state of the adsorbed PE.

PE chains with low charge density attain coiled conformation and the corresponding layers are porous and floppy. Fully charged PEs tend to adsorb in a strained conformation in order to maximize the inter-chain distance, which leads to the formation of thin and rigid layers. High charge density of PE chains results in a smaller adsorbed masses of PEs needed for complete neutralization of surface charge. Thus, the external conditions that affect the charging state of PE chains determine the structure and the amount of the adsorbed mass.

13 The salt level is an important parameters of PE adsorption and it can either increase or decrease the adsorption, depending on the conditions¹⁴. High concentration of electrolyte screens segment-segment repulsion in the PE chain, and PEs adsorb in coiled conformation, meaning that more PE chains will fit on the surface. At the same time the segment-surface attraction is also screened. Thus, in the process of PE adsorption added salt has two antagonistic effects and it depends on the balance between electrostatic and nonelectrostatic attraction whether or not the increase of the salt concentration leads to an increase or a decrease in the amount of adsorbed mass.

Variations of pH can equally affect the amount of adsorbed PEs. This parameter is particularly important for weak PEs, for which the charged fraction of segments is pH dependent. However, if the charge of the surface originates from dissociation of surface groups, pH of the solution affects the net surface charge as well. In certain systems pH- induced charging of PE chains is opposed by discharging of the surface.

The molecular mass dependence of the amount of PE adsorbed on a solid substrate depends on the type of the system studied. For the adsorption of PEs on uncharged surfaces an increase of adsorbed ammount is observed with the increase of molecular mass, while for charged substrates, the adsorbed mass appears to be independent from the PE molecular mass, which was theoretically predicted and observed experimentally^15,16. In systems of relatively weak interactions between PE chains and a substrate, energetic effects are small and adsorption is mainly controlled by adsorption entropy, meaning that chains will be mainly in

“loop and tail” conformation. Charged surfaces promote the adsorption of PE chains in flat conformation with the segments mainly forming the trains in direct contact with the surface and entropic contributions become less important.

1.3. Multilayer systems formed by layer-by-layer (LBL) technology

Since the introduction of LBL technique by Decher in 1992¹⁷, creation of different multilayered systems with programmed properties has become a rapidly developing field of surface chemistry. The driving force of multilayer assembling is the electrostatic attraction

14 between oppositely charged components, which can be PEs^18-20, proteins^20-22, enzymes^23-25, nanoparticles^25-27, and charged organic molecules^28,29. The procedure of preparation of multilayered systems is based on sequential adsorption of positively and negatively charged compounds on solid substrates, performed either by dipping the surface in the solution¹⁷ or spraying the compounds on the surface^30,31. The properties of multilayered systems are dictated by the components of the system as well as their spatial organization in the layer.

Due to the simplicity of the procedure and almost limitless variety of possible compounds, LBL technology has stimulated investigations of possible applications of the obtained multilayered systems, including modification of surfaces³², formation of biosensors^33-35, modification of chromatography columns and membranes^36-38, production of light-emitting and photovoltaic devices^39-41, and batteries⁴².

Study of adsorption of thin layers on solid substrates has become possible with the development of surface sensitive techniques, capable to detect adsorbed species at low adsorption densities. The following section is devoted to the description of physical bases of several surface sensitive techniques, most commonly used for adsorption studies.

1.4. Surface sensitive techniques 1.4.1. Quartz Crystal Microbalance (QCM)

Among surface sensitive techniques QCM is one of the most easy to handle. It belongs to the class of mechanical sensors and explores the principle of piezoelectricity. The phenomenon of piezoelectricity was first discovered by brothers Curie in 1880 via a pressure effect on quartz.

Almost a century later Sauerbrey showed the linear relationship between the oscillation frequency of the quartz crystal and the mass deposited on it⁴³. Quartz crystal became a sensing platform for detecting the adsorption initially from gas phase. First QCM sensors were used for detecting moisture, volatile organic compounds⁴⁴ and environmental pollutants⁴⁵. Later on, QCM has been shown to operate in contact with liquids, enabling its application for measuring the mass changes associated with liquid/solid interface phenomena.

Since then the number of QCM applications began to grow progressively, in particular in the field of biotechnology and immunosensing.

15 The heart of QCM sensor is a quartz plate trapped between two gold electrodes. A piezoelectric quartz crystal resonator is a slab of natural or synthetic crystal of quartz precisely cut under 35°10’ angle from the Z-axis (AT-cut)⁴⁶ (Fig. 1).

Fig. 1. AT-cut of a quartz crystal. The cut forms 35°10’ angle from the Z-axis (taken from ref 46).

The cut angle determines the mode of induced mechanical vibration. AT-cut provides a pure shear motion of the crystal once it is subjected to an external electric field. Moreover, AT-cut quartz plates show tremendous frequency stability and a temperature coefficient which is close to zero between 0°C and 50°C, making these particular cuts the most suitable for QCM applications.

Being a piezoelectric material, quartz undergoes mechanical strain once an external electrical field is applied. One can rationalize this effect by picturing the atoms in the crystal lattice as ordered dipole moments (Fig. 2a). When an electric field is applied across the crystal, the dipoles tend to align along the direction of this external field and this alignment induces a deformation of the crystal.

16

Fig. 2. a) Phenomenological explanation for the converse piezoelectric effect; b) Schematic representation of the quartz crystal sensor. The bottom diagram shows a section of the crystal (not to scale). Two oscillating standing waves corresponding to the fundamental and first overtone resonance frequencies are drawn on top of the crystal section.

The alternating electric field results in a generation of shear waves of opposite polarity at the electrodes on both sides of the quartz crystal with the shear displacement in plane with the crystal surface. Both waves traverse across the crystal plate and are reflected at the opposing crystal face (phase shift 180°) and then return to their origin⁴⁷. The resonance condition of the QCM sensor occurs when the generated standing wave between the electrodes is an odd integer of the thickness of the plate. The resonance frequency can be expressed as

0 2

f nv

= d

(21)

where ν is the speed of sound in the crystal, d is the thickness of the sensor, and n = 1, 3,

… is the overtone number. Overtones are resonant frequencies higher than the fundamental frequency of the crystal. In QCM sensing, the fundamental frequency is rarely used due to its substantial noise. Higher overtones of the QCM sensor are preferred since they are characterized with much lower noise. The reason is due to the fact that overtone vibrations are more concentrated in the centre of the crystal and probe smaller surface area⁴⁸. Resonant frequencies of typical QCM crystals are of the order of MHz, normally in the range of 5-20 MHz, with higher resonant frequencies corresponding to thinner crystals.

a) b)

17 When mass is deposited on the crystal surface forming a rigid and uniform layer, it leads to the increase of the thickness of the resonator and, as a consequence, decrease of the resonance frequency. The relation between the resonance frequency and the amount of deposited mass is described by Sauerbrey equation⁴³:

( )

2 2

2 / _{q f}

f mf A µρ C m

∆ = − ∆ = − ∆ (22)

where ∆f is measured resonance frequency decrease (Hz), f is the intrinsic crystal frequency, ∆m is the elastic mass change, A is the electrode area, ρ_q is the density of quartz, and µ is the shear modulus. C_f is the summarized sensitivity constant. For the QCM sensors purchased in Q-Sense and used in the present study C_f = 0.177 mg Hz^-1m^-2. It is essential to note that the Sauerbrey equation is valid only for rigid uniform layers adsorbed on the crystal surface and it breaks down once the deposited mass exceeds 2% of the mass of the quartz crystal.

Once the excitation is switched off, the generated oscillation of the quartz crystal decays. The decay rate of the oscillation depends on the properties of the quartz resonator as well as on the properties of the contact medium. Fitting the voltage of oscillatory decay yields an additional parameter that characterizes the properties of the adsorbed layer, which is dissipation D⁴⁹. The dissipation describes the damping in the system and is defined as:

2π

= ^diss

st

D E

E (23)

where E_st is the energy that is stored in the oscillating system and E_dissis the energy which is lost during one oscillating cycle.

Dissipation is used to characterize the viscoelastic properties of the surrounding liquid medium as well as of the adsorbed layer. Dense and highly viscous liquids cause the fast decay of quartz plate oscillation and high dissipation values. The same effect concerns thick and non rigid layers, which oppose the vibrational motion of the crystal, resulting in a faster

18 decay of the amplitude of vibration. For

106

D 0.25 f

∆ × >

∆ layers are considered to be non rigid. Voinova showed that viscous loss of energy in the layers causes the deviation from Sauerbrey behavior and results in a non-trivial reduction of the measured adsorbed mass⁵⁰. More complicated model should be applied in order to describe the system. This model was initially proposed by Rodahl and Kasemo. The quartz crystal is considered as a Voigt element and Navier-Stokes equations are used to calculate both the resonant frequency and dissipation for the viscous layer⁵¹. This model was further adapted by Voinova who derived the equations for frequency shift and dissipation for viscoelastic layers in a bulk liquid, assuming quartz crystal to be purely elastic, and the surrounding solution to be purely viscous. Other assumptions include the uniformity of the layer thickness and density, the frequency independence of the viscoelastic properties and absence of slip between the adsorbed layer and the crystal⁵². Within this model ∆f and ∆Dare given by:

Im( ), Re( )

2 _{q q q q}

f D

d f d

β β

πρ π ρ

∆ = ∆ = − (24)

where β is defined as

1 1

1

2 1 exp(2 )

2 1 exp(2 )

π η µ α ζ

β ζ π α ζ

− −

= +

f f f

f

f i d

f d

(25) The parameters α, ξ₁, ξ₂ are defined as

1 2 1 2

2 1

2

2 1

2

π η µ

ζ

ζ π η

α ζ π η µ

ζ π η

− +

= −

−

f f

l

f f

l

f i

f

f i

f (26)

2 1

(2 ) 2

π ρ

ζ = −µ π η +

f

f f

f i f

(27)

19

2

ζ 2π ρ

= η ^l

l

i f

(28)

In these equations f is the resonant frequency, ρ_f, η_f, µ_fand d_f are the density, viscosity, shear modulus and thickness of the layer, respectively, ρ_q and d_q are density and thickness of the quartz plate, respectively, and η_l and ρ_l are viscosity and density of the bulk liquid, respectively. Parameters of the adsorbed layer are determined by adjusting their values in order to find the best fitting of experimental data, once the parameters of the quartz plate and bulk liquid are known. The application Q-Tools from Q-Sense was developed specifically to solve this mathematical task.

Sensitivity of the QCM method is around 0.1 mg/m², which is sufficient for studying the biochemical interactions normally involving large molecules like proteins. However, this technique is several orders less sensitive compared to optical methods, which are going to be discussed further.

1.4.2. Optical surface sensitive techniques

Before starting the discussion about optical techniques we will first revise certain concepts of elementary optics.

Polarization of light

Light is an electromagnetic wave with electrical (E

) and magnetic (B

) components oscillating perpendicular to each other and perpendicular to the direction of wave propagation⁵³. Changes in the trajectory of electrical vector corresponding to a fixed point in the space with time determine the state of the polarization of light and is described by the amplitudes of x- and y- components of the electrical vector of light travelling in the positive z- direction in a right-handed x-y-z coordinate system with the magnitudes E_x and E_y and

20 their phases δ_x and δ_y. The most convenient way to describe the state of polarization of light is encoding it into the column vector with complex components known as Jones vector⁵⁴:

x

y

i

x x

i y

E e E

E Ey e E

δ δ

   

 

=   =  

 (29)

The state of the polarization of light thus can be:

- linear, when δ_x−δ_y =0( )π ;

- circular, when δ_x−δ_y = ±π / 2and E_x = E_y ;

- elliptical, whenδ_x≠δ_y and E_x ≠ E_y .

Linearly polarized light whose electrical vector executes simple-harmonic oscillations along the x-axes will be presented by the following Jones vector:

0

1 E E  0

=  

 

 (30)

where E₀ is the magnitude of the electrical vector. More generally, orthogonal linear polarizations with its direction of oscillation inclined to the x-axes by the angle α is:

0

cos E E sinα

α

 

=  

 

 (31)

Another pair of orthogonal polarizations are the right-handed circular polarization:

0

1 1 E E 2

i

=   

 

 (32)

and left-handed circular polarization:

0

1 2 1 E E  i

=  

 

 (33)

21 Finally, the following two normalized Jones vectors represent two orthogonal elliptic polarizations for any complex number χ.

0 2

1 1 1

E E

χ χ

=   

+  

 (34)

*

0 2

1 1 1

E E χ

χ

− 

=  

  +

 (35)

Fresnel equations and Abelés matrix formalism

Fresnel equations describe the reflection and transmission of light at the interface of two media with different refractive indices. Particularly, Fresnel equations determine the reflection r and transmission t coefficients for the p- and s- components of the reflected and transmitted light, respectively⁵³.

The light is incident at the interface of two media with refractive indices n₁ and n₂. The angle of incidence of light isϕ₁, the angle of transmittance is ϕ₂. These parameters are related according to the Snell’s law:

1sin 1 2sin 2

n ϕ =n ϕ (36)

The fraction of light reflected from the interface is determined by reflection coefficients for p- and s- components of the electrical vector of the light beam:

1 2 2 1

1 2 2 1

cos cos

cos cos

p

n n

r n n

ϕ ϕ

ϕ ϕ

= −

+ (37)

1 1 2 2

1 1 2 2

cos cos

cos cos

s

n n

r n n

ϕ ϕ

ϕ ϕ

= −

+ (38)

The fraction of transmitted light is determined in the similar manner by transmission coefficients:

22

1 2

1 2 2 1

2 cos

cos cos

p

t n

n n

ϕ

ϕ ϕ

= + (39)

1 1

1 1 2 2

2 cos

cos cos

s

t N

N N

ϕ

ϕ ϕ

= + (40)

The reflectance is further determined as:

2

p p

R = r (41)

2

s s

R = r (42)

which give the fraction of the total intensity of an incident plane wave that appears in the reflected wave for p and s polarization. As a consequence of energy conservation transmittance for non absorbing materials is determined as:

p 1 p

T = −R (43)

s 1 s

T = −R (44)

Fig. 3 shows the variation with the angle of incidence of reflectance R_p and R_s. Two cases are considered. In the first case the light passes from the medium of lower refractive index and strikes the denser medium with higher refractive index. The second case represents the inverse situation.

23

Fig. 3. Dependence of p- and s- components of reflectance on the angle of incidence. Two cases are presented:

the light passes from the medium of lower refractive index and strikes the denser medium with higher refractive index and vise versa.

While R_s increases monotonously with increasing angle of incidence, R_p passes through minimum at the Brewster angle Θ_B. When the light is incident from the side of high index of refraction, total internal reflection takes place at angles larger than a critical angle. As the incident angle ϕ₁ in denser medium increases, the refraction angle ϕ₂ in rarer medium also increases according to the Snell’s law (eq. 36). At critical angle ϕ_c, the angle of refraction becomes 90°. The light beam that has the incident angle higher than the critical angle is totally reflected form the interface.

Reflectometry and ellipsometry measurements are normally performed at an angle close to Brewster angle. Despite the lower intensity of the p-component of reflected light, one achieves maximum of sensitivity at this angle⁵³.

The case of particular importance, in which the substrate is covered by a thin film, will be now discussed in detail. Fig. 4 illustrates the system where the film with the thickness d₁ is sandwiched between the ambient and the substrate media. It is assumed that the film, the substrate and the ambient medium are homogeneous and optically isotropic, with refractive indexes n , ₁ n₀ and n₂, respectively.

24

Fig. 4. Reflection and transmission of the light beam by an ambient(0)-film(1)-substrate(2) system with parallel-plane boundaries. d₁ is the film thickness, ϕ₀ is the angle of incidence in the ambient and ϕ₁^, ϕ₂ ^are

the angles of refraction in the film and substrate, respectively

Once a plane wave passes through the medium 0 and is incident on the substrate covered with a thin film at the incident angle ϕ₀, part of it is reflected in the same medium, and part is refracted in the film, according to the Fresnel equations (eqs. 37-40). The refracted light inside the film further undergoes multiple reflections at the interfaces between the interfaces film/ambient medium and film/substrate, which are not perfectly reflecting. It leads to the partial refraction of the wave each time it strikes the interface⁵³. The phase change that the multiply-reflected wave inside the film experiences as it traverses the film from one boundary to another is given by Airy’s formula⁵³:

1 1 1

2πd n cos

β ϕ

= λ (45)

where λ is the wavelength of the light beam, d₁ is the film thickness. The overall complex- amplitude reflection and transmission coefficients for the ambient-film-substrate system are defined as follows:

2

01, 12,

2 01, 12,

1

i

p p

p i

p p

r r e

R r r e

β β

−

−

= +

+ (46)

2

01, 12,

2 01, 12,

1

i

s s

s i

s s

r r e

R r r e

β β

−

−

= +

+ (47)

25

01, 12, 2 01, 12,

1

i

p p

p i

p p

t t e

T r r e

β β

−

= −

+ (48)

01, 12, 2 01, 12,

1

i

s s

s i

s s

t t e

T r r e

β β

−

= −

+ (49)

where r_01,p, r_12,p,r_01,s, r_12,s, t_01,p, t_12,p, t_01,s, t_12,sare Fresnel reflection and transmission coefficients for the interface 0-1 and 1-2 which are calculated according to eqs. 37-40. The three angles ϕ₀,ϕ₁ and ϕ₂ between the directions of propagation of the plane waves in media 0, 1 and 2 respectively and the normal to the film are interrelated through the complex refractive indices of the media according to the Snell’s law⁵³ (eq. 36).

For multiple isotropic layer system with homogeneous layers, the calculation of reflection coefficients is more complicated. The most commonly used approach is Abelés matrix formalism⁵⁵. It relies on the 2 x 2 matrix representation of each layer. Thus, the j^th layer of the multilayer system is represented by two transfer matrices:

,

cos cos sin

P

sin cos

cos

j

j j

j j p

j

j j

j

i n i n

β ϕ β

β β

ϕ

 

 − 

 

=  

 

 

 

(50)

,

cos sin P cos

cos sin cos

j j

j j

j s

j j j j

in

in

β β

ϕ

ϕ β β

 

 

=  

 

 

(51)

where β_jis the phase shift for j^th layer, calculated using eq. (45), n_jis a complex refractive index of the j^th layer, and ϕ_j is the angle of refraction in the j^th layer, given by the Snell’s law (eq. 36).

26 The characteristic matrix, which describes the overall optical properties of the system of S films, is determined by:

0, , 2,

1

M ( )

S

p p j p p

j

χ P χ

=

=

∏

0, , 2,

1

M ( )

S

s s j s s

j

χ P χ

=

=

∏

Theχ₀ and χ₂ matrices are the characteristic matrices for the ambient and the substrate, respectively, and are given by:

0

2 0

0, 2, 2

0 0

1 cos cos

1 1 0

cos ,

2 2

1 1 0

p p

n n

n

ϕ ϕ

χ χ

ϕ

 

 

 

 

 

= −  =  

 

(54)

0 0

2 2

0, 2,

0 0

1 1 1

cos 0

1 1

, cos 1

2 2

1 1 0

cos

s s

n n

n

ϕ ϕ

χ χ

ϕ

 

 

 

 

 

=  −  =  

 

(55)

The complex reflection coefficients are then calculated from the elements of the characteristic matrices according to:

21, 11, p p

p

r M

= M (56)

21, 11, s s

s

r M

= M (57)

Abelés matrix formalism mathematically describes the relation between the reflectance and optical characteristics of the adsorbed film and is widely used in experimental data interpretation obtained by optical surface sensitive techniques.

27 Optical elements

Polarization of light can be achieved when the light beam passes through the special optical element called polarizer. Ideal linear polarizer is a device that transforms any state of light polarization into a linear state. There are three principal groups of linear polarizers selected according to the physical mechanism by which one of the two orthogonal components of light is rejected. These groups include double reflection polarizers, dichroism polarizers, and reflection polarizers⁵³.

Double reflection polarizers consist of transparent uniaxially or biaxially anisotropic crystals.

A common example of double reflection polarizer is a Wollaston prism, which consists of two calcite prisms that are cemented together by optically transparent cement. The optic axes of the two prisms are perpendicular to each other and perpendicular to the direction of propagation of the incident light. Light striking the surface of the prism is refracted in ordinary (o) and extraordinary (e) components, which however continue to propagate in the same direction. As optical axes of the prisms are perpendicular to each other, once the light reaches the interface between the prisms, the e-component of light becomes an o-component in the second prism and vice versa. Due to the difference in the refractive indexes for o- and e- components of light the beams diverge from the prism giving two spatially separated polarized rays (Fig. 5a).

28

Fig. 5. a) The principle of Wollaston prism. the arrows show the optical axes of each part of the prism, the inset shows the prism from the side. The optical axes of the right part is perpendicular to the page; b) Reflection polarizer. At Brewster angle the intensity of reflected light is minimal.

Another type is dichroic polarizer. In certain absorbing media, the attenuation is dependent on the direction of linear polarization (linear dichroism)⁵³. The absorption achieves its maximum and minimum when the direction of the vibration of electrical field of light is along two orthogonal directions. The light beam that comes out of the polarizer has the vibration of its electric component only in one plane which corresponds to the optical axes of the polarizer.

The third type of polarizers explores the Brewster angle principle. The light incident on the dielectric surface at Brewster angle is reflected with the minimal intensity of its parallel component towards the plane of incidence (Fig. 5b).

The existence of the Brewster angle (Θ_B) can be interpreted by the interaction of p-polarized light with the dielectric medium. Once the incident p-polarized light is absorbed by the surface, it is reradiated by the oscillating electric dipoles in the dielectric medium close to the interface. The oscillating dipoles produce the reflected light (Fig. 6). However, dipoles do not produce any light in the direction of the dipole moment. At certain angle of incidence Θ_B the direction of refracted light is perpendicular to the direction of light which is supposed to be reflected. The dipoles cannot generate the light energy, and the intensity of reflected light drops to zero.

a) b)

29 Besides linear polarizers one should also mention compensators (retarders), which are used to convert linear polarization of light into elliptical one. This is achieved by delaying one of the two orthogonal components of the electrical field of light. The compensator is a birefringent crystal with carefully chosen thickness and orientation of the optical axes along the crystal plane. Thus, the component of electric field of light parallel to the optical axes of the compensator travels inside the crystal with the different velocity than the orthogonal one. The anisotropy of the crystal causes the retardation of one of the components and creates a phase shift and, as a result, the change in the polarization state. The thickness of the crystal of the compensator is adjusted to get the desired phase shift. One can select quarter-wave plates which shift the phase by the quarter of the wavelength. By adjusting the direction of linear polarization of the incident light to 45° in respect to the optical axes of the compensator, one obtains circular polarization. The other common type of the compensators is half-wave plates, which retard one of the components of light by half of the wavelength. Such plates change the direction of polarization of linearly polarized light.

The convenient way to describe the effect of optical element on the state of polarization of light is the application of Jones calculus. Each optical element can be presented by 2 x 2 matrix. Examples include:

- linear polarizer with the azimuth along the x-axes:

1 0 0 0

 

 

  (58)

- linear polarizer with the azimuth along the y-axes:

0 0 0 1

 

 

  (59)

- linear retarder with the fast axes parallel to x-axes advancing the slow axes by δ_R:

1 0

0 e^−iδR

 

 

  (60)

30 - quarter-wave plate with the phase shift δ_R =π/ 2 and azimuth α :

1 cos 2 sin 2 1

sin 2 1 cos 2 2

i i

i

i i

α α

α α

 + 

−  −  (61)

The simple diagonal matrix is valid only in the coordinate system of the component. Once the transformation between the coordinate system is required one needs the rotation matrix

( ) Rα :

cos sin

sin cos

α α

α α

 

 − 

  (62)

where α is the angle of rotation. The polarization state of light after passing through the optical element can be expressed by multiplication of the corresponding Jones matrix and the Jones vector of the incident light.

Jones matrix can also describe the effect of the isotropic surface with the reflection coefficients r_p and r_s on reflected right:

0 0

p s

r r

− 

 

  (63)

Jones matrix formalism is a convenient mathematical tool to describe any optical systems.

The example will be given further for the ellipsometry setup.

1.4.2.1. Ellipsometry

Ellipsometry is an optical surface sensitive technique capable to probe the properties of thin layers⁵³. Ellipsometry was first developed by Drude in 1887, who also derived the ellipsometric equations. First ellipsometers were operated manually and the measurements were time-consuming. Only in 1970s the complete automation of ellipsometry was performed making this technique appropriate for routine analysis of thin layers.

31 Ellipsometry is based on the measurement of the state of polarization of light upon reflection from the surface. The reason of the change in the polarization state of reflected light is in the difference in currents induced by the p- and s-components of incident light in isotropic material. The change in polarization state can be described by two ellipsometric angles Ψ and ∆. Ψ describes the change in the ratio of the amplitudes of p- and s-components of electric field according to:

tan /

/

r i

p p

r i

s s

E E

E E

Ψ = (64)

where index i corresponds to the incident light and r – to the reflected light. Angle Δ describes the shift in phase between p- and s-components as:

(δ_p^r δ_s^r) (δⁱ_p δ_sⁱ)

∆ = − − − (65)

The reflectivity properties of the sample are described by the reflectivity coefficients r_p and r_s, which depend on the change in phase and amplitude of the reflected electric field E^r according to:

( ^r_p ⁱ_p) r

p i

p i

p

r E e

E

δ δ−

= (66)

( _s^r _sⁱ) r

s i

s i

s

r E e

E

δ δ−

= (67)

Thus, the basic equation of ellipsometry is obtained:

tan ^{i p}

s

e r

ψ ⋅ ^∆ = r =ρ (68)

where ρ is the complex reflectance ratio. Eq. (68) relates the experimentally measured ellipsometric angles with reflectivity coefficients of the optical system. Optical characteristics of the system including thickness and refractive index of the top layer of the sample are set as

32 adjustable parameters in order to find the best fit between experimentally measured and theoretically calculated reflectance predicted by Fresnel theory within Abelés matrix formalism. It is important to note, that in case of very thin films with the thickness less than 20 nm simultaneous determination of both of the optical parameters of the adsorbed film (d₁ and n₁ in Fig. 4) tends to be impossible due to insufficient sensitivity⁵⁶. In this case, one of the parameters has to be fixed. Normally, the thickness of the layer is fixed, and the adjustable parameter in the system is a refractive index of the film.

Typical ellipsometry setup is depicted on Fig.6. It consists of laser, polarizer, retarder (compensator), analyzer and detector (PCSA-configuration). The arm with laser, polarizer and compensator produces the known polarization state of light incident on the sample. The arm with the analyzer and detector is used to detect the change in the polarization state once the light is reflected from the surface.

Fig. 6. Polarizer-compensator-sample-analyzer (PCSA) configuration of an ellipsometer. Position of the polarizer and analyzer are adjusted in order to minimize the intensity of the light that reaches detector (null- ellipsometry mode).

The monochromatic light passes through the linear polarizer and obtains linear polarization state. It is further converted into circularly polarized beam by the quarter-wave plate used as compensator. The compensator is positioned in such a way, that the optical axes of the compensator crystal makes 45° with the plane of incidence. Elliptically polarized light strikes the surface and is reflected with the modified state of the polarization. After reflection the light beam passes through the analyzer and enters the detector. Ellispometric measurement involves the adjustment of the polarizer in such a way that elliptically polarized light beam incident on the surface is reflected with linear polarization state. The position of analyzer is