Lukas Felix Michalek MSc Materials Science Felix_Michalek... · 2020-02-02 · QUANTIFYING MACROMOLECULAR GROWTH AND HIERARCHICAL STRUCTURING ON INTERFACES Lukas Felix Michalek MSc

QUANTIFYING MACROMOLECULAR

GROWTH AND HIERARCHICAL

STRUCTURING ON INTERFACES

Lukas Felix Michalek

MSc Materials Science

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Chemistry, Physics and Mechanical Engineering

Science and Engineering Faculty

Queensland University of Technology

2020

To create is to live twice.

- Albert Camus

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces i

Abstract

For the development of the next generation of soft matter based functional

surfaces it is critical to obtain an in-depth molecular understanding of the processes

that govern the functionalization. There exists a multidisciplinary need for advanced

surface performance ranging from wear/chemical resistance to biomedical

applications. Especially the interplay between theoretical and experimental

assessments is of substantial value for generating an encompassing picture of the

macromolecular functionalisation process on the molecular level.

In the present thesis, the current methods for determining quantitative grafting

densities are critically assessed and a user's guide is provided to estimate maximum

chain coverage. More importantly, the most frequently employed approaches for

determining grafting densities, i.e., dry thickness measurements, gravimetric

assessment, and swelling experiments are examined. An estimation of the reliability

of these determination methods is provided via carefully evaluating the underpinning

assumptions, their simplicity as well as the stability of the deriving equations. The

assessment is concluded with a perspective on the development of advanced

approaches for determination of grafting density.

By functionalisation of a quartz crystal microbalance sensor via the ‘grafting-

to’ approach of poly(methyl methacrylate) (PMMA), one of the proposed possibilities

of precise grafting density determination was experimentally assessed. By virtue of

this experimental approach, it was demonstrated that grafting a distribution of polymer

chains onto a surface critically affects the shape of the distribution, with shorter chains

being preferentially attached. The preferred surface attachment of shorter chains was

unambiguously underpinned by single-molecule force spectroscopy measurements,

establishing a preferential grafting factor. The preferential grafting factor allows to

predict the molar mass distribution of polymers on the surfaces compared to the initial

distribution in solution. These findings not only have serious consequences for

functional polymer interface design, yet also for the commonly employed methods of

grafting density estimation.

Furthermore, the reaction conditions were found to influence the resulting

grafting density and molar mass distribution when grafting polymers onto surfaces.

Theoretically and experimentally the application of poor solvents is proven to be

beneficial for the ‘grafting-to’ approach. The effect is demonstrated by grafting

PMMA chains on silica nanoparticles in different solvents and comparison of the

molar mass distributions via size exclusion chromatography. The shorter polymer

chains are preferentially grafted onto the surface, leading to a distortion effect between

the molar mass distribution in solution and on surfaces. The molecular weight

distortion effect is significantly higher for better solvent quality than for poor solvents.

In summary, the current thesis is exploring theoretical and experimental

aspects of functionalising surfaces with macromolecules, focusing on the precise

determination of grafting densities. Especially for the ‘grafting-to’ approach novel

procedures of surface characterisations were introduced, enabling an advanced

understanding of the grafting procedure on the molecular level. The gained

information result in the introduction of a preferential grafting factor, which can be

used in any further investigation of surface grafting by the ‘grafting-to’ method.

ii Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

Published Papers Included in this PhD

Research Program

Chapter 2:

L. Michalek, L. Barner, C. Barner-Kowollik, Polymer on Top: Current Limits

and Future Perspectives of Quantitatively Evaluating Surface Grafting, Advanced

Materials, 2018, 30, 1706321(1-18). DOI:10.1002/adma.201706321 (IF = 25.809)

Chapter 3:

L. Michalek, K. Mundsinger, C. Barner-Kowollik, L. Barner, The Long and the

Short of Polymer Grafting, Polymer Chemistry, 2019, 10, 54-59.

DOI:10.1039/c8py01470a (IF = 4.760)

Chapter 4:

L. Michalek, K. Mundsinger, L. Barner, C. Barner-Kowollik, Quantifying

Solvent Effects on Polymer Surface Grafting, ACS Macro Letters, 2019, 8, 800-805.

DOI:10.1021/acsmacrolett.9b00336 (IF = 5.775)

The signed Statement of Contribution of Co-Authors for Thesis by Published

Papers for the three publications included in this PhD Thesis (as mentioned above) is

attached as Appendix A.

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces iii

Additional publications during candidature

M. M. Zieger, P. Müller, E. Blasco, C. Petit, V. Hahn, L. Michalek, H. Mutlu,

M. Wegener, C. Barner-Kowollik, A Substractive Photoresist Platform for Micro- and

Macroscopic 3D Printed Structures, Advanced Functional Materials, 2018, 28,

18014059(1-7). DOI:10.1002/adfm.201801405 (IF = 15.621)

B.T. Tuten, F.R. Bloesser, D. Marshall, L. Michalek, C.W. Schmitt, S.J.

Blanksby, C. Barner-Kowollik, Polyselenoureas via Multicomponent Polymerizations

Using Elemental Selenium as Monomer, ACS Macro Letters, 2018, 7, 898-903.

DOI:10.1021/acsmacrolett.8b00428 (IF = 5.775)

H. Woehlk, J. Steinkönig, C. Lang, L. Michalek, V. Trouillet, P. Krolla, A.S.

Goldmann, L. Barner, J.P. Blinco, C. Barner-Kowollik, K.E. Fairfull-Smith,

Engineering Nitroxide Functional Surfaces Using Bioinspired Adhesion, Langmuir,

2018, 34, 3264-3274. DOI:10.1021/acs.langmuir.7b03755 (IF = 3.683)

S. Bialas, L. Michalek, D.E. Marschner, T. Krappitz, M. Wegener, J. Blinco, E.

Blasco, H. Frisch, C. Barner-Kowollik, Access to Disparate Soft Matter Materials by

Curing with Two Colors of Light, Advanced Materials, 2019, 31, 1807288(1-5).

DOI:10.1002/adma.201807288 (IF = 25.809)

A.S. Goldmann, N.R. Boase, L. Michalek, J.P. Blinco, A. Welle, C. Barner-

Kowollik, Adaptable and Reprogrammable Surfaces. Manuscript in press Advanced

Materials, 2019, (IF = 25.809)

iv Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

Conference Contributions

L. Michalek, L. Barner, C. Barner-Kowollik, (November 2017) Polymer on

Top: Current Limits and Future Perspectives of Quantitatively Evaluating Surface

Grafting - Talk, Queensland-Annual-Chemistry-Symposium (QACS) 2017, QUT

Brisbane, Australia.

L. Michalek, K. Mundsinger, L. Barner, C. Barner-Kowollik, (July 2018) The

Limits of Grafting Processes - Talk, World Polymer Congress (MACRO) 2018, Cairns

Convention Centre, Australia.

L. Michalek, K. Mundsinger, L. Barner, H. Frisch C. Barner-Kowollik,

(September 2018) Investigating Light Gated Folding and Limits of Grafting Processes

via AFM - Poster, International Microscopy Congress (IMC) 2018, International

Convention Centre Sydney, Australia.

L. Michalek, L. Barner, C. Barner-Kowollik, (November 2018) Exploring the

Limits of Surface Grafting - Talk, Queensland-Annual-Chemistry-Symposium (QACS)

2018, Griffith University Brisbane, Australia.

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces v

Awards and Grant Funding Resulting from the

PhD Research Program

L. Michalek (2017-2019) QUT Postgraduate Research Award ($26,688 p.a.).

L. Michalek (November 2017), Excellent Student Presentation at Queensland-

Annual-Chemistry-Symposium (QACS) 2017.

L. Michalek (July 2018), MACRO Travel Bursary for World Polymer Congress

(MACRO) 2018.

L. Michalek (November 2018), Best Student Presentation at Queensland-

Annual-Chemistry-Symposium (QACS) 2018.

L. Michalek (April 2019), High Achiever Award from the Faculty of Science

and Engineering for outstanding PhD work.

C. Barner-Kowollik, L. Barner, H. Frisch, L. Michalek (2019-2020) Merck 350

Year Anniversary Grant for Photochemistry in Mechanical Fields: The Next Frontier

in Dynamic Material Design ($320,000) as co-investigator.

vi Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

Table of Contents

Abstract ..................................................................................................................................... i

Published Papers Included in this PhD Research Program ...................................................... ii

Additional publications during candidature ............................................................................ iii

Conference Contributions........................................................................................................ iv

Awards and Grant Funding Resulting from the PhD Research Program ................................. v

Table of Contents .................................................................................................................... vi

List of Figures ....................................................................................................................... viii

List of Tables ........................................................................................................................... ix

List of Abbreviations ................................................................................................................ x

Statement of Original Authorship ........................................................................................... xi

Acknowledgements ................................................................................................................ xii

Chapter 1: Introduction ...................................................................................... 1

1.1 Surface functionalisation ................................................................................................ 3

1.2 Polymers on Surfaces ..................................................................................................... 5

1.3 Surface Characterisation ................................................................................................ 7

1.4 Thesis Outline .............................................................................................................. 11

Chapter 2: Current Limits and Future Perspectives of Quantitatively

Evaluating Surface Grafting ......................................................... 13

2.1 Abstract ........................................................................................................................ 13

2.2 Introduction .................................................................................................................. 13

2.3 Recapping the Theory of Polymer Brushes ................................................................. 15

2.4 Characterization of Grafting Densities ......................................................................... 26

2.5 Conclusion and future Perspectives ............................................................................. 40

Chapter 3: The Long and the Short of Polymer Grafting .............................. 45

3.1 Abstract ........................................................................................................................ 45

3.2 Introduction .................................................................................................................. 45

3.3 Grafting Densities evaluated via QCM ........................................................................ 46

3.4 Preferantial Surface Grafting ....................................................................................... 49

3.5 Conclusion ................................................................................................................... 53

Chapter 4: Quantifying Solvent Effects on Polymer Surface Grafting ........ 55

4.1 Abstract ........................................................................................................................ 55

4.2 Introduction .................................................................................................................. 56

4.3 Theoretical evaluation of MMDs on Surface Grafting ................................................. 57

4.4 Experimental Solvent Effects on Polymer Grafting ..................................................... 60

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces vii

4.5 Conculsion ....................................................................................................................62

Chapter 5: General Discussion ......................................................................... 65

5.1 Summary of Studies and Key Outcomes ......................................................................65

5.2 Future Direction ............................................................................................................67

5.3 Conclusions ..................................................................................................................69

Bibliography ............................................................................................................. 73

Appendices ................................................................................................................ 89

Appendix A Statements of Contribution of Co-Authors for Thesis by Published Paper ........89

Appendix B Supporting Information Chapter 2 ......................................................................93

Appendix C Supporting Information Chapter 3 ....................................................................104

Appendix D Supporting Information Chapter 4 ...................................................................115

Appendix E Calculations of MMDs and grafting densities for surface grafting via dry

thickness method. ..................................................................................................................123

viii Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

List of Figures

Figure 1: Schematic representation on AFM based SMFS measurements of surface

grafted polymer chains.

Figure 2: Streamline of PhD thesis.

Figure 3: Density profiles (box like-like and parabolic brush) of polymer brushes

using AG or SCF theory.

Figure 4: Thickness vs. grafting density, scaling behaviour in all grafting regimes.

Figure 5: Schematic representation for crystalline polymer brush confirmation for

maximum grafting density calculation.

Figure 6: Maximum grafting density in dependency of C-atoms in side chain and

molecular weight for poly--olefines.

Figure 7: Scheme for grafting density limitations of polymer functionalised

surfaces via ‘grafting-to’ approach.

Figure 8: Grafting density limitations for ‘grafting-to’ approach as a function of

radius of gyration and degree of polymerisation.

Figure 9: Schematic illustration of the three most common grafting‐density

determination methods.

Figure 10: Grafting density as a function of the dry thickness of a polymer brush

determined via the dry thickness approach.

Figure 11: Grafting‐density dependency of the polymer weight fraction of a polymer

brush calculated via the gravimetric assessment.

Figure 12: Grafting‐density dependency of the swelling ratio of a polymer brush

evaluated via swelling experiments.

Figure 13: MMDs of PMMA polymer library and grafting densities determined via

in-situ QCM experiments.

Figure 14: Experimental and calculated time dependent frequency change for

polymer mixtures.

Figure 15: Mass weighted, number weighted and predicted surface MMDs and

rupture length determined via SMFS.

Figure 16: Simplified presentation of the preferred attachment of shorter PMMA

polymer chains onto SiO2 nanoparticles.

Figure 17: Calculated shift of MMD on surface depending on solvent quality

Figure 18: Deconvoluted peak for MMD calculations with calculated shift of MMD

on surface and solution depending on solvent quality.

Figure 19: Experimentally observed shifts in DMAc and EA with percentage

difference of Mw on surface and solution after grafting.

Figure 20: Difference of grafting density in dependency of solvent quality for a

polymer graft evaluated via dry thickness method.

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces ix

List of Tables

Table 1: Literature values for estimated grafting densities of polymer systems

characterized by dry thickness analysis.

Table 2: Determined maximal grafting densities for polymers considered here.


characterized by gravimetric measurements.


characterized by the swelling method.

Table 5: Literature values for estimated grafting densities of polymer systems for

less‐used characterization methods

Table 6: Samples used to investigate grafting bias, their composition, theoretical

Mn in solution Mnsol and on the surface Mn

sur with calculated preferential

grafting factor.

x Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

List of Abbreviations

acronyms name acronyms name

AFM atomic force microscope PMMA poly(methyl

methacrylate)

AG Alexander–de Gennes theory PS polystyrene

ALD atomic layer deposition PVD physical vapour

deposition

BET Brunauer–Emmett–Teller QCM quartz crystal

microbalance

C14 tetradecane RAFT

reversible addition

fragmentation chain

transfer

CTA chain transfer agent RDRP reversible deactivation

radical polymerization

CVD chemical vapour deposition SAM self-assembled

monolayers

DMAc N,N-dimethylacetamide SCF self‐consistent field

approximation

DP degree of polymerisation SEC size exclusion

chromatography

EA ethyl acetate SEM scanning electron

microscopy

FJC freely jointed chain SERS surface-enhanced Raman

spectroscopy

FRC freely rotating chain SIMS secondary ion mass

spectroscopy

GPC gel permeation chromatography SMFS single molecule force

spectroscopy

IR infrared spectroscopy SPM scanning probe

microscope

LEED low-energy electron diffraction STM scanning tunnelling

microscope

LFM lateral-force measurements TEM transmission electron

microscopy

MMD molar mass distribution TERS tip-enhanced Raman

spectroscopy

MS microsphere THF tetrahydrofuran

NP nanoparticle ToF time‐of‐flight

NR neutron reflectivity UHV ultra-high vacuum

PDMAEMA poly(2‐(diemethylamino)ethyl

methacrylate) UPS

ultraviolet photoelectron

spectroscopy

PEG poly(ethylene glycol) WLC worm-like chain

PEO poly(ethylene oxide) XPS x-ray photoelectron

spectroscopy

PHEMA poly(2‐hydroxyethylmethacrylate)

XRR X‐ray reflectivity

PIBA poly(isobornyl acrylate)

Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces xi

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

QUT Verified Signature

xii Quantifying Macromolecular Growth and Hierarchical Structuring on Interfaces

Acknowledgements

First, I want to thank my principle supervisor Prof. Dr. Christopher Barner-Kowollik

for the possibility of undertaking my PhD studies in his Soft Matter Materials Lab at

the Queensland University of Technology. Giving me – a material scientist with

previous focus on materials physics – the opportunity to do research in his polymer

chemistry group is a perfect example of how Christopher welcomes any new

researcher sincerely and is open to any scientific idea. The constant support and trust

he shows towards my research is incredible and means a lot to me. Furthermore, he

gave me the possibilities of being part of new research proposals, which resulted

successfully in the Merck 350 Year Anniversary Grant.

A special thanks goes to my secondary supervisor Prof. Dr. Leonie Barner also from

the Soft Matter Materials Lab at the Queensland University of Technology. She gave

me always constructive feedback to my research and had always time to discuss the

latest results of my work. I especially want to thank her for giving me the opportunity

of conducting QCM measurements by organizing an instrument on which I could carry

out the experiments.

I especially want to acknowledge the support by the QUTPRA PhD scholarship.

Another thanks go to Kai Mundsinger who did most of the synthetic work included in

this PhD thesis. Even though he has his own research topics he finds always the time

to help me out and collaborate on additional projects. Without the help of Kai

Mundsinger I wouldn’t have been able to progress successfully through my PhD

research in the same time frame. I want to thank the whole Macroarc team for the

special atmosphere and working conditions. The collaborative environment and spirit

of helping each other is impressive. Here I especially want to name Dr. Hendrik Frisch,

Dr. Tim Krappitz and Sabrina Bialas for amazing collaborative work and

companionship.

Last but not least, I want to thank my whole family and friends for supporting me

during the time of research in Australia. Even though it was tough being apart from

my family for such a long time, I could still feel their support at any moment of my

journey. Without them it would have been never possible to perform my research in

such a way.

Chapter 1: Introduction 1

Chapter 1: Introduction

Surfaces are the part of any matter we perceive and interact with constantly.

However, the precise definition and analysis of such surfaces is a rather complex task.

From the early beginnings of philosophy to the introduction of modern science,

philosophers vigorously debated the precise definition of matter and its interactions

with its environment.1 One of the founders of modern science, René Descartes, defined

matter – “essence” of a material substance – as the extension in length, breadth, and

depth which constitutes the space occupied by a body, is exactly the same as that which

constitutes the body.2 In other words, the sole fundamental property of a body/matter

is its three-dimensional expansion – which is essentially the surface of matter. This

rather simple and fundamental explanation of matter was revised in the early 1970ies

by the Bulgarian philosopher Panayot Butchvarov, addressing the perception of

surfaces and bodies in his book The Concept of Knowledge. Butchvarov implies that

the surface of a body and the inside are not the same and have no direct correlation

necessarily.3 For being able to understand the properties of matter the surface and the

inside (bulk) have to be investigated separately. Transferred to the field of surface

science, the concept of Butvhvarov can be compared to the often significant

differences in surface and bulk properties of materials and the need of separately

investigating each.

The pioneers in the field of surface science were Paul Sabatier (Nobel Laureate

Chemistry 1912) and Fritz Haber (Nobel Laureate Chemistry 1918), who established

the principles of heterogeneous catalysis.4-5 In both cases, metals were employed to

catalyse high temperature and high pressure chemical reactions for making them

industrially relevant. Today, the Haber-Bosch process generates multi-million tons of

ammonia every year, which is primarily used as fertilizer.6 A further important founder

of the field of surface science is Irving Langmuir (Nobel Laureate Chemistry 1932),

who introduced the concept of mono(molecular) layers.7 The asymmetry of chemical

binding at surfaces – compared to the inner bulk of a material – lead to the conclusion

that the surface atoms (and their properties) have to be different. The work of Langmuir

redefined our understanding of surfaces and interfaces, thus his name stands for the

2 Chapter 1: Introduction

American Chemical Society journal Langmuir, which focus on the research areas of

surface chemistry and surface characterisation.

The disparity of the surface atoms to the bulk atoms – specifically their impact

on the properties of a material – is enhanced by miniaturising feature sizes of matter.

The increased impact of the surface atoms can be explained by the example of the

surface to volume ratio of a spherical particle, which is increasing asymptotically by

decreasing the radius of a particle. In 1959, the renowned physicist Richard Feynman

(Nobel Laureate Physics 1965) gave his seminal lecture Plenty of Room at the Bottom

at the American Physical Society annual meeting, in which he explored the large

potential of miniaturization.8 His vision is the foundation of modern nanotechnology

and a perspective for the direct manipulation of atoms. One of the defining

characteristics of nanotechnology is the nanometre magnitude (at least in one

dimension) of feature size of a material/component. The increased amount of surface

to volume is changing the properties and performance of the material and therefore has

to be characterised thoroughly.

One of the major breakthroughs in the emerging field of nanotechnology was the

development of the scanning tunnelling microscope (STM) by Gerd Binnig and

Heinrich Rohrer (Nobel Laureates Physics 1986).9 In addition, precise imaging of

surfaces (atomic resolution for conducting materials), and accurate manipulation of

atoms arose with the invention of the STM. Further development steps from Binnig

resulted in the design of an atomic force microscope (AFM) which could also image

non-conductive materials to a resolution of a few angstroms.10 In addition,

(interaction) forces between a sample and a nano-sized probe could be recorded. The

development of both scanning probe microscopes (SPM) enabled an entirely new level

of understanding of surfaces and the concomitant design of precision functional

surfaces.

Gerhardt Ertl (Nobel Laureate Chemistry 2007) reformed the field of surface

science by establishing the in-depth characterisation of surface chemical reactions.11

His teaching redefined our understanding of heterogeneous catalysis and revealed the

nature of surface chemical bonds.12-13 The elucidation of fundamental principles of

chemical reaction on surfaces was feasible due to advanced surface characterisation

tools i.e. low-energy electron diffraction (LEED), ultraviolet photoelectron

spectroscopy (UPS) and STM. Additional surface characterisation techniques

including surface-enhanced Raman spectroscopy (SERS) and tip-enhanced Raman


spectroscopy (TERS) were frequently employed by Gerhardt Ertl.14-15 With such

techniques, the resolution of chemical imaging was considerably improved and

therefore revised our knowledge of surface chemistry and surface properties.

A further recent advance in the application of surface science is the topological

insulator (Nobel Prize Physics 2016), which is a material that behaves like an insulator

in bulk (interior), yet is conductive on the surface. Early calculations predicted time-

reversal symmetry-protected surface states which are leading to a surface conductive

behaviour of insulating binary compounds (containing bismuth).16-17 The first

experimental proof of such symmetry-protected surface states was carried out by

Hasan and co-workers, investigating Bi0.9Sb0.1 bulk material.18 Further studies show

that the concept of a topological insulator could be employed in low-power-

consumption electronics, which may be adopted in quantum computing.19 The

difference between the surface and bulk properties of the topological insulator is a

good example for how far (surface) science has progressed and how important it is to

investigate and tailor surfaces with precise material properties.

An advanced way to tailor surface properties of a material is by decorating the

surface with atoms or (macro)molecules, which include a specific function.20-21 In this

scenario the bulk material is merely a substrate (structural element) and the surface

properties are defined by the attached functionality. The design and characterisation

of such a functional surface is critical for many application fields of (soft matter)

materials. After all, the surface/interface is the most crucial point of interaction

between matter.

In the next three subchapters a short introduction into Surface Functionalisation,

Polymers on Surfaces and Surface Characterisation is provided, followed by a Thesis

Outline to position the progress of the present PhD project in the field of surface

science. The research which was conducted in this PhD thesis is exclusively focusing

on the functionalisation and characterisation of solid surfaces with polymers.

Therefore, no deeper introduction into any kind of liquid or gaseous interfaces will be

provided.

1.1 SURFACE FUNCTIONALISATION

For the majority of materials and their applications, their surface possesses

particular and well-defined functions. The surface can feature functionalities ranging

from wear resistance for structural materials,22 conductive or insulating layers for


micro-electronics,23 wettability,24 release of small molecules for drug delivery

systems,25 chemical resistance up to biocompatibility for medical devices26-27. The

surface decoration with specific functionalities can be achieved by attaching atoms,

ions, molecules or macromolecules on surfaces via adsorption processes, which are

commonly subdivided into chemisorption and physisorption.28 Both processes can be

defined by the interaction forces between the substrate and the adsorbent. Nevertheless

it is rather complex to define the difference between both processes, similar to the

separation between chemical and physical interactions in general. For the

chemisorption the substrate and the adsorbent encounter strong interaction forces

comparable to chemical valence forces (analogous to the ones in forming chemical

compounds). Therefore, also strong hydrogen bonding of the adsorbent to the surface

is called chemisorption by IUPAC definition.29 In contrast, no significant change in

the electronic orbital pattern occurs in physisorption, where the interaction forces are

van der Waals forces and significantly lower.30

One of the simplest forms of designing surfaces with defined properties is by

depositing a layer on top of a substrate. The thickness of the deposited layer can range

from a monolayer (a few Ångstrom) to several micro-meters.31 Generally, the

deposition methods are divided into two major categories, the physical- and the

chemical-deposition. In the later approach, a precursor is subject to a chemical change

at a solid substrate surface, whereas for the physical deposition, the deposition occurs

via physical impacts such as mechanical or thermodynamical processes.32 It is

important to note that the nature of the deposition method (chemical or physical

deposition) does not have to be necessary the same as the adsorption phenomenon

(chemisorption or physisorption). In other words, films which are deposited via

chemical methods (i.e. spin coating) can be adsorbed via physisorption and vice versa

(i.e. physical vapour deposition (PVD) sputtering techniques). Common techniques

for the chemical deposition include chemical solution deposition (sol-gel method),33

spin/dip coating (polymer coating),34-35 chemical vapour deposition (CVD)36 and

atomic layer deposition (ALD).37 Examples for the physical deposition involve

thermal evaportation,38 molecular beam epitaxy,39 sputtering techniques (PVD),40 and

electro-hydrodynamic deposition.41 The aim of the aforementioned deposition

methods is to form uniform (thin) films on top of a substrate with a precisely desired

function. For guaranteeing reliable surface properties, the attachment of the film to the

substrate has to be ensured. The macroscopic interaction between the film and the


substrate is defined as adhesion and is often a mixture between chemisorption and

physisorption on the microscopic scale. Adhesion is defined as the tendency of a

substance to attach to the surface of another substance – a process that requires energy

(adhesion energy), which can come either from physical and/or chemical linkage.42

There are primarily five mechanisms to describe the phenomenon of two materials

adhering: mechanically by mechanical interlocking (velcro principle),43 dispersive by

intermolecular van der Waals forces (Keesom and London forces),44-45

electrostatically by difference in electric charge,46 diffusive by inter-diffusion between

both materials (interdigitating chains on polymer surfaces)47 and chemically by

covalent bond formation (chemisorption).

The functionalisation of a surface by chemical adhesion is especially durable due

to the strong interaction forces between surface and substance. Nowadays

sophisticated chemical procedures allow to equip almost any surface with any kind of

function/property on the nanoscale domain, which makes this process highly

versatile.48 One way to achieve such uniform chemical surface modification is through

formation of self-assembled monolayers (SAM).49-50 The functional group of the

chemisorbed molecules – often small organic molecules with a head and tail structure

– have a strong chemical affinity to the surface. The driving force for the SAM

formation is the minimization of surface energy from the substrate, which occurs in

three phases defined by the density of the self-organized layer (low – intermediate –

high density).51 In addition, the tail of the SAM molecule can be equipped with specific

chemical end-groups for further functionalisation of the surface, for example by

surface initiating polymerisation groups.52 Functionalisation of the surfaces with

polymers rather than small organic molecules is increasing the range of accessible

surface properties even further.

1.2 POLYMERS ON SURFACES

Since the advent of polymer science and the macromolecular hypothesis by

Hermann Staudinger53 (Nobel Laureate Chemistry 1953), the fields of polymer

chemistry, physics and technology are vibrantly growing research areas. The need for

polymeric materials in almost any application field is increasing and therefore the

production is at an all-time high.54 Even though a large amount (by mass) of polymeric

material is still employed in the area of packaging, fields of more advanced

applications i.e. surface modification are growing to a great extent. Covering materials


surfaces with polymers is of fundamental interest in many application fields.55 As

described already briefly in the previous Sub-Chapter 1.1, there are in general two

ways to attach polymers to surfaces by weak physisorption (e.g. via processes such as

spin/dip coating)56 or via chemically tethering polymer chains onto surfaces. The later

approach is leading to strongly durable long living surface functionalisation and can

be achieved either by ‘grafting-to’ or ‘grafting-from’ approach.57 The process of

covalently attaching polymers on surfaces is referred to as (polymer) grafting and will

be explained in greater detail in Chapter 2 of the current PhD thesis. The current sub-

chapter is focusing on the properties of polymers on surfaces.

The properties of polymers on surfaces can be divided into single chain and

densely grafted chain properties. In the latter case, the interaction between many

polymer chains are resulting in specific properties of a polymer graft. For the possible

inter-molecular interaction of polymer chains on a surface, the single chains have to

be tethered sufficiently dense. The polymer is then typically referred to as polymer

brush.58 A further explanation of polymer brushes and how to define different grafting

density regimes can be found in Chapter 2 of the current thesis. Other ‘many’ chain

properties are the thickness, and the density profile of the formed polymer graft, which

depend strongly on the grafting density and the contour length of the single polymer

chains. The length of a single polymer chain is defined by the product of the monomer

size and the degree of polymerisation. Nota bene that the degree of polymerization and

therefore the length of the polymer chains are not the same for each single chain. The

dispersity describes the extent of dissimilarity between all degrees of polymerizations

and can be calculated by dividing the weight averaged molecular weight with the

number weighted averaged molecular weight.59 Especially for the ‘grafting-to’

approach the effect of dispersity on the formed polymer graft is critical and is

investigated in detail in Chapters 3 and 4.60-61 A further important property of a

polymer chain on a surface is the chain elasticity, which describes the flexibility of the

polymer backbone. Depending on the model used for describing the physics of a

polymer chain, the chain elasticity is defined as persistence length (worm-like chain

(WLC) model), Kuhn length (freely jointed chain (FJC) model) or rotating unit (freely

rotating chain (FRC) model).62 The determination of all above mentioned properties is

important for the performance of a polymer graft. To be able to tailor the performance

of such functionalized surfaces, an in-depth understanding of the polymers on the

surface has to be achieved.


1.3 SURFACE CHARACTERISATION

For understanding growth processes and hierarchical structuring of grafted

polymers on surfaces – specifically at the molecular level – in-depth characterisation

methods have to be employed. As already mentioned in previous subchapters, such

knowledge is the essential key for designing and tailoring surfaces with specific

properties. Surface characterisation can be grouped into three major categories,

chemical characterisation, surface imaging and physical properties.63 All three

categories will be introduced in the following paragraphs, focusing on the techniques

applied in the experimental work of the current PhD thesis.

1.3.1 Chemical Characterisation of Surfaces

There are several characterization techniques available for analysing the

chemical composition of surfaces i.e. x-ray photoelectron spectroscopy (XPS), Raman

Spectroscopy, secondary ion mass spectroscopy (SIMS), infrared spectroscopy (IR)

and many more.64 Nevertheless, samples prepared in the current work were solely

analysed via XPS, due to the conclusive results achieved with this method. In XPS,

the sample are irradiated with soft X-rays in an ultra-high vacuum (UHV) chamber.65

The x-ray irradiation leads to the emission of core electrons (photoelectrons) with

specific (measureable) kinetic energy, which can be used for the calculation of the

binding energy. For each element the binding energy of a photoelectron is

characteristic. Therefore, the signal intensity (usually given in counts of photoelectrons

per seconds) can be correlated to the elemental composition of the sample.65

Additional slight shifts (a few eV) of the binding energy can give insights into the

chemical vicinity and/or the oxidation states of the element. The possibility of not only

obtaining a quantitative result for the elemental composition but also a quantitative

result for chemical moieties (chemical shifts) is of specific interest for chemical

characterization of organic molecules and polymers on surfaces.56, 66 An additional

benefit of chemical characterisation via XPS – which is commonly mentioned – is the

non-destructive nature of the method. However, there are reported cases of x-ray

damaging samples, implying that care should be taken when conducting such

experiments.67-68 Standard x-ray photoelectron spectrometers are averaging over an

area of a few square millimetres, which allows representative chemical analysis of the

surface even if microscopic inhomogeneities are present. New developments in

spectrometer design allow XPS mapping (chemical imaging) of a surface with a lateral


resolution of roughly 50 m. XPS mapping not only offers the possibility to

chemically analyse the average surface composition, yet also provides spatial

resolution (limited to a few ms) to the chemical information.69 It is important to

mention that even though the ratio of elemental composition and binding energies of

specific elements can be readily determined quantitatively via XPS, a quantitative

correlation with the exact amount of atoms or (macro)molecules is challenging.

Therefore, a direct determination (calculation) of grafting densities via standard XPS

measurements is not feasible.

1.3.2 Imaging of Surfaces

There are many imaging tools which can spatially resolve surfaces/interfaces

such as confocal microscopy, scanning electron microscopy (SEM), transmission

electron microscopy (TEM), scanning probe microscopy (SPM) and many more.64, 70

Most of these characterisation methods can investigate additional material/surface

properties or the chemical composition. The imaging method of choice is mainly

depending on the material properties of the substrate and surface layer. For example,

in the case of STM the surface and the measurement probe have to be conductive for

being able to measure a tunnelling current.71 The samples in the present work are solely

investigated via AFM due to the unique flexibility of the measurement procedure.

Samples can be characterized via the tip surface interaction in environments ranging

from ambient to harsh conditions (even in organic solvents) along with a minimal

effort in sample preparation. Via AFM, unmatched quantitative atomic resolution in

lateral direction (z-height profile) can be achieved,72 although the resolution in the x-

y-plane is restricted by the extent of the tip radius of the employed cantilever. The

shape and uniformity of the tip can critically affect the resolution/quality of the

recorded surface topography, which can lead to artefacts such as shadowing effects,

broadening of surface features and double imaging.73 In addition to recording the

surface topography, AFM can investigate other materials’ characteristics including

electronic properties, magnetism, thermal conductivity or nano-mechanical properties

(storage/loss modulus, friction, adhesion and stiffness).74 Further chemical contrast

(qualitative) in AFM imaging can be achieved by phase contrast via operating the

AFM with an oscillating cantilever (commonly mentioned as ‘tapping-mode’) in either

non- or semi-contact mode. Depending on the chemical interaction between the sample

and the AFM cantilever tip the phase will vary with a resolution down to a few nm.75


One way to reach quantitative chemical surface imaging with an AFM is by coupling

it with additional spectroscopic methods including IR (AFM-IR) or Raman

spectroscopy (TERS). For the latter example, the Raman scattering is enhanced by a

metallic AFM tip which leads to quantitative chemical analysis. The spatial resolution

in the chemical imaging is limited by the AFM tip radius and can be as low as a few

nm (single dye molecule ~ 15nm).76-77

1.3.3 Physical Properties of Surfaces

When discussing physical properties of surfaces or surface coatings, literature

often refers to mechanical properties (i.e. stiffness, hardness and viscoelastic

behaviour). Although other properties including wettability, conductivity, density and

layer thickness are equally important for the performance of materials surfaces.

Investigation of the underlying samples in the current work – with respect to their

physical properties – was performed via AFM, quartz crystal microbalance (QCM)

and Ellipsometry. As already mentioned in the previous Sub-Chapter 1.3.2, AFM can

be used to map the nano-mechanical properties of a surface.74 The force modulation

via nano-mechanical imaging can be used for the precise distinction between disparate

soft matter materials on surfaces.56 An alternative way to characterize these physical

properties is via force spectroscopy. The term ‘force spectroscopy’ describes all

measurements on the response of a material to an externally applied mechanical force.

There exist several ways to perform force spectroscopic measurements, e.g. single

molecule force spectroscopy (SMFS),78 lateral-force measurements (LFM),79 or

colloidal probe measurements.80 For SMFS, the force response of a single polymer

chain to an applied pulling event is recorded. SMFS measurements on single chains

can be conducted using optical tweezers, magnetic tweezers and AFM.78 Nevertheless,

these methods are all based on the same principle of stretching a molecule – which is

tethered with one end to a substrate – by attaching the free end of the chain to a probe

and applying a force (pulling). One of the most common techniques is AFM based

SMFS.81-83 The measurement process can be divided into four stages, which are

illustrated in Figure 1. The process commences with pressing the AFM cantilever on

the surface (Figure 1 A) with a specific trigger force and dwell time. During stage (A)

a polymer can attach to the tip of the cantilever. By retracting the cantilever from the

surface with a specific velocity (Figure 1 B), the chain starts to uncoil. If the distance

between the substrate and the cantilever tip is further increased, the polymer is


stretched. This can be seen as an increase of the force (Figure 1 C) in the force distance-

curve, due to the change in deflection of the cantilever beam. When reaching a force

exceeding the rupture force, the attachment of the chain to the cantilever cleaves and

the force will decrease immediately to zero (see Figure 1 D). With the data acquired

from the third stage, it is possible to determine the length and elasticity of a single

chain quantitatively by fitting the data to a physical model of the polymer chain.62 The

precision of force recording is in the low pico-Newton range and can even detect the

conformational change between cis to trans isomers of alkenes in polymers.84 Even

though the SMFS shows high precision in investigating single chain properties, the

method becomes exponentially difficult in high grafting density regimes when

multiple chains are attached to the cantilever. A direct quantitative correlation of

grafting density by counting the amount of rupture events cannot be achieved.

Figure 1: Measured deflection versus Z position of the cantilever and calculated force-distance

curve of a force measurement on a single molecule chain. Sketches of the four stages of

the experiments: (A) pressing the tip of cantilever onto the sample, (B) retraction from

the surface, (C) stretching of the polymer chain and (D) rupture of the chain.

Another important analytical technique for precise quantitative surface

characterisation – addressing the number of tethered molecules – is QCM where

surface mass uptake can be recorded by the change of resonance frequency of an

oscillating quartz crystal. The strengths of a QCM setup is the high (mass) sensitivity,

in-situ operation and straightforward quantification.85-87 The analysed surface area is

in the range of several square millimetres up to centimetres, therefore inhomogeneities

are not greatly affecting results with regard to their statistical reproducibility.

Combined with the determination of the thickness of the surface layer (e.g.

ellipsometry), further calculation of densities can give a complete picture of the

dimensions and densities of a polymer functionalised surface.


1.4 THESIS OUTLINE

As the title – Quantifying Macromolecular Growth and Hierarchical Structuring

on Interfaces – of the current PhD thesis implies, a precise quantification of

macromolecules (polymers) attached onto interfaces (surfaces) is targeted as the final

goal. As outlined in the introduction, there is a multidisciplinary need for surface

functionalisation especially by tethering polymers covalently onto solid substrates. To

enable the design of new advanced functional surfaces, the understanding of the

processes governing surface attachment at the molecular level is critical.

One of the main challenges which commonly arises after grafting polymers on

surfaces relates to the question of how many polymer chains are attached to the surface

over a certain area? Or in other words, how dense are the chains tethered on the

surface? The physical quantitative measure is the grafting density. The frequent

employed procedures of grafting density determination are often flawed.

Reproducibility and statistical significance of such methods are questionable.

Critically, for most of the common evaluation methods physical assumptions are

invoked which influence the quality of quantitative determination significantly. For

understanding existing characterization procedures and their physical limitations, a

comprehensive assessment of existing procedures was undertaken resulting in the first

major publication of the current thesis – Chapter 2: Current Limits and Future

Perspectives of Quantitatively Evaluating Surface Grafting. In Chapter 2, not only a

precise investigation of existing methods is carried out, but also calculations for

physical limitations of the grafting process are shown. Furthermore, future

perspectives on the quantitative evaluation of grafting densities are provided.

One of the proposed evaluation methods is the evaluation via QCM

measurements. As mentioned in the previous Sub-Chapter 1.3.3, a QCM is able to

detect the amount of surface grafted polymers via the frequency change of an

oscillating crystal with highly reproducible precision. Therefore, QCM was selected

as a precise method of characterising surface grafting, resulting in the second major

publication of the current thesis – Chapter 3: The Long and the Short of Polymer

Grafting. By grafting PMMA chains (with different molecular weight) on a silica

QCM sensor via the ‘grafting-to’ process, the grafting density limitations could be

correlated to the radius of gyration. Additional ratio experiments detected the effect of


preferential surface grafting of shorter polymers chains on surfaces allowing the

introduction of the preferential grafting factor.

Further investigations of solvent effects on the change (shift) of molar mass

distribution on surfaces compared to solutions resulted in the third major publication

– Chapter 4: Quantifying Solvent Effects on Polymer Surface Grafting. By

calculations and experiments, the significance of solvent quality on the preferential

surface grafting was investigated. The experimental results (and calculations) are

suggesting the use of poor solvents for grafting via the ‘grafting-to’ method.

The graphical representation of the streamline/outline of present PhD thesis can

be found in Figure 2.

Figure 2: Streamline of PhD thesis with three major outcomes (publications) for quantifying

macromolecular growth and hierarchical structuring on interfaces.

Chapter 2: Current Limits and Future Perspectives of Quantitatively Evaluating Surface Grafting 13

Chapter 2: Current Limits and Future

Perspectives of Quantitatively

Evaluating Surface Grafting

2.1 ABSTRACT

Well‐defined polymer strands covalently tethered onto solid substrates

determine the properties of the resulting functional interface. Herein, the current

approaches to determine quantitative grafting densities are assessed. Based on a brief

introduction into the key theories describing polymer brush regimes, a user's guide is

provided to estimate maximum chain coverage and — importantly — examine the

most frequently employed approaches for determining grafting densities, i.e., dry

thickness measurements, gravimetric assessment, and swelling experiments. An

estimation of the reliability of these determination methods is provided via carefully

evaluating their assumptions and assessing the stability of the underpinning equations.

A practical access guide for comparatively and quantitatively evaluating the reliability

of a given approach is thus provided, enabling the field to critically judge

experimentally determined grafting densities and to avoid the reporting of grafting

densities that fall outside the physically realistic parameter space. The assessment is

concluded with a perspective on the development of advanced approaches for

determination of grafting density, in particular, on single‐chain methodologies.

2.2 INTRODUCTION

The design of functional interfaces is critical for almost every application, as the

majority of materials' interface with their environment.88 In particular, interfaces

decorated with (functional) macromolecules fulfill critical functions in a range of

devices and applications, covering areas from medical implants,89-90 3D cell

scaffolds,91-92 optoelectronics,93 and sensors94 to coatings.95-97 The functionality of the

interface and its interactions with the environment are critically determined by the type

of polymer and, critically, the density with which the strands are tethered to the surface.

Thus, the modification of surfaces with synthetic—and to some extent natural—

polymers has been, and is, currently a highly active research field.98 The covalent

modification of surfaces with polymers has been significantly affected by the advent

14 Chapter 2: Current Limits and Future Perspectives of Quantitatively Evaluating Surface Grafting

of polymerization methods that afford fine control over the polymer length and the

dispersity of the resulting molecular weight distribution. These advanced synthetic

methods, which have been developed over the last two decades, include reversible

deactivation radical polymerization (RDRP) processes,99-101 yet also advanced cationic

and anionic macromolecular growth mechanisms.102-103 For the first time, the length

of a synthetic polymer chain ensemble can be effectively controlled for a wide array

of functional monomers, presenting notable opportunities for surface modification.103

The process of growing chains from a surface, termed “grafting‐from” and discussed

in the following sections in more detail, exploits covalently tethered small‐molecule

initiation sites on the surface. Of similar impact on the field of covalent surface

modification has been the development of mild and efficient ligation protocols not

only for the construction of complex macromolecules in solution by linking

prefabricated polymer strands, but also the covalent attachment of such strands onto a

rich array of surfaces. Some of these linkage protocols adhere to the stringent click

chemistry criteria pioneered by Sharpless and co‐workers104 and refined in the context

of polymer chemistry by Barner‐Kowollik et al.105 The postattachment approach is

often termed “grafting‐to.” The chemistry of the employed polymers has been explored

in substantial depth, and a plethora of studies has been devoted to assess the resulting

surface properties, both on the macroscopic and molecular levels. For example, the

ability to protect surfaces from biological impact,26, 106-107 the fine control over surface

hydrophilicity and hydrophobicity,108-109 the possibility of releasing small

molecules,110-111 the fluorescence, as well as emission properties112 and tribological

behavior113-114 have been studied. On the molecular level, e.g., probing the surface

chemistry with photoelectron emission spectroscopy115 or time‐of‐flight secondary‐

ion mass spectrometry (ToF‐SIMS)116-117 has been explored in depth. However, one

property‐defining characteristic is typically not as critically examined, the grafting

density. Nearly all published studies recognize the importance of the chain grafting

density, and an array of methods and models on how polymer chains behave on

surfaces has been published.118-119 Despite this progress, the literature abounds with

grafting densities that challenge physical limits or that have been assessed based on

incorrect assumptions and thus feature limited accuracy. Thus, we herein provide—

after a concise overview of the most important polymer brush models—a discussion

on the limits of physical grafting densities by applying a simple model. More

importantly, however, we examine the three most employed methods for


experimentally estimating grafting densities, i.e., dry thickness measurements,

gravimetric assessment, and swelling experiments. Critically, we provide for each of

these a clear explanation of the inherent assumptions and assess the stability of the

underlying equations for each method by mathematical error propagation. We

demonstrate that key methods used for accessing grafting densities have inherent

weaknesses that lead to considerable errors, which, in some cases, can be minimized

by selecting the correct experimental assessment, while some methods are

nonapplicable in certain grafting regimes. Finally, we make recommendations on how

determinations of grafting density should ideally be carried out and provide a

perspective into future methods. We hope that this work will serve as a blueprint and

a reference guide for future measurements of grafting density, while concomitantly

serving as a user's guide for estimating the expected errors.

2.3 RECAPPING THE THEORY OF POLYMER BRUSHES

By tethering one end of a polymer chain to a surface or interface, a polymeric

brush surface can be constructed. The term “polymer brush”, if used correctly (to

describe the brush regime) describes a sufficiently dense (covalent) attachment of

polymer chains. Nevertheless, polymer systems with a lower‐ or a higher‐grafting‐

density conformation are also often denoted as “polymer brush” in the literature.58

Here, this term will also be used to refer to the different dense grafting conformations.

In the brush regime, the polymer chains interact with each other so that a stretched

configuration of the polymer chains (in the absence of an external field) is present. The

behaviour of a stretched configuration differs significantly to that of free polymers in

the random‐walk configuration.119 The behaviour of polymer chains in low‐density

brushes is also significantly different to that of free polymers in solution.120

While there exists a variety of possible polymer brush configurations, such as

polymer micelles,121 adsorbed (co)polymers (on solid or liquid–liquid interface),122-123

diblock copolymer melts,124 and covalently grafted polymers on a solid surface, here,

we will exclusively address the latter: end‐grafted polymers on a solid substrate. As

noted in the introduction, polymer brushes are accessible by two main approaches, i.e.,

“grafting‐to” and “grafting‐from” methods. In addition, some surfaces are modified by

“grafting‐through” approaches.125 As also noted, the synthetic process of grafting

polymers on surfaces is well established and summarized in an abundant number of

excellent reviews.52, 119, 126-128 Nonetheless, a brief reiteration—focusing on some


critical points—of these approaches is crucial for our current report. As the “grafting‐

to” and “grafting‐from” approaches are the primary methods for covalently attaching

polymers on solid surfaces, we will solely focus on these mechanisms.

In the case of “grafting‐to”, an end‐functionalized polymer chain is tethered

covalently to a suitable substrate by undergoing a chemical reaction under appropriate

conditions.99 The key advantage of the “grafting‐to” approach is that polymer chains

can be synthesized with a narrow molecular weight distribution incorporating the

desired functionalities before the attachment to the surface. Critically, the degree of

polymerization can be determined before tethering the polymer on the surface.

Knowledge of the chain length of the polymer is important for several polymer brush

characterization methods, which we will assess in depth in the following sections. A

prefunctionalization (by self‐assembled monolayers (SAMs) or coupling agents) of the

substrate surface is necessary for introducing functional groups that can react with the

end‐functionalized polymer chain. An often‐suggested general disadvantage of the

“grafting‐to” approach is that only a smaller number of chains can be attached per unit

area, due to slow diffusion of macromolecules.119 To reach a reactive site on the

surface, a polymer chain has to diffuse through the previously attached polymer coils.

If the thickness of the polymer layer is increasing, this barrier will become more and

more significant. Therefore, the grafting density and the thickness of a polymer brush

with the “grafting‐to” approach should be limited. The “grafting‐from” method can

circumvent these drawbacks.

During “grafting from”, a polymer chain is synthesized in situ on a surface‐

functionalized substrate.128 Surface‐confined macromolecular growth is achieved by

immobilization of a polymerization initiator onto the surface by covalent ligation.

Most of the known polymer syntheses can be carried out with this grafting mechanism.

In addition, controlled growth can be achieved, for example, via surface‐initiated

RDRP.129-130 As the polymer chain is constructed starting from the surface of the

substrate, only a monomer, and not an entire macromolecule, has to diffuse to the

active site, which is a much faster process. Therefore, thicker polymer brushes and

higher grafting densities may be accessible. The major disadvantage of the “grafting‐

from” approach is that the degree of polymerization, N, the number average molecular

weight, Mn, and the dispersity, Đ, of the polymers at the surface are unknown, posing

a significant challenge for quantitative brush surface characterization, as we will


discuss in the following sections. To determine the unknown characteristics of the

polymer chains, the polymers have to be either cleaved after synthesis or a sacrificial

initiator has to be added into the polymerization solution during synthesis.131-132

Cleaving of the polymer is a rather complicated and laborious task because a

sufficiently large surface area must be coated to obtain sufficient polymer chains for

common characterization methods such as size‐exclusion chromatography. On the

other hand, by adding a sacrificial initiator to the polymerization mixture and

determining the unknown characteristics of the polymer by measuring the free

polymers synthesized in solution, only a weak and indirect link to the polymers on the

surface is available. In the literature, the correlation between free polymer in solution

and on the surface is a controversially discussed topic and no clear statement regarding

the quality of the correlation can be made.133-134 In this context, we note a recent study

of Spencer and co‐workers, who investigated the influence of the grafting density on

the polymer growth rate on surfaces and solution, concluding that the difference

between the surface and solution‐generated molecular weight is a function of the

grafting density.135

In addition to the intramolecular properties such as the degree of polymerization,

N, and molecular weight, the critical properties such as the grafting density, σ (amount

of polymer chains per certain surface area), brush thickness, h, and density profile,

Φ(h) (progression of segment density perpendicular to the surface) determine the

properties of a polymer brush.136 These parameters are a result of the interactions of

the polymer chains with each other, the solvent and the surface. To obtain a clearer

picture of the distinctive properties of a polymer brush, several notable theoretical

models have been introduced in the past, which we will briefly review.119, 137-151

The simplest description of a polymer brush rests on free‐energy‐balance

arguments. In general, there are two counteracting driving forces affecting the polymer

chain: the maximization of the configurational entropy, achieved by a random‐walk

configuration, and the tendency for a polymer chain to be surrounded by solvent. With

increasing grafting density, the polymer brush has to balance the increase of Helmholtz

free energy, A, due to a reduction of configurational entropy and the overlapping of

neighbouring polymer chains (excluded volume effect).119 One of the first models

providing an elaborate analytical description of the polymer brush properties on solid

planar surfaces is the Alexander–de Gennes (AG) theory,137-139 for which several

assumptions are made. First, the segment density profile of the polymer brush


perpendicular to the surface is a step‐like function, featuring a constant value for the

monomer concentration through the entire brush with each chain behaving in the exact

same manner. The step‐like density profile and the corresponding integral are shown

in Figure 3 with black solid lines. Second, all free ends of the tethered polymer chains

are placed in one single plane with the distance, h (brush thickness), parallel to the

planar substrate surface. The “Flory approximation” is invoked to give an explicit

expression for the free energy of the polymer brush in the AG theory.140 This

approximation evaluates the reduction in configurational entropy caused by the

constraint of not being an ideal random‐walk polymer in solution and instead being

tethered on a surface with the free‐end of the polymer situated at the brush thickness,

h. As a result, the equilibrium brush thickness h for a polymer brush in a good solvent

can be obtained by Equation 1

ℎ ≈ 𝑁𝑎4 3⁄ 𝜎1 3⁄ (1)

with the degree of polymerization N, the monomer size a, and the grafting density .

In contrast to a free random‐walk polymer, where the dimension of the polymer (radius

of gyration) is scaling in an 𝑅g ~ 𝑁3 5⁄ relation, the dimension of the polymer brush

scales linearly with the degree of polymerization (ℎ ~ 𝑁), leading to the conclusion

that the densely grafted polymer chains are deformed (in comparison to the random‐

walk polymers). The linear scaling behaviour is also valid for poor solvents.119

Figure 3: Density profile of a box‐like brush (AG theory, in black solid lines) and a parabolic brush

(analytical SCF theory, in red dashed lines) with the corresponding integral of the segment

density.

As noted, the AG theory is only applicable for planar samples. For a theoretical

model of a polymer brush on a curved substrate (e.g., a nanoparticle (NP) or a

microsphere (MS)), the AG theory has to be adapted.143-144 Daoud and Cotton


implemented the radial distance dependency of the excluded volume effect into the

AG theory, leading to a more accurate description of polymer brushes on curved

substrates.143 Their extension is a simple geometrical calculation having a smaller

excluded volume (more space for polymer chains and, therefore, higher

configurational entropy), the further away the polymer is from the surface. The theory

of Daoud and Cotton was elaborated for star‐shaped polymers, yet can also be directly

used for polymer brushes on curved surfaces.

In a numerical study of Hirz (using a self‐consistent field (SCF)

approximation), their simulations show a quite different behaviour with regard to the

density profile of a polymer brush, in comparison to the AG theory (no step‐like

profile).145-146 Thus, a more accurate description of a polymer brush was suggested by

Milner et al.147-149 based on the idea that the free‐chain ends can be located anywhere

in the polymer brush. Specifically, an analytical approximation to solve an SCF theory

of polymer brushes in an external potential is invoked. The external potential in this

approximation is generated by replacing the monomer interactions with a position‐

dependent monomer chemical potential originating from the self‐consistent segment

density profile. For the calculations of the aforementioned potential, the excluded‐

volume effect is balanced with the decrease of configurational entropy. In the SCF

theory suggested by Milner et al., the self‐consistent field for equal length polymer

chains is a parabolic potential and is therefore called the “parabolic field”

approximation. The density profile and the corresponding integral are depicted in

Figure 3 by the red dashed lines. The key difference in comparison with the AG theory

manifests itself in the overall brush thickness, h. For the AG theory, the predicted value

is an underestimation for polymer brushes with similar grafting density, although an

exact value is needed for the quantitative determination of the grafting densities. The

underestimation of grafting densities is problematic for the different characterization

methods and will be examined comprehensively below.

In general, it should be noted that the parabolic density profile of the SCF

theory affords a better description of moderately densely grafted polymer brushes in a

good solvent. In an experimental result by Field et al., the measured density profile of

a polystyrene (PS)–poly(ethylene oxide) (PEO) brush was well described by a

parabolic profile (SCF theory).152 The step‐like profile of the AG theory seems to fit

better for strongly stretched (high‐grafting‐density) polymer chains on surfaces.


Nevertheless, both theories are failing in their description of polymer brushes with

very low or very high grafting densities.141 For a more accurate description of a

polymer brush, numerical calculations and simulations have to be adopted, including

Monte Carlo methods,150 molecular dynamic simulations,151 Brownian dynamics, or

numerical SCF calculations.153

The polymer brush architecture (conformation) is strongly influenced by the

grafting density of the polymer brush, as indicated earlier. By solvent interactions the

effect of grafting density on brush conformation can be even more pronounced, with a

good solvent leading to a swollen state, whereas a poor solvent will cause a collapse

of the polymer brush.154 Depending on the grafting density, the conformation of a

polymer brush can be subdivided into four different groups, i.e., the pancake‐,

mushroom‐, brush‐, and high‐density brush regime.120, 155-156 A schematic

representation of their analogous scaling law (thickness vs grafting density predicted

by AG theory) is presented in Figure 4. Here the effect of increasing grafting density,

σ, on brush thickness, h, for polymer chains with the same constant length in a good

polymer solvent is shown. The thickness, h, is proportional to the grafting density, σ,

to the power of the exponent n, which is dependent on the spacing, d, between the

chains (usually, σ = d−2). If the distance d is far larger than the radius of gyration Rg

(radius of polymer in random‐walk conformation) of the polymer chains, no

interaction between the chains will occur. Therefore, brush conformations can either

result in a pancake or mushroom conformation (Regime I in Figure 4) and an increase

of the grafting density σ will result in a constant brush thickness h (n = 0 for very low

grafting densities). The substrate–polymer interaction defines if the chain is flat

(attractive interaction between substrate and polymer) or erect (repulsive interaction

between substrate and polymer) on the substrates surface. By further increasing the

grafting density, σ, the distance between the chains, d, becomes smaller than twice the

radius of gyration, Rg, and the chains start to interact with each other, which is also

known as the moderate‐density brush (Regime II in Figure 4).155 Here, an increase of

the polymer thickness, h, by increasing the grafting density σ takes place (n = 1/3, see

also Equation 1). The high‐density brush regime (Regime III in Figure 4) is reached

by increasing the grafting density σ even more, which drives the thickness h to an even

steeper increase (n > 1/3 for very high grafting densities).156


Figure 4: Schematic representation of the different grafting regimes (pancake, mushroom, brush,

and high‐density brush) and their scaling laws for the brush thickness depending on the

grafting density in a good solvent.

One essential question associated with the schematic representation of the

scaling laws in Figure 4 is how far the grafting density can be increased and where the

theoretical maximum lies. To address this issue, a rather simple approach can be

followed for deriving an absolute theoretical limit for grafting densities. The

theoretical limit has to be the brush conformation where the polymer chains are aligned

perfectly and as densely as possible, resembling a crystalline polymer brush

(physically no higher density is possible). In this case, the brush thickness, h, would

be exactly the contour length, lc, of the polymer chains, and the distance, d, between

the nearest neighbouring polymer is constant and periodical throughout the entire

brush. Such a crystalline brush conformation is illustrated in Figure 5. The chains in

the shown case are perfectly aligned as a simple cubic crystal in plane (top view) and

perpendicular (side view) to the substrates interface. When the crystalline cross‐

section Axtal (red squares in Figure 5) of a polymer chain is known, the maximum

grafting density, σmax, can be calculated by

𝜎max =1

𝐴xtal≈

1

𝑑2 (2)

which gives the maximum amount of polymer chains per certain surface area (usually

denoted as chains per nm2).


Figure 5: Simplified representation of a crystalline polymer‐brush conformation in the side‐ and

top views and the related crystal cross‐sectional area of a polymer chain within the red

squares.

The size of the side groups of a polymer chain mainly influences the size of the

crystal cross‐sectional area. With the literature‐known crystal cross‐sectional areas of

several poly‐α‐olefins (refer to Table S1 in the Supporting Information), a perfect

linear dependency (R2 = 0.998) of the crystal cross‐sectional area to the length of side

groups (the number of carbon atoms in linear side chain NC,SC) is observed (Figure 6,

left‐hand side inner small graph). The literature values of the crystal cross‐sectional

area are represented by the black dots in the graph.157 Using Equation 2, this fit can be

converted into a prediction of the maximum grafting density 𝜎maxfit = (0.1579𝑁C,SC +

0.1526)−1

chains·nm−2, indicating that already for polybutylene a maximum grafting

density of 𝜎max ≈ 2 chains·nm−2 cannot be exceeded. For longer and, therefore, more

sterically demanding side chains, the maximum grafting density decays exponentially.

Furthermore, the crystal cross‐sectional area is plotted against the monomer molecular

weight m0 (Figure 6, right‐hand side small graph). A good linear fit of the crystal

cross‐sectional area to the monomer molecular weight can readily be explained by the

linear mass increase by adding more and more carbons in the side chain. Analogous to

the length of the side group, the maximum grafting density as a function of the

monomer molecular weight is plotted (Figure 6, right‐hand side). Here, the maximum

grafting density for poly(methyl methacrylate) (PMMA) and PS (based on the same

literature values),157 marked with a green and a blue star, respectively, and calculated

in the same way as for the polyolefins is shown, and deviates from the maximum

grafting‐density prediction of poly‐α‐olefins. This deviation demonstrates that side

chains can be heavier but not as sterically demanding as the linear chain (benzene ring

or methacrylate group). Thus, the prediction of the maximum grafting density by the

monomer molecular weight (for polymers different than polyolefins) is only an


estimation. For a more reliable determination of the maximum grafting density, the

length of the side chain (longest linear chain of atoms, NSC) has to be compared to the

length of the linear side chains of the polyolefins. If, for instance, the length of the side

chain of PMMA is compared to the length of the linear chain of a polyolefin

(methacrylate group of similar length to side chain of 1‐pentene), the actual maximum

grafting density 𝜎max ≈ 1.57 chains·nm−2 is certainly close to the predicted value of

𝜎maxfit = 1.60 ± 0.16 chains·nm−2.

Figure 6: Crystal cross‐sectional area as a function of the length of side groups (number of carbon

atoms in the linear side chain, NC, SC) and monomer molecular weight m0 with a linear fit

provided by the red dashed line for several poly‐α‐olefins in the inset graphs. Prediction

of the maximum grafting density in the main graphs with additional values for PMMA

and PS (green/blue stars).

Assuming perfect synthetic processes and thus full surface coverage of initiators

on the surface, as well as perfectly controlled polymerization (no chain termination

and a perfect monodisperse polymer weight distribution), it may be possible to reach

this maximum grafting density for “grafting‐from” polymer brushes. However, note

that the polymer brush has to be in a crystalline conformation, which is testable with

common characterization methods such as X‐ray diffraction.

The maximum grafting‐density limit suggested above is too high for the

“grafting‐to” approach and has to be lower. The maximum grafting density for the

“grafting‐to” approach is strongly dependent on the degree of polymerization—or

more accurately on the radius of gyration Rg —which is the degree of polymerization

scaled by the solvent conditions. By applying a simple geometrical model of filling the

surface with polymer coils of the size of the radius of gyration (𝑅g = 𝑁3 5⁄ random‐

walk polymer in coil),119 a first limit for grafting densities can be derived. A simplified

sketch of this model is depicted in Figure 7.


Figure 7: Representation of polymer coils filled densely on a planar surface for a “grafting‐to”

polymer brush. The small red circles represent the active centres of the polymer chain for

the surface reaction.

The grafting density can be calculated by

𝜎 ≈1

𝐴coil=

1

𝜋𝑅g2

=1

𝜋(𝑏𝑁𝑛)2 (3)

with the area of the polymer coil covering the surface Acoil, the radius of gyration Rg,

the smallest freely rotating unit in the polymer chain b (chain elasticity), the degree of

polymerization N, and the exponent n, which describes the scaling law (depending on

the solvent quality).119, 155-156 The results of Equation 3 are depicted in Figure 8. For

the graphs showing the degree of polymerization, a chain elasticity of b = 0.153 nm

(length of a C–C bond as smallest possible freely rotating unit) and exponent of n =

1/3 (for poor solvents) leads to the smallest possible polymer coil for a given degree

of polymerization. The data in Figure 6 show a rapid exponential decay; for a radius

of gyration of Rg = 1 nm, the grafting density is merely σ = 0.32 ± 0.6 chains·nm−2

(Figure 8, left) and for a degree of polymerization of N = 100 just

σ = 0.63 ± 0.33 chains·nm−2 (Figure 8, right).


Figure 8: Grafting density as a function of the radius of gyration Rg (left) and degree of

polymerization N (right) (with b = 0.153 nm and n = 1/3 for a smallest possible polymer

coil) in linear and double‐logarithmic scales.

The limit of the grafting density is a rather rough estimation of the maximum

grafting density (more specifically, the description of transition between Regimes I

and II in Figure 4).155 For a more accurate calculation, several effects have to be

included in the model: i.e., those effects that favour a higher grafting density. Like, a

reduction of the total energy in the system by binding the active centre of the polymer

chain to the surface, due to the lower energy in the binding state. The decrease of total

energy has to be balanced with the decrease of configurational entropy by stretching

the polymer chain in order to get higher grafting densities. Furthermore, a high

dispersity, Đ, favours a higher grafting density. Here, a variety of different radii of

gyration are possible, leading to the possibility of filling smaller spaces with smaller

polymers, which merge into a polymer brush with higher grafting densities. On the

other hand, there are effects that drive the polymer brush to even lower grafting

densities. The diffusion of the polymer coil to an empty unfilled spot on the surface is

one effect, which is a diffusion‐controlled process leading to a lower grafting density

if the surface reaction merely takes place over a certain period of time. In addition, the

probability that a polymer chain (polymer coil) does not bind covalently but instead

adsorbs onto the substrate surface will lower the grafting density. For attaching the

polymer covalently to the surface, the active group of the polymer chain must be at the

surface. As the polymer coil is a 3D object, there is the possibility that this coil

approaches the surface with the active group far away from a position close to the

surface. Therefore, the polymer coil could be attached only via van der Waals

interactions and may be washed away afterward, leading to a significant decrease of

the grafting density. The balance of all these effects will give an accurate picture of


the polymer brush on the surface. The values given by Equation 3 serve as a guideline.

If grafting densities that are far higher than these values were obtained experimentally,

the results have to be critically questioned and an appropriate explanation has to be

found.

A critical assessment of the common characterization methods for grafting

densities must consider the above theoretical descriptions of end‐grafted polymer

brushes on solid surfaces. The aim here is to combine the theory, calculations, physical

limitations, and error determination and use them to judge the accuracy of the

commonly used characterization methods for grafting densities. Thus, a prediction for

novel and more accurate quantitative characterization methods can be made.

2.4 CHARACTERIZATION OF GRAFTING DENSITIES

The quantitative characterization of grafting densities of polymer brushes is a

rather challenging process, which is exemplified by the large amount of diverse

experimental methods, and attempts that have been made to solve this issue.

Nevertheless, three major characterization methods have been firmly established for

determining grafting densities on solid substrates. These three established methods are

dry thickness measurements (by invoking the dry thickness, hdry; density, ρ0; and the

number‐average molecular weight, Mn),58, 134, 158-180 gravimetric assessment (by

knowing the amount of polymer on the substrate, wpoly, and the number‐average

molecular weight, Mn),172, 181-190 and swelling experiments (applying the thickness ratio

of dry and swollen polymer brushes, αSR).162, 191-196 Therefore, we will focus on the

aforementioned characterization methods. A schematic illustration of these three

determination methods is depicted in Figure 9.


Figure 9: Illustration of the three most common grafting‐density determination methods: dry

thickness, gravimetric measurements, and swelling experiments.

In the discussion of each method, we commence our exploration with a

derivation of the underpinning equations. This entails a discussion of the assumptions

that affect each equation. In the following, we will illustrate how these methods, via

their governing equations, are affected by errors in the input parameters. Here, our

approach is not focused on using individually determined errors for each input

parameter, but to provide the user with (i) a quantitative tool (i.e., equations) for

accurately reporting errors using their own individual errors and (ii) demonstrate that

even with relative modest errors in the input parameters, the stability of the

underpinning equations is limited. The error calculations are carried out via series

expansion of measurement errors of the independent input variables xi in the following

way

𝜎 = 𝜎(𝑥𝑖) → ∆𝜎 = ∑ |𝛿𝜎

𝛿𝑥𝑖| ∆𝑥𝑖

𝑛

𝑖=0 (4)

with the error of grafting density Δσ calculated by the sum of the derivative of the

grafting density to the input variables xi multiplied by the individual errors of the input

parameters Δxi. To be able to compare the different characterization methods, the

individual input parameter errors were chosen to be identical for all calculations. As

modest example errors, we assume a 4% (black error bars and lines in all following

graphs) and 10% (red error bars and lines in all following graphs) global value for the

input variables for the error propagation calculation. Note that the errors for an

individual example will be different from the herein employed 4% and 10%, and –

importantly – in most cases larger. As we will discuss in the following sections, for


one method (dry thickness) at least, one of the input parameters (the bulk polymer

density) is only valid for a certain grafting‐density regime and the method's outside

use of this regime leads to very high errors as it is no longer applicable. The derived

equation for the error propagation can be found in Appendix B Supporting Information

(Equation S1 – S6).

2.4.1 Dry thickness

In the first method we scrutinize, i.e., determination by dry thickness, the

grafting density is calculated based on the AG theory, which provides the relation

between thickness, h, of the polymer brush and the distance, d, between the polymer

chains. 58, 134, 158-180 As the name already implies, the thickness of a dry polymer brush

is invoked in the calculations and is measured, e.g., with ellipsometry, X‐ray

reflectivity (XRR), or neutron reflectivity (NR). A polymer brush in the dry states

behaves similar to the one in a poor solvent (collapse of polymer chains) with regard

to the scaling laws. Thus, the following equation for calculating the grafting density

holds

ℎdry𝑑2 = 𝑁𝑎3 (5)

𝜎 =1

𝑑2=

ℎdry

𝑁𝑎3 (6)

𝜌0 =1

𝑎3≈

𝜌𝑏𝑁A

𝑚0 (7)

𝜎 =ℎdry𝜌𝑏𝑁A

𝑚0𝑁=

ℎdry𝜌𝑏𝑁A

𝑀n (8)

with the thickness of the collapsed polymer brush hdry, the average distance d between

two polymer chains, the degree of polymerization N, the size of the monomer a (mesh

size in the lattice model of AG theory, based on Flory–Huggins theory), the grafting

density σ, the segment density ρ0, the bulk density ρb of the corresponding polymer,

the Avogadro constant (NA = 6.022 × 1023 mol−1), the molecular mass of the monomer

m0, and the number‐average molecular weight, Mn, of the polymer chain. For

Equation 5, the scaling behaviour of the dry thickness within the brush regime based

on the AG theory is implemented. In other words, Equation 5 describes the volume of

a polymer chain calculated in two ways. The volume of the polymer chain calculated

by the dry thickness, hdry, times the square of the distance, d, is equal to the degree of

polymerization, N, times the cube of the monomer size, a. By transforming Equation 5

and noting that σ = d−2, Equation 6 can be derived. Based on the assumption


(Equation 7) that the segment density, ρ0, can be approximated by the bulk density, ρb,

and the monomer molecular weight, m0, of the corresponding polymer, the final

equation (Equation 8) results. For calculating the grafting density with this method,

the dry thickness, hdry, the bulk density, ρb, and the number‐average molecular weight,

Mn, of the polymer chain must be known. The dependency of dry thickness on grafting

density for a polymer brush with Mn = 100 kg·mol−1 and ρb = 1.15 g·cm−3 is illustrated

in Figure 10. A linear increase of thickness with increasing grafting density is depicted

in the left graph. The percentage error (calculated as previously described and

illustrated on the right‐hand side of Figure 10) appears to be constant and independent

of the value of the grafting density. The same graphical representation of the

dependency between number‐average molecular weight, Mn, and grafting density can

be made and is shown in Figure S1 in Appendix B.

Figure 10: Dry thickness approach: grafting density as a function of the dry thickness of a polymer

brush with Mn = 100 kg mol−1 and ρb = 1.15 g cm−3 on the left‐hand side. Absolute and

percentage errors for these grafting densities on the right‐hand side. An error of 4% (black

error bars and lines) as well as 10% (red error bars and lines) of the value for the input

variables is used for the error propagation calculation as example values to illustrate the

stability of the method.

Although the percentage error of this characterization method appears low and

stable compared to the other methods (refer to the following sections) merely based on

a mathematical analysis, the method entails several critical disadvantages that

jeopardize is general applicability. The key drawback is the assumptions invoked in

Equation 7, which equates the segment density, ρ0, with the bulk density, ρb, of the

polymer. As noted in the section on error calculation, by deducing the grafting density

via the dry thickness method an already preset density is used, even though the density

in the polymer brush can be significantly lower than in a bulk polymer. Thus, even the

assumption of 10% error for the bulk density is certainly too low for cases where the


grafting density is low, and the surface does not display a brush regime. A similar high

error, yet in the other direction, is possible in the high‐density brush regime.

Furthermore, the bulk density assumption implies that the segment density is equal at

any position of the entire brush (AG theory), although we have already discussed in

the previous section that the density profile, especially for lower grafting densities, can

vary rather strongly (SCF theory). In addition, the absolute brush thickness is different

for the AG and SCF theories. Thus, an incorrect estimation of the grafting density can

result, as the observed polymer brush would be correctly described by the SCF theory,

yet the calculations are based on the AG theory, which lead to an overestimation of the

grafting density. Critically, a prefactor to relate the SCF brush height, hSCF, to the AG

theory, hAG, can be calculated with hSCF = (4/3)·hAG.196 Another critical point

concerning the dry thickness characterization method is that the number‐average

molecular weight, Mn, or the degree of the polymerization, N, of the polymer chains

must be known to calculate the grafting density. In the case of the “grafting‐to”

approach, this is easily possible, as the polymer chains are presynthesized and can

therefore be readily characterized. However, for the “grafting‐from” approach, this is

a rather complicated issue, as described in the theory section here.

The determination of grafting density via the dry thickness approach is by far the

most commonly used quantitative characterization method. It is used for the “grafting‐

to” as well as for the “grafting‐from” approach. A selection of several polymer systems

with a large variety of determined grafting densities is collated in Table 1. An even

more detailed description of these polymer systems and additional explanations for all

acronyms can be found in Table S2 and Table S3 of the Supporting Information in

Appendix B.

Table 1: Literature values for estimated grafting densities of several polymer systems

characterized by dry thickness analysis.

Ref. grafting mechanism Substrate polymer ℎdry

[nm]

𝑀n [kg mol‒1]

𝜎 [chains nm‒2]

158 from ATRP Si PAAm 0 - 3.5 17 0.06 - 0.2

159 from ATRP Si (100) PDMAEMA 68 39 0.84

160 from ATRP Si PHEMA 2 - 7 --- 0.26

161 from ATRP Si (100) PMAA 8 - 20 106 0.01 - 0.12

162 from ATRP Si (100) PMMA 13 - 71 38 - 151 0.23 - 0.31

160 from ATRP Si PMMA 2 - 5 --- 0.19

163 from ATRP Si (111) PMMA 5 - 28 8 - 35 0.43 - 0.61



[nm]

𝑀n [kg mol‒1]


179 from ATRP Si PMMA 12 - 102 23 - 171 0.4

164 from ATRP Si / SiO2 PMMA 5 - 55 57 0.07 - 0.70

58 from ATRP Si PMMA 40 - 140 50 - 200 0.517±0.012

160 from ATRP Si PMPC 4 - 10 --- 0.17

178 from ATRP Si PMPC 6 - 18 13 - 34 0.28 - 0.32

165 from ATRP Si (111) PMPC 9 - 39 29 - 140 0.22

166 from ATRP Au PNIPAM --- 33 - 152 0.063

166 from ATRP Si PNIPAM

--- 52 - 230 0.08 - 0.21

167 from ATRP Si / quartz PS 10 - 32 15 - 44 0.40 - 049

168 from NMP Si PS 10 - 47 12 - 28 0.55 - 1.10

134 from NMP Si PS 8 - 30 15 - 16 0.3 - 1.2

169 from free radical LASFN9 PS 134 500 - 1,000 0.11 - 0.25

170 from ATRP Si PS 25 - 29 34 - 38 0.44 - 0.49

171 from free radical LASFN9 PVP 23 - 170 1,100 0.11 - 0.25

171 from free radical LASFN9 (Me)PVP 15 - 557 1,800 - 2,500 0.005 - 0.261

172 to thiol - Au Au on Si (110) C14 1.5 0.2 3.3

173 to GPS+

carboxyl Si (100) PBA

0.58 -

0.94 6.5 0.048 - 0.075

173 to GPS+

carboxyl Si (100) PBA + PS

1.95 -

3.61 6.5 + 4.2 / 9.7 0.26 - 0.43

174 to SAM Si (100) PDMS 0.8 - 14 11 0.046 - 0.79

172 to thiol - Au Au on Si (110) PEO45 2.5 2.1 0.9

174 to SAM Si (100) PPG/PEG 6 - 10 2 2.41 - 4.01

180 to GPS+

carboxyl Si PS 1 - 7 17 0.14 - 0.22

175 to GPS+

carboxyl Si PS --- 3 - 75 0.045 - 0.24

176 to GPS+

carboxyl Si PS 5 48 0.062

174 to SAM Si (100) PS 6 - 12 8 0.071 - 0.14

177 to disulfide Au PS 0.6 12 - 17 0.005

172 to thiol - Au Au on Si (110) PS19 1.8 2.0 0.6

172 to thiol - Au Au on Si (110) PS125 0.9 13.3 0.04

180 to GPS+

carboxyl Si PS + PVP 5 - 8 17 + 39 0.021 - 0.06



[nm]

𝑀n [kg mol‒1]


176 to GPS+

carboxyl Si PS + PVP 6 - 7 --- 0.092

175 to GPS+

carboxyl Si PVP --- 42 0.030 - 0.053

176 to GPS+

carboxyl Si PVP 5 41 0.075

Most of the grafting densities determined above (Table 1) are within the limit of

the physically possible (maximum) grafting densities. The calculated maximum

grafting density for each of the polymer systems is given in Table 2. Even though the

grafting‐density values of tetradecane (C14)172 and poly(ethylene glycol) (PEG)174

seem to be high, the grafting densities may even be higher, as for these “polymer

chains” no sterically demanding side chains are present within the molecule.

Nevertheless, the grafting densities for the poly(2‐(diemethylamino)ethyl

methacrylate) (PDMAEMA)159 (σ = 0.84 nm−2 to 𝜎maxfit = 0.91 nm−2) and PS134, 168 (σ

= 1.2 nm−2 to 𝜎maxfit = 1.28 nm−2 or σmax = 1.43 nm−2) system shown in Table 1 are

close to the maximum grafting density (given in Table 2). If for the maximum grafting

density, 𝜎maxfit , a perfect crystalline polymer is expected, the degree of crystallinity, D,

of the polymer brush may be provided by the ratio 𝜎 𝜎maxfit⁄ , leading to the finding that

the degree of crystallinity of the polymer brush for PDMAEMA159 is D ≈ 89% and for

PS134 is D ≈ 94%. These calculated degrees of crystallinity are unusually high, and

therefore the accuracy of the determined grafting densities is questionable.

Table 2: Determined maximal grafting densities for several polymers considered here.

Polymer Ref. 𝑁SC

𝜎maxfit

[chains nm‒2]

C14 172, 184 0 6.55

PAA 185 2 2.14

PAAm 158 2 2.14

PAEA 185 5 1.06

PBA 173 6 0.91

PDHA 185 6 0.91

PDMAEMA 159 6 0.91

PDMS 174 2 2.14

PEMA 192-193 4 1.28


Polymer Ref. 𝑁SC

𝜎maxfit

[chains nm‒2]

PEO/PEG 172, 174, 181, 184,

186-187 0 6.55

PFOEMA 192 12 0.49

PHEMA 182 5 1.06

PHFMA 192 6 0.91

PIBA 188 6 0.91

PMAA 161 2 2.14

PMMA 160, 162-164, 194 3 1.60

PMPC 165, 178 12 0.49

PNIPAM 166, 185 4 1.28

PPG 174 1 3.22

PS

134, 167-170, 172,

174-176, 180, 188-

189, 195-196

4 1.28

PTFMA 192-193 4 1.28

PVP 171, 175, 180 4 1.28

(Me)PVP 171 5 1.06

2.4.2 Gravimetric Measurements

For determining the grafting density via gravimetric measurements, the grafting

density is calculated by the ratio of mass between polymer brush and substrate. 172, 181-

190 The method rests on the idea that the total number of chains and surface area can

be estimated, and accordingly the grafting density (Equation 9). The amount of

polymer is determined either via thermogravimetric analysis, elemental analysis, or

“before and after” weighing of the substrate covered with the polymer brush. In

addition, to track the weight increase or decrease due to the formation or

decomposition of the polymer brush at the substrate's surface, a certain amount of

polymer has to be present. Therefore, the gravimetric measurements are only

applicable for polymer films on NPs, MSs, and highly porous materials, due to the

high surface area and the thereby trackable amount of polymer. The surface area can

be determined via, for example, electron microscopy or N2 adsorption isotherms

(Brunauer–Emmett–Teller (BET)). For the calculation of the grafting density, the

following equations then hold


𝜎 =𝑁chains

𝐴sub (9)

𝑁chains =𝑚poly𝑁A

𝑀n (10)

𝑚poly = (𝑤poly

𝑤sub) 𝑚sub = (

𝑤poly

100 − 𝑤poly) 𝑚sub (11)

𝑚sub = 𝜌sub𝑉sub (12)

𝜎 =

(𝑤poly

100 − 𝑤poly) 𝜌sub𝑉sub𝑁A

𝑀n𝐴sub

(13)

with the grafting density σ, total amount of chains on the substrate Nchains, surface area

of the substrate Asub, absolute mass of the polymer mpoly, the Avogadro constant NA,

the number‐average molecular weight of the polymer chain Mn, the polymer weight

fraction wpoly, the weight fraction of substrate material in system wsub, the absolute

mass of the substrate msub, the density of the substrate ρsub, and the volume of the

substrate Vsub.

For the calculation of the grafting density, the total number of chains on a

substrate with a certain surface area, Asub, must be determined (see Equation 9). The

number of chains can be calculated via Equation 10, which equates the absolute mass

of the polymer divided by the number‐average molecular weight with the total number

of polymer chains, Nchains. The absolute mass of the polymer can be expressed by the

percentage amount of polymer in the system and the substrate density and volume

(Equations 11 and 12). By combining Equations 9–12, the final term for calculating

the grafting densities can be derived (Equation 13). For calculating the grafting density

with the gravimetric measurement method, the percentage amount of polymer in the

system wpoly, the substrate density ρsub, area Asub, and volume Vsub, as well as the

number‐average molecular weight, Mn, of the polymer chain must be known. An

additional simplification of Equation 13 for spherical particles can be invoked by

𝜎 =

(𝑤poly

100 − 𝑤poly) 𝜌sub𝑟sub𝑁A

3𝑀n

(14)

with the radius of a spherical particle as a substrate rsub. The dependency of the

percentage amount of polymer on the grafting density for a polymer brush with

Mn = 10 kg·mol−1 and rsub = 3.5 nm is described in Figure 11. The increase of polymer

weight fraction with increasing grafting density is depicted on the left‐hand side of


Figure 11. The percentage error (calculated as previously described and illustrated on

the right‐hand side of Figure 11) increases drastically with increasing grafting density.

Note that in a real experiment, the error of the input parameters, such as of the polymer

weight fraction, is not necessarily constant. The same graphical representation of the

dependency of number‐average molecular weight, Mn, and particle radius, rsub, to

grafting density is shown in Figures S2 and S3 in Appendix B.

The nature of the equation underpinning the gravimetric assessment leads to a

very strong increase of the percentage error with increasing grafting densities. Under

our scenario, for instance, grafting densities exceeding a value of σ > 2 chains·nm−2

have a percentage error of Δσ > 65% by assuming a general input value error of

Δxi = 10%. However, an advantage of this characterization approach is its overall

assumption‐free nature. The gravimetric measurement method is only based on a

simple mass balance. Nevertheless, the number‐average molecular weight, Mn, of the

grafted polymer chains must be known. As described earlier, for the “grafting‐to”

approach this is a straightforward task, whereas for the “grafting‐from” approach,

evaluating the number‐average molecular weight, Mn, is challenging.

The determination of grafting density via gravimetric measurements is most

commonly adapted for polymer brushes on NPs, MSs, or other nanostructures and

applied to the “grafting‐to” as well as the “grafting‐from” approach. A list of selected

polymer systems on different substrates is shown in Table 3. A more detailed

description and definitions for all acronyms can be found in Table S2 and Table S3 of

the Supporting Information in Appendix B.


characterized by gravimetric measurements.

Ref. grafting mechanism substrate polymer ℎdry

[nm]

𝑀n

[kg mol‒1]

𝜎

[chains nm‒2]

181 from anionic DVB MS PEO 7 -

10 14 1.65 - 2.09

182 from RAFT IIPs PHEMA --- 16 - 19 1.06 - 1.43

183 from ATRP PS latex PNIPAM --- 47 - 838 0.027 - 0.079

172, 184 to thiol - Au Au NPs C14 --- 0.2 4.35

185 to thio bromo, alkoxysilane Si NPs PAA --- 1.4 0.50

185 to thio bromo, alkoxysilane Si NPs PAEA --- 2.1 0.23 - 0.65

185 to thio bromo, alkoxysilane Si NPs PDHA --- 3.4 0.41

186 to melt grafting Fe3O4 NPs PEG --- 5 3.5

185 to thio bromo, alkoxysilane Si NPs PEG(MA) --- 3.8 0.31



[nm]

𝑀n

[kg mol‒1]

𝜎

[chains nm‒2]

187 to silane silica NPs PEO --- 5 0.58 - 1.18

172, 184 to thiol - Au Au NPs PEO45 --- 2 1.15

188 to cycloaddition DVB MS PIBA --- 6 - 26 3.6 - 7.3

185 to thio bromo, alkoxysilane Si NPs PNIPAM --- 1.7 0.30 - 0.58

189 to nitroxides Ag NPs PS --- 13 - 14 2.0 - 5.9

188 to cycloaddition DVB MS PS --- 4 31.1

190 to SAM Silica NPs PS --- 17 - 272 0.01 – 0.1

172, 184 to thiol - Au Au NPs PS19 --- 2 3.45

172, 184 to thiol - Au Au NPs PS125 --- 13.3 0.94

Applying the gravimetric measurement method, several calculated grafting

densities for different polymer systems (Table 3) exceed the limits of the physically

possible maximum grafting density. The estimated maximum grafting‐density limit

for each of the polymer systems is given in Table 2. For poly(2‐

hydroxyethylmethacrylate) (PHEMA)182 (σ = 1.43 nm−2 to 𝜎maxfit = 1.06 nm−2),

PS19172, 184 (σ = 3.45 nm−2 to 𝜎max

fit = 1.28 nm−2), PS189 (σ = 5.9 nm-2 to 𝜎maxfit = 1.28

nm−2), poly(isobornyl acrylate) (PIBA)188 (σ = 7.3 nm−2 to 𝜎maxfit = 0.91 nm−2), as well

as PS188 (σ = 31.1 nm−2 to 𝜎maxfit = 1.28 nm−2), the limit is massively exceeded. As

already mentioned by Barner‐Kowollik and co‐workers,188 these unfeasible high

values for grafting density may be caused by the incorrect determination

(underestimation) of the surface area of the substrate. In addition, the inaccuracy of

the method increases with increasing grafting densities based on the nature of the

equation underpinning the method (Figure 11, right‐hand side), thus compounding the

uncertainty in these values. Although the grafting densities of C14172, 184 and PEO181,

186 seem to be high, the maximum grafting density is not exceeded and can be

theoretically even higher, since for these “polymer chains”, no sterically demanding

side chain is present, as mentioned before.


Figure 11: Gravimetry: grafting‐density dependency of the polymer weight fraction of a polymer

brush with Mn = 10 kg mol−1 and rsub = 3.5 nm on the left‐hand side. Absolute and

percentage errors for these grafting densities on the right‐hand side. An error of 4% (black

error bars and lines) as well as 10% (red error bars and lines) of the value for the input

variables is used for the error propagation calculation as example values to illustrate the

stability of the method

2.4.3 Swelling Experiments

In the case of the swelling experiments, the grafting density is determined by

measuring the thickness ratio of the polymer film in the dry, hdry, and swollen state,

hswell.162, 191-196 The thickness of the polymer brush is measured, for example, by

ellipsometry, XRR, or NR. From this thickness ratio, a direct estimation of the grafting

density is possible by knowing the monomer size, a, and the scaling exponent, n, of

the swollen brush. Generally, the ratio of the dry, hdry, to the swollen state, hswell, is the

ratio of the polymer brush described by two different scaling laws, as the polymer

brush in the dry states behaves similar to the one in a poor solvent. The characterization

method is consequently also based on the AG theory and can be established by the

following procedure

ℎdry =𝑁𝑎3

𝑑2 (15)

ℎswell = 𝑛blob𝐷blob =𝑁

𝑔𝐷blob (16)

𝐷blob = 𝑎𝑔𝑛 ≈ 𝑑 (17)

𝛼SR =ℎswell

ℎdry=

𝑁 (𝑑𝑎)

−1 𝑛⁄

𝑑

𝑁𝑎3

𝑑2

= (𝑑

𝑎)

3−1 𝑛⁄

(18)

𝜎 =1

𝑑2= 𝑎−2𝛼SR

2𝑛1−3𝑛 (19)


with the dry thickness of the polymer brush hdry, the degree of polymerization N, the

size of the monomer a, the average distance between two tethering sites d, swollen

brush thickness in solvent hswell, number of blobs per polymer chain nblob, blob size

(diameter) Dblob, number of monomer units in a blob g, the exponent n (which describe

the solvent quality) and the swelling ratio of the polymer brush αSR. Equation 15

describes the scaling law of the dry thickness based on the AG theory and is a

rearrangement of Equation 5. In the AG theory, the swollen polymer brush is described

by a sequence of blobs, in which the polymer behaves like a free polymer in a solvent

(Equation 17). By the number of blobs per polymer chain, nblob, multiplied by the blob

size, Dblob, the swollen thickness can be calculated (see Equation 16). For calculating

the grafting density, the diameter of a blob, Dblob, and the distance, d, between two

tethering sites are assumed to be equal (approximation), which is provided by Equation

17. By combining Equations 15–17 to afford the swelling ratio, αSR, and transforming

Equation 18, the grafting density can be calculated as a function of the swelling ratio

αSR, the monomer size a, and the scaling exponent n (Equation 19). The exponential

decay of the swelling ratio by increasing the grafting density for a polymer brush with

a = 5 Å and n = 0.5 is depicted in Figure 12. The percentage error (calculated as

previously described and illustrated on the right‐hand side of Figure 12) drastically

increases for decreasing grafting densities, merely based on the mathematical nature

of the equation.

Figure 12: Swelling experiments: grafting‐density dependency of the swelling ratio of a polymer

brush with a = 0.5 nm and n = 0.5 on the left‐hand side. Absolute and percentage errors

for these grafting densities on the right‐hand side. An error of 4% (black error bars and

lines) as well as 10% (red error bars and lines) of the value for the input variables is used

for the error propagation calculation as example values to illustrate the stability of the

method.


The exponential decay of the percentage error with increasing grafting densities

leads to an enormous uncertainty for low grafting densities within our scenario. For a

grafting density of a value lower than σ < 0.5 chains·nm−2, the percentage error already

exceeds Δσ > 80% based on the assumption of a general input value error of Δxi = 10%.

Similar to the dry thickness measurements, the same approximations were made for

the swelling experiments, as the AG theory was used as well. Therefore, an advantage

of the swelling experiments is that the wrong estimation of the brush thickness (due to

incorrect use of the AG theory) can be neglected because the prefactor is eliminated

by the calculation of the ratio. Another benefit of the characterization method is that

the molecular weight of the polymer chain is not required for the evaluation of the

grafting density, which therefore makes it a suitable method for the “grafting‐from”

approach.

A selection of several polymer systems produced via the “grafting‐from”

approach with a variety of determined grafting densities by swelling experiments is

collated in Table 4. A more detailed description and definitions for all acronyms can

be found in Table S2 and Table S3 of the Supporting Information section of

Appendix B.


characterized by the swelling method.

Ref. grafting mechanism substrate solvent polymer ℎdry

[nm]

𝑀n

[kg mol‒1]

𝜎

[chains nm‒2]

191 from ATRP mica water PNIPAM 10 - 215 475 0.02 - 0.42

192 from ATRP Si (111) acetone, HFP,

NFE, FC-40 PEMA 75 74 - 94 0.53 - 0.68

193 from ATRP Si (111) acetone PEMA 6 - 206 26 - 167 0.16 - 0.82


NFE, FC-40 PFOEMA 80 360 - 1,140 0.09 - 0.22


NFE, FC-40 PHFMA 78 115 - 330 0.19 - 0.55

194 from PMPP Si THF PMMA 7 - 110 341 - 631 0.033 - 0.13

162 from ATRP Si (100)

methanol,

(cyclo)hexane,

acetone,

ethyl acetate,

THF, toluene

PMMA 13 - 71 38 - 151 0.24 - 0.35

195 from anionic glass toluene PS 16.6 - 18 40 - 41 0.28 - 0.31

196 from ATRP Si (100) toluene PS 31 - 116 147 - 420 0.16 - 0.19


NFE, FC-40 PTFEMA 80 74 - 100 0.57 - 0.69

193 from ATRP Si (111) acetone PTFEMA 9 - 140 69 - 137 0.09 - 0.73


All of the mentioned grafting densities in Table 4 are within the physical limit

of the maximum grafting density (Table 2). Nevertheless, all of the given values are in

a rather low‐grafting‐density regime. As explained above, the accuracy of low grafting

densities determined from swelling experiments may be questionable. However, the

concept of the swelling method is based on a direct measurement of one property

(swelling ratio), which is in a direct relation to the polymer brush.

The estimated maximum grafting densities of the polymer systems used here are

listed in Table 2 and are determined by comparing the length of the side chains of the

polymer to the length of the linear side chains of poly‐α‐olefins NSC. The procedure is

described in more detail in the previous theory section.

2.5 CONCLUSION AND FUTURE PERSPECTIVES

To assess which of the previously mentioned methods is most accurate and in

which range of grafting densities they are applicable, all three characterization

methods have to be compared with regard to their inherent assumptions, simplicity

(direct or indirect measure of grafting density), and error propagation. With regard to

a priori assumptions, the dry thickness and swelling experiments are based on the AG

theory and therefore all assumptions that are used in this theory also apply to these

characterization methods. In the case of dry thickness measurements, the additional

assumption of using a preset density (bulk density ρb of the polymer) for describing

the polymer brush introduces significant uncertainties. Only for the gravimetric

measurements, no general assumption is made, as the grafting density is solely

calculated by the mass change and the surface area. However, the molecular weight of

the polymer chains has to be known for the calculation by gravimetric measurements,

which is a critical disadvantage. The simplicity of a method is directly correlated with

the number of additionally required input parameters. For dry thickness and

gravimetric measurements, several parameters have to be known, including the dry

thickness, hdry, the number‐average molecular weight, Mn, the surface area (surface

volume), and mass change, wpoly. All of these values have to be estimated via different

experiments, leading to a high inaccuracy of these methods. Especially the problem of

access to exact number‐average molecular weights for polymer brushes generated via

“grafting‐from” increases this error. In the case of swelling experiments, only the ratio

of swelling (ratio between dry hdry and hswell) has to be measured, which can be


achieved by in situ swelling experiments. Nevertheless, the swelling experiments are

merely a “semidirect” characterization method, because the exponent n must still be

known. However, this exponent is mostly well established for a large number of

polymer solvents/systems in the literature.197 Based on a purely mathematical analysis

of the nature of the underpinning equations and a constant error of the input variables,

close inspection of Figures 8-10 (right hand side) indicates that the propagating error

is constant and lowest for the dry thickness method, whereas the error increases with

increasing grafting density for the gravimetric measurements, and increases with

decreasing grafting density for the swelling experiments. However, it is very important

to note that this merely reflects the mathematical behaviour of the methods and that

considerable additional uncertainties are introduced when operating the method

outside its application regime. This is particularly true for the often‐hailed dry

thickness method, where an a priori determination of the grafting regime – confirming

its brush nature – is essentially required before the bulk polymer density is inserted

into the equation. However, such an approach cannot be found in the literature.

Critically, the grafting densities have to be reported with an appropriate uncertainty

(error), which has to be calculated, as we have demonstrated above. For most of the

literature‐known values, no error is reported at all (see Table S2 in the Supporting

Information in Appendix B).

It is mandatory to note that several other (uncommon) methods exist for the

quantitative characterization of grafting densities.80, 171, 198-203 A few examples are

listed in Table 5. Some of these characterization methods are even more indirect and

inaccurate, such as estimating the grafting density by the assumption that only 10% of

the initiator grafted on the surface will result in a grafted polymer chain198 or the

determination of the grafting density via decomposition kinetics of an azo‐initiator

grafted on the surface.171, 199-200 However, some other methods measure the grafting

density in a more direct fashion, such as the fluorescamine‐ and ninhydrine‐based

assay labeling201 or colloidal probe measurements.80 For instance, a colloidal probe

experiment measures the force response of polymer brush by compressing it, which

can be directly converted into the grafting density by fitting the measured force–

distance curve.


Table 5: Literature values for estimated grafting densities of several polymer systems for less‐used

characterization methods


[nm]

𝑀n

[kg mol‒1]

𝜎

[chains nm‒2] method

96 from ATRP Si (100) PDMA 176 - 339 815 0.29 10% of init.

density

96 from ATRP Si (100) PHMA 44 - 113 272 0.29 10% of init.

density

199 from free

radical

Si,

LASFN9 PMAA 144 1,500 0.056

initiator

decomposition

kinetics

200 from free

radical

Si,

LASFN9 PMAA 23 - 160 177 - 3.733 0.005 - 0.16

initiator

decomposition

kinetics

80 from ATRP Si PMAA 60 - 100 --- 0.053 colloidal probe

measurements

202 from ATRP

porous

glass filter

s

PMMA --- 5 - 40 0.6 BET - FTIR

96 from ATRP Si (100) POMA 86 - 207 498 0.29 10% of init.

density

171 from free

radical LASFN9 PVP 23 - 170 1,100 0.015 - 0.105

initiator

decomposition

kinetics

171 from free

radical LASFN9 (Me)PVP 15 - 557

1,800 –

2,500 0.007 - 0.216

initiator

decomposition

kinetics

201 to ligand

exchange

Au

nanostruct

ure

PEG --- 3 - 20 0.14 - 2.21

fluorescamine-

and ninhydrine

assay

203 to SAM glass

PDMAEMA -

b-

PTMSPMA

--- --- 0.01 - 0.05 QA - method

For obtaining reproducible and comparable values for grafting densities, the

method must be direct and ideally entail no assumptions. As noted, the first attempts

of developing such determination procedures have been made. Such a method has to

either measure a property of the polymer brush that is directly related to the value of

grafting density, or the number of the single polymer chains has to be

counted/determined, and the exact surface area has to be known. In the following

section, we explore five possible approaches for developing a more precise and direct

way to quantitatively establish grafting densities. These methods have not yet been

developed into functioning approaches, yet hold – in our view – the potential to

significantly increase the accuracy and precision of grafting‐density determination.

While some of the following approaches are currently “science fiction”, all of them


have the potential to be viable and powerful avenues to an accurate grafting‐density

determination.

(i) The quantitative determination of the grafting density via advanced atomic

force microscopy (AFM) is a viable option. AFM is frequently adopted for the precise

evaluation of intramolecular chain properties, including contour length, rupture force,

and chain elasticity via single‐molecule force spectroscopy (SMFS), which is an

already well‐established technique.81, 204-205 By exploiting the SMFS experimental

concept and expanding the method from measuring single polymer chains to multiple

polymer chains (multiple‐molecule force spectroscopy)206, the exact number of chains

bound to the AFM cantilever can be determined. By knowing the exact area of indent

(depending on radius of the AFM cantilever) and the number of polymer chains bound

to the cantilever, the grafting density can be calculated in a direct and exact manner.

By scanning over a defined area of the surface (force map), even the spatial distribution

of the grafting density should be accessible. For this experimental approach, advanced

characterization methods and calculation protocols have to be developed.

(ii) An alternative possibility for an even more accurate process for quantitative

grafting‐density determination is the combination of the previously mentioned

colloidal probe measurements with swelling experiments. The colloidal probe

measurement already provides a direct relation of the force response to the grafting

density by compressing the polymer brush.80 By an additional variation of the solvent

(different brush conformations, poor solvent for collapsed, and good solvent for

swollen conformation) and adjustment of the calculation procedure, the grafting

density for several different brush conformations can be evaluated. The grafting

density of each of the different brush conformations has to be identical. If the values

of the grafting density are nonidentical, an adjustment of the calculation protocols must

be carried out. This gives a good measure for the accuracy of the characterization

method and would improve the concept of simple colloidal probe measurements of a

single brush conformation.

(iii) An alternative method can be realized by labelling each of the polymer

chains in a polymer brush with a distinctive marker. Using spectroscopic

characterization methods (such as X‐ray photoelectron spectroscopy and ToF‐SIMS),

which can detect the marker molecules, the number of chains can be estimated by

simple counting.207-208 In addition, the exact scanning area (analysed using a

spectrometer) is needed for the quantitative characterization of the grafting density.


For this method, the precision of the spectroscopic characterization tools must be of

highest standard.

(iv) In a rather unconventional suggestion, the process of grafting polymer

chains (synthesis of polymer brushes) on a surface could be investigated in situ,

especially for the “grafting‐from” approach. In this context, “in situ” implies that the

polymer brush growth is monitored “live” without any disruption of, or interference

with, the grafting process. Such an approach may become possible by following the

changes in mass, energy, or thickness of a growing polymer brush traced over time,

starting from time t0 until the final state of growth.209-211 With either an extrapolation

of the data to the starting point or the possibility of directly measuring the initial step

of brush growth, the value of the grafting density should be accessible, if the exact

surface area is known.

(v) Finally, a visionary concept for determining grafting densities can be realized

by using laser‐assisted atom probe tomography (APT).212-214 APT is a characterization

tool mainly used for conductive materials where atoms are field evaporated (from a

sharp tip r < 100 nm) and accelerated to a time‐of‐flight detector. Using the recorded

information by the detector, a complete 3D reconstruction of the tip can be simulated.

The recent progress toward laser‐assisted APT allows the characterization of

nonconductive material.215-216 With a further development of this technique, the

imaging of polymers grafted on such sharp tips could be realized.217 Thus, a 3D

reconstruction of the polymer brush can potentially be established and based on this

image, the grafting density may be determined.

We hope to have demonstrated that for the reliable, comparable, and quantitative

determination of grafting densities, a substantial number of problems have to be

addressed. We further submit that the herein‐reported first‐time comprehensive

quantitative assessment of the limits and errors of the most popular methods to

determine grafting densities will, on the one hand, lead to a critical revaluation of

literature‐reported densities, as well as—importantly—accelerate the development of

methods that are more powerful than the existing ones, possibly along the above

outlined lines. At stake is nothing less than the precision design of one of the most

critical points in any device resting on soft matter technology, the interface.

Chapter 3: The Long and the Short of Polymer Grafting 45

Chapter 3: The Long and the Short of

Polymer Grafting

The need for novel techniques for reliable grafting density estimation was

demonstrated in Chapter 2. Five possible ways of approaching the problem of grafting

density estimation were proposed. For demonstrating that these approaches can lead

to an improved precise quantitative evaluation of surface grafting the 4th approach was

chosen as a proof of concept. However, instead of measuring in-situ the growth of a

polymer brush via the ‘grafting-from’ method a ‘grafting-to’ polymer graft was

investigated with a flow QCM setup. In addition to the precise quantification of the

grafting density, a preferential grafting of short polymer chains could be observed.

3.1 ABSTRACT

We demonstrate that grafting a distribution of polymer chains onto an interface

critically affects the shape of the distribution, with shorter chains being preferentially

attached. This distortion effect is herein quantified for the first time, exploiting a quartz

crystal microbalance – underpinned by single-molecule force spectroscopy – on the

example of grafted poly(methyl methacrylate) (PMMA) chain distributions of

different molar mass. ‘Grafting-to’ of different ratios of number average molecular

weight of PMMA distributions unambiguously establishes the preferred surface

grafting of shorter polymers, which can be correlated to their smaller radius of

gyration. Our findings allow to establish a preferential grafting factor, 𝜅, which allows

to predict the molar mass distribution of polymers on the surfaces compared to the

initial distribution in solution. Our findings not only have serious consequences for

functional polymer interface design, yet also for the commonly employed methods of

grafting density estimation.

3.2 INTRODUCTION

Polymers covalently attached to surfaces can exhibit different confirmations,

primarily depending on the distance amongst tethering sites described by the grafting

density σ.119 For densely grafted polymers, also referred to as ‘polymer brushes’,155

the interactions between the individual polymer chains are influencing essential

46 Chapter 3: The Long and the Short of Polymer Grafting

surface properties including wettability,108 adhesion,218 tribological behaviour114 and

biocompatibility.219 For surface functionalization, two main methods can be

employed, i.e. the ‘grafting-to’ and the ‘grafting-from’ approach. The latter method is

generally suggested to lead to higher grafting densities as polymers are grown in situ

from the surface in a process driven by monomer diffusion.128 Thus, surface induced

reversible deactivation radical polymerization (RDRP) techniques can yield dense

polymer brushes.135, 220-221 Nevertheless, the ‘grafting-from’ method suffers from lack

of information about the degree of polymerisation (DP) and molar mass distribution

(MMD) of the grafted polymer. In the ‘grafting-to’ approach, pre-fabricated polymers

with suitable end-groups are tethered onto a surface.99 A critical advantage of the

‘grafting-to’ approach is that the bulk polymers can be characterized post-synthesis,

prior to their surface attachment.222 Assuming the properties determined in solution,

the DP and MMD, accurately reflect the properties of the surface-tethered polymers,

‘grafting-to’ provides comprehensive knowledge about the functionalized surface.119

However, this general assumption has never been experimentally tested and is

questionable, as the preferential attachment of shorter over longer polymer chains

appears possible due to diffusion, probability and geometrical effects.223

3.3 GRAFTING DENSITIES EVALUATED VIA QCM

In the current study, we scrutinize this hypothesis by conducting quartz crystal

microbalance (QCM) measurements of ‘grafting-to’ experiments and corroborate the

results by atomic force microscopy (AFM) based single-molecule force spectroscopy

(SMFS). QCM measurements enable the investigation of surface mass uptake by

recording the frequency change of an oscillating piezo-electric quartz crystal induced

by a mass change in the crystal's mass (attachment of polymer chains). These

measurements can be performed with high precision, even in liquid environments.85-87

Most of the conducted QCM studies on polymer systems investigated grafting

kinetics209, 211, 224-227 or conformational changes in the polymer brush228-229 with

additional dissipation energy recording. The current work solely focuses on the overall

mass uptake resulting from ‘grafting-to’ of end-functionalised poly(methyl

methacrylate) PMMA chains on a silicon dioxide (SiO2) coated QCM sensor. The

mass uptake Δm on a QCM sensor for rigid thin films in vacuum and air can be

calculated by the measured change of resonance frequency Δf using the Sauerbrey

Equation 20,230


∆𝑚 = −𝐶QCM

∆𝑓

𝑛 (20)

with the mass sensitivity constant CQCM = 17.7 ng·cm−2·Hz−1 (for an AT cut sensor

with f0 = 5 MHz), defined by the fundamental frequency f0, thickness and density of

the piezo crystal,87 and the overtone number n (= 1, 3, …). In the case of grafting

polymers to a QCM sensor from solution, two additional effects are influencing the

frequency change, i.e. the frequency change induced by the solvent and the additional

damping of the frequency due to viscoelasticity of the attached film. Both effects are

not covered by the Sauerbrey equation, Equation 20. To account for the effect of

frequency change induced by the solvent covering the sensor, the baseline can be

directly recorded in the solvent and used as reference for frequency changes.226

However, compensating the effect of a viscoelastic film is more challenging, yet it can

be achieved with rather complex models.231 Nevertheless, the work of Höök et al.

shows by theoretical simulations that for films less than 100 nm, no overestimation of

mass uptake was noticed using the Sauerbrey relation (the film thickness in our study

was approximately 8 nm, Table S10 in Appendix C).87 Therefore, the calculations of

the mass uptake are herein based on the Sauerbrey equation, Equation 20.

A reversible addition fragmentation chain transfer (RAFT) polymerization

agent (CTA) functionalized with a silyl ether (refer to the Supporting Information in

Appendix C CTA1)232 was employed to synthesize polymers P1 to P5 via RAFT

polymerization (Supporting Information in Appendix C P1–P5).233 The hydrolysis

sensitive end group allowed tethering of the polymers to SiO2 coated QCM sensors

under mild conditions. The attachment takes place via the R-group of the CTA,

ensuring (almost) all chains carry the silane function. RAFT was employed to establish

control over the molar mass and MMD as well as a high end group fidelity, rendering

the need for post-functionalization obsolete. For the current study, polymers with five

different number average molar masses Mn were synthesized (P1: Mn = 8,240 g·mol−1,

P2: Mn = 48,000 g·mol−1, P3: Mn = 106,000 g·mol−1, P4: Mn = 133,900 g·mol−1 and

P5: Mn = 216,100 g·mol−1, the MMD are depicted in Figure 13 A).


Figure 13: (A) Normalised weight MMD of P1 (black), P2 (red), P3 (blue), P4 (green) and P5 (grey).

(B) Time dependent grafting density σ of samples P1–P5 calculated by normalising the

mass uptake. Mass uptake Δm was determined from Δf for n = 9 via Equation 20. The

small non-linearity in the black curve is due a gas bubble that formed during the

experiment. A plateau in the grafting density was reached after 60 min and no change was

subsequently observed. (C) Grafting density σ in dependency of the number averaged

molecular weight Mn plotted in a double logarithmic graph and fitted to a power law. The

exponent of the power law (slope in double logarithmic graph) can be related to the

solvent interaction parameter n* (for PMMA in toluene at 50 °C).57

The polymer samples were dissolved in toluene and introduced into a QSense

QCM flow cell with a flow rate of 15 μl·min−1. The flow cell temperature was

controlled to 50 °C and flushed with pure toluene to record a stable baseline. The

frequency change associated with grafting of the polymer distributions onto the QCM

sensor was recorded for all overtones n = 1, 3, 5, 7, 9, 11 and 13. However, the

fundamental frequency (n = 1) was not taken into account for further evaluation, due

to the insufficient energy trapping. Subsequent calculation of the absolute mass change

Δm was performed according to Equation 20. Further normalization of the mass

change by the respective number average molecular weight Mn of each polymer

resulted in the grafting density σ = Δm/Mn. The resulting time dependent grafting

density σ for each polymer sample (P1–P5) at an overtone number of n = 9 is shown


in Figure 13 B. Before comparing the different grafting densities of P1 to P5, it is

important to note that for our system and conditions the surface saturation results in

one single plateau (regime) after approximately one hour. This is in contrast to

literature results, which suggest the existence of up to four different regimes in the

grafting kinetics.234 Such single regime kinetics occur at high temperatures and

concentrations, when the energy barrier is sufficiently small to directly form a densely

grafted layer in one step, as also observed in other studies.224

The quantitative comparison of the grafting densities for polymer samples P1 to

P5 sizes is depicted in Figure 13 C. The grafting densities estimated from overtones 3

to 13 (after 5,000 s) were averaged and plotted vs. the number average molar mass Mn.

The calculated data was subsequently fitted to a power-law. Thus, the slope represents

the scaling behaviour ν of the grafting density as a function of the number averaged

molar mass Mn. In a recent publication, we demonstrated that the grafting density σ for

the ‘grafting-to’ approach is likely related to the radius of gyration Rg,57

𝜎 ~1

𝑅g2 ~

1

(DP𝑛𝑛∗

)2 ~𝑀n

−2𝑛∗ (21)

with the degree of polymerisation DP and the solvent interaction exponent n*, which

describes the polymer solvent interaction at a given temperature (here PMMA in

toluene at 50 °C). The almost perfect fit of the measured data to the power law with a

solvent interaction parameter of n* = 0.47 ± 0.02 evidences the previously postulated

relation of grafting density σ to the radius of gyration Rg.57 Nevertheless, this result

merely suggests that a higher grafting density is achieved by grafting shorter polymers

– as already demonstrated235-236 – and no information on the preferential grafting of

shorter polymers within a given distribution are provided. P1, for instance, comprises

– despite its Mn of 8,240 g·mol−1 and narrow dispersity of 1.06 – chains ranging from

approximately 3,500 g·mol−1 to almost 17,400 g·mol−1.

3.4 PREFERANTIAL SURFACE GRAFTING

To elucidate the bias towards lower molar mass chains during ‘grafting-to’,

defined mixtures of polymers were immobilized on QCM sensors. By comparison of

the molar ratio of the polymer samples in solution with the ratio of the polymers on

the surface, we establish and quantify this bias. The aforementioned mixtures were

prepared from polymers P1 (Mn = 8,240 g·mol−1) and P3 (Mn = 106,000 g·mol−1) as

their MMDs do not overlap (Figure 13 A). The ratios chosen for the mixtures are


R1 = 1 : 0 (P1 : P3), R2 = 1 : 1, R3 = 1 : 4, R4 = 1 : 9 and R5 = 0 : 1 (in molar

equivalents). The evaluation of the ratios R2 to R4 is not as trivial as for the individual

polymer solutions, where the frequency change (or mass uptake) was simply

normalized by their number averaged molar mass in solution 𝑀nsol. If the frequency

change is normalized by a hypothetical average molar mass calculated according to

the employed ratios from the solution, the results show disparate ratios of short to long

polymers on the surface compared to the solution. Figure 14 A shows the progress of

the normalized frequency of the mixed samples. The black (R1) and grey (R5) curve

represent the lower and upper limit for the normalized frequency changes for all other

ratios as they represent the grafting of solely short and long chains, respectively. R3

(blue) and R4 (green) exceed these boundaries, indicating that the Mn in solution does

not reflect the average molar mass of the grafted polymer. Instead, the Mn on the

surface (𝑀nsur) is substantially lower than in solution, which clearly demonstrates that

shorter chains are preferentially grafted.

Figure 14: (A) Time dependent frequency change for polymer mixtures R1 to R5 normalized by

their average molar mass in solution 𝑀nsol. (B) Time dependent frequency change for R1

to R5 normalized by 𝑀nsur to match expected frequency behaviour.

While this finding provides a qualitative description of the size bias in ‘grafting-

to’, we also quantitatively establish to which extent shorter chains are preferentially

grafted. If the molar mass – or rather the radius of gyration – influences the grafting

density as demonstrated above, an equimolar mixture of two polymer samples should

ideally result in an averaged grafting density of the individual polymer samples. The

same should be true for mixtures of different ratios, i.e. the grafting density of a

mixture should be the weighted average based on the molar ratio of its components.

We introduce the value 𝑀nsur (Table 6) to describe the molar mass average of

the polymers on the surface to establish the expected grafting densities (Figure 14 B).

The molar mass averages of the surface attached polymers are well below their known


corresponding mass averages in solution 𝑀nsol (Table 6). Thus conclusively indicating

a preferential attachment of shorter chains. When we compare the ratio of the surface

mass average to the mass average in solution obtaining a preferential grafting factor

κ = 3.36 ± 0.58, we demonstrate that P1 (Mn = 8,240 g·mol−1) is grafted preferentially

to the surface compared to the P3 (Mn = 106,000 g·mol−1) sample. Interestingly, the

experimentally established preferential grafting factor κ is in excellent agreement with

the polymer samples’ ratio of their radii of gyration of 3.32 Equation 21, establishing

a comprehensive physical explanation for the grafting bias.

𝑅gP3

𝑅gP1

= (𝑀n

P3

𝑀nP1

)

𝑛∗

= (106

8.24)

0.47

= 3.32 (21)

with n* = 0.47 as determined above (Fig. 11C). Further defined mixture experiments

with P2 and P3 (Fig. S10 in the Supporting Information in Appendix C) were

conducted and resulted in a preferential grafting factor of κ = 1.40 ± 0.10, which

corresponds to an Rg ratio of 1.45, in excellent agreement with the above results. These

findings provide clear evidence that the ‘grafting-to’ method is heavily biased towards

lower molar masses, which is a result of their smaller radius of gyration. Thus, they

exhibit greater mobility and faster diffusion. A smaller Rg also increases the probability

of the end group to be located on the surface of the polymer coil, which critically

enhances the likelihood of a surface attachment. Finally smaller polymers simply

occupy less space and can therefore realize higher grafting densities.

Table 6: Samples used to investigate grafting bias, their composition, theoretical Mn in solution

Mnsol and on the surface Mn

sur with calculated preferential grafting factor.

Sample Composition

[P1:P3]

Norm. Mnsol

[g·mol-1]

Norm. Mnsur

[g·mol-1] adjustment factor

R1 1:0 8,240 8,240 ---

R2 1:1 57,100 17,500 3.26 ± 0.39

R3 1:4 86,400 20,200 4.29 ± 0.96

R4 1:9 96,200 37,900 2.54 ± 0.38

R5 0:1 106,000 106,000 ---

avg.: 3.36 ± 0.58

The short-chain grafting preference, however, is not merely relevant when samples of

different molar masses are to be grafted. It is just as important when well-defined

narrow disperse polymers are grafted. Even the most sophisticated contemporary

polymerization methods yield dispersities far from unity. Thus, polymer chemists


effectively always employ mixtures of polymers with different molar masses, in fact

even in a fairly narrow MMD, such as P1, only approximately 1.5% of the polymer

chains correspond to the Mn of 8,240 g·mol−1 and the chains within the sample cover

a range of molar masses from 3,500 g·mol−1 to 17,400 g·mol−1. Based on our current

results, we can now quantitatively predict how the MMD is affected upon

immobilization onto the substrate, taking the ratios of Rg within a given polymer

sample into account. The preferential grafting factor κ expresses by how much the

molar mass of interest Mi is preferred or unfavoured over the molar mass average Mn

of the sample.

𝜅 = (𝑀n

𝑀i)

𝑛∗

(21)

with the number averaged molecular weight Mn, the single molar mass Mi of the

polymer distribution, scaled by the solvent interaction parameter n*. Exemplary

preferential grafting factor κ curves for different number averaged molar masses are

shown in Appendix B Fig. S11.†

To predict the grafting behaviour of a polymer sample, it is necessary to know

the number weighted molar mass distribution MMDn of the sample, which can be

obtained by size exclusion chromatography (SEC) employing concentration sensitive

detectors (Fig. S3–S7 in the Supporting Information in Appendix C).

The number and mass weighted distributions of P1, P3 and P5 are depicted in

Figure 15 A/B. One important observation in the MMDn is that the majority of chains

(62%) for sample P5 (Figure 15 B) is below the Mn, which is not obvious when

inspecting the mass weighted distribution.

To estimate the molar mass distribution on the surface, the MMDn in solution is

multiplied with κ, which shifts the distribution further towards lower molar masses

(Figure 15 C) – under the provision that each chain in the distribution has a reactive

end group. For rather small polymers below 100,000 g·mol−1 this shift is almost

negligible, however for large polymers it is substantial. For P1, for instance, κ is 1.5

at 3,500 g·mol−1 and 0.7 at 17,500 g·mol−1 implying that the lower end of the

distributions is approximately twice as likely to be grafted as its upper end. For P5, κ

is 2.6 at 28,000 g·mol−1 and 0.4 at 1,240,000 g·mol−1, implying the lower end is more

than six times probable to be grafted than the upper end of the distribution. ‘Grafting-

to’ with high molar mass samples therefore requires special attention to address the

mismatch of the SEC derived molar mass distribution with the actually grafted chain


distribution (see Figure 15 D shift of solid to dotted line). To confirm the prediction of

preferential surface grafting of shorter polymer chains, AFM based SMFS

measurements were conducted. The normalized Gaussian distributions of the recorded

rupture length lR are depicted in Figure 15 E for sample P1 (lR = 26 ± 3 nm),

P3 (lR = 150 ± 9 nm) and P5 (lR = 154 ± 15 nm). The expected shift from different

peak maxima in solution (see Fig. 13A) to the same peak maximum on a surface of P3

and P5 (see Figure 15 C) is confirmed by the same average rupture length. A detailed

description of these experiments can be found in the Supporting Information in

Appendix C.

Figure 15: (A) Normalized mass weighted distribution in solution 𝑀𝑀𝐷wsol, (B) normalized number

weighted distribution in solution 𝑀𝑀𝐷nsol and (C) normalized number weighted

distribution on surfaces 𝑀𝑀𝐷𝑛surfor P1 (black curves), P3 (blue curves) and P5 (grey

curves). (D) Shift of the molar mass distributions 𝑀𝑀𝐷wsol (solid line), 𝑀𝑀𝐷n

sol (dashed

line) and 𝑀𝑀𝐷𝑛sur (dotted line) of P5. (E) Normalized Gaussian distribution of the rupture

length determined via SMFS experiments from sample P1, P2 and P3.

3.5 CONCLUSION

In summary, we demonstrate and quantify to which extent a polymer distribution

is distorted when grafted from solution onto a surface. We furthermore evidence that

this distortion can be quantitatively related to the radius of gyration, thereby providing

a physical explanation for the phenomenon. These results will critically affect methods

for the determination of the grafting density that rely on information of the number

average molecular weight of the grafted polymers, including the ‘dry thickness’


method or ‘gravimetric measurements’.57 We suggest a simple, yet quantitative

measure, κ, to predict the likelihood of certain polymer masses within a sample to be

grafted. The preferential grafting factor κ allows for the prediction of the grafted

distribution, effectively enabling quantitative access to grafted chain size distribution.

Chapter 4: Quantifying Solvent Effects on Polymer Surface Grafting 55

Chapter 4: Quantifying Solvent Effects on

Polymer Surface Grafting

As demonstrated in the previous chapter, shorter polymer chains are

preferentially attached to surfacers, and a preferential grafting factor, was

introduced. The observed shift of MMD could be correlated to the radius of gyration

of different-size polymer chains. Via SMFS the surfaces were investigated and a

drastic difference between surface and solution MMD for larger molecular weights

was detected. The result of this strong surface shift raised the question if the shift can

be minimized by ideal reaction conditions. By grafting PMMA chains on silica NPs

the difference between solution distributions after grafting and surface distribution can

be investigating. With theoretical calculations and experimental results the effect of

solvent quality on the shift of the surface distribution was investigated.

4.1 ABSTRACT

When grafting polymers onto surfaces, the reaction conditions critically

influence the resulting interface properties including the grafting density and molar

mass distribution (MMD) on the surface. Herein, we show theoretically and

experimentally that application of poor solvents is beneficial for the ‘grafting-to’

approach. We demonstrate the effect by grafting poly(methyl methacrylate) (PMMA)

chains on silica nanoparticles (NPs) in different solvents and comparison of the MMD

of the polymer in solution before and after grafting via size exclusion chromatography

(SEC). The shorter polymer chains are preferentially grafted onto the surface, leading

to a distortion effect between the MMD in solution and on surfaces. The molecular

weight distortion effect is significantly higher for ethyl acetate (better solvent quality,

difference in Mw surface to solution 14%) than for N,N-dimethylacetamide (poor

solvent quality, 6%). The difference in MMD on the surface to the solution

significantly affects both the surface properties (e.g. the grafting densities) and their

determination.

56 Chapter 4: Quantifying Solvent Effects on Polymer Surface Grafting

4.2 INTRODUCTION

Designing and tailoring functional surfaces is one of the key endeavours in soft

matter materials science. Applications for functional interfaces range from 3D cell

scaffolds,91-92 optoelectronics237-239 and coatings96, 240-241 to sensors.94, 242-243 One way

to tailor properties such as hydrophobicity,109, 244-245 tribology114, 246 or anti-

biofouling26, 97, 106, 247-249 is to covalently tether polymers onto surfaces. The surface

attached polymers can be additionally equipped with specific functional groups to fine-

tune the desired surface properties.108, 250 Two important properties of surface grafted

polymers are the grafting density (tethering distance between individual polymeric

chains) and the molar mass distribution (MMD) of the surface attached polymers, as

these determine the surface performance. The main approaches to achieve polymer

functionalized surfaces are the ‘grafting-to’ and ‘grafting-from’ method.52, 119, 126-127,

130, 251 In the latter approach, the polymers are synthesized in-situ, growing from the

surface, and giving access to high grafting densities. The ‘grafting-to’ approach uses

polymer chains which are synthesized prior to grafting and are equipped with

endgroups which easily react with the surface generally leading to lower grafting

densities compared to the ‘grafting-from’ approach.128 However, the ‘grafting-to’

method allows for a plethora of characterization methods of the polymer chains to be

applied prior to surface attachment. Polymer properties determined as such are often

assumed to reflect the polymer properties on the surface and are even used as factual

data in further characterization.119, 222 Recently, we demonstrated this assumption to

be incorrect.60 Due to a preferred attachment of shorter polymer chains, a significant

shift of the MMDs to lower molar masses was observed on the surface. A schematic

representation of the preferred attachment of short polymer chains on surfaces is

depicted in Figure 16 on the example of silica nanoparticles, which we based the

herein introduced method on. This method allows for the simple and instrumentally

non-demanding determination of MMD shifts during grafting-to of polymers.


Figure 16: Schematic and simplified presentation of the preferred attachment of shorter PMMA

polymer chains onto SiO2 nanoparticles (NPs). The coloured lines represent the short

(red) and long (blue) polymer chains.

4.3 THEORETICAL EVALUATION OF MMDS ON SURFACE GRAFTING

Earlier, the preferred attachment of shorter polymer chains was quantified via

quartz crystal microbalance (QCM) measurements and correlated with the difference

in radius of gyration, Rg. The quantification of this shift, corroborated by atomic force

microscopy (AFM) based single-molecule force spectroscopy (SMFS) measurements,

led to the establishment of a preferential grafting factor, κ. The preferential grafting

factor applied on a number based molar mass distribution predicts the shift of the

distribution upon surface attachment. The shift is more pronounced for higher molar

masses and broad distributions.60 In addition, a solvent dependency of the shift is

expected due to the power law of the preferential grafting factor, κ, with the solvent

interaction exponent, n*,

𝜅 = (𝑀n

𝑀i)

𝑛∗

(21)

and the ratio of number average molecular weight, Mn, and single molar mass, Mi, at a

certain point of the polymer distribution. In the case of n* = 0.5 (θ-conditions) the

polymer chain behaves like an ideal chain following a random walk. At θ-conditions,

the polymer-polymer interactions are energetically equal to polymer-solvent

interactions, and therefore the excess energy of mixing is zero. In good solvents


(n* > 0.5), the polymer-solvent interactions are advantageous, causing the chain to

expand. Whereas for bad solvents (n* < 0.5) the polymer chain contracts due to

favourable polymer-polymer interactions.252

To evaluate and quantify the effect of preferred surface attachment (difference

of MMDs from solution to surface) with varying solvent quality, we herein conduct a

calculation, showing a larger difference for good solvents. Symmetrical Gaussian

distributions are employed as approximation for MMDs to simplify the calculations.

Furthermore, the distributions employed in the calculations are considerably narrow

(dispersity of Ɖ = 1.05 for a number average molar mass of Mn = 5.00·105 g·mol-1) to

illustrate the relevance for polymers synthesized via controlled polymerization

techniques. The prediction of the MMD on a surface are conducted by multiplying the

preferential grafting factor (Equation 21), κ, with the MMD in solution. The calculated

shifts of MMD on surfaces for the described distribution with a solvent interaction

parameter from n* = 0.1 (very poor solvent) to n* = 1.0 (very good solvent) are

depicted in Figure 17.The dotted lines in Figure 17 A show the normalized MMD of

the calculated surface distributions. An increase of the solvent quality results in a

stronger shift of the MMD towards lower molar masses (difference in percentage of

Mn plotted in Figure 17 B) whereas bad solvents show a negligible shift. The difference

in radius of gyration, Rg, of polymer coils – with low versus high molar mass – is

distinctly lower for contracted chains. In a good solvent, the increased size difference

of the expanded polymer coils results in a preferred attachment of the lower molar

mass fraction of the MMD and therefore results in a more pronounced shift of the

distribution. For an increase of the solvent quality the impact on the surface

distribution is noticeably stronger, suggesting that preferably poor solvents should be

employed for the ‘grafting-to’ approach to guarantee a similar surface MMD in

comparison to the solution MMD. We herein demonstrate this effect.


Figure 17: (A) Calculated shift of normalized MMD on surfaces (dotted lines) for different solvent

interaction parameters, n* (ranging from 0.1 ‘poor solvent’, over 0.5 ‘θ-condition’, to 1.0

‘good solvent’). Starting with a narrow Gaussian MMD (Ɖ = 1.05) with a number and

weight average molecular weight of Mn = 5.00·105 g·mol-1 and Mw = 5.24·105 g·mol-1

(solid black line). (B) Percentage shift of Mn on surface MMD depending on the solvent

quality extracted from the MMDs in Figure 17A.

A shift in MMD from higher to lower molar masses on the surface implies that a

shift of the same magnitude but towards higher molar masses has to occur in the

solution. The sum of the shifted MMD on the surface and the MMD of the remaining

reaction solution has to be equal to the distribution of the starting reaction solution

(principle of mass conservation). The above raises a critical question: Why are there

no reports for such MMD shifts of the reaction solution? Typically in the ‘grafting-to’

approach, a large excess of polymer chains to reactive surface sites is employed to

drive the reactions to high grafting densities.180, 253 Especially for smooth planar

substrates the excess is commonly several orders of magnitude. In the case of our

previous QCM study, the experiments were conducted in a flow setup (very high

excess of end-functionalized polymers) to insure homogenous and full coverage of the

surface.60 The enormous excess of polymer guaranteed exclusion of any concentration

related effects on the grafting density, however, it is also the reason why a shift in the

MMD of the remaining reaction solution cannot be detected. The change in the

polymer concentration is therefore infinitesimal small and thus virtually non-

observable. To be able to detect a change in the MMD of the remaining polymer

concentration either the number of endfunctionalized polymers in the reaction solution

has to be strongly decreased or the number of reactive surface sites has to be

sufficiently high. In an ideal case, the concentration of polymer chains in the reaction

solution will be reduced by fifty percent after grafting. In other words, half of the total

amount of polymers is tethered to the surface and the other half remains in solution.


The sum of the two MMDs must result in the distribution of the reaction solution before

grafting. For a symmetrical Gaussian distribution (same as in Figure 17 with a solvent

interaction parameter of n* = 0.9) the surface and solution MMD would look like the

deconvoluted peak in Figure 18 A. The area of the surface distribution (red curve) and

the remaining reaction solution distribution (blue curve) is equal and the sum is

resulting in the start solution distribution (black curve). The different shape of the two

distributions can be explained by the power law relation (see Equation 21) of the

preferential grafting factor, κ. The increased weighting of smaller molar masses will

lead to slight peak broadening of the surface distribution. In Figure 18 B, the

normalized MMDs on surfaces (red curves) and in solution (blue curves) for varying

solvent quality can be seen. The sum of the surface and remaining reaction solution

distribution always results in the solution distribution before grafting (black curve).

Figure 18: (A) Deconvoluted peak for a starting reaction solution MMD (reduction in polymer

concentration of 50%) with the surface distribution (red curve) and solution distribution).

(B) Calculated shift of normalized MMD on surfaces (red dotted lines) and solutions (blue

dotted lines) for different solvent interaction parameters, n* (ranging from 0.1 ‘poor

solvent’, over 0.5 ‘θ-condition’, to 1.0 ‘good solvent’). Starting with a narrow Gaussian

MMD (Ɖ = 1.05) with a number and weight average molecular weight of

Mn = 5.00·105 g·mol-1 and Mw = 5.24·105 g·mol-1 (solid black line)

4.4 EXPERIMENTAL SOLVENT EFFECTS ON POLYMER GRAFTING

To elucidate the complementary shift of MMD in solution, a sufficient amount

of polymer must be tethered to the surface to induce measurable changes in the

remaining reaction solution. Theoretically, this may be achieved by very low sample

concentrations, yet in practice one would have to use such low concentrations that

most characterization methods would not yield meaningful data. Instead, we chose to


use common procedures and instruments. A concentration of 2 mg·mL-1 of polymer is

common and can be readily analyzed via SEC. To observe a change in this solution,

sufficient polymer must be removed from the solution, requiring a large surface for

grafting-onto.

Spherical nanoparticles (NPs) are ideal substrates due to their ratio of surface to

volume that increases with decreasing particle size. Poly(methyl methacrylate)s with

hydrolysable silane end groups were used as this enables the application of silica

nanoparticles as substrate. The minimal nanoparticle size is limited by the radius of

gyration of the grafted polymers. The particles diameter needs to be large enough to

rule out excluded volume effects.144 Therefore, a difference between the diameter of a

silica NP before and after grafting should be as low as possible. With a radius of

gyration of Rg ≈ 7.1 nm (refer to the Supplementary Information in Appendix D), the

difference between a 460 nm silica NP before and after grafting is less than 3%. Silica

NPs of 460 nm diameter were synthesized according to a published procedure.254 The

next consideration is the amount of NPs to be used as substrate, ideally 50 % of the

polymer in solution should be immobilized. In our hands solutions containing

4 mg·mL-1 polymer and dispersions containing 400 mg·mL-1 silica particles result in

the desired 50 % reduction. The solutions were prepared in different solvents (S1 Ethyl

acetate (EA), S2 N,N-dimethylacetamide (DMAc) containing 0.08 wt.-% LiBr).

Subsequently, 2.5 mL of polymer solution and 2.5 mL of nanoparticle dispersion were

combined and stirred for 7 d at 50 °C. The particles were separated via centrifugation

and washed twice with THF to remove physisorbed polymer. Prior to SEC analysis,

the supernatants were combined and the solvent evaporated under reduced pressure.

The SEC traces before and after grafting indicate a change in concentration of

50 ± 3 % (peak area), which is the desired range and equal to the shift assumed in the

calculated data (see Figure 18 A). DMAc and EA were chosen as they represent a near

θ-solvent (DMAc) and a good solvent (EA) under the current conditions. The

difference in the apparent hydrodynamic radii was determined via dynamic light

scattering to be 0.3 nm (7.3 nm in DMAc, 7.6 nm in EA, see Figure S7 in the

Supplementary Information in Appendix D). All SEC samples were run multiple times

to exclude errors due to shifts in retention volume caused by potential fluctuations in

temperature or eluent composition. Retention volumes were referenced to an internal

standard. Per sample three virtually identical traces were averaged to simplify the

further calculations. Figure 19 illustrates the shift of the MMD in solution from the


dissolved polymer before grafting in black (solution start) to the MMD after grafting

(solution DMAc/EA) in blue and the complementary MMD on the particle surface in

red (Surface DMAc/EA). The surface distributions were calculated by the difference

of the MMD in solution before and after grafting (principle of mass conservation).

Clearly evident is a distinct difference in the observed shift depending on the solvent

quality, with the shift being significantly larger for good solvents. Especially for the

higher molar mass region this effect is clearly visible. The distortion effect results in a

difference of the weight averaged molar mass Mw between in solution (after grafting)

and on surfaces of 14% for EA and 6% for DMAc (shown in Figure 19 B). Using

DMAc, the impact on the preferential grafting of shorter polymer chains on surfaces

was reduced and therefore the MMD on the surface is closer to the MMD in solution.

Figure 19: A) Experimentally observed shift of normalized MMD – recorded via SEC – on surfaces

(red curves) and solutions (blue curves) for grafting in DMAc (dashed lines) and EA

(dotted line). B) Weight averaged molecular weight Mw of the MMD and the difference

between surface and solution for DMAc and EA functionalization.

4.5 CONCULSION

Why are these information of relevance and how does the preferential grafting

change the properties of the functionalised surface? For an accurate calculation of

grafting densities (tethering density of polymer chains on surfaces), a precise

knowledge of the molar mass and MMD on the surface is critical. The general used

procedures for such grafting density calculations are the dry thickness method, the

gravimetric assessment and swelling experiments. The precise knowledge of MMD on

the surface is significantly increasing the exact quantification of surface grafting for

the first two methods.57 By changing from good solvents to θ-conditions, the precision

of these quantification methods can be improved and the error of the calculated value

can be reduced by a factor of two (difference in molecular weight in Figure 19 B).

Additionally, previous studies conducted via XPS and AFM showed that the grafting


density itself can be improved by decreasing the solvent quality,255 showing that a

decrease in solvent quality (by changing the ratio of a bad and good solvent in a binary

solvent mixture) is increasing the grafting densities.

In summary, we demonstrate that solvent-polymer interaction has a critical

impact on the resulting surface functionalization via the ‘grafting-to’ approach. Using

DMAc (near θ-condition) instead of EA (good solvent), the preferential grafting of

shorter polymer chains was reduced by a factor of two. If surfaces with a similar MMD

in solution and on surface are desired, a rather poor solvent must be employed,

accompanied by a concomitant increase of the grafting density.255 The choice of a poor

solvent is in contrast to the usual choice of good solvents for surface functionalization,

because the polymer must remain soluble to achieve surface functionalization. Solvent

systems with low solvent interaction parameters (n* << 0.5) are unable to sufficiently

solvate the polymer and can lead to agglomeration and precipitation. The ideal

conditions of surface grafting by the ‘grafting-to’ approach are therefore close to

θ-conditions, as demonstrated by the grafting of PMMA chains on silica NPs in DMAc

at 50°C. We submit that the herein introduced simple nanoparticle and SEC based

method for determining shifts in MMDs during surface grafting is applicable to other

polymer-solvent systems.


Chapter 5: General Discussion 65

Chapter 5: General Discussion

Throughout the entire PhD thesis, the need of in-depth surface characterisation

for designing interfaces with defined functions was pointed out multiple times. As

knowledge is the key for tailoring and designing advanced surface properties, a precise

understanding of the molecular processes which govern surface functionalisation has

to be established. Especially functionalisation of surfaces with molecular chain

structure serves a multidisciplinary need as described in Chapter 1. By evaluating the

covalent attachment of polymer chains on solid surfaces theoretically and

experimentally, this current study specifically establishes the molecular process of

tethering macromolecules via the ‘grafting-to’ approach. The study resulted in three

peer-reviewed publications, which are all showing the importance of a detailed

quantitative characterisation. In subsequent subchapters the key outcomes and future

directions of the current research work are summarized, followed by a general

conclusion of the PhD thesis.

5.1 SUMMARY OF STUDIES AND KEY OUTCOMES

Before any experimental work was conducted, a careful evaluation of existing

quantification methods was carried out. In the process of assessing the commonly used

grafting density characterisation methods – dry thickness method, gravimetric

assessment and swelling experiments – disadvantages concerning the quality of each

of the determination methods were identified. The assessment was divided into three

areas: (i) number of assumptions invoked in determining grafting densities (ii)

simplicity of determination (how many separate measurements have to be undertaken)

and (iii) stability in error propagation. Especially for the most frequently employed

determination method – the dry thickness method – questionable assumptions are

made, which critically influence the quality of the determined results. One can

conclude that all of the commonly used methods have their disadvantages with

approximately 12% of the reported values above or close to the physical limitations

introduced in Chapter 2. However, most strikingly, only around 6% of the reported

grafting densities (included in the current PhD thesis) conducted proper error

estimations for their grafting density calculations. For future research in this area, a

guide for precise error analysis is given, which will critically aid to judge the quality

66 Chapter 5: General Discussion

of the quantitatively determined value. Physical limitations of maximum grafting

densities are provided for both grafting approaches and should be used as a guide for

researchers employing any kind of grafting density determination. In cases for

calculated grafting densities close to – or even exceeding – the physical limitations,

the choice of determination method has to be reviewed and additional evidence for

such high grafting densities has to be provided. Alternative routes for a reliable

evaluation of grafting densities are proposed at the end of Chapter 2. For improving

the quality and precision of grafting density determination a simple direct assumption

free method has to be employed.

One of the proposed methods is the evaluation via in-situ QCM measurements

of the surface grafting. The experimental work resulted in the exact evaluation of

surface grafting densities on a library of different sized macromolecules (all PMMA

based) on a silica QCM sensor, which are in good agreement with the theoretical

calculations. The grafting density scales with the size of the polymer, with higher

grafting densities achieved with shorter polymer chains. It is important to note that in

the conducted experiments the QCM was operated in a flow setup, which implies that

always fresh polymer solution is streaming over the sensor surface. Unreacted polymer

chains are flushed away in this scenario, leading to a static ultimate state (no change

in resonance frequency of the quartz crystal and therefore no further mass uptake) of

maximum grafting density for applied conditions. The final state of polymer graft is

likely driven by the diffusion kinetics of the polymer coil, with faster diffusivity of

smaller macromolecules towards the surface. Further investigations via ratio

experiments demonstrated that shorter polymer chains are preferentially grafted (in

comparison to longer polymer chains) onto surfaces. The observed phenomenon of

preferential surface grafting can be physically explained by the difference in the ratio

of gyration, establishing a preferential grafting factor. Additional AFM based SMFS

measurements are congruent with the QCM observed phenomenon of preferential

surface grafting. A significant shift of MMD on surfaces in comparison to solution

distributions was discovered. The difference of MMD is especially crucial for any

process which involves surface functionalisation with high molecular weight polymers

via the ‘grafting-to’ approach.

Further investigations of functionalizing silica NP surfaces were conducted to

investigate the effect of solvent quality on the shift of MMD. Instead of examining the

surface MMD directly on the substrate, the large surface area of the NPs was employed


to alter the distribution of the polymer solution after polymer grafting (easy SEC

readout). It is important to note that the amount of surface had to be significantly high

to observe a preferential surface grafting of shorter polymers without grafting the

entire concentration of polymer onto the NPs. For investigating the shift of MMDs

before and after grafting, a change of 50% of polymer concentration was targeted. The

effect of solvent quality on the shift of the MMD on the surface was investigated by

calculations and experimentally. The experiments were conducted over a reaction time

of 7 days (slow stirring for minimal shearing of polymer chains between NPs) to

achieve a similar static ultimate state comparable to the QCM experiments. For surface

functionalisation via the ‘grafting-to’ approach, the choice of a rather poor solvent

results in a less pronounced shift of MMD on the surface and in solution, which implies

to choose poor or theta conditions for surface functionalisation via the ‘grafting-to’

approach if similar surface to solution MMDs are desired. The application of poor

solvents will additionally result in a higher grafting density due to the decreased radius

of gyration.

5.2 FUTURE DIRECTION

As already proposed in Chapter 2, there are numerous possibilities of increasing

the quality of quantitative evaluation of grafting densities. In the current PhD thesis

the approach via in-situ QCM flow measurements was employed for grafting

macromolecules on silica surfaces by the ‘grafting-to’ method. The high sensitivity of

mass detection is an ideal tool for determining the mass increase per area due to the

surface functionalisation. The QCM additionally gave even more detailed insights into

the grafting process and allowed the implementation of a preferential grafting factor.

The experiments conducted in throughout the entire PhD thesis focused on

understanding a static ultimate state of surface grafting. For a full understanding of the

process of surface grafting via the ‘grafting-to’ approach a precise investigation on the

energetic and kinetic contributions has to be investigated. One way to investigate the

kinetic equilibrium of adsorption/desorption process (physisorption) on a QCM sensor

surface can be achieved by employing polymers without reactive end groups. The size

difference on the sorption behaviour will drive the kinetic equilibrium in either static

or flow (with varying flow rate) experiments and can be recorded via the difference in

frequency change. A variation of the reaction temperature for the surface grafting

process (chemisorption) will provide additional insights into the energetic contribution


to the static ultimate state of the surface graft. Additional variation of the chemistry

employed in the chemisorption process (varying surface reactions) can increase or

decrease the energetic contribution and will change the thermodynamic effects on the

ultimate static state of the polymer brush. Another approach to increase the precision

of the performed QCM experiments – especially for the performed ratio experiments

– is by employing monodisperse macromolecules. The preferred attachment of shorter

polymers inside a distribution gives an additional distortion effect on the results, which

can be prevent by using i.e. monodisperse DNA. In particular the application of

monodisperse macromolecules could ease the determination of kinetic and energetic

contribution on surface grafting via the ‘grafting-to’ approach.

In a next step, the evaluation of grafting densities via QCM for the ‘grafting-

from’ approach has to be investigated. The evaluation of the data is not as straight

forward as for the ‘grafting-to’ method and has to be modelled to growth kinetics of

the polymer brush. Extrapolation towards the starting point of the polymer growth

should result in an accurate value of the grafting density. The investigation of changing

segment density inside the in-situ growing brush will give additional insights into

polymer growth kinetics of the brush and initiator efficiency. It is important to note

that especially for dense polymer layers which exceed a certain thickness, the simple

correlation between frequency change and mass uptake (refer to Equation 20 in

Chapter 3.3) cannot be employed. In the case of strong visco-elastic behaviour of the

polymer brush elaborate calculations have to be conducted, which take dissipation of

the surface grafts into account. By combining the QCM experiments with other surface

characterisation techniques e.g. neutron reflectometry (density profile in Z-direction)

an absolute understanding of the polymer brush – down to the molecular level – can

be achieved.

The grafting of PMMA chains onto silica NPs resulted in a significant change of

the MMD of the solution after grafting (in comparison to the initial solution). The shift

to higher molecular masses in solution after surface grafting is in accordance with the

preferential surface grafting of shorter polymer chains on surfaces observed via QCM

and AFM based SMFS. The performed experiments could only be conducted in a

narrow condition window (solvent, polymer concentration and NP concentration) due

to the fact that all components had to be soluble/disperse in the reaction solution. By

using rather highly porous membranes or monoliths (large surface area) the problems

of NP aggregation can be overcome and the condition window can be extended. The


simple SEC readout of the solution after grafting provides a new opportunity in

determining the polymer solvent interaction. The shift in MMD between starting

solution and solution after grafting can be translated into the solvent interaction

parameter, possibly developing into a new and easy read out method for determining

the solvent interaction parameter of specifically challenging solvent polymer systems.

A further promising proposed method for high precision grafting density

evaluation is via AFM based colloidal probe measurements, as already mentioned at

the end of Chapter 2. Instead of trying to directly counting the polymers on a surface

via multiple rupture events recorded in a force spectroscopic measurement, the force

response of the polymer brush upon compression with a colloidal probe can be related

to the grafting density of the surface graft. Especially by using the effect of different

force responses of the polymer brush in varying solvents, the grafting density can be

directly related. The accuracy in load/force determination in AFM is high and can even

determine the grafting densities in low density regimes. Due to the possibility of

recording a large amount of single force spectroscopic measurements, the statistical

significance of the captured data is high. AFM therefore is one of the most ideal

methods for the precise quantitative evaluation of polymer surface functionalisation.

5.3 CONCLUSIONS

As a general conclusion, the interplay of theoretical/computational work and

experiments resulted in the detailed understanding of surface functionalisation with

polymers via the ‘grafting-to’ approach. The interplay of theoretical and experimental

research is the only way for successfully understanding processes at the molecular

level. By investigating the process of grafting PMMA chains onto silica surfaces (via

QCM), for the first time quantitatively the preferential surface grafting of short

polymer chains was observed. The observed phenomenon resulted in the

implementation of a preferential grafting factor, with a huge impact on how to design

functional surfaces with surface attached polymers. To design and tailor specific

surface functionalities a precise control over every aspect of the grafting procedure is

needed. By controlling the solvent quality in which the grafting process takes place,

the distribution of polymers on the surface can be controlled. The drastic effects the

preferential surface grafting has on the commonly used determination methods for

grafting densities and the following understanding of application performance can be

comprehend by the following example.


In the case of grafting a polymer distribution with a number average molar mass

of Mn = 17,000 g·mol-1 and a broad distribution of Ɖ = 1.63 (comparable to reference

180) a dry thickness of hdry ~ 7 nm can be determined. Using the most commonly

employed dry thickness method for evaluating the grafting density (see Equation 8 in

Chapter 2.4.1), a value of σ = 0.22 chains·nm-2 is estimated using the number average

molar mass from solution. If now a preferential grafting of shorter polymer chains on

the surface would be taken to account, the number averaged molar mass would be

shifted to a lower value, which will result in a shift to higher grafting densities. For

increased solvent qualities (n* = 1), the shift can exceed 60% difference in grafting

density value, resulting in a value of σ = 0.36 chains·nm-2 (example from above). The

influence on the percentage difference on the grafting densities as a function of the

solvent interaction exponent is illustrated in Figure 20. The green and red area show

the regimes of where the polymer can be grafted (soluble – green area) or will be

precipitated (insoluble – red area). The complete calculations can be found in

Appendix E. If such a surface graft should be applied for the release of small

molecules (small molecule attached to the polymer at the free end-group) in a drug

discovery application, a miss calculation of the released small molecule concentration

from the surface (of up to 60 % difference in shown example) could lead to false

interpretation of the resulting performance.

Figure 20: Difference of grafting density in dependency of solvent quality for a polymer distribution

with Mn = 17,000 g·mol-1 and Ɖ = 1.63, evaluated via the dry thickness method

(comparable to reference 180).


The results of the current thesis highlight the necessity of investigating processes

at the molecular level. Especially for processes in the field of surface functionalisation

with polymers a full understanding of the grafting process must be achieved. Further

investigations and application of discovered molecular effects (e.g. solvent interaction)

are crucial for the design of novel surface with unique surface properties. The surface

of any material is nothing less than the point of interaction in everyone’s daily life.

Therefore, it should be one of our highest aims to fully understand this crucial aspect

of matter.


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Appendices 89

Appendices

Appendix A

Statements of Contribution of Co-Authors for Thesis by Published Paper

The signed Statement of Contribution of Co-Authors for Thesis by Published Papers

for:

Chapter 2: L. Michalek, L. Barner, C. Barner-Kowollik, Polymer on Top:

Current Limits and Future Perspectives of Quantitatively Evaluating Surface Grafting,

Advanced Materials, 2018, 30, 1706321(1-18).

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Long and the Short of Polymer Grafting, Polymer Chemistry, 2019, 10, 54-59.

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Quantifying Solvent Effects on Polymer Surface Grafting, ACS Macro Letters, 2019,

8, 800-805.

90 Appendices


Appendices 91


92 Appendices


Appendices 93

Appendix B

Supporting Information Chapter 2

94 Appendices

Appendices 95

96 Appendices

Appendices 97

98 Appendices

Appendices 99

100 Appendices

Appendices 101

102 Appendices

Appendices 103

104 Appendices

Appendix C


Appendices 105

106 Appendices

Appendices 107

108 Appendices

Appendices 109

110 Appendices

Appendices 111

112 Appendices

Appendices 113

114 Appendices

Appendices 115

Appendix D


116 Appendices

Appendices 117

118 Appendices

Appendices 119

120 Appendices

Appendices 121

122 Appendices

Appendices 123

Appendix E

Calculations of MMDs and grafting densities for surface grafting via dry

thickness method.

For the simulations of the MMD in solution (comparable to Reference 180 with

Mn = 17,000 g·mol-1 and Ɖ = 1.63) a log-normal distribution with tailing to higher

molecular weight was used:

𝐼 =1

𝑥𝜎√2𝜋exp (−

(ln 𝑥−𝜇)

2𝜎2

2) with 𝜎 = 0.7 and 𝜇 = 9.5

Figure A1: MMD of polymer in solution (black curve) and polytmer on surface (red curve – for an

solvent interaction parameter of n* = 1)

The shifted surface distributions where calculated by multiplying the upper

distribution with κ from Equation 21. For different solvent interaction parameter

following grafting densities can be calculated via the dry thickness approach (Equation

8):

n* Mn [g·mol-1] σ [chains·nm-2]

1 10,457 0.36

0.9 10,982 0.35

0.8 11,533 0.33

0.7 12,113 0.31

0.6 12,721 0.30

0.5 13,360 0.28

0.4 14,031 0.27

0.3 14,735 0.26

0.2 15,475 0.25

0.1 16,252 0.23

1000 10000 100000

norm

. M

MD

Molar Mass / g mol-1

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Lukas Felix Michalek MSc Materials Science Felix_Michalek... · 2020-02-02 · QUANTIFYING MACROMOLECULAR GROWTH AND HIERARCHICAL STRUCTURING ON INTERFACES Lukas Felix Michalek MSc