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THE EFFECTS OF NORMAL LOAD ON HYDROGEL TRIBOLOGY
By
JUAN MANUEL URUEÑA VARGAS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2018
© 2018 Juan Manuel Urueña Vargas
To my parents
4
ACKNOWLEDGMENTS
I would like to thank my parents Amparo and Randolfo, and my sister Adriana for all
their unconditional love and support over the years. I am grateful to my adviser, mentor, and
friend Prof. W. Gregory Sawyer.
5
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ...............................................................................................................4
LIST OF TABLES ...........................................................................................................................7
LIST OF FIGURES .........................................................................................................................8
LIST OF ABBREVIATIONS ........................................................................................................10
ABSTRACT ...................................................................................................................................12
CHAPTER
1 INTRODUCTION ..................................................................................................................14
2 BACKGROUND AND PRIOR RESEARCH ........................................................................15
Lubrication Regimes ...............................................................................................................15
Biotribology ............................................................................................................................18
The Eye and Tear Film ...........................................................................................................19
Ocular Mucins ........................................................................................................................23
Contact Lenses ........................................................................................................................24
Hydrogels ................................................................................................................................26
3 TRIBOLOGICAL EXPERIMENTS WITH HYDROGELS ..................................................29
Friction Transition in Surface Hydrogel Layers .....................................................................29
Gemini Hydrogels. ..................................................................................................................31
Superlubricity in Soft Matter Interfaces .................................................................................37
4 NORMAL LOAD EFFECTS ON HYDROGEL TRIBOLOGY ...........................................40
Hydrogel Contact Mechanics .................................................................................................40
Hydrogel Indentations ............................................................................................................43
Normal Load Effects on Friction ............................................................................................43
5 CONCLUSIONS ....................................................................................................................49
APPENDIX
A SOMMERFELD NUMBER ...................................................................................................50
B HYDROGEL SYNTHESIS ....................................................................................................51
Poly(N-isopropylacrylamide) (pNIPAM) Hydrogels .............................................................51
Polyacrylamide Hydrogels ......................................................................................................51
6
Sample Preparation .................................................................................................................51
C TRIBOLOGICAL EXPERIMENTATIONS ..........................................................................53
D ELASTIC MODULUS CACLULATIONS OF HYDROGELS USING CONTACT
MECHANICS THEORIES ....................................................................................................55
E METHODS FOR MESH SIZE MEASUREMENTS .............................................................58
Small-angle X-ray Scattering .................................................................................................58
Microrheology ........................................................................................................................58
F SOFT EHL CALCULATION FOR HYDROGELS ..............................................................60
G JKR THEORY OF CONTACT MECHANICS .....................................................................61
LIST OF REFERENCES ...............................................................................................................64
BIOGRAPHICAL SKETCH .........................................................................................................71
7
LIST OF TABLES
Table page
B-1 Constituents for each of the samples in Urueña et al. AAm. .............................................52
8
LIST OF FIGURES
Figure page
2-1 Stribeck curve.. ..................................................................................................................15
2-2 Fluid film lubrication. ........................................................................................................17
2-3 The eye and tear film. ........................................................................................................20
2-4 The tear film. ......................................................................................................................21
2-5 Schematic of a soft contact inserted in the ocular system. ................................................22
2-6 Mucins genes found in the tear film.. ................................................................................24
2-7 Schematic of a blink during contact lens wear. .................................................................26
2-8 Polymer chains of a hydrogel network swollen in a good solvent. ...................................28
3-1 Lubricity of surface gel layers. ..........................................................................................30
3-2 Contact conditions in tribological testing. .........................................................................31
3-3 Friction comparison for all contact conditions. .................................................................32
3-4 Friction behavior of pNIPAM across four orders of magnitude in sliding speed.. ............33
3-5 Friction coefficient versus sliding speed for five different concentrations.. ......................34
3-6 Friction coefficients versus mesh size. ..............................................................................36
3-7 Universal curve that shows the friction behavior for all five polymer concentrations. .....37
3-8 Friction coefficient vs mesh size........................................................................................38
3-9 Superlubricity of hydrogels................................................................................................39
4-1 Schematic of a hemisphere indenting an elastic half-space. ..............................................43
4-2 Rotary microtribometer and representative cycle data. .....................................................44
4.3 Friction behavior of pAAm at different normal loads. ......................................................45
4-4 Friction behavior of pAAm at different normal loads. ......................................................46
4-4 Soft EHL calculations for friction coefficient and shear stress. ........................................48
C-1 Schematic of pin-on-disk microtribometer.. ......................................................................53
9
D-1 Contact area measurements using a confocal microscope. ................................................56
D-2 Normal force vs indentation depth at different dwell experiments. ...................................57
G-1 Variations of contact radius and load .................................................................................61
10
LIST OF ABBREVIATIONS
NOMENCLATURE
µ Friction Coefficient
Fn Normal Force
S Sommerfeld Number
E Elastic Modulus
Poisson’s Ratio
R’ Composite Radius
E’ Composite Elastic Modulus
T Thermal Energy
Π Osmotic Pressure
Solvent Viscosity
Mesh Size
kB Boltzmann’s Constant: 1.3806488x10-23 J/K
r Relaxation Time
E’ Composite Elastic Modulus
µo Coefficient of Friction in the Speed-independent Regime
ABBREVIATIONS
APS Ammonium Persulfate
BIS N,N’-Methylenebisacrylamide
EHL Elasto-hydrodynamic Lubrication
JKR Johnson-Kendall-Roberts Contact Mechanics Theory
LCST Lower Critical Solvation Temperature
11
MSD Mean Square Displacement
pAAm Polyacrylamide
SAXS Small-angle X-ray Scattering
TEMED Tetramethylethylenediamine
12
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE EFFECTS OF NORMAL LOAD ON HYDROGEL TRIBOLOGY
By
Juan Manuel Urueña Vargas
May 2018
Chair: W. Gregory Sawyer
Major: Mechanical Engineering
Hydrogels are excellent synthetic materials for biotribological studies for their high
degree of tunability that can be used to replicate the biological interface. The body is an exquisite
system capable of providing lubrication in a wide range of sliding speeds and contact pressures.
Recently high water-content hydrogels have been studied over a wide range of sliding speeds and
revealed a friction behavior unlike any other engineering system. Under a constant pressure the
friction behavior varied with sliding speed from high friction at the slowest speed, to very low
friction in the speed-independent regime, then increased again with increasing speed after a
transition point. This work aimed to determine the effects of contact pressure and contact area in
the friction behavior of these interfaces by changing the normal load two orders of magnitude
(0.1 to 20 mN) and the sliding speed four orders of magnitude (0.01 to 100 mm/s).
Microtribological experiments in ultrapure water were performed using a pin-on-disk tribometer
with a polyacrylamide hydrogel (> 90% water) under a Gemini configuration. The friction
coefficient decreased for all sliding speeds as the applied load was increased. This is consistent
with predictions of the contact area scaling non-linearly with the applied normal load and contact
pressure-independent surface shear stresses. Hydrogels under these conditions follow Hertzian
13
contact mechanics theory, supporting the scaling of friction with normal load to the negative 1/3
power in the speed-independent regime.
14
CHAPTER 1
INTRODUCTION
Tribology is the study of friction, lubrication, and wear at the interface of two materials
sliding in relative motion [1]. Tribology is an interdisciplinary field that exists at the intersection
of mechanics, materials science, physics, chemistry, interfacial engineering, and mechanical
engineering. It is simultaneously at the forefront of, and older than, science itself. While new
understandings of interfacial interactions are currently shaping the design of new materials, the
earliest attempts to reduce friction were performed many millennia ago and involved the
application of natural and widely available lubricants (e.g. water, sand, oils, fats, etc.) [2, 3].
While its utility was well known, the earliest recognition of the role of friction in mechanics was
not until the 4th century B.C. in the Mo Ching, which discussed resistance to motion [4]. Around
the same period, Aristotle was developing similar theories [5]. Several centuries later, Leonardo
da Vinci [6], through experimentation, recorded the first observations that would later become
the first laws of friction. Leonardo da Vinci concluded that (1) friction is independent of apparent
contact area, and (2) the resistance of friction is directly proportional to applied load. Roughly
two centuries later (1699), Amontons independently arrived at da Vinci's conclusions and
published the first laws of friction [6], which were later verified and expanded upon by Coulomb
[7]. Coulomb also introduced friction dependence on time in contact (time of repose) and on
sliding velocity [7]. The coefficient of friction is defined as the ratio of the friction force to the
applied normal force as shown in Equation 1-1.
µ =𝐹𝑓
𝐹𝑛 (1-1)
15
CHAPTER 2
BACKGROUND AND PRIOR RESEARCH
Lubrication Regimes
Friction between two surfaces is a dynamic process. When a fluid is present, friction
coefficients can vary by several orders of magnitude depending on film-thickening parameters
(sliding speed, viscosity, and inlet geometry) and the normal load or contact pressure, often
described by a dimensionless parameter called the Sommerfeld number. Successful
hydrodynamic lubrication is dependent upon a delicate balance between these parameters and is
characterized by very low friction coefficients and fully separated surfaces. Stribeck’s work on
journal bearings in the early 1900’s revealed multiple lubrication regimes (Figure 2-1).
Stribeck’s work showed how friction changes as the Sommerfeld number changes (Appendix A)
[8, 9].
Figure 2-1. Stribeck curve. The curve shows the friction regimes of two lubricated impermeable
surfaces in relative motion. A schematic of the interface when the two surfaces are
experiencing fluid film lubrication is shown. Adapted from Pitenis et al. [10].
16
The traditional way to illustrate the wide range of friction coefficients for most
engineering materials is the Stribeck curve. This curve is a plot of friction coefficient versus the
Sommerfeld number or versus one of its controlling parameters (Figure 2-1). From this curve,
clear transitions in frictional behavior can be observed. These are named based on the driving
mechanisms of friction: “dry” – no fluid is present at the sliding interface (high friction, asperity
contact) [11–13]; “boundary lubrication” –fluid films adsorb on the surfaces, thereby reducing
friction. It generally occurs at slow sliding speeds and high pressures; “mixed lubrication” – a
transition between boundary and fluid film lubrication; “elasto-hydrodynamic lubrication
(EHL)” – the thinnest fluid film lubrication in which the fluid pressures cause elastic
deformations; and “hydrodynamic lubrication” – a condition of full fluid film lubrication
(Figure. 2b). In this regime, friction forces are dominated by viscous drag of the shearing fluid at
the interface [9, 14, 15]. Fluid lubrication theory is only valid when the fluid films are thick
enough to completely separate the surfaces; typically, this occurs when the film is many times
thicker than the surface roughness. Although useful to predict the friction behavior of rigid,
impermeable materials, the Stribeck curve does not accurately model the lubrication regimes of
aqueous, soft, and permeable materials.
17
Figure 2-2. Fluid film lubrication. A) Schematic of a lubricated asperity contact showing a single
asperity in contact. B) Schematic of asperity contact for all lubrication regimes.
Adapted from Pitenis et al. [10].
18
Biotribology
Our bodies are composed of thousands of sliding interfaces [16] and their integrity is of
major importance, for its impact on our health and quality of life; inadequate lubrication can lead
to joint replacement. The study of direct contact, shearing stresses, and friction at the biological
interface can lead to better understanding of lubrication in biology and provide better tools for
the design of implantable devices in the body.
The word “Biotribology” was first introduced by Dowson and Wright (1973) to describe
the study of lubrication mechanisms in biological interfaces [2]. The early focus of biotribology
was the study of load-bearing natural sliding interfaces in the body, including articular joints and
cartilage, for the development of total joint replacements [17]. Today, there are tremendous
opportunities in biotribology to directly address the gap in the knowledge of physiological
conditions at the sliding interface between soft biomedical implants and devices and surrounding
natural surfaces- often, mucinated epithelial cell layers. In our bodies, epithelial cells line all
moist sliding interfaces [18] and are classified as squamous, columnar, or cuboidal epithelial
cells. Epithelial cells protect the natural sliding interface by (1) regulating the transport and
absorption of nutrients, (2) sensing stress, and (3) secreting mucins [19–33]. Implants may
increase the normal forces, contact pressures, and shear stresses against epithelial cells beyond
physiological norms and may cause unintended effects on soft biological surfaces. Informed
implant design to reduce contact pressures and shear stresses will undoubtedly improve patient
outcomes during medical implant use. An excellent system for biotribological studies is a soft
contact lens directly contacting and sliding against the cornea. Currently over 30 million people
use contact lenses in the United States, and a significant percentage of users discontinue contact
lens wear due to reported discomfort despite several decades of contact lens material
development and increasing sophistication in contact lens design [34]. While the exact etiology
19
of “discomfort” during contact lens wear is currently unknown, recent hypotheses have emerged
that lubricity may play a defining role [35]. There is therefore a great need to fully characterize
the lubrication of this synthetic/natural sliding interface, beginning with characterizing both sides
of the tribological pair: (1) the ocular environment and (2) the soft contact lens.
The Eye and Tear Film
The human eye is a sophisticated and highly complex organ that provides the ability to
process visual detail (Figure 2-3). Our eyes detect light and convert it into electro-chemical
impulses that are sent to the brain for processing. The cornea forms a dense, transparent,
connective tissue barrier that protects the organ. The cornea-eyelid interface is subjected to
10,000 - 20,000 of blinks per day (total sliding distance between 200 – 400 m) with contact
pressures ranging in magnitude from Pa to single kPa and sliding speeds ranging from ~10 µm/s
to ~100 mm/s [36]. The eye survives and provides clear vision, despite these demanding
conditions thanks to the tear film. [19, 20, 37]. Among its many functions, the tear film moistens
the surface of the cornea, transports proteinaceous waste away from the eye, and provides
protection against wind and airborne debris and contaminants. [38]. Perhaps its most crucial
function is the tear film’s ability to fully separate the surfaces of the eyelid and the cornea during
a blink and thus reduce shear stresses that could otherwise cause cellular damage [29, 38–50].
The tear film’s role of moistening the cornea has been known since the late 17th century
thanks to James Keill (1698) [44, 45]. It was not until Eugene Wolff over 200 years later that the
structure of the tear film was described: a trilaminar layer composed of mucus, water and lipid,
assembled in that order [46]. Our current understanding is that the tear film is a highly complex,
aqueous, gel-spanning network of mucins (~5 µm in thickness) with a waxy lipid layer (50-100
nm in thickness) at the air interface (Figure 2-4) [51–55].
20
Figure 2-3. The eye and tear film. During a blink the sliding speed of the upper eyelid is
approximately 100 mm/s. On the right is a schematic of the tear film during a blink.
Each blink serves as a way to re wet the surface of the cornea and provide a way to
get rid of debris, dust particles, pollen, and many small particulates. The velocity
profile and high viscosity zones of the tear film is shown. The thickness of the tear
film is 1-2 µm during a blink [56].
The tear film anchors to the epithelial cells thanks to specialized microvilli and
microplicae that form the ultrastructure of the squamous corneal epithelium. The microvilli
increase surface area and provide an anchor for the secreted components of these epithelia, such
as mucins, shown in Figure 2-4 [57]. Ocular mucins (Figure 2-6) contribute to the homeostasis of
the ocular surface, maintain the clarity of the cornea, and provide protection against foreign
debris while permitting the passage of selected gases, fluids, ions, and nutrients [19, 20, 23, 25,
29–33].
21
Figure 2-4. The tear film. Schematic of corneal epithelia, ultrastructure of corneal epithelial and
tear film and associated secretory and tear film mucins. (1a) Schematic of a vertical
section of human cornea epithelium showing Bowman’s membrane, corneal epithelia,
tear film. The tear film covers the entire ocular surface [58]. (Left) Schematic of the
tear film (~5 µm in thickness) and associated secretory and tear film mucins. The
lipid rafts (~50-100 nm in thickness) are believed to prevent evaporation of the tear
film [49, 59, 60]. (1a) Schematic of the ultrastructure of the corneal epithelium the
protrusions are called microvilli. The microvilli act to increase the surface area for
anchoring membrane-associated mucins. MUC20 is secreted and can be found
between the cells [61]. (1b) Schematic of the microvilli in more detail. (2)
Membrane-associated mucins (MUC1 [31], MUC4 [62–64], and MUC16 [65])
anchored the secreted and soluble mucins to form the glycocalyx [19].
For normal and healthy eyes, the tear film is capable of providing remarkably robust
lubrication. The insertion of a contact lens into the ocular system, however, completely disrupts
this delicate system (Figure 2-5). Firstly, a soft contact lens is roughly 20-50x thicker than the
tear film itself, which is contrary to the commonly held idea that the contact lens somehow
“floats” upon the tear film. Secondly, the contact lens creates two entirely different sliding
conditions in the eye: (1) between the eyelid and the contact lens, and (2) between the contact
lens and the cornea. The eyelid – contact lens sliding interface is subjected to a wide range of
sliding speeds (~100 µm/s to 100 mm/s), while the contact lens – cornea interface has much
22
lower sliding speeds (generally lower than 10 mm/s). These low speeds increase the probability
of direct contact against corneal epithelial cells. The corneal epithelium is the most innervated
surface of the entire body [66] - over 30 times more sensitive than tooth pulp, and over 300 times
more sensitive than skin [67]. Higher contact pressures due to the insertion of a contact lens
likely increase shear stresses on the cells of the corneal epithelium, which then may report pain
through the highly innervated network beneath them. The contact lens, which is a flexible
polymer network of aqueous gel, must therefore be designed to interface with both the eyelid and
the cornea through improved surface science with an eye towards lubricity under higher normal
loads and over a wide range of sliding speeds.
Figure 2-5. Schematic of a soft contact inserted in the ocular system. To scale the contact lens is
about 20 times bigger than the tear film. Contact pressures in the eyelid/contact lens
range from 10 – 20 kPa and in between the contact lens/ cornea 1-5 kPa. The sliding
speeds during a blink range from 0.1 to 100 mm/s between the eyelid and the contact
lens, while the motion of the lens against the cornea is much lower and ranges from
0.01 to 10 mm/s. A contact lens can be seen schematically represented as a flexible
polymer network with a mesh size on the order of nanometers.
23
Ocular Mucins
During contact lens wear, the soft hydrogel network of the contact lens directly interacts
with the tear film mucins. There are two types of mucin families: membrane-associated and
secreted mucins. Both types of mucin form the gel layer that maintains the hydration and
lubrication of the corneal surface shown in Figure 2-4 [51–54]. Table 2-1 lists all the mucins
found in the tear film.
Membrane-associated mucins (MUC1, MUC4, MUC16, and MUC20) are anchored to
the apical cell membrane before they are shed into the tear fluid. These mucins are believed to
interact with intracellular proteins. The act of shedding these mucins may induce changes in the
neighboring membrane domain and play a role in transferring signals and information to the cell
interior [68]. These mucins are generally lower molecular weight. MUC1, MUC4, and MUC16
are associated with the microvilli and microplicae of the corneal and conjunctival epithelium,
while MUC20 typically resides between epithelial cells in the epithelium.
The secreted mucins (MUC2, MUC5AC, MUC5B, MUC7, and MUC19) have been
found at the ocular surface and in the tear fluid. These mucins are high molecular weight
molecules with oligosaccharides and are gel-forming. Secreted mucins are responsible for the
rheological properties of the mucin layer which enables the eye to clear foreign bodies and
pathogens from the ocular surface and transport them into the nasolacrimal duct with each eye
blink [52, 54]. MUC5AC is a gel-forming mucin produced by goblet cells in the conjunctival
epithelium. The gel-forming mucins are not produced by the corneal epithelium.
The different molecular weights and functions of the tear film mucins likely do not
represent a redundancy, but instead a resiliency. The mesh size of the mucinated biopolymer gels
in the tear film have been estimated to be on the order of 100’s of nanometers [19]. Soft contact
24
lenses, which typically have mesh sizes on the order of 5 nm and less, must be designed to
interface with the natural tear film mucins on both the eyelid and the cornea side.
Figure 2-6. Mucins genes found in the tear film. Mucin genes and their associated human
genome nomenclature identification numbers (HGNC IDs) and their corresponding
location in the epithelium. There are two types of mucins: Membrane-associated
mucins (MUC20 [61], MUC1 [31], MUC4 [62–64], and MUC16 [65]) and secretory
mucins (gel forming: MUC2 [69, 70], MUC5AC [53, 64, 71], and MUC5B [72, 73];
soluble mucins originate in the lacrimal gland MUC7 [53, 72]).
Contact Lenses
Contact lenses have a long spanning history that covers about 500 years. Leonardo Da
Vinci was the first person to introduce the idea of a contact lens as a way to correct vision in the
Codex of The Eye, Manual D. Da Vinci in 1508 described changing the cornea power by
submerging a person’s head in a bowl full of water or wearing a glass hemisphere full of water
on top of the eye. In 1636 René Descartes described a hollow glass tube filled with water, several
years later (1801) Thomas Young developed on the idea to correct his own vision. In 1887, a
German glass blower, F. E. Muller, developed the first successful vision correction device that
could be tolerated in the eye. A year later, German ophthalmologist Adolf Gaston Eugen Fick,
developed the first successful fitting of a glass lens. Glass-blown sclera lenses then became the
only available technology until 1940, when a German optometrist Heinrich Wöhlk produced the
25
first plastic lens composed of poly(methyl methacrylate) (PMMA). This development cleared the
path for lenses to sit on the cornea and not on the sclera, making them more comfortable and
smaller. Although these lenses were more comfortable, they impeded oxygen diffusion into the
cornea leading to cornea edema. Softer and more comfortable contact lenses were developed
later in the 1970’s of poly(hydroxyethyl methacrylate) (pHEMA), but like PMMA, they also
suffered from low oxygen permeability. It was not until the development of silicone contact
lenses in the late 1990’s that oxygen permeability problems were largely resolved. Soft silicone
contact lenses proved to be popular, though surprisingly, the percentage of contact lens drop-out
(i.e., the number of people that discontinue contact lens wear) did not change, and most cited
end-of-day discomfort as the primary reason. This could be due to the fact that when a soft
contact lens is inserted within the corneal-eyelid wiper interface, it disrupts the ocular system
(Figure 2-5 and Figure 2-6). On one side, the contact lens is in contact with the eyelid wiper.
Here, the sliding speeds range from 0.1 – 100 mm/s and pressures range from 1 – 10 kPa. On the
other side, the contact lens is in contact with the cornea. Here, ocular motions move the contact
lens at much slower sliding speeds (0.01 – 10 mm/s) with contact pressures ranging from 1 – 5
kPa. This complex interface has been addressed in modern contact lens design by anchoring a
soft, highly aqueous surface gel layer on both the apical and bottom surfaces of the contact
lenses aiming to lower the contact pressures and provide improve lubricity by increasing water
content at the sliding interface.
Dunn et al. developed a model that took into account the contact pressures of the contact
lens – cornea eyelid wiper and cornea contact lens – cornea interfaces and corresponding sliding
speeds. Their model predicted hydrodynamic lubrication at the highest sliding speeds during a
26
blink and boundary lubrication on the contact lens cornea interface during slow ocular
movements.
Figure 2-7. Schematic of a blink during contact lens wear. The contact lens separates the
interface into two different lubrication zones. A) contact lens against eyelid wiper.
During a blink the eyelid wiper slides at about 0.1 – 100 mm/s with contact pressures
ranging from 1 – 10 kPa. and B) Contact lens against corneal epithelium. Ocular
motion moves the contact lens at speeds ranging from 0.01 – 10 mm/s with contact
pressures ranging from 1 – 5 kPa.
Hydrogels
Biological systems bring tremendous complexities to tribological measurements;
biological samples are alive, dynamic, sensitive to environmental conditions, suffer from sample
variability, and are fragile. While the eye is the most accessible sliding interface for
biotribological studies [33], the frictional dissipation mechanisms responsible for the durability
and lubricity of biological sliding interfaces are difficult to investigate in vivo. Biological
samples present challenges beyond their inherent complexity. The total volume of mucin in the
tear film is low, which makes collection and physical measurements difficult, if not impossible,
and there are significant experimental and handling challenges associated with studying natural
27
biopolymers extracted from the body [69]. Fundamental biotribological studies of these
interfaces can greatly benefit from synthetic materials analogous to natural biomaterials in
stiffness and water content but with high tunability, repeatability, and control of polymerization,
such as hydrogels [48, 74–77].
Hydrogels are three-dimensional, cross-linked networks of hydrophilic polymer chains
with a defined characteristic distance between polymer chains (mesh size, ξ), typically between
1-10 nm. Figure 2-7 shows a 2D representation of a hydrogel mesh. [78–80]. From a polymer
physics perspective, a hydrogel is a type of polymer network that exists in the semi-dilute
regime. In this regime, the concentration of polymer chains in a solution reaches a critical or
overlap concentration, c*, where the polymer chains begin to form a network of monomers and
cross-linkers whose mesh size (defined as the average distance between neighboring polymer
chains) is less than or equal to the radius of gyration. For hydrogels in the semi-dilute regime, the
mesh size is of the same order of magnitude as the average spacing between the chemical
crosslinks [78]. The mesh size can be measured in many ways, including by small-angle x-ray
scattering (SAXS), microrheology (Appendix E).
Hydrogels derive their mechanical and transport properties from the random motion of
the polymer mesh driven by thermal energy. Mesh size is the single parameter that controls both
the mechanical attributes of a hydrogel (e.g. elastic modulus and fluid permeability) and the
dynamic behavior of its individual polymer chains (e.g. polymer relaxation time). [80]. The
elastic modulus of flexible polymers scales with mesh size like an inverse cubed power, where
kBT is thermal energy and ξ is mesh size:
𝐸 =𝑘𝐵𝑇
𝜉3 (2-1)
28
The hydraulic permeability of fluids to flow through the hydrogel is driven by a pressure
gradient, and is determined by Equation 2-2, where, s is the kinematic viscosity of the solvent
(water):
𝑘 =𝜉2
𝜂𝑠 (2-2)
The polymer relaxation time of the hydrogel, r, is determined by:
𝜏𝑟 =𝜉3𝜂𝑠
𝑘𝐵𝑇 (2-3)
Figure 2-8. Polymer chains of a hydrogel network swollen in a good solvent. A) 2D schematic of
polymer chains in a solvent thermally fluctuating at the overlap concentration, c*.
The zero shear viscosity scales inversely with the mesh size. B) 2D schematic of a
semi-dilute flexible polymer network showing detail of polymer chains, chemical and
physical crosslinks, contour length, and chain free-ends. The mesh size is shown in
red dashed circles and is defined as the average distance between neighboring
polymer chains. Adapted from Urueña et al. [81]
29
CHAPTER 3
TRIBOLOGICAL EXPERIMENTS WITH HYDROGELS
Friction Transition in Surface Hydrogel Layers
Hydrogels are widely used as biomedical implants, like contact lenses, due to their high
water-content, optical transparency, oxygen permeability, and biocompatibility. The original soft
contact lenses contained roughly 40% water when hydrated, but most modern contact lenses
contain between 50-60% water with some containing over 70%. Contact lens design has
continually evolved in an effort to accommodate the physiological conditions (e.g. water content,
stiffness) in the cornea and tear film and to decrease the number of users that stop contact lens
wear due to end of day discomfort. Dunn et al. characterized a hydrogel contact lens that
featured a high water content, graded surface gel layer with a thickness of about 6 µm and an
elastic modulus of E = 25 ± 7 kPa [82].
The authors performed tribological experiments against these lenses using a rigid,
spherical borosilicate glass probe. Under physiologically-relevant contact pressures (between 1-8
kPa), the friction behavior of the surface hydrogel layers shows smooth sliding and a low
average friction coefficient of µ ~ 0.02. However, at higher contact pressures, a drastically
different friction behavior was observed (average friction coefficient of µ ~ 0.5) and the interface
exhibited clear indications of stick-slip (Figure 3-1e).
Dunn et al. theorized that at physiologically-relevant pressures, the surface gel layer of
these contact lenses supports the normal load and the polymer chains of the gel layer are allowed
to thermally fluctuate. However, at higher pressures, the surface gel layer collapses (Figure 3-
1c), thereby reducing the motion of the polymer chains and reducing the ability to support the
normal load, yielding much higher friction coefficients.
30
Figure 3-1. Lubricity of surface gel layers. A) Schematic of new generation soft hydrogel contact
lens depicting a graded surface gel layer. B) Schematic of hydrated surface layer at
low pressures. C) Friction loop at high pressures showing stick slip and high friction
coefficients. D) Schematic of collapsed hydrated surface gel layer due to high
pressures direct contact pressures. E) Friction loop at pressures lower than
physiologically relevant pressures. It shows smooth sliding and low friction
coefficients. Adapted from [82].
31
Gemini Hydrogels.
A fundamental understanding for friction at soft, aqueous sliding interfaces will likely not
be achieved until matched tribological pairs of soft materials are tested at physiologically-
relevant sliding conditions. In the body the sliding interfaces (e.g. the eyes, the digestive system,
and the articulating joints) are soft, permeable, are always found in biologically-similar
tribological pairs, and are lubricated and protected by a hydrophilic biopolymer network called
mucin. Historically tribological studies have typically been performed with probes with high
stiffness, low roughness, and impermeable. Hydrogels’ ease of polymerization enable
fundamental measurements of a matched sliding interface.
Figure 3-2. Contact conditions in tribological testing. Migrating contact is described as an
impermeable probe sliding on a hydrogel flat. Stationary contact is described as a
hydrogel probe sliding on an impermeable flat glass surface and finally, Gemini
contact is described as a self-mated interface where both probe and flat are made of a
hydrogel.
Dunn et al. tested all possible sliding configurations (Figure 3-2) and their effects on
hydrogel lubricity [48]. Dunn et al. described remarkable differences in polyacrylamide (pAAm)
lubrication behavior between stationary (soft-on-hard), migrating (hard-on-soft), and twinned,
self-mated, or “Gemini” (soft-on-soft) sliding interfaces (Figure 3-2). In migrating contacts, the
friction coefficient is speed-dependent and controlled by two rates: (1) the rate that the
impermeable probe indents the hydrogel and (2) the rate the impermeable probe slides to an un-
32
indented region. When the sliding speed increases, the indentation depth decreases and so does
the contact area consequently lowering the friction coefficient. In stationary contact, the friction
coefficient is time-dependent; under a constant normal load, the friction coefficient increases
monotonically over time at all sliding velocities (Figure 3-3C). In contrast, Gemini hydrogel
lubrication is neither time- nor speed- dependent (Figure 3-3A). Interestingly, the friction
coefficient of a Gemini contact remains low across an order of magnitude of sliding speeds.
Figure 3-3. Friction comparison for all contact conditions. A) Gemini contact shows no
dependence on time nor speed. B) migrating contact shows a speed dependence with
friction. C) stationary contact shows a time dependence with friction.
To determine the extent over which Gemini hydrogels’ friction coefficient remains
insensitive to sliding speed, Pitenis et al. constructed a lubrication curve spanning four orders of
magnitude in sliding speed using a pin-on-disk microtribometer (Figure H-1) on a poly(n-
isopropylacrylamide), (pNIPAM) probe and on a pNIPAM countersurface. Pitenis et al.
33
observed two distinct friction behaviors as the sliding speed increases (Figure 3-4) [10]. The
transition in pNIPAM Gemini hydrogel friction transition was postulated to arise from two
competing lubrication mechanisms. At slow sliding speeds, the friction coefficient was at its
lowest and driven by thermal fluctuations of the polymer chains at the hydrogel interface.
Thermal fluctuations of the polymer provide low friction because the shear strain induced by the
relative sliding of rapidly relaxing polymer chains does not accumulate (Figure 3-4).
Figure 3-4. Friction behavior of pNIPAM across four orders of magnitude in sliding speed.
Friction versus sliding speed of pNIPAM hydrogel at two different temperatures and
four orders of magnitude in sliding.
34
The concept of hydrogel lubrication being driven by the thermal fluctuations of polymer
chains at the sliding interface naturally led to the question of how lubrication could be affected
by restricting the motion of polymer chains, in particular by changing the mesh size. Since all
fundamental materials and transport properties are driven by the mesh size and the mesh size of a
polymer is dependent on the polymer concentration Urueña et al. synthesized five different
polymer concentrations of pAAm hydrogels to interrogate how the mesh size of hydrogels
affects friction. As described in Urueña et al., four orders of magnitude of sliding speed were
tested using a pin-on-disk tribometer for all five hydrogel compositions.
Figure 3-5. Friction coefficient versus sliding speed for five different concentrations: pAAm. The
solid lines emphasize the difference in friction behavior between the low (3.8, 7.5,
and 10.0 wt%) and high (12.5 and 17.5 wt%) concentrations. The dash lines represent
the average friction coefficient, µo, in the speed-independent regime for all hydrogel
concentrations.
Several trends emerged (Figure 3-5). First, hydrogels with the highest polymer
concentration (12.5 and 17.5 wt% pAAm) showed a nearly constant friction coefficient across
35
several orders of magnitude in sliding speed, suggesting a speed-independent friction regime.
The lower polymer concentrations (3.75-10 wt % pAAm) showed a constant friction coefficient
before it began to rise as the sliding speed increase. This zone is called the speed-independent
regime and is characterized by a low friction coefficient across a range of sliding speeds. In fact,
the average friction coefficient in the speed-independent regime called, µo, decreased as the
polymer concentration decreased. As the mesh size increased the friction coefficient decreased.
For these hydrogels (3.75-10 wt%) a friction transition emerged in which the friction coefficient
remained low at low sliding speeds and increased with νs0.5 above a transition speed, ν* (Figure
3-5). The lower polymer concentrations samples (3.75-10 wt % pAAm) follow the curve in
Figure 3-5, where µo, is the average friction coefficient in the speed independent regime, a is a
fitting coefficient, vs is the sliding speed and p is an exponent of ½. Equation. 3-1 represents the
friction behavior, µ, for the hydrogel compositions that exhibit a friction behavior transition.
µ = µ𝑜 + 𝑎𝑣𝑠𝑝 (3-1)
This friction transition correlates with the mesh size and relaxation time of these
hydrogels (Figure 3-6). Solving for the transition speed, ν*, gives:
𝑣∗ =𝜉𝜏𝑟⁄ (3-2)
Knowing that the relaxation time, τr, is given in Equation 2-3, thus, the transition speed is:
𝜈∗ =𝑘𝐵𝑇
𝜉2𝜂 (3-3)
The transition criterion occurs when the time to traverse a single polymer mesh at the interface,
kick time, tk, is faster than your relaxation time, τr. The transition criterion is the ratio of the
polymer relaxation time to the kick time.
𝜏𝑟
𝑡𝑘=
𝜉2𝜂𝑣
𝑘𝐵𝑇 (3-4)
36
The crossover point from low-speed to high-speed friction behavior is predicted by this criterion.
Friction is governed by polymer chain thermal fluctuation at low speeds due to negligible non-
Newtonian shear effects. At high speeds, non-Newtonian mechanics play a dominating role in
shearing across the sliding interface and the passing frequencies (kicks) of the surface chains
exceed the fluctuation frequencies, resulting in an accumulation of strains in the polymer mesh
and consequently higher friction coefficients. When the sliding speed from Figure 3-7 is rescaled
by v*, the resulting dimensionless group is given by Equation. 3-5:
𝜓 =𝜆𝜉2𝜂𝑣𝑠
𝑘𝐵𝑇 (3-5)
Where λ is a fitting coefficient of 40 used when collapsing the data.
Figure 3-6. Friction coefficients versus mesh size. Plot shows the drastic difference between a
low polymer concentration hydrogel with a mesh size of 10 nm with a high polymer
concentration hydrogel with a mesh size of ~1 nm. Plot shows that friction scales
with mesh size to the -1 power.
37
Figure 3-7. Universal curve that shows the friction behavior for all five polymer concentrations.
The dash line shows the normalized friction data scales with ½ with sliding speed.
When the friction coefficient is normalized by the average friction coefficient in the
speed-independent regime, µo, and plotted as a function of the dimensionless speed parameter,
ψ, all datasets collapse onto a single curve as shown in Figure 3-7. The origins of mesh size-
dependent friction are indicated in the scaling of µo with mesh size, ξ (Figure 3-7). In the speed-
independent regime for the lower polymer concentration samples, friction scales to the power -1
with mesh size.
Superlubricity in Soft Matter Interfaces
The discovery that the larger the mesh size, the lower the friction behavior in self-mated
Gemini hydrogels naturally provoked the question of how long this trend would be true for the
38
softest hydrogels (Figure 3-8) [83]. Large mesh size hydrogels (ξ >10 nm) are so soft that they
push the limits of what is considered to be a solid. To perform tribological experiments on such
soft materials and to overcome handling challenges, a relatively stiff hydrogel (composed of 7.5
wt.% pAAm, ξ =7 nm and E ~ 20 kPa) was polymerized while exposed to open air. The oxygen
inhibited polymerization at the air interface and created a large mesh size at the surface of a stiff,
easily handled hydrogel.
Tribological experiments were performed on a custom-built microtribometer (described
in Appendix C) at an applied load of 500 µN, a sliding speed of 200 µm/s revealed friction
coefficients between 0.001 < µ < 0.005 which fall within the regime of superlubricity (Figure 3-
9).
Figure 3-8. Friction coefficient vs mesh size. Five different polymer concentrations of
polyacrylamide (pAAm) were polymerized and equilibrated in ultrapure water for
39
over 24 hours. Friction coefficient scales to Fn-1. Adapted from Urueña et al. to show
the limit for superlubricity for high mesh size hydrogels [81].
Figure 3-9. Superlubricity of hydrogels. A) A representative friction loop for Gemini interface
with a normal load of 500 µN, track length of 800 µm and sliding speed of 200 µm/s.
The friction coefficient was calculated during free-sliding away from the reversals to
avoid errors associated with the change in direction. B) Friction coefficient vs mesh
size values from Urueña et al. with the addition of two data with different mesh sizes.
C) Friction loop showing the projection of the area of contact for one reversal. As the
probe moves away from the reversal (1) the probe moves away from its original
footprint (2) and (3). D) Histogram from friction coefficients of µ = 0.001-0.002
reported in the book Superlubricity [84].
40
CHAPTER 4
NORMAL LOAD EFFECTS ON HYDROGEL TRIBOLOGY
Normal load and contact pressure are frequently invoked as strong drivers of friction. In
previous lubrication models, softer, larger mesh size hydrogels actually retain water and drain
more slowly under applied pressure. Larger mesh size hydrogels tend to be lower friction [81],
which suggests that the larger mesh size hydrogels are more likely to be low friction and stay
low friction. This is a fascinating area of research as this is likely how biology solves the
aqueous lubrication challenge.
In 2005, Rennie et al. reported the first study on the effects of the applied normal load on
friction for hydrogels countersurfaces against a smooth spherical glass probe [85]. This work
reported a decrease in friction coefficient with an increase in normal load (μ ~ Fn-1/2). This
scaling originated from Winkler-like mechanics of contact area scaling like the square-root of the
applied load [85]. Work from Gong et al. on hydrogels under different applied normal loads and
chemical compositions strongly implicated surface chemistry in the origins of friction [76, 77,
86]. Work from the groups of Archard [87], de Gennes [78], Persson [88], Muser [89],
Israelachvili [90], Klein [91], and Salmeron [92] suggest a relatively simple and straightforward
hypothesis regarding the scaling of friction for gels: the friction coefficient will be proportional
to the real area of contact.
Hydrogel Contact Mechanics
Hydrogels are excellent candidate materials for interfacing with biological interfaces (e.g.
implants such as contact lenses, catheters, etc.) because of their tunability, rapid polymerization,
and repeatability. However, most of these biological interfaces are composed of mucinated,
moist epithelia, with large, open mesh networks with elastic moduli on the order of Pa.
Understanding the contact mechanics of these soft interfaces is critical for designing successful
41
biocompatible implants. Previous studies have clearly demonstrated the consequence of
exceeding the design constraints of soft hydrogel layers, which under high forces applied by a
glass probe can exhibit high friction and even stick-slip behavior [82]. Performing experiments
with a “twinned” (Gemini) hydrogel-hydrogel contact configuration may provide a better model
of the hydrogel implant’s tribological behavior in vivo under physiologically-relevant normal
loads, contact pressures, and sliding speeds in vitro.
Indentations experiments are useful for determining hydrogel mechanics. One of the first
Gemini hydrogel indentations was performed using pNIPAM [10]. These temperature-sensitive
hydrogels were of interest because of their ability to undergo reversible polymer chain collapse
at temperatures above the lower critical solvation temperature (LCST) (~32 °C). The unloading
portion of the indentation curve was fit to the JKR model (Appendix G) to determine the elastic
modulus, because of the non-negligible force of adhesion, Fadh, after the LCST transition. Below
the LCST the hydrogel displayed negligible adhesion, and an elastic modulus of E ~25 kPa. In
contrast, the indentation curve above the LCST exhibits adhesion of about 1 mN and an elastic
modulus of ~ 50 kPa. This experiment was limited by the lack of contact area measurements
during indentations.
There are many experimental challenges associated with imaging the contact area of
transparent hydrogels submerged in water with matching index of refraction. One way to reveal
the contact area during indentation measurements to image the contact using a confocal
microscope. Recent work by Lee et al. and Schulze et al. [93, 94] demonstrated the method of
using a confocal microscope to image the real area of contact and experimentally determined the
elastic modulus of hydrogels. Schulze et al. [94] determined that Hertzian contact mechanics
theory is effective for determining the stiffness of hydrogels, shown in Appendix D.
42
Recent work by Schulze et al. [94] has revealed that hydrogels follow Hertzian contact
mechanics models over short and long timescales. From the classical Hertzian contact mechanics
theory for a sphere-on-flat contact (Figure 4-1), the applied normal load, Fn, scales with
penetration depth, d, and contact area, A, like 𝐹n~𝑑3/2 ~𝐴3/2. This is because the contact radius, a,
scales with penetration, d, like 𝑎~𝑑1/2, or 𝑎2~𝑑, and therefore contact area scales with normal
load like 𝐴~𝐹2/3 [95]. For a spherical hydrogel probe in contact with a flat, thick hydrogel
countersample of identical polymer composition and water content, the area of contact scales like
𝐴~𝐹2/3 from Equation 4-1 below, where R’ is the composite radius of curvature and E’Hertz is the
composite elastic modulus from Hertzian theory:
𝐴 = (3𝐹𝑛𝑅′
4𝐸′𝐻𝑒𝑡𝑧)
23⁄
(4-1)
The friction coefficient, μ, is the ratio of the friction force, Ff, and the applied normal
load, Fn. Assuming that frictional dissipation is linked to a shear stress, τ, at the surface and is
independent of pressure, Fn/A, the friction coefficient can be described in Equation 4-2.
µ =𝜏𝐴
𝐹𝑛 (4-2)
The area of contact scales with applied normal force as 𝐴~𝐹2/3 and using Equation 4-2,
one solves for the friction coefficient, Equation 4-3, and after simplifying, Equation 4-4.
µ =𝜏𝐹𝑛
23⁄
𝐹𝑛 (4-3)
µ = 𝜏𝐹𝑛−1
3⁄ (4-4)
Hydrogels are aqueous and optically transparent materials. In Gemini contact, the
average surface roughness is approximately on the order of their mesh size; thus, the real contact
43
area is approximately equal to the nominal area determined from Hertzian contact analysis.
Therefore, the expected scaling of friction coefficient with applied load is to the -1/3 power.
Figure 4-1. Schematic of a hemisphere indenting an elastic half-space. Where Fn is the applied
normal force, R1 is the radius of curvature of the probe, 2a is the contact diameter, d
is penetration depth, and R2 is the radius of curvature of the elastic half-space.
Hydrogel Indentations
Hydrogels present experimental challenges when trying to determine the contact area,
because they are optically transparent and index-matched when submerged in water. Schulze et
al. and Lee et al. [93, 94] overcame this challenge by polymerizing these hydrogels with
fluorescent particles and imaging them under contact in a confocal microscope. The area of
contact was measured by processing the image stack (Appendix D). These experiments found the
behavior of hydrogels to follow Hertzian mechanics as long as the contact pressure did not
exceed the osmotic pressure.
Normal Load Effects on Friction
To evaluate how normal load affects the friction behavior of Gemini hydrogel
lubrication, eight different experiments were performed by changing the applied load over two
orders of magnitude (100 µN to 20 mN) and the sliding speed over four orders of magnitude
(0.01 to 100 mm/s). These experiments were performed in a Gemini configuration completely
submerged in ultrapure water. These hydrogels (7.5 wt% pAAm) were polymerized as described
in Urueña et al. [81] with a mesh size of = 7 nm.
These experiments were performed as described in Appendix C. Figure 4-2 shows a
schematic of the hydrogel disk and hydrogel gel probe in contact. The hydrogel disk reciprocated
44
over 46º resulting in a track length of 8 mm per reversal and 16 mm per reciprocating cycle. The
friction coefficient was calculated as described in [96]. A representative cycle is shown in Figure
4-2 and an inset of the friction force which is two orders of magnitude lower than the normal
force.
Figure 4-2. Rotary microtribometer and representative cycle data. A) Illustration of the Gemini
hydrogel configuration. The hydrogel probe (2 mm radius of curvature) is mounted
onto a double leaf cantilever and loaded against an oscillating hydrogel disk (40 mm
in diameter and about 5 mm in thickness). The sliding path is represented by the dark
line from 0 º to 46 º and back to 0 º for one cycle. B) Normal and friction forces in
mN for a single sliding cycle. C) Friction force loop for the plot in B), which is two
orders of magnitude lower than the normal force yielding a friction coefficient of µ ~
0.01. The free sliding region is highlighted in black and represents the region that
over where the friction coefficient is calculated.
Figure 4-3 shows average friction coefficient for all eight of the normal loads tested.
Three clear trends can be seen in these data. First, for each of the normal loads (0.1 to 20 mN), a
speed-dependent friction coefficient behavior is in agreement as seen in previous investigations
[10, 81]. Second, a speed-independent regime (v ~ 0.1 to 5 mm/s) was identified in these data
45
(Figure 4-4) and agreed with findings in the literature [10, 48, 81] and third, the friction
coefficient drops for all normal loads and across the whole range in sliding speed, as would be
predicted from considerations of contact area scaling non-linearly with applied load.
To evaluate the scaling of the friction behavior across all normal loads and to eliminate
the influence of sliding speed the average friction coefficient at the speed-independent regime
was used (Figure 4-5).
Figure 4.3. Friction behavior of pAAm at different normal loads. Average friction coefficient, μ,
versus sliding speed for eight different normal loads from 0.1 to 20 mN. The average
normal forces and respective standard deviations over each experiment are reported in
the adjacent legend. It appears that the friction coefficient decreases with increase in
normal force for each corresponding sliding speed. In addition, the friction behavior
of each normal force displays three distinct lubrication regimes from the lowest to
highest sliding speeds tested. At the lowest speeds, the friction coefficient is initially
high, then falls with increasing speed to a minimum in the speed-independent regime
(generally ~0.1 to ~5 mm/s) and rises again with increasing sliding speed. The error
bars represent the standard deviation over the 20 reciprocating cycles per experiment.
46
Figure 4-4. Friction behavior of pAAm at different normal loads. The average friction coefficient
in the speed-independent regime, μo, plotted against normal force, Fn, scales like a -
1/3 power. The error bars represent the standard deviation in the speed-independent
regime.
The recent finding that high water content hydrogels can be described by Hertzian
contact mechanics is consistent with the discovery in this document that the scaling of friction
coefficient in the speed-independent regime follows µ ~ Fn -1/3. It is interesting that in the speed-
independent regime, the shear stress is constant because both the contact area and the friction
coefficient do not change. The shear stress was calculated to be = 18.4 3.5 Pa. The energy
dissipation mechanism in hydrogels may be linked to shear stress, which could originate from
the solvent shearing within the hydrodynamic penetration zone at the surface. The hydrodynamic
penetration zone is defined as the effective depth into the hydrogel that solvent is shearing as a
result of a fluid shear at the surface. In fact, Milner [97] predicted the hydrodynamic penetration
to be in the order of the mesh size. But this assumption fails to predict a speed-independent
47
regime, in fact, it predicts an increase of shear stress as the speed increases. Using a hydrogel
with a mesh size = 7 nm and the viscosity of water, the shear rate ranges from 104 to 106 s-1
with shear stress = 10 to 1,000 Pa. This discrepancy might result from the possible existence of
a fluid film under the contact that is much bigger than the hydrodynamic penetration and the
shearing is occurring in this fluid film. To test this hypothesis, soft elasto-hydrodynamic
lubrication (EHL) models of Hamrock and Dowson [98] were used to calculate the fluid film
thickness (Appendix F) and calculate the friction coefficient and shear stress as a function of
applied load and sliding speed. Figure 4-4 shows the weak relation of normal load and friction
coefficient as seen by how closely packed the lines are for the predicted friction coefficients
(dark lines), in fact, EHL fails to predict the friction coefficient at slower speeds (v ~ 0.01to 1
mm/s) and the friction at higher speeds (v ~ 1 to 100 mm/s) is much higher than predicted by soft
EHL calculations. When comparing the calculated and measured shear stress, it also fails to
predict the measured values. Soft EHL predicts the shear stress to monotonically increase with
sliding speed, in contrast to the measured values which showed a collapse of the data over the
whole range of sliding speeds. The models and experiments combined suggest that friction is
lower than predictions from fluid shear within a single polymer mesh, and yet greater than
predictions from soft EHL theory.
48
Figure 4-4. Soft EHL calculations for friction coefficient and shear stress. A) The average
friction coefficient, μ, plotted versus sliding speed. Solid lines are the theoretical
predictions from soft EHL theory at the corresponding loads and speeds. B) Shear
stress, τ, as predicted assuming Hertzian contact mechanics and uniform shear stress
is plotted versus sliding speed for eight different normal loads from 0.1 to 20 mN.
Solid lines again are based on the soft EHL theory under the corresponding loads and
speed.
49
CHAPTER 5
CONCLUSIONS
The design of soft, robust, and comfortable biomedical implants and devices that
interface with, as opposed to interrupt, the natural biological surfaces in the body is an important
area of research. The ability to provide a fundamental understanding of the effects of normal load
on the lubrication of soft aqueous gels over a physiological range of sliding speeds is a first step
towards informed implant design. In this study, polyacrylamide hydrogel samples with over 90%
water content were slid in self-mated (“Gemini”) contact over eight different normal loads (0.1
to 20 mN) and thirteen different sliding speeds (0.01 to 100 mm/s). The friction coefficient
decreased with increasing normal load across all sliding speeds. The lubrication curves appeared
to have three distinct regimes: (1) from 0.01 to 0.1 mm/s, the friction coefficient was initially
high and decreased to (2) speed-independent friction regime from about 0.1 to 5 mm/s before (3)
increasing again until 100 mm/s. In the speed-independent friction regime, the average friction
coefficient, called µo, scaled with the normal force, Fn, like µo ~ Fn-1/3. This is in strong
agreement with previous studies that suggested that these same materials follow Hertzian contact
mechanics. Efforts to characterize the lubrication behavior with soft elastohydrodynamic
lubrication theory failed to predict the shear stress, indicating that isoviscous fluid shear is not
the likely mechanism of energy dissipation for Gemini hydrogels.
50
APPENDIX A
SOMMERFELD NUMBER
The Sommerfeld number is defined as:
𝑆 = (𝑟
𝑐)2 µ𝑁
𝑃 (A-1)
Where S is the bearing-characteristic number, r is the journal radius, c is the radial
clearance, µ is the absolute viscosity, N is the relative speed between the journal bearing,
P is the load per unit of projected bearing area [8].
51
APPENDIX B
HYDROGEL SYNTHESIS
There are three main pathways to hydrogel synthesis: (1) physical interactions (e.g.,
entanglements, electrostatics, and crystallite formation), (2) linking polymer chains through
chemical reactions, and (3) using ionizing radiation to produce main-chain free radicals which
can recombine as crosslinking junctions. [61].
All hydrogels in this work were synthesized by polymerizing a monomer (acrylamide
(AAm) or N-isopropylacrylamide (NIPAM)) in the presence of small amounts of a bifunctional
crosslinker, typically methylenebisacrylamide (MBAm). Polymerization began with the
production of free radicals by adding ammonium persulfate (APS) in the presence of the
tetramethylethylenediaminine (TEMED). Aliquots for each of the constituents were prepare in
large volumes to limit the errors associated with weight measurements with a resolution of 1 mg.
Poly(N-isopropylacrylamide) (pNIPAM) Hydrogels
Hydrogel samples of pNIPAM used in Pitenis et al. [63] were polymerized using the
following components, reported as mass per total mass of the solution: NIPAM (7.5 wt.%),
MBAm (0.3 wt.%), APS (0.6 wt.%), TEMED (0.06 wt.%) in ultrapure water and in an oxygen-
starved environment.
Polyacrylamide Hydrogels
Polyacrylamide hydrogel samples used in Urueña et al. [81] were polymerized in five
different polymer concentrations as shown in Table 2. The polymerized samples were
equilibrated in ultrapure water for ~ 40 hours prior to tribological experiments.
Sample Preparation
The Gemini hydrogel interface is composed of a hydrogel probe against a flat hydrogel
disks sliding against each other. The hydrogel probes were made by casting them in a diamond-
52
turned polyolefin mold with a roughness of Ra = 10 nm and a radius of curvature of ~2 mm
around a 4-40 set screw. The hydrogel disks were cast in 60 mm polystyrene dishes. The lid of
the dish was used as the bottom countersurface and the bottom dish was used at the top surface
of the hydrogel disk. After polymerization the disk was reduced in diameter to ~45 mm with a
thickness greater than 4 mm to eliminate sub-surface shear effects.
Table B-1. Constituents for each of the samples in Urueña et al. AAm: acrylamide monomer,
MBAm: N'N'-methylenebisacrylamide crosslinker, TEMED:
tetramethylenthylenediamine catalyst, APS: ammonium persulfate initiator with the
remember of the solution to the ultrapure water.
53
APPENDIX C
TRIBOLOGICAL EXPERIMENTATIONS
Tribological experiments on hydrogels present unique experimental challenges, because
these materials are very soft, aqueous, and difficult to handle. To overcome these challenges,
tribological experiments using Gemini hydrogel configurations were performed on a custom-
built high-speed pin-on-disk microtribometer (Figure C-1) [10, 81]. The hydrogel disk was
attached to a polystyrene dish and fixed to a piezoelectric rotary stage (Physik Instrumente M-
660.55 (4 μrad resolution), capable of speeds from 100 nm/s to 100 mm/s. The hydrogel probe
was threaded to the dual flexure titanium cantilever (161 μN/μm and 75 μN/μm normal and
tangential stiffness, respectively). The probe was brought into contact with the hydrogel disk
using a micrometer z-stage.
Figure C-1. Schematic of pin-on-disk microtribometer. The Gemini hydrogel configuration
consists of a hydrogel probe (4 mm diameter, 2 mm radius of curvature) mounted to a
cantilever, slid against a rotating hydrogel disk, that is attached to a petri dish and
filled with enough water to fully submerged both the hydrogel dish and hydrogel
probe. Capacitance sensors measure the deflections of the cantilever and output
normal (Fn) and friction (Ff) forces. The uncertainty of the force measurements is ~ 2
µN for the normal force and ~ 1 µN for the tangential force.
54
The hydrogel probe is brought into contact to the hydrogel disk with a 2 mN normal load
using a vertical positioning micrometer stage. Once the desired load was reached the sample was
allowed to equilibrate at that load and adjusted for any deviations in load. The experiment was
started when the system showed no fluctuations in the normal load due to handling of the sample
and the positioning stage. To reduce normal force variations associated with misalignments, the
hydrogel disk was leveled normal to the axis of rotation by adjusting the disk holder. The normal
and tangential forces acting on the interface were measured with capacitance sensors. Each
sensor has a sensitivity of 5 μm/V and a range of 20 V. The friction coefficient (μ) was computed
using the measured normal (Fn) and friction (Ff) forces. To limit evaporation the petri dish holder
was machined to be able to hold a polystyrene lid with a hole where the hydrogel probe can fit
through for easy access to the hydrogel disk. The pin-on-disk microtribometer experimental
configuration (capacitance probes and cantilever) produces a friction coefficient noise floor of μ
= 0.002 and uncertainty in friction measurements were calculated following the methods in
Schmitz et al. [99].
55
APPENDIX D
ELASTIC MODULUS CACLULATIONS OF HYDROGELS USING CONTACT
MECHANICS THEORIES
The real area of contact is paramount to compute the elastic modulus of the hydrogels
used in this work using classical contact mechanics equations. To measure the contact area
during indentation experiments, the hydrogel sample is synthesized with a 4% solution of 20 nm
fluorescent polystyrene beads to allow the visualization of the real area of contact. Indentations
are performed with a borosilicate glass sphere attached to the cantilever. The tribometer was
mounted to an inverted confocal microscope. Slow indentations were performed to eliminate any
strain rate effects (v = 1 μm/s) with a maximum normal load of 2 mN (Figure D-1).
Contact area measurements are relatively easy in the confocal. The X and Y resolution is
much higher than the Z resolution allowing more accurate measurements of the contact radius.
The contact radius is determined from the point where the hydrogel surface diverges from the
known indenter geometry Figure D-1.
Hertzian contact mechanics has geometrical considerations for determining the contact
half width, a, when an ideal sphere indents an infinite half space, by penetration depth, d, and
composite radius, R’ as given in Equation D-1:
𝑑 =𝑅′
𝑎2 (D-1)
The composite radius is given as the following equation. Where R1 and R2 are the radii
of the probe and the countersurface respectively.
1
𝑅′=
1
𝑅1+
1
𝑅2 (D-2)
The composite elastic modulus can now be calculated using the following equation:
𝐸′ =4𝑅
12⁄ 𝑑
32⁄
3𝐹 (D-3)
where E*, is the composite elastic modulus and F, is the normal force.
56
Finally, the elastic modulus is calculated using the following equation:
1
𝐸′=
1
𝐸1+
1
𝐸2 (D-4)
Figure D-1. Contact area measurements using a confocal microscope. A tribometer was mounted
onto the turret of the confocal microscope. A sapphire probe with a 1.6mm radius of
curvature was coated in F127 Pluronic and brought into contact with a 7.5 wt%
polyacrylamide (pAAm) hydrogel (1 mm in thickness) while a z stack revealed the
area of contact. The hydrogel and probe were submerged in ultrapure water 18.2 M.
a) The processed imaged is an azimuthal average of the confocal stack, showing the
deformation of the hydrogel surface at normal loads of 500, 1000, 2000, 3000 μN.
The un-deformed surface is noted by the horizontal dashed line, and the profile of the
sapphire probe is represented by the dotted-and-dashed line. b) Schematic of a
hemispherical probe of radius R1 indenting an elastic half-space with a normal force,
Fn. The red region of interest shows the contact half width, a. c) A surface profile
ratio is computed by dividing the known hemispherical indenter profile by the
indented gel surface profile. The edge of contact is determined from the point where
the profile ratio rapidly rises [94]
57
Figure D-2. Normal force vs indentation depth at different dwell experiments. The penetration
depth follows a Hertzian like behavior by scaling to the 3/2 power with normal load.
The composite modulus for all these experiments averaged E* = 29 kPa. The dashed
line (red) represents the fit to the all the data.
58
APPENDIX E
METHODS FOR MESH SIZE MEASUREMENTS
Small-angle X-ray Scattering
Polymer conformations were studied by small-angle X-ray scattering (SAXS). SAXS is a
non-destructive method based on wave-diffraction phenomena that can quantify the structural
details of the sample to determine the mesh size. To measure the mesh size an X-ray beam
elastically interacts with the sample and scatters onto a detector. The 2D scattering data is
integrated and fitted with the Lorentzian of the form 𝑆(𝑞) = Γ 4𝑞2 + Γ2 [78]. Knowing that the
mesh size is 𝜉 = 2 Γ, mesh size can be determined.
Microrheology
Classical particle tracking is a micro-rheological technique that allows simultaneous
tracking of several micrometer-sized particles using time-lapse microscopy [100]. This type of
microscopy records images of micro-sphere undergoing Brownian motion at a known frame rate.
These frames are analyzed using an image processing code which provides the individual
trajectories of individual particles. Some particles may move out of focus, which ends their
tracking, and return back into focus which begins a new trajectory. The individual trajectories of
the particles are used to calculate the mean square displacement (MSD) using the following
equation:
𝑀𝑆𝐷(𝜏) = 2𝑑𝐷𝜏 = ⟨(𝑥(𝑡 + 𝜏) − 𝑥(𝑡))⟩ (E-1)
Where a Brownian particle’s trajectory x(t) is parameterized by D, its self-diffusion
coefficient, and d is the number of dimensions of trajectory data for τ, time. A thermodynamic
average over many starting times, t, for a single particle or over many particles for an ensemble
is indicated by angle brackets.
The mesh size is calculated from the MSD of each particles using the following equation:
59
𝜉 = (𝜋𝑎(Δ𝑟)2)12⁄ (E-2)
Where the is the mesh size and a is the particle radius.
60
APPENDIX F
SOFT EHL CALCULATION FOR HYDROGELS
The fluid film thickness was calculated using the Soft EHL calculations from Hamrock
and Dowson [98] and recast in Rennie et al. [85] as the following expression
ℎ ≅ ℎ𝑚𝑖𝑛 = 2.8𝑅′0.77(𝜂𝑜𝑉)0.65𝐸′−0.44𝐹𝑛
−0.21 (F-1)
From Equation F-1, h can be solved for a range of normal loads, Fn = 0.1 - 20 mN, over a range
of sliding velocity, V = 0.01 - 100 mm/s. Solvent (water) viscosity is assumed to be η = 8.9 x 10-4
Pa-s, the composite radius of curvature is R’ = 2 mm, the composite modulus of elasticity is E’ =
26.67 kPa (from a matched hydrogel interface, each with an elastic modulus of E = 20 kPa).
Knowing the fluid film thickness, the shear stress, τ, can be solved using the following equation,
and plotted versus sliding speed in the figure below:
𝜏 =𝑉𝜂
ℎ (F-2)
The area of contact can be solved using Hertz equations:
𝐴 = 𝜋 (3𝐹𝑛𝑅′
4𝐸′𝐻𝑒𝑟𝑡𝑧) (F-3)
Where E’ is the composite modulus based on the Hertzian equation below, assuming the
Poisson’s ratio to be = 0.5 and E = 20 kPa.
𝐸′𝐻𝑒𝑟𝑡𝑧 =𝐸′
2(1 − 𝜈2) (F-4)
Finally, the friction coefficient, µ, is calculated using Equation F-5 and the calculated shear
stress from Equation F-2.
µ =𝜏𝐴
𝐹𝑛 (F-5)
61
APPENDIX G
JKR THEORY OF CONTACT MECHANICS
Hertzian contact mechanics assumes that there are no adhesive forces when two bodies
are in contact. For this reason Johnson-Kendall-Roberts (JKR) developed a contact mechanics
theory to incorporate the effects of adhesion [101] when two non-conforming surfaces are in
contact.
Figure G-1. Variations of contact radius and load between adhesive and non-adhesive contact.
During an indentation the adhesive forces pull the surfaces into contact the resulting
area of contact is much bigger than that predicted by Hertzian mechanics. When the
normal load is zero the area of contact is given by point C. As one keeps unloading
the sample the contact radius keeps shrinking until one reaches point B. At point A
the adhesive contact breaks.
The indentation of an elastic solid is shown in Figure G-1 must satisfy Equation G-1 for a
normal displacement produce by a described normal load.
𝑤1 + 𝑤2 = 𝛿 − 𝑟22𝑅⁄ (G-1)
Where w1 is the z-displacement of the probe surface and w2 is the z-displacement of the surface
of the elastic-half space, is the z-axis displacement, and R is the composite radius of curvature
given by (1/R) = (1/R1 + 1/R2).
The general solution for the pressure distribution is described in Equation G-2
62
𝑝(𝑟) = 𝑝𝑜 (1 −𝑟2
𝑎2⁄ )
12⁄
+ 𝑝𝑜′ (1 − 𝑟2
𝑎2⁄ )
12⁄
(G-2)
Where po = 2aE*/R. In the presence of adhesion, however, a negative value of po’ cannot be
omitted.
The total free energy, UT o the system is expressed by Equation G-3:
𝑈𝑇 = 𝑈𝐸 + 𝑈𝑆 (G-3)
Where UE is the stored elastic strain energy and US is the surface energy due to adhesive forces.
At equilibrium
[𝜕𝑈𝐸𝜕𝑎
]𝛿=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
= 0 (G-4)
[𝜕𝑈𝐸𝜕𝑎
]𝛿= −[
𝜕𝑈𝑆𝜕𝑎
]𝛿
(G-5)
The elastic strain energy stored in the system can be calculating using Equation (G-6):
𝑈𝐸 =𝜋2𝑎3
𝐸∗(2
15𝑝𝑜2 +
2
3𝑝𝑜𝑝𝑜
′ + 𝑝𝑜′ 2) (G-6)
and the deformation, can by calculating using Equation (G-7)
𝛿 = (𝜋𝑎 2𝐸∗⁄ )(𝑝𝑜 + 2𝑝𝑜′ ) (G-7)
Using Equation G-8
[𝜕𝑈𝐸𝜕𝑎
]𝛿=𝜋2𝑎2
𝐸∗𝑝𝑜′ 2 (G-8)
Considering adhesion forces a surface energy term is introduce US. US increases when the
two surfaces separate and decreases when the two surfaces come into contact. This can be
written as:
𝑈𝑆 = −2𝛾𝜋𝑎2 (G-9)
Where is the surface area per unit area for each of the two surfaces.
63
When in equilibrium:
𝜋2𝑎2
𝐸∗𝑝𝑜′ 2 = −4𝜋Υ𝑎 (G-10)
The first term is the strain energy while the other term is the change in energy due to adhesion;
solving for po’:
𝑝𝑜′ = −(
4𝛾𝐸∗
𝜋𝑎⁄ )12⁄
(G-11)
Therefore, the contact force is given by:
𝐿 = 16𝜋𝛾𝐸∗𝑎3 (G-12)
Rewriting it:
𝑎3 =3𝑅
4𝐸∗(𝐿 + 3𝜋𝛾𝑅 + √6𝜋𝑅𝛾𝐿 + (3𝜋𝑅𝛾)2) (G-13)
64
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71
BIOGRAPHICAL SKETCH
Juan Manuel was born and raised in Bogotá, Colombia. He immigrated to the United States
in 1999. He attended the University of Florida to pursue studies in mechanical engineering. Driven
by a fascination of experimentation, he joined the University of Florida Tribology Laboratory under
the mentorship of Prof. Sawyer and dove into the soft and slippery world of biotribology.
