in fulfilment of the requirements of the PhD Degree in Physics

Mechanisms of irreversible adsorption

Nonequilibrium vs finite size effects

Thesis submitted by Weide HUANG

in fulfilment of the requirements of the PhD Degree in Physics

“Docteur en Physique”

Academic year 2019-2020

Supervisor: Professor Simone NAPOLITANO Laboratory of Polymer and Soft Matter Dynamics

Experimental Soft Matter and Thermal Physics

Thesis jury:

Mustapha TLIDI (Université libre de Bruxelles, Chair) Patricia LOSADA-PÉREZ (Université libre de Bruxelles, Secretary) Daniele CANGIALOSI (CSIC Donostia, Spain)

Michael WÜBBENHORST (KULeuven)

Simone NAPOLITANO (Université libre de Bruxelles)

ACKNOWLWDGEMENTS

I would like to express my special appreciation and thanks to my promoter Professor Dr. Simone Napolitano, you have been a tremendous mentor for me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist. Your advice on both research as well as on my career have been invaluable.

I would also like to thank my committee members, professor Mustapha Tlidi, professor Patricia Losada- pérez, professor Daniele Cangialosi, and professor Michael Wübbenhorst, for serving as my committee members even at hardship. I also want to thank you for letting my defense be an enjoyable moment, and for your brilliant comments and suggestions, thanks to you.

I would especially like to thank colleagues in the laboratory of polymer and soft matter dynamcs at Université libre de Bruxelles. Pascal Pirotte, Françoise Van Eycken, Dr. Daniel Martinez-tong, NataliaPerez-de-Eulate, Dr. David Nieto Simavilla, Dr. Anna Panagopoulou, all of you have been there to support me when I processing experiments and collected data for my Ph.D. thesis. Dr. Cristian Rodríguez-Tinoco, Dr. Allen Mathew, Dr. Kai Betlem and Zijian Song, thanks for encourage and valuable suggestions for defense. I would like to thanks the company of my friend, Qian Chen, Dong Yang, Yingli Chen, Huan Gao, Kainan Wang, Jiayin Gu, It’s your company make my life more wonderful. There are still many friend not listed here, but I would like to say thank you, thank you for all your help and suggestions.

I also would like to thanks the Chinese Scholarship Council for providing me the financial support, and helpful advice on my consultation.

Finally, but by no means least, I would like to my family. Words can not express

how grateful I am, the sacrifices that you’ve made on my behalf. I could never

graduate without all your self-giving supporting. I would also like to thank to my

beloved wife, Dr. Jing Li. Thank you for supporting me for everything, and

especially I can’t thank you enough for encouraging me throughout this

experience. To my beloved son Chuanyi Huang, I would like to express my

thanks for being such a good boy always cheering me up. You are the most

important people in my world and I dedicate this thesis to you.

Index

INDEX ... 5

CHAPTER 1 INTRODUCTION ... 7

1.1POLYMERS ... 7

1.2INTERFACIAL ENERGY AND WORK OF ADHESION ... 9

1.3THE HAMAKER CONSTANT ... 13

1.4MODELS OF POLYMER IRREVERSIBLE ADSORPTION KINETICS ... 15

1.4.1GUISELIN BRUSHES ... 16

1.4.2“PARKING” MODEL ... 17

1.4.3IRREVERSIBLE PHYSISORPTION MODEL ... 18

1.4.4TWO KINETIC REGIME MODEL ... 19

REFERENCES ... 21

CHAPTER 2 EXPERIMENTAL TECHNIQUES ... 23

2.1ELLIPSOMETRY ... 23

2.1.1BASIC EQUATIONS IN ELLIPSOMETRY ... 23

2.1.2BREWSTER ANGLE ... 26

2.1.3AN EXAMPLE ... 28

2.2ATOMIC FORCE MICROSCOPY ... 29

2.3SPIN-COATING TECHNIQUE ... 30

REFERENCES ... 32

CHAPTER 3 POLYMER ADSORPTION MECHANISMS ONTO SILICON SUBSTRATES ... 33

3.1INTRODUCTION ... 33

3.2POLYMERS AND SAMPLE PREPARATION ... 34

3.3RESULTS ... 36

3.4POLYMER ADSORPTION KINETICS: RESULTS AND DISCUSSION ... 38

3.4.1TOWARDS A NEW EXPRESSION FOR THE KINETICS OF ADSORPTION ... 39

3.4.2EFFECT OF TEMPERATURE ... 41

3.6CONSERVATION OF MONOMER ADSORPTION RATE ... 46

3.7CONCLUSION ... 47

CHAPTER 4 LONG RANGE INTERACTION AT INTERFACE ... 53

4.1MATERIALSANDMETHODS ... 59

4.1.1PREPARATION OF SINGLE POLYMER LAYER SYSTEMS ... 59

4.1.2PREPARATION OF POLYMER BULAYER SYSTEMS ... 60

4.1.3ADSORPTION EXPERIMENTS ... 60

4.1.4DETERMINATION OF L AND HADS VIA ELLIPSOMETRY ... 61

4.1.5DETERMINATION OF THE ADSORBED AMOUNT Γ∞ ... 62

4.1.6KINETICS OF IRREVERSIBLE ADSORPTION, DATA HANDING AND FITTING TO THE KINETIC MODEL ... 63

4.2NANOCONFINEMENT AT INTERFACE ... 64

4.2.1EFFECT OF NANOCONFINEMENT ON THE KINETICS OF IRREVERSIBLE ADSORPTION ... 64

4.2.2CALCULATION OF THE EFFECTIVE HAMAKER CONSTANTS ... 69

4.2.3RETARDATION EFFECTS ON THE EFFECTIVE HAMAKER CONSTANT ... 72

REFERENCES ... 78

CHAPTER 5 CONCLUSION ... 83

7

Chapter 1 Introduction

In my Thesis I investigated the physics of irreversible adsorption, focusing on the formation of the interface between a polymer melt and a solid substrate. In this chapter, I will provide an introduction to interfacial interactions, polymer melts and the existing models of the kinetics of adsorption.

1.1 Polymers

A polymer chain is composed of N repeating units called monomers, where N is the degree of polymerization. In this study, we have used linear polymers, as those sketched in Figure 1.1.1 This long chain structure is obtained via a series of chemical reactions called polymerization, which define the value of N of the synthesized polymer.

Figure 1.1.1 (a) Polymer coil, polymer chain of length 𝑙 is composed of 𝑁 monomers with monomer length 𝑏; (b) A long polymer chain in the form of a coil in 3D.

Real samples, however, contain chains of different size, characterized by a distribution in chain lengths, and consequently a distribution in molecular weight.

The distribution of molecular weight is frequently analysed via the polydispersity index (PDI), which refers to the ratio between the number average 𝑀𝑛 and weight average of molecular weight 𝑀𝑤 . By using the sample mass ( 𝑀 ) distribution function 𝑝(𝑀), the latter quantities can be evaluated as

𝑀

= 𝑝 𝑀 𝑀 𝑑𝑀 (eq1.1) 𝑀

=

(eq1.2)

(a) l

(b) b

!"

8

The limiting case of 𝑃𝐷𝐼 = 𝑀𝑛 𝑀

= 1 is reached when all the chains have the same length and, hence, the same mass. In this study, we used polymers of narrow molecular weight distribution with a PDI as small as ~ 1.03.

To characterize the average size occupied by a polymer of N monomers of length 𝑏, we introduce the concept of the ideal chain. In such object, the orientation of a given monomer is totally independent on the orientation of its consecutive monomers. An ideal chain, hence, corresponds to a random walk of N steps of length 𝑏 , and its end-to-end distance 𝑙 , see in figure 1.1.1a, is given by the relation [1]:

𝑙 = 𝑏𝑁

(eq1.3) This model, also known as freely rotating chain [2], is valid in the case of polymer melts considered in this Thesis.

While directly applicable in the case of linear chains, the end-to-end distance is not of practical use, just consider other common macromolecular architectures as those in Figure 1.1.2

Figure 1.1.2 Illustration of 4 typical polymer architectures: linear, star, ring, and comb shape.

To overcome this issue, in polymer physics we consider another characteristic length, given by the radius of gyration 𝑅

defined as

𝑅

=

∆𝑅

(eq1.4) where 𝛥𝑅

is the distance from a given monomer to the 𝑖

random monomer.

In the case of a linear chain of N monomers of length b, 𝑅

is given by:

𝑅

=

=

(eq1.5)

or a polymers of 10000 repeating units of length 1 nm, 𝑅

is on the order of 10 nanometers.

Most of the existing polymers interact with solid substrates by means of weak, non-covalent interactions known as van der Walls (vdW) or dispersive forces.

Unlike the interaction between atoms in a polymer chain is mainly covalent bonds interaction, at molecular scale, the distance between atoms or molecules play an important role in affecting the vdW interactions – molecules or atoms nearby will cause the polarizations of particles fluctuating [3], which in turn attract or repulse each other. The repulsion component of Van de Waals force is due to the existence of Pauli exclusion principle when particles approaching too close, otherwise, the interaction will appear attraction, which decreases fast when enlarging the distance between interacting molecules. The Lennard - Jones potential [4] is used to describe this interaction as an approximate model, shown as Figure 1.1.3.

Figure 1.1.3 The illustration of Lennard Jones Potential 𝑉

is presented as a function of distance 𝑟 between two particles, 𝑉

and 𝑟 is nondimensionalized with energy depth 𝜀 and minimum energy distance 𝑟

. According to the polarity of two interacting objects, VdW interaction are classified into London dispersion, Debye, Keesom interactions [3].

1.2 Interfacial energy and work of adhesion

An interface is a boundary between any two contacting phases. Surface is the

name commonly given to an interface with vacuum or air. To understand the

energy involved in the formation of a surface, we consider that in order to bring

10

a molecule from the bulk to the interface, a work to break some of the bonds is required, see Fig 1.2.1

Fig 1.2.1 Sketch on molecular interaction between molecules in liquid or bulk state.

Because of the energy conservation principle, the excess of free energy earned by molecules at the surface, the surface free energy (𝛥𝐺), is equal to the work (𝛥𝑊) done on splitting the bulk (or creating the surfaces). This relationship is given as 𝛥𝐺 = 𝛥𝑊 = 2𝛾𝐴, where surface free energy (surface tension) 𝛾 is half of free energy per unit area, A. Commonly the term ”surface energy” is indicated by the symbol 𝛾, and in the SI is expressed in 𝑚𝐽 𝑚

, which replaces the outdated unit 𝑒𝑟𝑔 𝑐𝑚

.

In a similar way we can define the interfacial tension between two phases A and B, and indicate it with 𝛾

. Good and Girfalco proposed an equation based on a semi-empirical model to describe 𝛾

, which successfully expressed the adhesion free energy, by referring to the cohesion free energy necessary to separate the phases [5]. For example, considering a system with phase A and phase B in contact at an interface, the interfacial free energy (surface tension) 𝛾

was given as:

𝛾

= 𝛾

+ 𝛾

− 2𝜑 𝛾

𝛾

(eq1.6) where 𝜑 is a Good-Girifalco interaction parameter [6]. 𝛾

and 𝛾

are their cohesion free energy of separate phases (or surface energy in the vacuum), and 𝛾

and 𝛾

are their dispersive components.

However, dispersion interactions alone are not sufficient to describe interfacial interactions in a complex liquid. Fowkes suggested that the total surface free energy is the superimposition of all intermolecular forces, including hydrogen bonding and polar interactions [6, 7-9]. In other words, the total surface free energy is the sum of all intermolecular forces acting on an interface. Similarly, from the energy point of view, surface energy(γ) and work of adhesion (W

) can be parameterized as the sum of components associated to different types of interactions. For example, for a surface with dispersion interaction, polar

Surface Tension

interaction, and hydrogen-bonding interaction, the surface free energy and work of adhesion are written as

𝛾 = 𝛾

+ 𝛾

+ 𝛾

+ ⋯ (eq1.7)

𝑊

= 𝑊

+ 𝑊

+ 𝑊

+ ⋯ (eq1.8) Where superscripts 𝑑, 𝑝, ℎ, are refer to the dispersion, polar, and hydrogen- bonding interaction components.

Fowkes’s theoretical assumption were confirmed by experiments by Wendt [10]

and Kaelble [11], who measured the surface free energy by using the so-called contact-angle method (Figure 1.2.2)

Figure 1.2.2 A sketch of Contact angle method. A liquid drop spread on solid reaches an equilibrium state in the air at contact angle 𝜃

. 𝛾

, 𝛾

, and 𝛾

are the surface tension between liquid and gas, solid and liquid, solid and gas respectively.

The contact angle method determines the surface tension by measuring the contact angle between solid and referenced liquids (Figure1.2.2). This approach is based on Young equation, which is used to describe the interface of a drop of liquid at equilibrium with a (its) vapor, on a solid surface:

𝛾

− 𝛾

= 𝛾

𝑐𝑜𝑠𝜃 (eq1.9) where 𝛾

is the surface energy between solid and vapor, 𝛾

is the surface energy between solid and liquid, and 𝛾

is the surface energy between liquid and vapor. We should notice that considering a liquid drop onto a solid substrate (Figure 1.2.3), wetting happens when there are interactions between the liquid and the solid, in which case the contact angle 𝜃

< 90° and the surface tensions are in relation that 𝛾

> 𝛾

+ 𝛾

∙ cos 𝜃

; otherwise, non-wetting occurs, in which case 𝜃

> 90° is observed, the surface tensions are in relation that 𝛾

<

𝛾 + 𝛾 ∙ cos 𝜃

Figure 1.2.3 Shape of a liquid drop over a solid substrate for (a) wetting and (b) non-wetting.

To understand the correlation between wetting and interfacial interactions, we need to consider Dupré equation for the work of adhesion [12]: 


𝑊

= 𝛾

+ 𝛾

− 𝛾

(eq1.10) Combining Eq 1.9 and Eq 1.10, the magnitude of the adhesion force can be presented as a function of the contact angle 𝜃 between liquid and solid, via the Young-Dupré equation:

𝑊

= 𝛾

1 + 𝑐𝑜𝑠𝜃 (eq1.11) This is the Young-Dupré equation. More details can be found in textbooks [13].

This equation is widely used in experiments to obtain the surface tension of at solid. By combining equation1.11 with equations 1.7, eq1.8, and eq1.9, the relationship between contact angle and different bonding components is determined. This is the Young-Good-Girifalco-Fowkes equation.

𝛾

1 + 𝑐𝑜𝑠𝜃 = 2 𝛾

𝛾

+ 2 𝛾

𝛾

+ 2 𝛾

𝛾

(eq1.12)

The superscripts +/− indicate the acid and base, in the definition of Lewis,

components. According to this equation, by measuring the angles between several

(at least 3) reference liquid drops on the solid surface, the three components of

the equation for solid surface energy can be obtained. Finally, the surface tension

of a liquid at a solid surface is derived. In chapter 4, I will verify that such

approach is actually not valid in the case of very thin films.

1.3 The Hamaker constant

The experiment in contact angle method has proved that surface free energy is additive [6-9] for two flat surfaces. Actually, those interactions could exist between two bodies in any other geometry. Hamaker calculated the free energy of interactions between two bodies of some common geometries, which are given in terms of a parameter, known as Hamaker constant [13].

Figure 1.3.1 Non-retarded van der Waals interaction free energy between two atoms calculated on the basis of pairwise additive.

We considering two atoms interacting with a vdW potential (Figure 1.3.1), as 𝑤 𝑟 = − 𝐶 𝑟

, where 𝐶 is a parameter indicating the strength of the VdW pair potential, r is the separation between two atoms. Assuming that the interaction between two bodies is non-retarded and additive, the Hamaker constant 𝐴 is given by [14]

𝐴 = 𝜋

𝐶𝜌

𝜌

(eq1.14) where 𝜌

and 𝜌

are the number of atoms per unit volume in the two bodies,.

More rigorous methods of calculating the Hamaker constant in terms of macroscopic properties of the media are reported in the literature [13]. However, the additivity is still an assumption that lacks a precise approach. Lifshitz theory [14] validated Hamaker approach [15], which consider van der Waals interactions as essentially electrostatic, arising from the dipole field of an atom ‘reflected back’

by a second atom that has been polarized by this field.

Figure 1.3.2 Illustration of interaction between media 1 and 2 in medium 3 Based on Hamaker approach, the interaction potential between a sphere a radius R and a flat surface (Figure 1.3.3), placed at distance d from the center of the sphere is given by

Figure 1.3.3 Illustration of interaction between a sphere and a surface.

This equation is obtained by geometrically integrating (that is, via a volume integral) all the possible interactions between the atoms of the sphere and the substrate.

Similarly, in the case of two flat surfaces (Figure 1.3.4)

Figure 1.3.4 Illstration of interaction between two surfaces.

The latest example is the one I have widely investigated in my thesis, because it can permit to evaluate the interaction between a polymer film and a substrate.

Following Lifshitz theory, the expression of the Hamaker constant for the interaction of two media 1 and 2 in a third medium 3 (Figure1.3.2) is given as,

𝐴 ≈

𝑘𝑇

Kv„_…

„_Lw„_…

„_Lv„_…

+

K D‡ v„_… D‡

„_L D‡ w„_… D‡

„_L D‡ v„_… D‡

𝑑𝑣

‰

‡_K

(eq1.14)

where 𝑘 and ℎ are Boltzmann's and Planck's constants respectively, 𝑇 is the temperature of the system, and 𝜀

is the dielectric constant of the material. In equation 1.14, the imaginary part represents the absorption frequency of media.

After some deduction and assuming absorption frequencies are similar, the Hamaker constant for two macroscopic phases is simplified as,

𝐴

≈

•

𝑘𝑇

„_Kv„_…

„_Lw„_…

„_Lv„_…

+

2_K^Lv2_…^L 2_L^Lv2_…^L 2_K^Lv2_…^L v 2_L^Lv2_…^L

(eq1.15) where 𝑛

is the refractive index, 𝜐

is main electronic absorption frequency in the UV typically around 3×10

𝑠

.

1.4 Models of polymer irreversible adsorption kinetics

The two-surface case of Eq 1.15 is the one I have widely investigated in my thesis,

because it can permit to evaluate the interaction between a series of thin films

placed, or not, in contact. In particular, I have studied the interaction of a thin

polymer film placed on top of a solid substrate. To study interactions at

fundamental level, I have used a method developed by my supervisor, which

permits to vary the interfacial potential via irreversible adsorption. This method

is based on the fact that the work of adhesion is directly proportional to the number of contacts formed per unit surface between the two phases, and consequently the number of molecules adsorbed at the polymer/substrate interphase. To understand interfacial interactions, it is thus necessary to study the physics of adsorption. In the following sections I will present the most relevant models of adsorption of polymer chains onto solid substrates.

1.4.1 Guiselin brushes

The Guiselin brush model, was developed by O. Guiselin [16] in the early 90’s.

This model considers the adsorption of a long linear polymer chain onto an impenetrable wall. The polymer chains are either in a melt or in a semi-dilute solution. This adsorption is regard as irreversible, that is, the probability of desorption is very small. It is convenient to introduce the three way in which a chain can adsorb (Figure 1.4.1): 1) train, a series of consecutive monomers adsorbed; 2) loop, a section of non-adsorbed monomers in between two non- consecutive adsorbed monomers; 3) tail, a chain end including non-adsorbed monomers bonded to an adsorbed monomer.

Figure 1.4.1 Sketches of a loop, a tail and a pseudo-tail. (Reproduced from Eur.

Phys. Lett., 17, 225-230 (1992))

According to Guiselin, when chains are irreversible adsorbed, their conformations are unaltered upon exposure to pure solvent. Adsorbed chains adopting the conformation predicted by scaling arguments previously derived by Alexander and de Gennes [17] as experimentally verified [19]. The protocol/experiment designed by Guiseliin to prepare an adsorbed layer is given by a series of simple steps, easily reproducible in a laboratory:

i) A melt or a semi-dilute solution is put into contact with the immiscible surface.

ii)

iii) The residual free polymer chains are washed out with pure solvent.

1.4.2 “Parking” model

Granick and coworkers developed an interesting theory on the adsorption process [20], based on their experiments on poly(methyl methacrylate), PMMA. By means of infrared spectroscopy (IR), they obtained information on the bound fraction, that is, the average number of monomers adsorbed per chain. This allowed them to build up a histogram of chain conformations of PMMA adsorbing in dilute solutions onto silicon substrate (Figure. 1.4.2). The adsorption chains information was achieved based on the perpendicular or parallel orientation of the main chain with respect to the surface. The ratio between those two components is the dichroic ratio, which can determine the adsorbed amount.

The results are interestingly concluded as “parking problem” and random sequential adsorption.

In the model by Granick polymer chains diffuse and reach the substrate monomer by monomer and adsorb onto random locations, with exception of the occupied ones. Chains adsorb on the substrate in two ways. The proportions of chains arrive earlier can flatten on the surface, since there are plenty of free surfaces sites when they hit the surface. Because of the fewer free sites to pin to, later-coming chains can only attach to the substrate by “standing” perpendicular to the substrate, with only few of their chain being directly adsorbed.

Figure 1.4.2 Histogram of bound fractions determined, for fractions separated by

Δp′ = 0.02 . p = Γbound/Γ , bound mass adsorbed relative to total mass

adsorbed. Inset shows schematic illustrations of chains with large and small

bound fractions. Reproduced form Langmuir 12(4):994

1.4.3 Irreversible physisorption model

O’Shaughnessy and D. Vavylonis developed a theory to describe chemisorption and strong physisorption [21, 22]. Here we will focus on physisorption, that is, adsorption due to vdW interactions. In this model, the adsorbed polymer layer system is regard as non-equilibrium system. When evaluating density profile and loop distribution of the polymer/surface system, the authors consider as if they are same as equilibrium layers. In this framework, the conformation of adsorbed layer would be considered as two part, the inner part is a tight packed structure, and the outer part makes only few surface contacts, in line with the work of Granick.

The phenomenon of physisorption irreversibly happens even for weak attractive surface and polymer interactions [23].

Irreversible adsorption is here achieved when the free energy between monomer and surface 𝜖 reaches values of a few 𝑘𝑇. The free energy of system is give as figure1.4.3.

Figure 1.4.3: Scheme of free energy as a function of distance between monomer and surface. In physic-sorption there is no activation barrier and monomer stick on surface immediately once come into contact.

The irreversible adsorption process was regarded as two stages in their model.

i) early stages: monolayer formation

Initially, the surface is empty and thus the attachment of chains is diffusion controlled and monolayer of flattened chain starts to develop on surface. The monomers of chains are behavior corresponding to the zipping mode. After this, new chains arrival surface must form loops to join disconnected empty sites and consequently enters new stage of building up a diffuse layer.

ii)later stages: diffuse outer layer 


!

"

#

For time longer than a character time 𝑡

[20, 21], a continuously decreasing happens on density of surface available free sites for adsorption. 


1.4.4 Two kinetic regime model

Figure 1.4.4 (a) illustration of PS film adsorbed onto silicon substrate, the corresponding adsorbed layer thickness is ℎ

. (b) The adsorbed layer thickness of PS film is plotted as a function of time in logarithmic scale. This figure is reproduced from Macromolecules (2014), 47, 3390 [24]

Housmans et al. recently proposed two regimes for the kinetics of irreversible adsorption of polystyrene onto silicon substrates [24]. Using ellipsometry, they studied a large amount of material adsorbed at different annealing times (Figure 1.4.4(b)). They find a complex adsorption kinetics, with a linear regime followed by a logarithmic one. To understand their data, first they assumed an adsorption reaction dominated by a zero-th order mechanism. At time 𝑡 (after the start of adsorption 𝑡 = 0 ), the adsorbed amount Γ , and thus the thickness of the adsorbed layer ℎ

, increases linearly with time, ℎ

~𝑡 . Second, they assumed that chains obey a reflected random walk in proximity to a non-repulsive wall. This implies that at any given time, ℎ

~𝑁

~𝑅

, where 𝑅

is the gyration radius. As the linear growth progresses as a zero-th order reaction mechanism, due to reduced space available for pinning in the later regime, late coming chains need to stretch before diffusing through the layer of molecules that arrived first at the surface. The adoption of these polymer conformations result in entropy loss, as soon as the adsorbed chains begin to overlap significantly and to stretch [25]. Under these conditions, the growth rate follows a logarithmic growth on 𝑡. This picture concurs with other analysis [26] and experiments [20, 27].

With these considerations, the two-kinetic regime model is proposed. The description of the kinetics of irreversible adsorption constitutes a crossover between a linear and a logarithmic growth.

PS

Silicon Substrate

ℎ_"#$

(a) (b)

ℎ

𝑡 = ℎ

+ 𝑣𝑡

ℎ

+ Π𝑙𝑜𝑔𝑡 𝑡 < 𝑡

𝑡 > 𝑡

(eq1.16) where 𝑣 and Π express the growth rates in the different regimes [21,22], ℎ

is the value of the thickness at the crossover time 𝑡

and the value of 𝑡 in the argument of the logarithm is normalized by 𝑡

= 1s, to ensure correct dimensionality. This equation contains information on macromolecular nature.

These models are valid, but they lack in a connection between interfacial forces

and the kinetics of adsorption. In my thesis I worked on this missing point.

References

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[4] Lennard-Jones, J. E.; "On the Determination of Molecular Fields", Proc. R.

Soc. Lond. A, (1924) 106, 738.

[5] Girifalco, L. A.; Good, R. J.; “A Theory for the Estimation of Surface and Interfacial Energies. I. Derivation and Application to Interfacial Tension”. J.

Phys Chem. (1957) 61, 904. 


[6] Pocius, A. V.; Adhesion and adhesives: an introduction, 2002 Carl Hanser Verlag GmbH & Co. KG publisher, Munich (2012)

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[8] Fowkes, F. M.; “Dispersion Force Contributions to Surface and Interfacial Tensions, Contact Angles, and Heats of Immersion”. Adv. Chem. Ser (1964) 43,99. 


[9] Fowkes, F. M.; “Chemistry and Physics of interfaces”, SRossed. American Chemical Society. (1971) 


[10] Ownes, D. K.; Wendt, R. C.; “Estimation of the surface free energy of polymers”. J. Appl. Polym. Sci. (1969) 13, 1741. 


[11] Kaelble, D. H.; “Dispersion-Polar Surface Tension Properties of Organic Solids”. J. Adhesion (1970) 2, 66. 


[12] de Gennes, Pierre-Gilles; Brochard-Wyart, Francoise; Quere, David;

Capillarity and Wetting Phenomena. Springer (2004)

[13] Israelachvili, J. N.; intermolecular and surface forces. Academic Publisher:

San Diego, (2011) 


[14] Lifshitz, E. M.; “The theory of Molecular A ractive Forces between Solids”.

Journal of Experimental Theoretical Physics USSR. (1954) 29: 94-110.

[15] Hamaker, H. C.; “The London—van der Waals attraction between spherical particles”. Physica (1937) 4, 10, 1058-1072. 


[16] Guiselin, O.; “Irreversible Adsorption of a Concentrated Polymer Solution”.

Europhys. Lett (1991) 17, 225.

[17] Cohen Addad, J. P., Polymer, (1989) 30,1821.

[19] Binder, K.; Kremer K., Scaling Phenomena in Disordered Systems, edited by R. Pynn and A. Skjeltro (Plenum Press, New York, N.Y.) 1985. 


[20] Hildegard M. Schneider, Peter Frantz, and Steve Granick; “The Bimodal Energy Landscape When Polymers Adsorb” Langmuir (1996) 12 994-996

[21] O’Shaughnessy B., and Vavylonis D., “Irreversible adsorption from dilute

polymer solutions”, Eur. Phys. J. E (2003) 11, 213-230

[22] O’Shaughnessy B., and Vavylonis D., “Irreversibility and Polymer Adsorption”. Phys. Rev. Lett. (2003) 90, 056103.

[23] de Gennes, Pierre-Gilles; Brochard-Wyart, Francoise; Quere, David; 2004

“Capillarity and Wetting Phenomena”. Springer

[24] Housmans, C.; Sferrazza, M.; Napolitano, S. “Kinetics of Irreversible Chain Adsorption”. Macromolecules (2014), 47, 3390.

[25] Ligoure, C.; Leibler, L. “Thermodynamics and kinetics of grafting end- functionalized polymers to an interface” J. Phys. France (1990), 51, 1313−1328.

[26] Linse, P. “Effect of solvent quality on the polymer adsorption from bulk solution onto planar surfaces”. Soft Matter (2012), 8, 5140−5150.

[27] Gin, P.; Jiang, N.; Liang, C.; Taniguchi, T.; Akgun, B.; Satija, S. K.; Endoh,

M. K.; Koga, T. “Revealed Architectures of Adsorbed Polymer Chains at Solid-

Polymer Melt Interfaces”. Phys. Rev. Lett. (2012) 109, 265501.

Chapter 2 Experimental techniques

This chapter presents the main techniques used in the work discussed in the thesis:

1) Ellipsometry, the technique employed to determine the thickness of the adsorbed films [1, 2]; 2) Atomic Force Microscopy (AFM) [3-5] used to verify characterize surface morphology and 3) spin-coating, the technique used to prepare thin polymer films.

2.1 Ellipsometry

The principle of ellipsometry is based on the change in the phase and intensity of the components of a polarized light beam upon reflection on the investigated sample [1, 2].

2.1.1 Basic equations in Ellipsometry

First, let’s consider a light beam reflected by a surface, for example the interface between air (𝑛

= 1 ) and a medium. Part of the beam will be reflected, the incident angle and reflection angle are equal (Figure 2.1), and a part will be transmitted in the medium. The relation between incident light beam and transmitted light beam follows the Snell’s law, given by:

¤¥¦ §_K

¤¥¦ §_¨

=

K

(eq.2.1) Where 𝜃

is the angle of incidence/reflection, 𝜃

the angle of transmission, 𝑛

is the refractive index of air (= 1), or of the material in contact with the surface, and 𝑛

is refractive index of the medium composing the surface.

Figure 2.1 Reflection of a light beam on a surface. 𝜃

is the angle of

incidence/reflection, 𝜃

is the angle of transmission.

Figure 2.2 Linear polarized light reflected on substrate covered with film of 𝑑 thickness. The incident light and its components are marked with subscript ‘i’, the reflected light and its component are marked with subscript ‘r’. The refractive index for ambient, film and substrate are marked with 𝑛

, 𝑛

and 𝑛

. The incident, reflection, refraction and norm of surface are all parallel to the plane of incident (POI), 𝜃

is the angle of incidence/reflection on surface 01, 𝜃

is the angle of refraction on surface 01, 𝜃

is the angle of refraction on surface 12.

Figure 2.2 illustrates a thin film thickness 𝑑 of refractive index 𝑛

deposited onto a substrate of refractive index 𝑛

. The incident light beam (the incident electric field 𝐸

) reflects from the different interfaces, as illustrated in the figure 2.2. In order to simplify the problem, the plane including the incident light beam (i), reflection beam (r) and film surface normal N are denoted as the plane of incident (POI). Due to the film properties, the components parallel (𝐸

) and perpendicular (𝐸

) of reflection to the POI will reflected differently [6]. 


Ellipsometry measurements are based on the determination of two quantities : the

phase difference of the components of the 𝐸 before and after the reflection,

labeled by Δ, and the change in the ratio of their amplitudes before and after

reflection, given by tan Ψ . We denote with 𝜌

and 𝜌

the reflection

coefficients of light, which is polarized in two orthogonal direction, respectively

parallel and perpendicular to the POI. Following a convention based on German

language, all components parallel or perpendicular to the POI will be denoted

with subscript 𝑝 or 𝑠 correspondingly, see Figure 2.2.

The complex ratio 𝜌 allows visualizing the way in which the ellipsometric angles, Δ and Ψ, are obtained:

𝜌 =

°

= 𝑡𝑎𝑛Ψ𝑒

(eq2.2)

In a reflecting film surface (Figure 2.2), the electric component of light beam can be written in general form

𝐸

= 𝐴

∙ 𝑒

(eq2.3) where subscript 𝑚 could be 𝑖, 𝑟, 𝑖𝑝, 𝑖𝑠, 𝑟𝑝, 𝑟𝑠, 𝑡 (Figure2.2). Here the symbols 𝐴

and 𝛿

are the amplitude and phase of the component 𝑚 , respectively.

Assuming linear polarization of the incident light beam (that is 𝛿

− 𝛿

= 0,

__·¯

__·°

= 1), the expression of 𝛥 and 𝛹 are:

∆= 𝛿

− 𝛿

(eq2.4) 𝑡𝑎𝑛Ψ =

¹°

(eq2.5) The components expression for the reflected part of Fresnel coefficients [7] only consider the interface between the ambient and film (𝑟

), which are derived as:

𝑟

=

º_·°

=

2_¨¡‹Ÿ§_¨v2_K¡‹Ÿ§_K

(eq2.6) 𝑟

=

·¯

=

K¡‹Ÿ§_¨v2_¨¡‹Ÿ§_K

(eq2.7) where 𝑟

and 𝑟

are the Fresnel reflection coefficients for the components parallel (p) and perpendicular(s) of reflected light beam on surface 01. The reflection coefficients (𝑅) (with refractive index 𝑛

) at the incident angle of 𝜃

, are given as:

𝑅

= 𝑟

(eq2.8)

𝑅

= 𝑟

(eq2.9)

for two interfaces system as illustrated as Figure 2.2, the reflection occurs on both interfaces between surface 01 and surface 12, the reflectance of system is given by:

𝑅

=

>v¢_¨K¯¢_KL¯‘ ^»·L¼

(eq2.10) 𝑅

=

¨K°¢_KL°‘ ^»·L¼

(eq2.11) where β is:

𝛽 = 2𝜋

𝑛

𝑐𝑜𝑠 𝜃

(eq2.12) where d is the film thickness.

2.1.2 Brewster angle

Figure 2.3 Illustration of reflection and refraction happens on an interface between transparent medium1 (𝑛

) and medium2 (𝑛

).

By further analyzing the ellipsometric equations, one can ask for the conditions

permitting optimization of experimental conditions. To obtain the best

called the Brewster angle, 𝜑

, where the p-polarized reflectivity vanishes. Such angle corresponds to the condition:

𝜑

= arctan 𝑛

𝑛

(eq2.13) where the 𝑛

and 𝑛

are the refractive indexes of the two media.

The measurements are performed at an incident angle as close as possible to the Brewster angle: in these conditions a maximum difference can be achieved for 𝛥 and a minimum for 𝛹 . For example, according to the reflection 𝑅

and 𝑅

(given in equation 2.8 and 2.9), considering a silicon substrate (with refractive index 𝑛 = 3.87 − 0.02𝑖 ) [8], the reflection intensity for the 𝑝 and 𝑠 components as a function of incident angle is shown in figure 2.4. The minimum value achieved on reflect intensity corresponding to the Brewster angle. As observed a minimum value of p-polarized is achieved at around 75 degrees.

Figure 2.4 The reflection intensity of electromagnetic wave reflected on a silicon substrate system of refractive index n=3.87-0.02i.

The ellipsometer used for experiments in this thesis is the spectroscopic ellipsometer Jobin Yvon Horiba MM-16. This is a photometric rotating analyzer ellipsometer (RAE) that utilizes both static and dynamic measurements [6,9-11].

The light source is a combination of a halogen tungsten lamp and a blue-LED to

provide stable and persistent illumination, which across the visible spectra range

in 400 − 850𝑛𝑚. Besides, wavelength resolution is better than 2𝑛𝑚. The light

is polarized by the polarizer in two identical input and output head, and the phase

of light is adjusted by the ferroelectric liquid crystals and retardation plate. The light from output head is measured at CCD detector and analyzed by a spectrograph. The analyzing result will be send in form of 16-element Mueller matrix rapidly and precisely.

Figure 2.5 Illustration of a poly(4-chlorostyrene) film on silicon substrate, with a SiO

lay (~2nm).

2.1.3 An example

We consider a film of poly(4-chlorostyrene) deposited on a substrate of silicon covered by a 2 nm thick oxide layer, see Figure 2.5

Figure 2.6 Fitted Ellipsometry curves of polar polymer monolayer film. Ψ and

∆ as a function of variable annealing time on polar monolayer via Ellipsometry.

The blue line is data acquainted from experiment, while the red line is a fitting

line of the experiment.

By ellipsometry we measure 𝛹 and 𝛥 as a function of the wavelength, see figure 2.6. By fixing the refractive index of the different layers to literature data, the experimental data of Ψ and Δ are fitted using the equation 2.12. The thickness of the film can be extracted by minimizing the difference between model values obtained via equation 2.12 and the experimental values. According to the model, the measurement data are in blue color and the fitted one in the red.

We could see these two curves match very well.

2.2 Atomic Force Microscopy

AFM was used to verifies the thickness of the sample and to monitor the surface morphology.

An Atomic Force Microscope can be characterized as two-part system: the optical system and the electronic system [3-5]. The optical system consists of two components: a cantilever with a very sharp tip at the end and a laser, focused onto the back of the cantilever. The electronic part also has one main component that is a photodiode. When the tip is very close to the sample’s surface, the van der Waals force between the tip and surface causes a deflection on the cantilever. The contour of the surface, and the variations in the position of the cantilever is detected by a change in the angle of the laser beam that is reflected by the cantilever and detected by the photodiode. This change in the cantilever deflection relates to the different topographical features. The force applied on the endpoint of the cantilever can be approximated to 𝐹 = 𝑘𝛥𝑍, where 𝛥𝑍 is the deflection of the cantilever and 𝑘 is the elastic constant of the cantilever. The changes of force applied on cantilever are all detected in this way. Then, a feedback mechanism of AFM would be optionally used in different working mode. There are three main different modes [4,5] possible for an AFM experiment: contact, non-contact and tapping mode.

In our case, sample surfaces are scanned in the tapping mode, the probe (cantilever and tip) is oscillating at a high resonance frequency above the sample surface. When the tip of probe hits the sample surface, the repulsive interactions between tip and samples are not that destructive, compare to non-contact mode.

It avoids the sample damage by intermittently contacting the surface, at the mean time, it prevents the tip from being trapped by adhesive forces from the sample surface.

The property and geometry size of cantilever plays an important part in deciding

the sensitivity and resolution of AFM. In tapping mode, for Si

N

cantilever, the

cantilever is sensitive enough to detect extremely weak applied force, and high

enough resonance frequency to avoid un-stable affection from oscillation. Since

SiO

cantilever, which is much harder than Si

N

one, has a higher resonance frequency and bigger elastic constant [12].

2.3 Spin-coating technique

Spin-coating is a technique that permits to prepare thin polymer films uniform at the nanoscale level [13-14]. To fabricate nano-films we start by cleaning the substrate where we will deposit the film: the impurities and dust on top of substrate are washed in a set of solvents (typically acetone followed by isopropanol and finally toluene, or in the case of PMMA samples benzene).

Figure 2.7 Illustration of spin-coating procedure

Spin coating process commonly involves the spreading of a solution of desired material on a substrate followed by a very fast rotation.

First a polymer solution of determined mass concentration is deposited onto a silicon substrate to cover the surface. Second, the substrate is rotated at high speed (typically 2-3000 rotations per minute), during which the polymer is spread onto the surface, solvent is fast evaporated and the the majority of the solution is sliding out from the side. [15,16]

Third, the residual of solution is evaporated while the substrate is still rotating at high speed, leaving at the end of the process a uniform film. We prepared films with a spinning time of 60 seconds.

The thickness of the film can be modified by adjusting parameters such the

concentration of the polymer in the solution and rotation speed [17]. Figure 2.8

illustrates the variation of the thickness as a function of the polymer concentration

for a spin-coated P4MS system, as an example.

Figure 2.8 Film thickness as a function of PMMA weight concentration. All

solutions are spin-coated at 3000 rpm for 60 seconds.

References

[1] Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized light. North- Holland Publisher: Amsterdam (1977) 


[2] Richards, R. W.; Peace, S. K.; Polymer Surfaces and Interfaces III. Wiley Publisher: New York (1999) 


[3] Binnig, G.; Quate, C. F.; Gerber, C.; “Atomic Force Microscope”. Phys. Rev.

Lett. (1986) 56, 930 


[4] Binnig, G.; Gerber, C.; Stoll, E.; Albrecht, T. R.; Quate, C. F. “Atomic Resolution with Atomic Force Microscope”. Europhys. Lett. (1987) 3, 1281 
 [5] Albrecht, T. R.; Quate, C. F. “Atomic resolution imaging of a nonconductor by atomic force microscopy”. J. Appl. Phys. (1987) 62, 2599 


[6] Jellison , G. E.; Data Analysis for spectroscopic ellipsometry, handbook of Ellipsometry,William Andrew Publisher (2005) 


[7] Azzam, Rasheed M. A.; “Fresnel's interface reflection coefficients for the parallel and perpendicular polarizations: global properties and facts not found in your textbook”. Proc. SPIE. (1994) 2265, 120

[8] Butt, Hans-Jürgen, Kh Graf, and Michael Kappl. "Measurement of Adsorption Isotherms." Physics and Chemistry of Interfaces. Weinheim: Wiley-VCH, 2006.

206-09. Print.

[9] Ord, J. L.; Wills, B. L. “A computer operated following ellipsometry” Appl.

Opt. (1967) 6, 10 


[10] Mathieu, H. J.; McClure, D. E.; Muller, R. H.; “Fastself-compensating ellipsometer”, Rev. Sci . Instrum. (1974) 45, 6 


[11] Rothen, A.; “The Ellipsometer, an Apparatus to Measure Thicknesses of Thin Surface Films”. Rev. Sci. Instrum. (1945) 16, 26 


[12] Albert Folch, Mark S. Wrighton, and Martin A. Schmidt, “Microfabrication of Oxidation-sharpened Silicon Tips on Silicon Nitride”. Journal of Microelectromechanical Systems, (1997) 6, 4

[13] Croll, S. G.; “The origin of residual internal stress in solvent - cast thermoplastic coatings”. J. Appl. Polym. Sci. (1979) 23, 847 


[14] Karpitschka, S., Weber, C.M., Riegler, H.: “Physics of Spin Casting Dilute Solutions”. http://arxiv.org/abs/1205.3295 


[15] Scriven, L.E. (1988). "Physics and applications of dip coating and spin coating". MRS Proceedings. 121: 717

[16] Jose Danglad-Flores, Stephan Eickelmann, Hans Riegler, “Deposition of polymer films by spin casting: A quantitative analysis” Chemical Engineering Science 179 (2018) 257-264

[17] José Danglad-Flores, Stephan Eickelmann, Hans Riegler, “Deposition of

polymer films by spin casting: A quantitative analysis”. Chem. Eng. Sci. (2018)

Chapter 3 Polymer Adsorption Mechanisms onto Silicon Substrates

This chapter will present the results on the mechanisms governing irreversible polymer chains adsorption onto the silicon substrate process. After reviewing current prevailing models and interrelated concepts, we will present our recent experimental results on four atactic polymers (PS, P4MS, PtBS and PMMA) for which the formation of an irreversible adsorption layer was investigated under isothermal annealing conditions. Two dominating mechanisms were observed during the adsorption process, one is the molecular rearrangement mechanism which is the main driving force making monomer-solid contact happen, another one is the potential driven adsorption mechanism which explains how surface systems reach the adsorption equilibrium (saturation state).

Specifically, we focus on how thermal energy affects the non-equilibrium component of the adsorption processes, and on the impact of the interaction potential on the equilibrium adsorbed amount.

We find that the adsorption process is 
 thermally activated, with activation energy comparable to that of local non-cooperative processes. On the other hand, the final adsorbed amount depends on the interface interaction. We observe that the monomer pinning mechanism is independent of surface coverage at short time, while the adsorption rate is limited when more surfaces are occupied by the chain.

3.1 Introduction

The behavior of materials confined at the interface at nanoscale level results on the material properties far from those exhibited by the bulk material [1-8]. Among those applications, the adsorption of macromolecules at the interface plays a critical role and it is intensively investigated.

Glynos et al. [4] demonstrated the glass transition temperature reduction ∆𝑇

of star-shaped polystyrene is directly proportional to the number of chains adsorbed onto supporting substrate. Napolitano and Wübbenhorst [3] demonstrated that for confined polystyrene thin film on top of aluminum layers, the glass transition temperature 𝑇

is adjustable when prolong the annealing time above 𝑇

, which implies that thin films are intrinsically non-equilbrium systems.

The process of thickening of irreversible adsorbed layer onto the supporting

substrate simultaneously accompanies with reduction of ∆𝑇

, suggesting that

after a series of the metastable states during prolonging the annealing time, the

nature of confinement effects is erased toward stable state upon adsorption [3].

Furthermore, Housmans et al. [9] demonstrated that the number of chains attached onto supporting substrate does not depend on the polymerization degree.

To further understand this phenomenon, we have investigated the kinetics of four polymers (polystyrene, poly(4-tert-butylstyrene), poly(4-methylstyrene) and poly(methyl methacrylate)) on silicon wafers covered by a thin SiO

layer. The polymer systems used and the experimental methods are presented in sect. 3.2.

The experimental results achieved under specific protocol in sect. 3.3. A discussion of the method will be briefly given in sect. 3.4. Section 3.5 will present the discussion of the results.

3.2 Polymers and sample preparation

Polymers used for experiments include both non-polar and polar systems, such as: PS, P4MS, PtBS and PMMA respectively. The polymers are all atactic and listed in table T1 that includes molecular weight (𝑀

), polydispersity index(PDI), and manufacturers.

Table T1 Information on the polymer materials used

All samples are strictly prepared following the same protocol. The substrate used is silicon wafer covered by thin oxidized layer (~2𝑛𝑚). This was measured for each sample prior the polymer deposition. Cleaning was performed on the substrate, a series of solvents are used to clean the wafer before spincoating, (typically acetone to clean the dust, then using isopropanol and finally toluene, or in the case of PMMA samples benzene instead of toluene). The solvent use was toluene for PS, P4MS and PtBS, and benzene for PMMA. The control the thickness was obtained by varying the polymer concentration in the solution as discussed in chapter 2.

Polymer 𝑀w [kDa] PDI Manufacturer PtBS 1210 1.4 Polymer Source Inc.

PS 955 1.08 Polymer Source Inc.

PS 560 1.11 Sigma-Aldrich

P4MS 72 2.61 Sigma-Aldrich

PMMA 320 - Sigma-Aldrich

Afterwards, the solutions were spin-coated onto the substrate at rotation speed 4000 rpm for 60 seconds. The measurement of this parameter is determined by Ellipsometry as previously discussed, and for some selected cases checked using atomic force microscopy (AFM). Since we are considering the geometry confinement of thin film from the bulk ones, the definition we adopted is that film thickness L > 10R

can be regard as a bulk film [5], considering that the confinement effects in thin film properties, are usually observed for thicknesses much smaller than 7R

[26]. Correspondingly, depending on the molecular weight, the values of film thickness 𝐿 employed were between 100 𝑛𝑚 to 500 𝑛𝑚.

In order to relax the films and remove the solvent, a pre-annealing at temperatures just above the polymer’s 𝑇

in vacuum for 10 minutes was performed. Then the system is annealed at constant 𝑇 above 𝑇

for different annealing time (up to 72 hours). Un-adsorbed polymer chains are washed away in the very same solvent used previously to prepare the solution (typically 1 hour for toluene, in the case of PMMA 3 hours in Benzene).

Note that, during the processing of annealing, we find that initial polymer film thickness around or below ∼ 3 nm is unstable (unstable films can be observed under optical microscopy or AFM, as shown as an example for a few systems in figure 3.2.).

a) PMMA/PS/SiO2/Si (32𝜇𝑚×24𝑢𝑚)

b) PS/SiO2/Si

(172𝜇𝑚×129𝑢𝑚)

c)PtBS/SiO2/Si

(172𝜇𝑚×129𝑢𝑚)

Figure 3.2 Unstable polymer films scanned under optical microscopy. (a)PMMA

on top of PS bilayer system onto silicon substrate is dewetted, (b)PS single layer

system onto silicon substrate is dewetted, (c) PtBS single layer system onto

silicon substrate is dewetted.

3.3 Results

Each sample was measured with ellipsometry at 3 ~ 5 different spots. Generally, each set of samples include more than 13 points (each corresponding to one annealing time) in order to get a satisfactory analysis of the kinetics of adsorption.

The results of the kinetics for the different systems are summarized in figures 3.3- 3.7. The adsorption kinetics of polymers (PS560k, PS995k, PtBS 1210k, PMMA 320k and P4MS) are in line with two-regime model by Housmans et al.

introduced in chapter 1. The adsorption layer thickness increases when annealed under isothermal condition for longer annealing time 𝑡

. The adsorption thickness ℎ

initially grows very fast following a linear fashion; after a crossover time the thickness growth rate becomes smaller while entering the logarithmic regime.

Figure 3.3 The kinetics of adsorption for polystyrene ( 𝑀𝑤 = 560𝑘𝐷𝑎 ) at

annealing temperature 433K, the thicknesses of adsorbed layer ℎ

are

presented as a function of annealing time 𝑡

for spincoated monolayer

samples of initial thicknesses ( 𝐿 = 307𝑛𝑚 ).

Figure 3.5 The kinetics of adsorption for poly-4-tert-bulty-styrene ( 𝑀𝑤 = 1210𝑘𝐷𝑎 ) at annealing temperature 433K, the thicknesses of adsorbed layer ℎ

are presented as a function of annealing time 𝑡

for spincoated monolayer samples of initial thicknesses (𝐿 = 40.5nm)

Figure 3.6 The kinetics of adsorption for poly(methyl methacrylate) ( 𝑀𝑤 =

320𝑘𝐷𝑎) at annealing temperature 433K, the thicknesses of adsorbed layer ℎ

are presented as a function of annealing time 𝑡

for spincoated monolayer

samples of initial thicknesses (𝐿 = 32𝑛𝑚).

Figure 3.7 The kinetics of adsorption for poly-4-methy-styrene (𝑀𝑤 = 72𝑘𝐷𝑎) at annealing temperature 433K, the thicknesses of adsorbed layer ℎ

are presented as a function of annealing time 𝑡

for spincoated monolayer samples of initial thicknesses (𝐿 = 9.4𝑛𝑚).

3.4 Polymer adsorption kinetics: results and discussion

Let’s consider a polymer chain adsorbing on a non-repulsive wall. The interface will be covered by a number of monomers: the enthalpy gain of adsorption for one repeating unit of long chain is less than 1 𝑘

𝑇 [27,28]. Since more monomers of a chain are interacting with the surface, the adsorption gain energy should be multiplied by the number of adsorbed monomers. The desorption of the whole chain requires cooperative detachment of the whole set of adsorbed segments. Thus, this mechanism of adsorption is considered irreversible.

Housmans et al. observed the time evolution of the thickness of the adsorbed layer ℎ

: a linear growth at short times followed by a logarithmic one[9]

ℎ

= ℎ

+ 𝑣𝑡

ℎ

+ Π𝑙𝑜𝑔 𝑡 𝑡

𝑓𝑜𝑟 𝑓𝑜𝑟

𝑡 < 𝑡

𝑡 > 𝑡

eq3.1a eq3.1b where 𝑡

and ℎ

represent the crossover point between cross regimes;

ℎ

is initial adsorbed thickness; and 𝜈 and 𝛱 are the growth rates, respectively,

in the linear and in the logarithmic regimes. The analysis of the adsorption kinetics of polystyrene (PS) on silicon oxide by Housmans et al. showed also that both adsorptions rate 𝜈 and 𝛱 scale with the square root of the molecular weight, implying that, at each given time, the number of adsorbed macromolecules does not depend on chain length. Additionally, it was shown that 𝜈 followed a thermally activated law, suggesting that the mechanisms of adsorption are dictated by non-cooperative rearrangements due to spontaneous fluctuations.

Further works [10,11] confirmed the validity of eq 3.1 for other polymer systems.

One limit of equation 3.1 is that it fails to describe the adsorption kinetics at the final stage, where a constant value is expected.

The adsorption rate is determined by the ensemble effect of those two mechanisms:

(1) the first-order thermal kinetics at the molecular level – molecular motion and conformational change,

(2) the effect of long-range (Van der Waals) forces in the adsorption.

In our experiments, alteration in annealing temperature 𝑇

results in a significant strong effect on molecular motion meanwhile has no influence on the interaction potential, which is approximate to 𝑘

𝑇 ≈ 2 − 5 𝑘𝐽 𝑚𝑜𝑙 [12]. The thermal activation energy 𝐸

is around 50 − 100 𝑘𝐽 𝑚𝑜𝑙 (see table T1).

3.4.1 Towards a new expression for the kinetics of adsorption

At the beginning of annealing times, the adsorption process is driven by a first-

order reaction mechanism [16-19], the superposition of molecular fluctuations

and interfacial potential determines the monomer adsorption rate 𝑞. Assuming

that the whole chain is irreversible adsorbed once a chain makes one contact with

surface [20], the monomer adsorption rate - counting the number of monomers

adsorbed in unit area and unit time, ∂Γ/ ∂t, can be obtained by calculate the

product between the number of available monomers and the polymerization

degree 𝑁. The expression of monomer availability can be obtained by multiply

the monomer density and ratio of partition functions between surface (Z

) and

bulk (Z

) [21]. This ratio is equivalent to counting the numbers of monomers of

a chain of N segments which sit on flat surface intersecting the random coil. Here,

we need to apply the statistics of polymer melts, then it is given by [21]: