DROPLET C PROTEIN (COLLAGEN AS · 1.3 viscosity ... figure 1.3: schematic of collagen type i...

SURFACE VISCOMETRY OF AN EVAPORATING

DROPLET CONTAINING A PROTEIN (COLLAGEN) AS

A FUNCTION OF TIME AND DEPTH

A Thesis Presented

By

Seyed Mohammad Siadat

to

The Department of Mechanical & Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in the field of

Mechanical Engineering

Northeastern University

Boston, Massachusetts

August 2015

i

ABSTRACT

There is a need for tissue engineers to recreate extracellular matrix that mimics the highly

organized extracellular structures as seen in vivo. In a previous study in the extracellular

matrix engineering research laboratory (EMERL), a micromechanical system was used to

create such structures by drawing fibers from a droplet of neutralized collagen monomers

at room temperature. For further investigation in the formation of highly aligned and

continuous fibers, the laboratory is interested in developing a more effective experimental

procedure. Therefore, to supply the proper concentration of collagen monomers for a

collagen fiber printing device, a good estimate of the collagen concentration in the

droplet surface is required. The goal of this study was to measure the concentration

variation as a function of thickness in the dense layer on the droplet’s top surface when a

fiber can be created.

To measure the concentration, we need to know the viscosity of the droplet under the

same experimental conditions. The viscosity of the droplet was estimated from the

measured velocity of magnetic microspheres distributed in the collagen solution. To

accelerate the microspheres the droplet was placed in the uniform region of a magnetic

field produced by a permanent magnet. Magnetic microspheres travelled in the direction

of the magnetic lines after quickly achieving with a constant velocity, which is related to

the viscosity based on Stokes Law.

The average relative velocities in the direction of the magnetic field of these

microspheres were measured using a custom MATLAB tracking algorithm at a depth of

ii

20, 40, 60, and 1000 µm below the droplet surface. The concentration of the collagen was

predicted based on a calibration curve relating the collagen concentration to the viscosity.

The initial solution was made at 4.4 mg/ml of collagen monomers. After 150 seconds, the

concentration inside the droplet (1000 µm below the surface) increased to 4.6 mg/ml,

while the surface concentration spiked to 14 mg/ml. As expected, the concentration

gradient is nonlinear from the droplet center to the surface. Between the surface to 20 µm

below the surface, the collagen monomer concentration dropped from 14 mg/ml to 8

mg/ml. Therefore, the layer of dense collagen solution is limited to less than 20 µm below

the surface.

The result of this study gave an important information in the critical surface

concentration where collagen fibers can be formed, which can lead to the design of a

more efficient and predictable methodology to produce highly organized collagen fibers.

iii

CONTENTS

1 INTRODUCTION AND BACKGROUND................................................................ 1

1.1 COLLAGEN ............................................................................................................... 1

1.2 COLLAGEN FIBER PRINTING FROM THE SURFACE OF A DROPLET OF COLLAGEN

MONOMERS.................................................................................................................... 5

1.3 VISCOSITY .............................................................................................................. 10

1.4 VISCOSITY MEASUREMENT .................................................................................... 12

1.4.1 Capillary Viscometers ...................................................................................... 13

1.4.2 Falling-Body Viscometer ................................................................................. 15

1.4.3 Rotational Viscometer ...................................................................................... 17

1.4.4 Oscillating-Body Viscometer ........................................................................... 19

1.4.5 Vibrating Viscometer ....................................................................................... 20

1.5 SURFACE VISCOMETRY .......................................................................................... 22

1.6 MOLECULAR ROTORS ............................................................................................ 23

1.7 SUMMARY .............................................................................................................. 25

2 BROWNIAN MOTION OF MICROSPHERES .................................................... 26

2.1 INTRODUCTION ....................................................................................................... 26

2.2 EXPERIMENTAL METHOD ....................................................................................... 27

2.3 VIBRATIONAL NOISE .............................................................................................. 28

iv

2.4 REDUCING ERROR .................................................................................................. 28

2.5 RESULTS ................................................................................................................ 30

2.6 DISCUSSION............................................................................................................ 32

3 MAGNETIC MICROSPHERES MOTION DUE TO AN APPLIED MAGNETIC

FIELD ............................................................................................................................ 33

3.1 FORCE BALANCE ON MAGNETIC MICROSPHERES ................................................... 33

3.2 MAGNETIC FIELD MEASUREMENT ......................................................................... 35

3.3 TERMINAL VELOCITY ............................................................................................. 35

3.4 MAGNETIC SUSCEPTIBILITY ................................................................................... 39

3.5 COLLAGEN SOLUTION PREPARATION ..................................................................... 41

4 IMAGE ANALYSIS AND RESULTS ..................................................................... 45

4.1 GROUPED MICROSPHERES ...................................................................................... 45

4.2 LOCAL FORCES ON THE MAGNETIC MICROSPHERES .............................................. 46

4.3 COLLAGEN VISCOSITY ........................................................................................... 50

4.4 CONCENTRATION OF THE DROPLET SURFACE ........................................................ 50

4.5 THICKNESS OF THE DENSE LAYER .......................................................................... 51

4.6 CONFIRMING FIBER PULL TIME .............................................................................. 52

4.7 CONTROL TEST ...................................................................................................... 54

4.8 SUMMARY .............................................................................................................. 54

v

5 APPENDICES ............................................................................................................ 56

6 REFERENCES ........................................................................................................... 76

vi

LIST OF FIGURES

FIGURE 1.1: SCHEMATIC OF PROCOLLAGEN AND TROPOCOLLAGEN MOLECULES.

PROTEINASES REMOVE THE PROPEPTIDES TO MAKE THE TROPOCOLLAGEN MOLECULE. 3

FIGURE 1.2: SCHEMATIC OF INTERMOLECULAR CROSS-LINKS OF TROPOCOLLAGEN

MOLECULE. .................................................................................................................. 4

FIGURE 1.3: SCHEMATIC OF COLLAGEN TYPE I STAGGERING STRUCTURE. ........................... 4

FIGURE 1.4: THE EXPERIMENTAL SETUP FOR COLLAGEN FIBER PRINTING. A DROPLET OF

COLLAGEN SOLUTION IS PLACED ON AN 8 MM COVER GLASS, ON A MICROSCOPE STAGE.

THE DROPLET IS SURROUNDED BY A NITROGEN DIFFUSING CHAMBER. THE GLASS

MICRO-NEEDLE CAN BE CONTROLLED BY A MICROMANIPULATOR SHOWN ON THE LEFT.

.................................................................................................................................... 5

FIGURE 1.5: THE TELO-COLLAGEN FIBER DRAWING FROM A DROPLET OF COLLAGEN

MONOMERS. ................................................................................................................. 6

FIGURE 1.6: A TELO-COLLAGEN FIBER PULLED FROM THE DROPLET. THE DIC IMAGE SHOWS

THE HIGHLY ALIGNED FIBRILLAR STRUCTURE OF THE TELO-COLLAGEN FIBER. ............ 7

FIGURE 1.7: DIC IMAGE OF PRINTED FIBER FROM A COLLAGEN DROPLET UNDER SILICON

OIL. COLLAGEN FIBRILS ARE PACKED DISORGANIZED AND IN RANDOM DIRECTIONS. ... 9

FIGURE 1.8: DIC IMAGE OF THE COLLAGEN DROPLET UNDER SILICON OIL AFTER 45

MINUTES. COLLAGEN FIBRILS ARE SELF-ASSEMBLED INSIDE THE DROPLET. ............... 10

FIGURE 1.9: COUETTE FLOW INDUCED BY RELATIVE MOTION OF TWO PLATES ................... 11

FIGURE 1.10: SCHEMATIC OF A CAPILLARY VISCOMETER ................................................... 14

vii

FIGURE 1.11: SCHEMATIC OF A FALLING-BODY VISCOMETER DESIGN. A SPHERICAL BODY IS

RELEASED AND ACCELERATED TO THE TERMINAL VELOCITY (VT). THE TIME THE

SPHERE TAKES TO TRAVEL A LENGTH L IS MEASURED. ............................................... 15

FIGURE 1.12: FALLING SPHERE THROUGH A LIQUID. THE DRAG FORCE, FD, ACTS ON THE

OPPOSITE DIRECTION OF THE GRAVITY FORCE, FG. ..................................................... 16

FIGURE 1.13: SCHEMATIC OF A CONCENTRIC CYLINDER VISCOMETER. THE INSIDE CYLINDER

ROTATES AT A CONSTANT ANGULAR VELOCITY, 𝜔, AND THE TORQUE, M, OF THE FLUID

IS MEASURED BY A STRAIN GAUGE ON THE FIXED CYLINDER ...................................... 18

FIGURE 1.14: SCHEMATIC OF A CONE-PLATE VISCOMETER. THE FLUID VISCOSITY CAN BE

CALCULATED BY MEASURING THE APPLIED TORQUE TO ROTATE THE CONE AT A

CONSTANT VELOCITY. ................................................................................................ 19

FIGURE 1.15: SCHEMATIC OF AN OSCILLATING-PISTON VISCOMETER. THE ABSOLUTE

VISCOSITY IS OBTAINED BY MEASURING THE TIME REQUIRED FOR THE PISTON TO MOVE

THE TRAVEL DISTANCE. .............................................................................................. 20

FIGURE 1.16: SCHEMATIC OF OSCILLATING SPHERE VISCOMETER. ..................................... 21

FIGURE 1.17: SCHEMATIC OF A TUNING FORK VISCOMETER ............................................... 22

FIGURE 2.1: THE TRAJECTORY OF 1 µM BEADS INSIDE A WATER DROPLET AT ROOM

TEMPERATURE. DIFFERENT BEADS ARE SHOWN WITH DIFFERENT COLORS. IT CAN BE

SEEN THAT IN THIS EXPERIMENTAL SETUP THE DRIFT FORCES ARE DOMINANT AND THE

BEADS DO NOT EXPERIENCE A RANDOM WALK. .......................................................... 29

viii

FIGURE 2.2: HISTOGRAM BAR CHART OF ONE-DIMENSIONAL DISPLACEMENTS. THE

BROWNIAN MOTION OF ONE-MICROMETRE BEADS IN WATER WERE RECORDED WITH

230 MS ACQUISITION TIME. ........................................................................................ 30

FIGURE 2.3: VISCOSITY OF KNOWN CONCENTRATION COLLAGEN SOLUTIONS. THE BLUE

DOTS ARE THE VISCOSITY-CONCENTRATION DATA PROVIDED BY ADVANCED

BIOMATRIX. THE RED SQUARES ARE MEASURED VISCOSITIES OF KNOWN

CONCENTRATION COLLAGEN SOLUTIONS. .................................................................. 31

FIGURE 3.1: THE MAGNETIC AND DRAG FORCES ACTING ON A MAGNETIC MICROSPHERE. .. 34

FIGURE 3.2: VARIATION OF THE MAGNETIC FIELD WITH DISTANCE FROM THE MAGNET. THE

SLOPE OF THE LINE FITTED ON THE DATA POINT AT 53 MM DISTANCE IS 29 GAUSS/MM

OR 2.9 T/M. ................................................................................................................ 36

FIGURE 3.3: TERMINAL VELOCITY OF THE MICROSPHERES IN A SOLUTION WITH THE

VISCOSITY OF 100 CP. ................................................................................................. 37

FIGURE 3.4: TERMINAL VELOCITY OF THE MICROSPHERES IN A SOLUTION WITH THE

VISCOSITY OF 1000 CP. ............................................................................................... 38

FIGURE 3.5: MAGNETIC RESPONSE OF DYNABEADS M-270 IN A FIELD RANGING FROM - 5 TO

5 TESLA. OERSTED (OE) IS THE UNIT OF MAGNETIC FIELD (H) IN CGS SYSTEM OF

UNITS AND IS EQUAL TO 103/4Π A/M. EMU/G IS THE UNIT OF MASS MAGNETIZATION

WHICH IS EQUAL TO A.M2/KG. .................................................................................... 40

ix

FIGURE 3.6: THE LINEAR REGION OF THE MAGNETIZATION CURVE FROM FIGURE 3.5. THE

SLOPE OF THE FITTED LINEAR LINE IS USED TO CALCULATE THE INITIAL

SUSCEPTIBILITY. ........................................................................................................ 41

FIGURE 3.7: THE EXPERIMENTAL SETUP. THE INVERTED MICROSCOPE, THE ZERO HUMIDITY

CHAMBER, AND THE PRESSURE CONTROLLER ARE SHOWN IN THE IMAGE. .................. 43

FIGURE 3.8: THE ZERO HUMIDITY CHAMBER ON THE MICROSCOPE STAGE. ......................... 44

FIGURE 4.1: SINGLE AND GROUPED MICROSPHERES IN THE PRESENCE OF THE MAGNETIC

FIELD.......................................................................................................................... 46

FIGURE 4.2: THE 2.8 µM MAGNETIC (M) AND 1.9 µM NON-MAGNETIC (N) MICROSPHERES

ARE DISTINGUISHED BY THEIR SIZE. ........................................................................... 47

FIGURE 4.3: TRAJECTORY OF MAGNETIC AND NON-MAGNETIC MICROSPHERES AT 1000 µM

BELOW THE DROPLET SURFACE. ................................................................................. 48

FIGURE 4.4: TRAJECTORY OF MAGNETIC AND NON-MAGNETIC MICROSPHERES AT 20 µM

BELOW THE DROPLET SURFACE. ................................................................................. 49

FIGURE 4.5: TRAJECTORY OF MAGNETIC AND NON-MAGNETIC MICROSPHERES ON THE

DROPLET SURFACE. .................................................................................................... 49

FIGURE 4.6: VISCOSITY OF COLLAGEN FOR CONCENTRATIONS LESS THAN 6 MG/ML. THE

POLYNOMIAL TREND LINE FITTED ON THE DATA IS USED TO PREDICT THE VISCOSITY OF

HIGHER CONCENTRATIONS. ........................................................................................ 50

FIGURE 4.7: CONCENTRATION OF THE DROPLET SURFACE OVER THREE MINUTES (N=5 PER

TIME POINT). .............................................................................................................. 51

x

FIGURE 4.8: CONCENTRATION OF THE DROPLET AFTER 150 SECONDS AT 0, 20, 40, 60, AND

1000 µM BELOW THE SURFACE (N=5 PER DATA POINT) ............................................... 52

FIGURE 4.9: DIC IMAGE OF COLLAGEN FIBER IN PBS. THE COLLAGEN FIBER WAS PULLED

AFTER 150 SECONDS IN THE NEW EXPERIMENTAL SETUP FOR VISCOSITY

MEASUREMENT. ......................................................................................................... 53

xi

LIST OF APPENDICES

APPENDIX 1: BROWNIAN MOTION OF THE MICROSPHERES ................................................ 57

APPENDIX 2: TERMINAL VELOCITY OF THE MICROSPHERES IN A SOLUTION WITH A

VISCOSITY OF 100 CP ................................................................................................. 64

APPENDIX 3: DETERMINING THE TERMINAL VELOCITY OF THE MICROSPHERES USING

RUNGE-KUTTA 4TH ORDER METHOD ......................................................................... 66

APPENDIX 4: RENAMING AND INVERTING OF FRAMES ....................................................... 68

APPENDIX 5: VELOCITY OF THE MAGNETIC MICROSPHERES .............................................. 69

APPENDIX 6: VELOCITY OF THE NON-MAGNETIC MICROSPHERES ...................................... 71

APPENDIX 7: VISCOSITY MEASUREMENT ........................................................................... 73

Chapter 1 1

Siadat - August 2015

1 INTRODUCTION AND

BACKGROUND

1.1 Collagen

Collagen is the most abundant protein in mammals and the main source of tensile

strength of connective tissues such as tendon, bone, cartilage, and skin [1-3].

Approximately 30% of all vertebrates’ body protein, including more than 90% of the

extracellular protein in the tendon and bone and more than 50% in the skin is comprises

collagen [4]. Up to the present time, more than 30 types of collagen and collagen-related

protein have been identified [5]. For instance collagen type I is dominant in tendon, bone,

skin, cornea, and blood vessel walls while cartilage is mostly comprises collagen type II

[2].

Collagen molecules are synthesized and secreted by fibroblasts cells in the endoplasmic

reticulum [6]. Since single collagen molecules are not stable in the physiological

condition of the vertebrates’ body, they go through several intracellular and extracellular

Chapter 1 2


stages to form bigger and more stable structures [7-9]. Fibroblasts control the collagen

based structures by secretion of regulatory molecules during synthesis, packing, and

growth of collagen [10].

Collagen primarily organizes into fibrils and then during matrix assembly mature fibrils

organize into higher structures called fibers and these are organized into bigger tissue

structures [8, 9, 11-13]. To accomplish these multistep process, the surface of collagen

molecules lose solvent molecules and assemble into larger structures with circular cross

sections, which minimizes the ratio of the surface area to the volume of the final

structure. This self-assembly and generation of larger structures is one of the hallmarks of

most living organisms and collagen type I, which is found in connective tissues of

vertebrates, is the best example of this structure [14].

The basic collagen molecules can be distinguished from other proteins by their unique

right handed triple helical structure, containing of three procollagen chains (α-chains) [1,

15]. Each procollagen chain consists of more than 1000 amino acids [4] with a repetitive

sequence of (glycine-X-Y)n, in which X and Y can be any amino acid but are frequently

the imino acids proline and hydroxyproline and n is 337–343 (depending on collagen

type) [15, 16].

Each procollagen chain contains of two terminal parts, known as the N- and C-

propeptides that do not have the glycine-X-Y repeat structure [17]. Contraction between

the C-propeptides, bring procollagen chains together and twist them to form the triple

helical structure [18]. Existence of glycine is necessary for folding of C-propeptides

domain and trimerization [1]. The presence of this triple helical structure is a necessary

condition to be a member of the collagen family (it is not a sufficient condition) [15].

Chapter 1 3


The existence of N- and C-propeptides slows down the assembly of procollagen to higher

structures. In the later stages of biosynthesis, removing the N- and C-propeptides by

specific metalloproteinase, leads to spontaneous assembly of collagen fibrils [19].

Figure 1.1 shows the configuration of procollagen with N- and C- propeptides and the

resulting tropocollagen molecule after separation of propeptides by proteinases [15].

Figure 1.1: Schematic of procollagen and tropocollagen molecules. Proteinases remove the

propeptides to make the tropocollagen molecule.

The tropocollagen molecule formed after trimming has a helical rod shape part and two

non-helical parts, called telopeptides. The chemical modification of collagen molecules

continues with intermolecular cross-links between the non-helical telopeptide part of one

tropocollagen molecule and the helical region of another [20-23] (Figure 1.2). The self-

assembly of tropocollagen molecules into fibrils are highly affected by the presence of

Tropocollagen

Procollagen N- Propeptides C- Propeptides

Proteinases Proteinases

Chapter 1 4


telopeptides while they only contain 3% of the total tropocollagen molecule’s mass. In

the absence of telopeptides, the self-assembly of tropocollagen molecules is slower [24].

Figure 1.2: Schematic of intermolecular cross-links of tropocollagen molecule.

X-ray analysis identifies that α-chains form left-handed helices with 3.3 residues per turn

and a pitch of 0.87 nm. In the next level, three chains form a right-handed triple helix

with a pitch of approximately 8.6 nm. The ultimate formed triple helix has a molecular

weight of approximately 300 kDa, a length of 300 nm, and a diameter of 1.5 nm [4]. The

triple-helices can organize into larger structures which are staggered by 67 nm with an

additional gap of 40 nm between molecules [25] (Figure 1.3).

Figure 1.3: Schematic of collagen type I staggering structure.

N

N C

C

67 nm 40 nm

300 nm

1.5 nm

Chapter 1 5


1.2 Collagen Fiber Printing From the Surface of a Droplet of

Collagen Monomers

In a previous study [26] in the extracellular matrix engineering research laboratory

(EMERL), a micromechanical system was used to create collagen fibers. Telo (intact

telopeptides) and atelo (lacking telopeptides) collagen fibers were drawn from the surface

of a droplet of neutralized collagen monomers at room temperature.

Figure 1.4: The experimental setup for collagen fiber printing. A droplet of collagen solution is

placed on an 8 mm cover glass, on a microscope stage. The droplet is surrounded by a nitrogen

diffusing chamber. The glass micro-needle can be controlled by a micromanipulator shown on

the left.

Chapter 1 6


The experimental setup on the stage of an inverted microscope is shown in Figure 1.4. To

create these fibers, a droplet of 125 µL volume of neutralized collagen solution (4.4

mg/mL) was pipetted on an 8 mm circular cover glass. The droplet was exposed to

nitrogen gas to increase the rate of evaporation from the droplet surface. Higher

evaporation rate helped to obtain a thin layer of highly concentrated collagen solution on

the top surface of the droplet. Then a glass micro-needle was inserted into the droplet and

pulled out vertically with a maximum speed of 240 µm/s. A collagen fiber formed on the

tip of the needle and grew as the needle was being pulled out (Figure 1.5).

Figure 1.5: The telo-collagen fiber drawing from a droplet of collagen monomers.

Chapter 1 7


A fiber can be drawn immediately after placing the droplet on the cover glass. The

created fiber has a small diameter and breaks before getting long enough for further

analysis. A long and highly organized fiber can be drawn only after 150 seconds of

nitrogen gas flowing over the droplet in the experimental setup.

Differential Interference Contrast (DIC) microscopy indicated a high degree of uniaxial

alignment within the collagen fibrils of the printed fibers (Figure 1.6). Transmission

electron microscopy (TEM) verified the presence of the fibrillar D-banding (65.4 ± 2.2

nm) in the printed fibers.

Figure 1.6: A telo-collagen fiber pulled from the droplet. The DIC image shows the highly

aligned fibrillar structure of the telo-collagen fiber.

Chapter 1 8


DIC microscopy showed no evidence of fibrillar structures inside the droplet or on its

surface at the time when the fiber was drawn. It has been proposed that drawing process

induces an elongational flow at the point where the fiber exits the droplet (the necking

region) and that the extensional strain field is responsible for aligning the molecules and

initiating the assembly of fibrils. In addition, TEM images showed highly aligned fibrils

in the shell and poorly packed fibrils in the core of the fibers. This suggests that although

the extensional strain can initiate the organized assembly from a concentrated layer of

collagen monomers, the strain rate is insufficient to align the lower concentration bulk

solution that gets entrained into the core of the fiber.

In a control experiment, we showed when the collagen droplet is surrounded by silicon

oil, printing a fiber takes more time (around 45 minutes) and the produced fiber is not

packed with aligned fibrils (Figure 1.7). This delay in formation of the fiber and

misalignment of fibrils happens since the surface concentration is not high enough to

initiate the formation of the fiber on the needle. In this situation self-assembly of collagen

fibrils happens in the bulk solution and these aggregated fibrils in random directions form

the fiber (Figure 1.8).

For further investigation in the formation of highly aligned and continuous fibers, the

laboratory is interested in developing the experimental setup. Therefore, to supply the

proper concentration of collagen monomers for a collagen fiber printer device, an exact

collagen concentration in the droplet surface is required. The goal of this study is to

measure the concentration and thickness of the dense layer on the droplet’s top surface

when a fiber can be pulled (150 seconds after the collagen droplet is placed on the cover

Chapter 1 9


glass). To measure the concentration, we need to know the viscosity of the droplet under

the same experimental conditions.

Figure 1.7: DIC image of printed fiber from a collagen droplet under silicon oil. Collagen fibrils

are packed disorganized and in random directions.

Chapter 1 10


Figure 1.8: DIC image of the collagen droplet under silicon oil after 45 minutes. Collagen fibrils

are self-assembled inside the droplet.

1.3 Viscosity

A laminar flow moves in smooth layers. When a layer of fluid moves in relation to

another layer, an internal friction force acts on these layers. The greater the friction, the

greater the amount of force required to cause this movement. Viscosity is a measure of

the internal friction of a fluid [27].

Chapter 1 11


The laminar flow in Plane Couette flow (Figure 1.9) is a simple flow that shows the

concept of viscosity. Couette flow is a fluid between two rigid plates where the lower

plate is fixed and the top plate moves with a constant velocity u0 in its own plane.

Assuming a fixed driving force, the velocity of the moving plate can be determined by

the viscosity of the fluid. The more viscous the fluid the slower the top plate moves.

According to Newton’s law of viscosity for laminar flows,

𝜏 = 𝜂𝑑𝑢

𝑑𝑦 , 𝜏 =

𝐹

𝐴 (1.1)

dynamic or absolute viscosity (η) is defined as the ratio of shear stress (τ) to the velocity

gradient (du/dy) between two plates. Fluids that follow Newton’s law are called

Newtonian fluids [27]. Newtonian fluids like water are homogeneous. In a non-

Newtonian flow, viscosity is dependent on shear stress, shear strain rate, and time. Some

biological samples like whole blood behave as non-Newtonian fluids.

Figure 1.9: Couette flow induced by relative motion of two plates

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑢0

X direction

Y Direction

U (y)

Chapter 1 12


The viscosity of a fluid is highly dependent on thermodynamic factors such as

temperature and pressure. Liquid viscosity decreases with temperature:

𝑙𝑛𝜂

𝜂0≈ 𝑎 + 𝑏 (

𝑇0𝑇) + 𝑐 (

𝑇0𝑇)2

(1.2)

where T is absolute temperature, η0 is known viscosity at known temperature T0, and a, b,

and c are constants. On the other hand, gas viscosity increases with temperature:

𝜂

𝜂0≈

{

(

𝑇

𝑇0)𝑛

𝑃𝑜𝑤𝑒𝑟 𝑙𝑎𝑤

(𝑇𝑇0)

32(𝑇0 + 𝑆)

𝑇 + 𝑆 𝑆𝑢𝑡ℎ𝑒𝑟𝑙𝑎𝑛𝑑 𝑙𝑎𝑤

(1.3)

where n and S are constants [28].

1.4 Viscosity Measurement

Measurement of viscosity is essential in many fields therefore several methods and

devises have been developed suitable for specific circumstances and materials. Sample

size, working conditions, time consumption, cost, continuous measurement, and fluid

phase are some of the important factors in selecting the appropriate viscosity

measurement method.

The SI (International System of Units) unit of dynamic viscosity is Pa.s or kg/(s.m) which

sometimes refer to as Poiseuille (symbol PI) Named after the French physician Jean

Louis Marie Poiseuille (1797 - 1869). The CGS (centimetre–gram–second system of

units) unit of dynamic viscosity is poise (symbol P) which is equivalent to 0.1 Pa.s. In

Chapter 1 13


most everyday applications, centipoise (symbol cP) which is 0.1 Pa.s is used as the

dynamic viscosity unit. Most of fluids have viscosities between 0.5 to 1000 cP [29].

Many viscometers measure kinematic viscosity υ which is define as

𝜐 =𝜂

𝜌 (1.4)

Therefore, the density of fluid must be known to measure the dynamic viscosity. The

CGS unit of kinematic viscosity is square centimetres per second or stokes (symbol St).

One stokes is equal to the dynamic viscosity in poise divided by the density of the fluid in

gram per cubic centimetres [27, 29].

A viscometer is a device used to measure the viscosity of a Newtonian fluid and a

rheometer usually used for fluids with viscosities varying with flow conditions. There are

two basic methods to measure the resistance of a fluid to flow. Either an object moves

through a stationary fluid, or the fluid flows through a stationary object. Here some types

of standard laboratory viscometers and rheometers are introduced.

1.4.1 Capillary Viscometers

The kinematic viscosity of Newtonian fluids can be determined using capillary

viscometers which is one of the earliest widely used methods due to their simple design

and process [29]. They cover the range from 0.2 to 300,000 cSt. Figure 1.10 shows a

schematic of a capillary viscometer. Kinematic viscosity can be determined by measuring

the flow rate of a known volume of a fluid through the capillary tube with a known

diameter and known length. The dynamic viscosity is equal to

Chapter 1 14


𝜐 =𝜋𝑎4𝛥𝑃

8𝑄(𝐿 + 𝑛𝑎)+

𝑚𝜌𝑄

8𝜋(𝐿 + 𝑛𝑎) (1.5)

where Q is the flow rate, a is the radius of the capillary tube, L is the length of the tube,

∆P is pressure drop along the tube, ρ is the fluid density, n and m are correction factors

[29-31]. The correction factors (viscometer constants) depend on the local gravity, fluid

density, surface tension [32], temperature, fluid volume, expansion coefficient of fluid

and glass [27].

Figure 1.10: Schematic of a capillary viscometer

Viscosity measurement using capillary viscometers is simple and cheap, though it has

some weaknesses. This method is time consuming and requires a large volume of fluid.

P2 P1

Chapter 1 15


Being limited to the Newtonian fluids is the biggest disadvantage of capillary viscometers

[27].

1.4.2 Falling-Body Viscometer

A falling-body viscometer can be used to measure the fluid viscosity by measuring the

elapsed time of a sphere or cylinder shaped body travelling inside a viscous fluid. The

more viscous the fluid the more time the body takes to travel a fixed distance [27]. A

simple schematic illustration is shown is Figure 1.11.

Figure 1.11: Schematic of a falling-body viscometer design. A spherical body is released and

accelerated to the terminal velocity (VT). The time the sphere takes to travel a length L is

measured.

Chapter 1 16


Based on Stokes’ law the force that slows down a sphere travelling through a viscous

fluid is related to the fluid viscosity and the velocity and radius of the sphere. This drag

force can be measured by

𝐹𝑑 = 6𝜋𝜂𝑅𝑢 (1.6)

where Fd is the drag force, R is the radius, and u is the velocity of the sphere [27, 33, 34].

The terminal velocity can be measured by the force balance between the Stokes drag

force and the force caused by gravity Fg (Figure 1.12):

𝐹𝑔 =4

3𝜋𝑅3(𝜌𝑝 − 𝜌𝑓)𝑔 (1.7)

Figure 1.12: Falling sphere through a liquid. The drag force, Fd, acts on the opposite direction of

the gravity force, Fg.

Fd

Fg

Chapter 1 17


where ρp and ρf are respectively the density of sphere and fluid and g is the gravitational

acceleration. Therefore, the resulting terminal velocity is given by

𝑉𝑇 =2

9

(𝜌𝑝 − 𝜌𝑓)

𝜂𝑔𝑅2 (1.8)

In this method, fluid viscosity can be measured only in a laminar and fully developed

flow with small Reynolds number [27].

1.4.3 Rotational Viscometer

Rotational viscometers measure the resistance of a fluid to torque. In these viscometers, a

rotor is immersed in a fluid and is rotated at a constant speed. The fluid viscosity is

proportional to the torque on the surface of viscometer. The working equation that relates

the torque to the viscosity is defined by the viscometer geometry [27, 33].

An example of these viscometers is the concentric cylinder viscometer. This viscometer

is made of two concentric cylinders containing a fluid between them. One or both of the

cylinders rotate (usually one cylinder is stationary and the other rotates with constant

velocity) and the torque on the stationary cylinder is measured. Figure 1.13 represents a

schematic of a concentric cylinder. The fluid viscosity can be calculated by equation

(1.9):

𝜂 = 2𝜋𝑅2𝜏 4𝜋 [(𝛼2𝑅2

𝛼2) (𝜔𝑜 − 𝜔𝑖)]⁄ (1.9)

where R is the radius of the inner cylinder, τ is the shear stress per unit length of the inner

cylinder, α is the ratio of the radius of the outer cylinder to the radius of the inner

Chapter 1 18


cylinder, ωo and ωi are respectively the angular velocity of the outer and inner cylinders

[35].

Figure 1.13: Schematic of a concentric cylinder viscometer. The inside cylinder rotates at a

constant angular velocity, 𝜔, and the torque, M, of the fluid is measured by a strain gauge on the

fixed cylinder

Another example of rotational viscometer is the cone-plate. In the cone-plate viscometer,

the test fluid is contained in a small gap between a flat cone (with reference angle less

than 3°) and a flat plate. The apex of the cone just touches the plate surface. The cone (or

the plate) rotates in a constant speed and the induced torque on the other surface is

measured. The main advantage of this design is the existence of a constant shear rate at

all locations between the cone and the plate. The reason is that as the distance between

Chapter 1 19


the cone and the plate increases, the linear velocity also increases. Figure 1.14 shows a

schematic of a cone-plate viscometer.

Figure 1.14: Schematic of a cone-plate viscometer. The fluid viscosity can be calculated by

measuring the applied torque to rotate the cone at a constant velocity.

The fluid viscosity can be calculated by equation (1.10):

𝜂 = 3𝐺𝜓 2𝜋𝑅3𝜔⁄ (1.10)

where G is the torque on the cone, ψ is the cone angle in radians, R is the radius of the

cone base, and ω is the angular velocity [36].

1.4.4 Oscillating-Body Viscometer

Oscillating-body viscometers contains a thermally controlled chamber and a magnetically

influenced piston (or blade). An electromagnetic field causes the piston to move back and

forth inside the chamber. The fluid viscosity can be determined according to Newton’s

Plate

Cone Liquid

Chapter 1 20


law of viscosity, and the piston oscillation time and distance. Figure 1.15 shows a

schematic of an oscillating-piston viscometer [27].

Figure 1.15: Schematic of an oscillating-piston viscometer. The absolute viscosity is obtained by

measuring the time required for the piston to move the travel distance.

The oscillating piston viscometers are widely used for small sample viscosity

measurement. It can be used with different size of pistons to cover a vast range of

viscosity from 0.1 to 2000 cP.

1.4.5 Vibrating Viscometer

Vibrating viscometers can determine the fluid viscosity by measuring the damping of an

oscillating electromechanical resonator [37]. Oscillating sphere and tuning fork are two

types of vibrating viscometers. The measurement principle of these viscometers is that

Piston

Chamber

Travel Distance

Electromagnets

Chapter 1 21


the product of the fluid viscosity and density is proportional to the viscous damping of

the oscillation amplitude.

The damping of the resonator can be usually measured by the required power to maintain

the oscillator vibration at constant amplitude. Another method is the measurement of the

decay time. The higher the viscosity, the shorter the decay time [29].

In the oscillating sphere viscometers, a stainless steel sphere oscillates with constant

amplitude and the viscosity is calculated based on the required power to maintain the

oscillation. In the tuning fork viscometers fluid viscosity and density can be measured

separately from the bandwidth and the frequency of the vibrating fork resonance [29].

Figure 1.16 and Figure 1.17 show schematics of these vibrational viscometers [38].

Figure 1.16: Schematic of oscillating sphere viscometer.

Transducer

Flow

Chapter 1 22


Figure 1.17: Schematic of a tuning fork viscometer

1.5 Surface Viscometry

The interpretation of surface viscometry is complicated due to the unbalance of attracting

forces by the nearby molecules [39]. Contribution of the airflow above the surface and

the three-dimensional solution subphase is not completely determined in the surface drag

forces [40]. The usual method for surface viscometry are based on two broad categories:

torque measurement [41] or particle traction [42].

Petkov et al. [42] measured the viscosity of a monolayer between sodium dodecyl sulfate

and hexadecyltrimethylammonium bromide. They used an external capillary force to

Temperature Sensor

Electromagnetic Unit

Direction of

Vibration

Oscillator

Sample

Chapter 1 23


move a submillimetre-size sphere. The equation of motion of the sphere was solved and

the viscosity was calculated based on the sphere drag coefficient.

Waugh [43, 44] used a continuum mechanical approach to measure the surface viscosity.

A fluid moved through a vesicle containing 20 - 65 µm tethers and the magnitude of the

force acting on the tether was measured using Stokes drag force. The surface viscosity

was calculated by knowing the rate of tether growth, and the force on the vesicle.

Ziemann et al. [45] used a magnetically driven bead micro-rheometer to measure the

local viscoelastic moduli of entangled actin networks. They used the viscosity of the fluid

and Stokes force to calibrate the driving force. Bausch et al. [46] used this micro-

rheometer to measure the viscoelastic parameters of the surface of adhering fibroblasts.

Particle tracking techniques used to track 4.5 µm paramagnetic microspheres. The

viscosity was determined using the viscoelastic response of the cells. A theory of

diffusion of proteins in a membrane coupled to a solid surface [47] was used to estimate

the viscosity of the cytoplasm.

1.6 Molecular Rotors

Molecular rotors are fluorescent molecules with the capability to go through twisted

intramolecular charge transfer (TICT). The TICT molecules consist of three parts: an

electron donor, an electron acceptor, and an electron-rich spacer unit which brings the

donor and the acceptor units together. The donor and the acceptor units rotate relative to

each other, and consequently the fluorescence emission changes with this rotation [48-

50].

Chapter 1 24


The emission intensity of the TICT molecules depends on the viscosity and polarity of

their solvent while it has been shown that the emission intensities of molecular rotors are

more sensitive to the solvent viscosity than polarity [51]. Molecular rotors come to be

fluorescent only when their rotation is inhibited and therefore they can be used in

microscopic scale for viscosity measurement of biological samples in vitro and vivo

when the commercial viscosity measurement instruments are not applicable [49, 52-56].

The dynamic viscosity of a solvent is related to its fluorescence quantum yield, Φ, based

on the Forster-Hoffmann equation:

𝑙𝑜𝑔𝜙 = 𝐴 + 𝐵(𝑙𝑜𝑔 𝜂) (1.11)

where A and B are respectively temperature and dye dependent constants [57-59].

Haidekker et al. [60] used fluorescent molecular rotors to measure blood plasma

viscosity. They successfully measured the viscosity of plasma-pentastarch mixtures in a

range of 1 to 5 cP. Kung et al. [55] used Benzylidene malononitriles (DCVJ) a molecular

rotor to investigate the polymerization of tubulin at concentrations less than 2.5 mg/mL.

They showed the molecular rotors bind to tubulin and their quantum yield increases with

increasing tubulin concentration. Haidekker et al. [51] showed the peak emission

intensity of DCVJ depends on the viscosity of solvent. They used mixtures of ethylene

glycol and glycerol as the solvent covering a vast range of viscosity up to 945 cP.

Fluorescence lifetimes of the molecular rotors are viscosity-dependent. Kuimova et al.

[61] used meso-substituted 4,4′-difluoro-4-bora-3a,4adiaza-s-indacene as a molecular

rotor in mixtures of methanol-glycerol and showed the fluorescence lifetime increases

from 0.7 ± 0.05 to 3.8 ± 0.1 ns with increasing the mixture viscosity from 28 to 950 cP.

Chapter 1 25


Using fluorescence-lifetime imaging microscopy (FLIM), intracellular viscosity can be

calculated by measuring the fluorescent lifetime of the molecular rotors [61-63].

1.7 Summary

Mechanical viscometers are usually time consuming and unable to measure the viscosity

of biological micro-samples. The new and powerful method based on the molecular

rotors is suitable for some biological samples in vivo and in vitro. However, more studies

are necessary to validate this method for other biological samples. In addition, using

molecular rotors requires accessibility to confocal microscopy for FLIM, which is not

practical for our experimental setup.

Brownian motion is a powerful way to measure fluid viscosity. In chapter 2, the

Brownian motion of microspheres in the collagen droplet is investigated and the

restriction of this viscosity measurement method for the fiber pulling experiment is

discussed.

To measure the viscosity of the micro-layer on top of the collagen droplet, an optical

method was used in the exact experimental situation. In chapter 3 and 4, the viscosity and

collagen concentration within the droplet were estimated from the measured velocity of

magnetic microspheres in the solution when the droplet was positioned in the uniform

region of a magnetic field produced by a permanent magnet. Each magnetic microsphere

travels in the direction of magnetic lines with a constant velocity, which is related to the

viscosity based on Stokes Law.

Chapter 2 26


2 BROWNIAN MOTION OF

MICROSPHERES

2.1 Introduction

Brownian motion is a classical experiment to measure the viscosity of a solution.

Suspended particles in a fluid randomly collide with the atoms or molecules in the fluid.

These collisions are thought to result in a random walk of particles in the fluid, called

Brownian motion. Robert Brown first reported this phenomenon in 1827 [64].

The measure of the speed of diffusion in these random movements is characterized by the

diffusion coefficient (D). The diffusion coefficient can be measured by the mean square

displacement (MSD) of particles and the lag time. For an isotropic and unrestricted

translational diffusion, the mean square displacement of particles can be expressed as

{

⟨(∆𝑟)2⟩ = 2𝐷. ∆𝑡, 𝑜𝑛𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛

⟨(∆𝑟)2⟩ = 4𝐷. ∆𝑡, 𝑡𝑤𝑜 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛

⟨(∆𝑟)2⟩ = 6𝐷. ∆𝑡, 𝑡ℎ𝑟𝑒𝑒 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛

(2.1)

where ⟨(∆r)^2⟩ is the mean square displacement, and ∆t is the lag time.

Chapter 2 27


In 1905, Albert Einstein predicted the diffusion coefficient of a spherical particle based

on the physical properties of the medium and particle

𝐷 =𝑘𝐵𝑇

6𝜋𝜂𝑟 (2.2)

where kB is the Boltzmann constant, T the temperature, η the viscosity of the medium,

and r the radius of particle [65].

2.2 Experimental Method

A back of the envelop calculation reveals that the one dimensional mean free path within

100 milliseconds acquisition time for a one micrometre diameter particle in water at room

temperature is about 300 nanometre while this number for a sample fluid with a viscosity

of 1000 cp is less than 10 nanometre. In order to measure these small motions, a Gaussian

can be fitted to the one dimensional displacement histogram. When the mean

displacement is zero, the variance of the Gaussian is equal to the MSD.

𝑉𝑎𝑟 (∆𝑥) = ⟨∆𝑥2⟩ − ⟨∆𝑥⟩2 (2.3)

In our implementation of the experiment, green-fluorescent beads (Flow Cytometry Sub-

micron Particle Size Reference Kit, LifeTechnologies, ranging in diameter from 0.02 µm

to 2.0 µm, Catalog number: F13839) were suspended in water at room temperature. To

simulate the collagen fiber pulling experimental setup, a 125 µl volume of water

containing one-micrometre green fluorescent beads was pipetted on an 8 mm cover glass

placed on the stage of a microscope (Nikon inverted microscope eclipse TE2000-E). The

Brownian motions of the fluorescent beads were recorded by a CoolSNAP HQ2 CCD

Chapter 2 28


camera and a Nikon Plan Fluor ELWD 20x/0.45 objective (0.32 µm/pix). Images were

taken at the rate of 4.3 frame per seconds for one minute (230 milliseconds acquisition

time).

2.3 Vibrational Noise

ImageJ (FIJI, TrackMate Plugin) software was used to find and track the suspended

microspheres. Figure 2.1 shows a sample of the trajectory of the microspheres. Because

of the fiber pulling experimental setup, the medium is not completely protected from

vibrational noises. So the microspheres do not move in a random walk and as it is

obvious in Figure 2.1, the mean displacement is not zero.

2.4 Reducing Error

To reduce the error of drift and vibrational noises, the average of one-dimensional

movement in X and Y directions were subtracted from the data. A MATLAB code was

used to post process the tracked beads by ImageJ software, obtain the new mean

displacement, and calculate the viscosity of the medium. Figure 2.2 shows the histogram

bar chart and the probability distribution of one-dimensional displacements.

The post processing of data resulted in a viscosity of 1.17 cp for water at room

temperature (n=3). Therefore, the same method was used to measure the viscosity of

collagen droplets with known concentrations and viscosities. Nutragen (Advanced

BioMatrix, Bovine Collagen Solution, Type I, 6 mg/ml, Catalog #5010-50ML) was

diluted using 0.01M hydrochloric acid to obtain collagen concentration of 0 to 6 mg/ml.

Chapter 2 29


For each concentration, a 125 µl volume of collagen solution was pipetted on an 8 mm

cover glass on the microscope stage.

Figure 2.1: The trajectory of 1 µm beads inside a water droplet at room temperature. Different

beads are shown with different colors. It can be seen that in this experimental setup the drift

forces are dominant and the beads do not experience a random walk.

Chapter 2 30


Figure 2.2: Histogram bar chart of one-dimensional displacements. The Brownian motion of one-

micrometre beads in water were recorded with 230 ms acquisition time.

2.5 Results

The Brownian motions of one-micrometre fluorescent beads were tracked on the surface

of the droplet for four minutes for each sample. The viscosity of collagen solution

samples were measured using ImageJ software and the MATLAB code as mentioned

before. The calculated viscosities were compared to provided viscosity-concentration

data by Advanced BioMatrix (Figure 2.3).

Chapter 2 31


Figure 2.3: Viscosity of known concentration collagen solutions. The blue dots are the viscosity-

concentration data provided by Advanced BioMatrix. The red squares are measured viscosities of

known concentration collagen solutions.

In the viscosities higher than approximately 40 cp, the noise in the system is dominant

and the Brownian motion of beads is not detectable. Therefore, this method of subtracting

the drift of beads does not work in collagen droplets with high concentrations.

0

20

40

60

80

100

120

140

160

180

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Vis

cosi

ty, c

p

Concentration, mg/ml

Chapter 2 32


2.6 Discussion

The diffusion of particles can be increased by using nanometre sized quantum dots

instead of microspheres. Even though, the noise in the system also increases due to

smaller size particles. In the other hand, using bigger particles can decrease the noise of

drift motions. Nevertheless, they cannot be used to measure the viscosity of the droplet

surface micro-layer.

Using a magnetic field to move particles can be a solution to this problem. A magnetic

field can be applied near the droplet and force some magnetic microspheres to move in

the direction of magnetic field. If the particles are big enough, the Brownian motion of

particles can be ignored in compared to their movement due to magnetic force. The

magnetic field and the viscosity measurement of collagen droplet are described in the

next chapter.

Chapter 3 33


3 MAGNETIC MICROSPHERES

MOTION DUE TO AN

APPLIED MAGNETIC FIELD

3.1 Force Balance on Magnetic Microspheres

There are two types of forces acting on a magnetic microsphere in a viscous solution and

in the presence of a magnetic field: the magnetic force in the direction of the magnetic

lines and the drag force in the opposite direction (Figure 3.1).

Chapter 3 34


Figure 3.1: The magnetic and drag forces acting on a magnetic microsphere.

The drag force on the magnetic microsphere can be calculated by Stokes Law:

𝐹𝑑⃗⃗⃗⃗ = −6𝜋𝜂𝑅�⃗� (3.1)

where η is the dynamic viscosity of the solution, R and u are respectively the radius and

velocity of the magnetic microsphere. The most common relationship used to measure

the magnetic force acting on a magnetic particle is

𝐹𝑀⃗⃗⃗⃗ ⃗ =𝑉∆𝑥

𝜇0(�⃗� . 𝛻)�⃗� (3.2)

where V is the particle’s volume, ∆x is the difference in magnetic susceptibility between

the particle and the solution, μ0 is the permeability of the vacuum which is equal to

𝜇0 = 4𝜋 ∗ 10−7, 𝑇𝑚𝐴−1 (3.3)

and B is the applied magnetic field [54].

Permanent Magnet

Cover

Collagen Droplet

Magnetic

Force Drag

Force

Chapter 3 35


The collagen droplet was placed close to the magnet and exactly on the centerline of its

surface, which has a uniform magnetic field. Neglecting gravitational and conventional

forces, the force balance for the magnetic microsphere is defined by

∑𝐹 = 𝐹𝑀⃗⃗⃗⃗ ⃗ + 𝐹𝑑⃗⃗⃗⃗ = 𝑚𝑎 (3.4)

where m is the magnetic microsphere mass and vector a is its acceleration.

3.2 Magnetic Field Measurement

The magnetic field was supplied by a permanent magnet (BY0Y08-N52, K&J Magnetics,

Inc.) and its strength and gradient were respectively measured as 593*10-4 T, and 2.9

T/m by a gauss meter at a distance of 53 mm from the center of the magnet (Figure 3.2).

3.3 Terminal Velocity

Considering the force balance on the magnetic microspheres in the direction of magnetic

field lines and neglecting the change of magnetic force throughout the droplet, magnetic

microspheres reach to a terminal velocity toward the center of the magnet. Since the

experiment happens in a short time, the accuracy of the viscosity measurement depends

on the time that takes particles to reach this terminal velocity.

Numerical methods were used to estimate the time before all the particles reach to the

constant terminal velocity. Microspheres velocity at each time point (ui) can be predicted

based on the initial velocity and the rate of change of velocity:

Chapter 3 36


Figure 3.2: Variation of the magnetic field with distance from the magnet. The slope of the line

fitted on the data point at 53 mm distance is 29 gauss/mm or 2.9 T/m.

𝑢𝑖 = 𝑢𝑖−1 +𝑑𝑢

𝑑𝑡|𝑖−1

𝛥𝑡 (3.5)

Also according to the equation (3.4), the rate of change of velocity can be written as:

𝑑𝑢

𝑑𝑡|𝑖= 𝑎𝑖 =

𝐹𝑚 − 𝛼𝑢𝑖𝑚

(3.6)

where constant α is equal to

𝛼 = 6𝜋𝜂𝑅 (3.7)

The terminal velocity was calculated using MATLAB (The MATLAB code is available

in appendix). The initial velocity was assumed equal to zero. The diameter, magnetic

-1400

-1200

-1000

-800

-600

-400

-200

0

30 40 50 60 70 80

Ma

gn

etic

Fie

ld, g

au

ss

Distance from the Magnet, mm

Chapter 3 37


susceptibility, and density of the magnetic microsphere where respectively presumed as

2.8 µm, 0.5, and 1600 kg/m3 based on the common magnetic microspheres in the market.

The results are shown in Figure 3.3 for a sample solution with the viscosity of 100 cp (~5

mg/mL collagen) and in Figure 3.4 for ~98% glycerol by volume at 20°C with a viscosity

of 1000 cp. Time steps of 0.1 ns were used in this numerical method. The Runge-Kutta

method was also used to solve the equation (3.5) and the result was the same.

Figure 3.3: Terminal velocity of the microspheres in a solution with the viscosity of 100 cp.

Figure 3.3 and Figure 3.4 demonstrate that the magnetic microspheres reach to their

terminal velocity in a few nanoseconds in our experimental situation. Therefore the

microspheres velocity throughout the experiment can be considered constant and the

equation (3.4) written as

0

50

100

150

200

250

300

350

0 10 20 30 40 50 60

Vel

oci

ty, n

m/s

Time, ns

Chapter 3 38


Figure 3.4: Terminal velocity of the microspheres in a solution with the viscosity of 1000 cp.

∑𝐹 = 𝐹𝑀⃗⃗⃗⃗ ⃗ + 𝐹𝑑⃗⃗⃗⃗ = 0 (3.8)

So the terminal velocity can be calculated by setting the magnitudes of the magnetic and

the drag forces equal to each other. Equation (3.9) shows the final form of the terminal

velocity,

𝑣 =2

9

𝑅2𝜒

𝜂𝜇0(𝐵𝑥

𝜕𝐵𝑥𝜕𝑥

) (3.9)

where the x-direction is in the same direction as magnetic force and χ is the initial

magnetic susceptibility of the particles (the magnetic susceptibility of the medium is

neglected).

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7

Vel

oci

ty, n

m/s

Time, ns

Chapter 3 39


The resulting velocity for the microspheres from equation (3.9) is 303 nm/s in a solution

with the viscosity of 100 cp and 30 nm/s in a solution with the viscosity of 1000 cp,

which is compatible with the results from the numerical methods. Therefore, in a

potential concentrated droplet surface with a viscosity of 1000 cp, the magnetic

microspheres with the assumed properties and specified magnetic field, travel 1800 nm

per each minute. This travel distance is detectable with a Nikon Plan Fluor ELWD

20x/0.45 objective (0.32 µm/pixel).

3.4 Magnetic Susceptibility

In order to calculate the viscosity of the collagen droplet, the equation (3.9) can be

written in the form of

𝜂 =2

9

𝑅2𝜒

𝑣𝜇0(𝐵𝑥

𝜕𝐵𝑥𝜕𝑥

) (3.10)

In this equation, the terminal velocity can be measured experimentally by tracking the

magnetic microspheres in the collagen droplet. In addition, the magnitude and gradient of

the magnetic field can be measured by a gauss meter as described in section 3.2.

However, the magnetic susceptibility of particles usually alternate between lots.

The magnetic susceptibility is the measure of magnetization of a material inside a

magnetic field. The bigger the susceptibility the faster the magnetic microspheres move.

Therefore to precisely measure the magnetic susceptibility of the particles, a

superconducting quantum interference device (SQUID) was used at room temperature,

300 K, to generate the magnetic response of Dynabeads M-270 Carboxylic Acid, Catalog

Chapter 3 40


number 14305D. These magnetic microspheres have a uniform diameter of 2.8 µm,

density of 1.6 g/cm3, and initial concentration of 2*109 beads/ml (~30 mg/ml). The

magnetization curve is shown in Figure 3.5.

Figure 3.5: Magnetic response of Dynabeads M-270 in a field ranging from - 5 to 5 Tesla.

Oersted (Oe) is the unit of magnetic field (H) in CGS system of units and is equal to 103/4π A/m.

emu/g is the unit of mass magnetization which is equal to A.m2/kg.

The susceptibility of the magnetic microspheres, χ = 0.512, is measured from the slope of

the line fitted to the initial part of the magnetization curve (weak fields). The slope of the

line is multiplied by 4π*ρ to dimensionless the magnetic susceptibility Figure 3.6.

-15

-10

-5

0

5

10

15

-60000 -40000 -20000 0 20000 40000 60000

Mo

men

t, e

mu

/g

Magnetic Field, Oe

Chapter 3 41


Figure 3.6: The linear region of the magnetization curve from Figure 3.5. The slope of the fitted

linear line is used to calculate the initial susceptibility.

3.5 Collagen Solution Preparation

A collagen solution was made by mixing 320 ml of TeloCol (type I, acid soluble bovine

collagen solution, 5.6 mg/ml, Part No. 5026-D), 40 ml of phosphate buffered saline (PBS,

10x solution, Fisher Bioreagents, Catalog No. BP3991), and 49 ml of Sodium hydroxide

(0.1 N standard solution, ACROS Organics, Catalog No. AC124190010) in room

temperature. A 2 μl volume of 4 times diluted Dynabeads M-270 Carboxylic Acid

(Invitrogen, 2.8 μm, Catalog No. 14305D) was added to the collagen solution. In

y = 0.0255x - 0.1541R² = 0.9995

-8

-6

-4

-2

0

2

4

6

8

-300 -200 -100 0 100 200 300

Mo

men

t, e

mu

/g

Magnetic Field, Oe

Chapter 3 42


addition, non-magnetic microspheres were also added to remove the motion of magnetic

microspheres caused by other forces acting on the microspheres in the droplet. The non-

magnetic microspheres were introduced to the solution by adding 2 μl of 1 part Fluoro-

Max green fluorescent polymer microspheres (Thermo Scientific, 1.9 μm, Catalog No.

G0200) to 9 part deionized water to the solution.

A 125 µl volume of the well-mixed collagen solution was transferred on an 8 mm circular

cover glass. The droplet was surrounded by an aluminium cylinder. Using this aluminium

cylinder and a dual valve pressure controller (Alicat Scientific), the humidity of the

chamber decreased to approximately zero percent. The pressure controller was set up on

0.022 psig. In order to not disturb the air inside the cylinder, the collagen droplet was

pipetted on the cover glass through a hole on the cylinder.

In the similar experimental condition as the fiber pulling experiment but with the

presence of the magnetic bar, DIC images of the surface of the droplet were captured at a

rate of 1 frame per second for 30 seconds after 0, 1, 2, and 3 minutes of nitrogen gas

flowing over the droplet. The nitrogen gas was stopped after the mentioned times and

DIC images were taken using a Nikon inverted microscope eclipse TE2000-E with a

CoolSNAP HQ2 CCD camera and a Nikon Plan Fluor ELWD 20x/0.45 objective (0.32

µm/pix). With the purpose of determining the thickness of the viscous layer on the

droplet’s surface, the same experiment was done in 20, 40, 60, and 1000 μm below the

droplet’s surface after 150 seconds of nitrogen gas flowing over the droplet. The

experimental setup is shown in Figure 3.7 and Figure 3.8.

Chapter 3 43


Figure 3.7: The experimental setup. The inverted microscope, the zero humidity chamber, and the

pressure controller are shown in the image.

The Zero Humidity Chamber

The Pressure Controller

Chapter 3 44


Figure 3.8: The zero humidity chamber on the microscope stage.

y = 0.0255x - 0.1541R² = 0.9995

-8

-6

-4

-2

0

2

4

6

8

-300 -200 -100 0 100 200 300

Mo

men

t, e

mu

/g

Magnetic Field, Oe

The collagen droplet

Cylinder holes for pipetting

the collagen droplet

The permanent Magnet

Chapter 4 45


4 IMAGE ANALYSIS AND

RESULTS

The Magnetic and non-magnetic microspheres were distinguished based on their size and

tracked using MATLAB. A MATLAB code was adapted from a method of digital video

microscopy by Crocker and Grier [66] and Gao and Kilfoil [67]. The code finds and

tracks microspheres and produces the trajectory of each microsphere as its output. The

features are identified by finding high intensity regions in each frame. Then the magnetic

and non-magnetic microspheres are recognized based on their shape, size, and intensity.

A feature can be reidentified when it disappears for a few frames.

4.1 Grouped Microspheres

During the experiment, some of the close by magnetic microspheres were aggregated and

form long chains. The corresponding magnetic and drag forces for these grouped

microspheres are different from the single ones. Therefore, a minimum distance of three

micrometre between identified features was set to exclude all of the grouped

Chapter 4 46


microspheres. An example of the aggregated microspheres in glycerol 50% is shown in

Figure 4.1.

Figure 4.1: Single and grouped microspheres in the presence of the magnetic field.

4.2 Local Forces on the Magnetic Microspheres

In addition to magnetic and drag forces explained in section 3.1, there are other local

forces acting on the magnetic microspheres. For instance, the evaporation of the droplet,

the flow of nitrogen gas, and surface tensions can complicate the understanding of force

balance on each microsphere. Therefore, non-magnetic microspheres were introduced to

the collagen solution to overcome this problem and measure the movement of magnetic

microspheres only due to the magnetic field.

Chapter 4 47


The non-magnetic microspheres were recognized by their size (1.9 µm). For each frame,

the velocity of individual magnetic microspheres were subtracted by the velocity of the

closest (maximum distance of 100 μm) non-magnetic microspheres. Then, the average of

the relative velocities in the direction of the magnetic field for all magnetic microspheres

was used to calculate the viscosity of the surface of the droplet (equation (3.10)).

Figure 4.2 shows the inverted DIC image of the magnetic and non-magnetic

microspheres inside the collagen droplet after 150 seconds.

Figure 4.2: The 2.8 µm magnetic (M) and 1.9 µm non-magnetic (N) microspheres are

distinguished by their size.

The trajectories of magnetic and non-magnetic microspheres at 1000 µm depth, 20 µm

depth, and on the droplet surface are shown respectively in Figure 4.3, Figure 4.4, and

Chapter 4 48


Figure 4.5. Sequences of 30 images (one-second intervals) were taken from the

microspheres at different depths in the droplet and post processed using ImageJ (FIJI,

TrackMate plugin). The initial positions of microspheres are represented by small circles.

The trajectory lines are color coded based on the microspheres velocity. Though the

magnet is on the right side, microspheres do not necessarily travel to the right side. This

local direction of flow in the droplet is dominant. However, as it can be seen more vividly

in Figure 4.3, some of the particles (magnetic microspheres) are slightly dragged to the

right compared to the other particles (non-magnetic microspheres).

Figure 4.3: Trajectory of magnetic and non-magnetic microspheres at 1000 µm below the droplet

surface.

Chapter 4 49


Figure 4.4: Trajectory of magnetic and non-magnetic microspheres at 20 µm below the droplet

surface.

Figure 4.5: Trajectory of magnetic and non-magnetic microspheres on the droplet surface.

Chapter 4 50


4.3 Collagen Viscosity

The collagen concentration was predicted based on data provided by Advanced

BioMatrix, Inc. for viscosity and concentration of PureCol (bovine collagen solution,

type I) in the range of 0.88 to 6 mg/mL (Figure 4.6).

Figure 4.6: Viscosity of collagen for concentrations less than 6 mg/ml. The polynomial trend line

fitted on the data is used to predict the viscosity of higher concentrations.

4.4 Concentration of the Droplet Surface

The change of surface concentration was measured over the first three minutes after the

droplet was placed on the cover glass and exposed to the nitrogen gas. Figure 4.7 shows a

y = 0.4988x3 + 1.451x2 + 1.1647x + 0.0256

R² = 0.9995

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6 7

Vis

cosi

ty, cp

Collagen Concentration, mg/ml

Chapter 4 51


linear increase of the surface concentration during the fiber pulling experiment. The data

demonstrates a concentration of approximately 14 mg/mL collagen after 150 seconds on

the droplet surface.

Figure 4.7: Concentration of the droplet surface over three minutes (n=5 per time point).

4.5 Thickness of the Dense Layer

Figure 4.8 shows the drastic change of concentration in a micro-layer on the droplet

surface. The concentration decreased to 7.9 mg/ml at only 20 µm below the surface. This

shows the thickness of the thin layer of dense collagen is less than 20 µm and the fiber

can only initiate in this region. The data also shows the concentration of bulk region does

0

5

10

15

20

25

0 50 100 150 200 250

Conce

ntr

ati

on, m

g/m

L

Time, sec

Chapter 4 52


not change markedly. The initial solution was made at 4.4 mg/ml and it increased to 4.6

after 150 seconds inside the droplet (1000 µm below the surface).

Figure 4.8: Concentration of the droplet after 150 seconds at 0, 20, 40, 60, and 1000 µm below

the surface (n=5 per data point)

4.6 Confirming Fiber Pull Time

Existence of the permanent magnet in the experiment chamber effects on the nitrogen gas

flow. Furthermore, the presence of magnetic microspheres might change the collagen

behavior in the droplet. Validation of the fiber pulling experiment in the new setup of

experiment was investigated by repeating the fiber pulling experiment. The fiber was

0

2

4

6

8

10

12

14

16

18

20

1 10 100 1,000

Conce

ntr

ati

on, m

g/m

L

Depth, µm

Chapter 4 53


pulled after the same wait time (150 seconds) and with same maximum pull speed (240

µm/s).

The pulled fibers were immersed in a dish of PBS. The DIC images of these fibers were

similar to the previous fibers. The organized structure of these fibers are shown in and

Figure 4.9. Comparing these images to Figure 1.7 which was pulled under silicon oil can

illustrate the organization of collagen fibrils in these fibers.

Figure 4.9: DIC image of collagen fiber in PBS. The collagen fiber was pulled after 150 seconds

in the new experimental setup for viscosity measurement.

Chapter 4 54


4.7 Control Test

The post processing method and experimental setup of viscosity measurement was

controlled by measuring the viscosity of glycerol 99% v/v. The glycerol 99% v/v was

made by adding a 4 μl volume of the magnetic microspheres (Dynabeads M-270

Carboxylic Acid, Invitrogen, 2.8 μm, Catalog No. 14305D, 1 part to 3 part deionized

water) and a 4 μl of non-magnetic microspheres (Fluoro-Max green fluorescent polymer

microspheres, Thermo Scientific, 1.9 μm, Catalog No. G0200, 1 part to 9 part deionized

water) to a 1040 ml volume of glycerol (Fisher Scientific, 99.8% assay, f.w. 92.09, CAS

56-81-5).

Viscosity of the bulk region of the droplet was measured with the same method that used

for collagen droplet. An average viscosity of 1120 cp was measured for glycerol 99% v/v

at 21°C temperature (n = 9, standard deviation = 121). The viscosity of glycerol 99% v/v

at 21°C temperature is reported as 1086 cp [68]. The approximation error of 3.13% shows

the high accuracy of the experiment.

4.8 Summary

In the present study, the viscosity of a collagen droplet was measured as a function of

time and depth. In the experimental setup, a magnetic field was set up near the collagen

droplet and the movements of some magnetic microspheres were tracked inside the

droplet. Then the viscosity of the droplet was measured using the Stokes drag force on

each magnetic microshere.

Chapter 4 55


The concentration of collagen was estimated based on the viscosity-concentration data

for the collagen source. The post processing of data showed a concentration of

approximately 14 mg/ml was attained at the top surface of the droplet after 150 seconds.

The thickness of this layer was measured less than 20 µm. The concentration dropped to

8 mg/ml when it measured at 20 µm below the surface.

The suspended particles on the top of the droplet may possibly experience a complicated

movement. For instance, surface tensions and airflow over the droplet can change the

force balance on the microspheres. In addition, the diameter and magnetic susceptibility

of the microspheres are not completely constant. These uncertainties are the most

important source of error in this viscosity measurement method.

The result of this study gave an important information in the critical surface

concentration where collagen fibers can be formed, which can lead to the design of a

more efficient and predictable methodology to produce highly organized collagen fibers.

Chapter 5 56


5 APPENDICES

APPENDIX 1: BROWNIAN MOTION OF THE MICROSPHERES ................................................ 57

APPENDIX 2: TERMINAL VELOCITY OF THE MICROSPHERES IN A SOLUTION WITH A

VISCOSITY OF 100 CP ................................................................................................. 64

APPENDIX 3: DETERMINING THE TERMINAL VELOCITY OF THE MICROSPHERES USING

RUNGE-KUTTA 4TH ORDER METHOD ......................................................................... 66

APPENDIX 4: RENAMING AND INVERTING OF FRAMES ....................................................... 68

APPENDIX 5: VELOCITY OF THE MAGNETIC MICROSPHERES .............................................. 69

APPENDIX 6: VELOCITY OF THE NON-MAGNETIC MICROSPHERES ...................................... 71

APPENDIX 7: VISCOSITY MEASUREMENT ........................................................................... 73

Chapter 5 57


APPENDIX 1: BROWNIAN MOTION OF THE MICROSPHERES

1. clear

2. clc

3. close all

4. % The image sequences of the beads on the surface of the collagen droplet was

post processed using ImageJ and FIJI

5. % Threshold = 22

6. % Linking Max Distance = 50 pix

7. % Closing Max Distance = 50 pix

8. % Gap-Closing Max Frame = 10

9. % The result of the beads tracking was extracted to an Excel file, “AllFrames”,

and imported into Matlab as “RawData”

10. filename = 'AllFrames.xlsx';

11. sheet = 1;

12. RawData = xlsread(filename);

13. % Column 1: Bead ID

14. % Column 2: Bead X Position

15. % Column 3: Bead Y Position

16. % Column 4: Frame Number

17. % 20X Objective (0.33 micrometer per pixle)

18. RawData(:,2) = 0.33.*RawData(:,2);

19. RawData(:,3) = 0.33.*RawData(:,3);

20. BeadID = RawData(:,1);

21. % The first column of RawData is Beads ID numbers

22. x = RawData(:,2);

23. % The second column of RawData is the X position of beads in micrometer

Chapter 5 58


24. y = RawData(:,3);

25. % The third column of RawData is the Y position of beads in micrometer

26. Frame = RawData(:,4);

27. % The fourth column of RawData is the frame number of beads

28. f = max(Frame);

29. [H,~] = size(RawData);

30. % The (delta x / delta frame) of each bead was calculated for all frames

31. for i=1:H-1

32. if BeadID(i) == BeadID(i+1)

33. RawData(i+1,5) = (x(i+1)-x(i))/(Frame(i+1)-Frame(i));

34. RawData(i+1,6) = (y(i+1)-y(i))/(Frame(i+1)-Frame(i));

35. else

36. RawData(i+1,5) = NaN;

37. RawData(i+1,6) = NaN;

38. end

39. end

40. for i=1:H

41. if RawData(i,5) == 0

42. RawData(i,5) = NaN;

43. end

44. if RawData(i,6) == 0

45. RawData(i,6) = NaN;

46. end

47. end

48. % Delta X and Delta Y movement of beads were classified according to the frame

number

49. FrameDataX = NaN;

50. FrameDataY = NaN;

Chapter 5 59


51. for i=1:f

52. k = 1;

53. for j=1:H

54. if Frame(j) == i

55. FrameDataX(i,k) = RawData(j,5);

56. FrameDataY(i,k) = RawData(j,6);

57. k = k+1;

58. end

59. end

60. end

61. [~,L] = size(FrameDataX);

62. for i=1:f

63. for j=1:L

64. if FrameDataX(i,j) == 0

65. FrameDataX(i,j) = NaN;

66. end

67. if FrameDataY(i,j) == 0

68. FrameDataY(i,j) = NaN;

69. end

70. end

71. end

72. % Average of Delta X and Delta Y movement of beads

73. FrameDataXMean = NaN(f,1);

74. FrameDataYMean = NaN(f,1);

75. for i=1:f

76. FrameDataXMean(i,1) = nanmean(FrameDataX(i,:));

77. FrameDataYMean(i,1) = nanmean(FrameDataY(i,:));

78. end

Chapter 5 60


79. % To reduce the error of drift or vibrational noise, the average of delta X and Y

were subtracted from original delta X and Y

80. FrameDeltaX = NaN;

81. FrameDeltaY = NaN;

82. for i=1:f

83. for j=1:L

84. if isreal(FrameDataX(i,j))

85. FrameDeltaX(i,j) = FrameDataX(i,j)-FrameDataXMean(i,1);

86. end

87. if isreal(FrameDataY(i,j))

88. FrameDeltaY(i,j) = FrameDataY(i,j)-FrameDataYMean(i,1);

89. end

90. end

91. end

92. for i=1:f

93. for j=1:L

94. if FrameDeltaX(i,j) == 0

95. FrameDeltaX(i,j) = NaN;

96. end

97. if FrameDeltaY(i,j) == 0

98. FrameDeltaY(i,j) = NaN;

99. end

100. end

101. end

102. SumFrameDeltaX = NaN(f,1);

103. SumFrameDeltaY = NaN(f,1);

104. for i=1:f

105. if isnan(FrameDeltaX(i,1))

106. SumFrameDeltaX(i,1) = NaN;

Chapter 5 61


107. SumFrameDeltaY(i,1) = NaN;

108. else

109. SumFrameDeltaX(i,1) = nansum(FrameDeltaX(i,:));

110. SumFrameDeltaY(i,1) = nansum(FrameDeltaY(i,:));

111. end

112. end

113. % The data have a zero mean displacement

114. figure(1)

115. subplot(2,1,1)

116. nbins = 200;

117. hist(SumFrameDeltaX,nbins);

118. hold on;

119. subplot(2,1,2)

120. hist(SumFrameDeltaY,nbins);

121. hold on;

122. % Seperation of all delta x and y data for each n frame

123. [H,L] = size(FrameDeltaX);

124. X = NaN;

125. k = 1;

126. for i=1:H

127. for j=1:L

128. if isfinite(FrameDeltaX(i,j))

129. X(k,1) = FrameDeltaX(i,j);

130. k = k+1;

131. end

132. end

133. end

134. [H,L] = size(X);

Chapter 5 62


135. for i=1:H

136. for j=1:L

137. if X(i,j) == 0

138. X(i,j) = NaN;

139. end

140. end

141. end

142. pd = NaN;

143. SD = NaN;

144. MSDX = NaN;

145. D = NaN;

146. Vis = NaN;

147. for i=1:L

148. figure(i+1)

149. nbins = 100;

150. hist(X(:,i),nbins);

151. % Creates a histogram bar chart of one-dimensional displacements (delta x) for

every 10 frames(4.35 fps)

152. pd = fitdist(X(:,i),'normal');

153. % Fit probability distribution object to the histogram bar

154. SD = pd.sigma;

155. % Standard Deviation

156. k = -2:0.01:2;

157. PDF = pdf(pd,k);

158. % Probability density function of the probability distribution object

159. PDF = PDF ./ max(PDF);

160. % Normalised

161. % scale to y axis

162. y=ylim;

163. PDF = PDF*y(2);

164. hold on;

Chapter 5 63


165. plot(k,PDF,'r-','LineWidth',2);

166. MSDX(i) = SD.^2;

167. % The variance of the Gaussian fit (pd) is equivalent to the one-dimensional

Mean Square Displacement if the mean displacement is zero [um/frame]

168. t = 233.493*10^-3;

169. % The time between data points [s]

170. D(i) = MSDX(i)./(2*t);

171. % Diffusion [um^2/s]

172. a = (1/2)*(10^-6);

173. % Bead radius

174. Vis(i) = (1000)*(1.3806488*(10^-23))*(294)/(6)/(3.14)/a./(D(i).*(10^-12));

175. % Viscosity [cp]

176. end

177. figure(L+2)

178. T = 5:10:10*L-5;

179. plot(T,Vis,'b.');

180. xlabel('Time [s]');

181. ylabel('Viscosity [cp]');

Chapter 5 64


APPENDIX 2: TERMINAL VELOCITY OF THE MICROSPHERES

IN A SOLUTION WITH A VISCOSITY OF 100 CP

1. clc; 2. clear all; 3. h = 0.1e-9; 4. % Time steps [s] 5. t = 0:h:5e-8; 6. % Time [s] 7. v = zeros(1,length(t)); 8. % Velocity of Microspheres [m/s] 9. v(1) = 0; 10. % Initial velocity 11. n = 100e-3; 12. % Viscosity [kg/(s.m)] 13. R = (2.8e-6)/2; 14. % Radius of Microspheres [m] 15. rho = 1.6e3; 16. % Density of Microspheres [kg/m^3] 17. V = 4/3*pi*R^3; 18. % Volume of Microspheres [m^3] 19. C2 = 6*pi*n*R/rho/V; 20. s = 0.5; 21. % Susceptibility of the Microspheres 22. B = 593e-4; 23. % Magnetic Field [T] 24. dB = 2.953; 25. % Magnetic Field's Gradient [T/m] 26. mu = 4*pi*10^-7; 27. % The Permeability of the vacuum [T.m/A] 28. C1 = s/rho/mu*B*dB;

Chapter 5 65


29. for i=2:(length(t)-1) 30. v(i) = v(i-1)+(C1-C2*v(i-1))*h; 31. end

32. filename = '100cp.xlsx'; 33. xlswrite(filename,t,'Time'); 34. xlswrite(filename,v,'Velocity');

Chapter 5 66


APPENDIX 3: DETERMINING THE TERMINAL VELOCITY OF

THE MICROSPHERES USING RUNGE-KUTTA 4TH ORDER

METHOD

1. clc; 2. clear all; 3. h = 0.1e-9; 4. % step size 5. x = 0:h:5e-8; 6. y = zeros(1,length(x)); 7. y(1) = 0; 8. % initial condition 9. n = 100e-3; 10. % Viscosity [cp] 11. R = (2.8e-6)/2; 12. % Magnetic Bead's Radius [m] 13. rho = 1.6e3; 14. % Magnetic Bead's Density [kg/m^3] 15. V = 4/3*pi*R^3; 16. % Magnetic Bead's Volume [m^3] 17. C2 = 6*pi*n*R/rho/V; 18. % Constant Number 19. s = 0.5; 20. % Bead's magnetic susceptibility 21. B = 593e-4; 22. % Magnetic Field [T] 23. dB = 2.953; 24. % Magnetic Field's Gradient [T/m] 25. mu = 4*pi*10^-7; 26. % The permeability of the vacuum [T.m/A] 27. C1 = s/rho/mu*B*dB; 28. % Constant Number 29. F_xy = @(t,r) C1-C2*r;

Chapter 5 67


30. for i=1:(length(x)-1) 31. % calculation loop 32. k_1 = F_xy(x(i),y(i)); 33. k_2 = F_xy(x(i)+0.5*h,y(i)+0.5*h*k_1); 34. k_3 = F_xy((x(i)+0.5*h),(y(i)+0.5*h*k_2)); 35. k_4 = F_xy((x(i)+h),(y(i)+k_3*h)); 36. y(i+1) = y(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; 37. % main equation 38. end

39. y' 40. figure(1) 41. plot(x*1e9,y*1e9) 42. title('Prediction of Terminal Velocity by Runge-Kutta 4th Order Method') 43. xlabel('Time [ns]') 44. ylabel('Velocity of Magnet Beads [nm/s]')

Chapter 5 68


APPENDIX 4: RENAMING AND INVERTING OF FRAMES

1. % Renaming a File in the Current Folder 2. a ='E:\Experiments\...\'; 3. A = dir( fullfile(a, '*.tif') ); 4. fileNames = { A.name }ν; 5. for iFile = 1 : numel( A ) 6. newName = fullfile(a, sprintf( 'fov1_%04d.tif', iFile ) ); 7. movefile( fullfile(a, fileNames{ iFile }), newName ); 8. end

9. % Inverting all the frames

10. clear 11. clc 12. close all 13. fovn = 1;

14. % ID# for the series of images (typically, one field of view) 15. pathin = ' E:\Experiments\...\'; 16. numframes = xlsread('Book1.xlsx','Experiment Data');

17. % The number of images you have in your series 18. for x = 1:numframes 19. strnam=[pathin 'fov' num2str(fovn) '\fov' num2str(fovn) '_' num2str(x,'%04i')

'.tif']; 20. img = imread(strnam); 21. img = abs(4096-img); 22. imwrite(img, strnam); 23. end

Chapter 5 69


APPENDIX 5: VELOCITY OF THE MAGNETIC MICROSPHERES

1. clear 2. clc 3. close all 4. load res_fovM; 5. A(:,1) = res(:,8); 6. A(:,2) = res(:,1); 7. A(:,3) = res(:,2); 8. A(:,4) = res(:,6); 9. % Column 1: Bead ID 10. % Column 2: Bead X Position 11. % Column 3: Bead Y Position 12. % Column 4: Frame Number

13. T = xlsread('Book1.xlsx','Recorded Data', 'B:B'); % The time at which the image

was recorded (Sorted by frame#) 14. % 20X Objective (0.32 micrometer per pixle) 15. A(:,2) = 0.32.*A(:,2); 16. A(:,3) = 0.32.*A(:,3); 17. [n,~] = size(A); 18. A(:,5) = NaN; 19. A(:,6) = NaN; 20. % The velocity of each bead was calculated for all frames

21. for i=1:n-1 22. if A(i,1) == A(i+1,1) && A(i+1,4)-A(i,4)==1 23. A(i+1,5) = (A(i+1,2)-A(i,2))./(T(A(i+1,4))-T(A(i,4))); 24. A(i+1,6) = (A(i+1,3)-A(i,3))./(T(A(i+1,4))-T(A(i,4))); 25. else 26. A(i+1,5) = NaN; 27. A(i+1,6) = NaN; 28. end

Chapter 5 70


29. end 30. save Magnet.mat A;

Chapter 5 71


APPENDIX 6: VELOCITY OF THE NON-MAGNETIC

MICROSPHERES

1. clear 2. clc 3. close all

4. load res_fovF; 5. B(:,1) = res(:,8); 6. B(:,2) = res(:,1); 7. B(:,3) = res(:,2); 8. B(:,4) = res(:,6); 9. % Column 1: Bead ID 10. % Column 2: Bead X Position 11. % Column 3: Bead Y Position 12. % Column 4: Frame Number

13. T = xlsread('Book1.xlsx','Recorded Data', 'B:B');

14. % The time at which the image was recorded (Sorted by frame#) 15. % 20X Objective (0.32 micrometer per pixle) 16. B(:,2) = 0.32.*B(:,2); 17. B(:,3) = 0.32.*B(:,3);

18. [n,~] = size(B); 19. B(:,5) = NaN; 20. B(:,6) = NaN;

21. % The velocity of each bead was calculated for all frames 22. for i=1:n-1 23. if B(i,1) == B(i+1,1) && B(i+1,4)-B(i,4)==1 24. B(i+1,5) = (B(i+1,2)-B(i,2))./(T(B(i+1,4))-T(B(i,4)));

Chapter 5 72


25. B(i+1,6) = (B(i+1,3)-B(i,3))./(T(B(i+1,4))-T(B(i,4))); 26. else 27. B(i+1,5) = NaN; 28. B(i+1,6) = NaN; 29. end 30. end

31. save Fluor.mat B;

Chapter 5 73


APPENDIX 7: VISCOSITY MEASUREMENT

1. clear

2. clc

3. close all

4. load Fluor;

5. load Magnet;

6. [n,~] = size(A);

7. [m,~] = size(B);

8. C = NaN(n,4);

9. T = xlsread('Book1.xlsx','Recorded Data', 'B:B');

10. % The time at which the image was recorded (Sorted by frame#)

11. for i=1:n

12. for j=1:m

13. L = sqrt((A(i,2)-B(j,2))^2 + (A(i,3)-B(j,3))^2);

14. if B(j,4)==A(i,4) && L < 3

15. A(i,5)=NaN;

16. A(i,6)=NaN;

17. end

18. end

19. end

20. for i=1:n-1

21. for j=i+1:n

22. L = sqrt((A(i,2)-A(j,2))^2 + (A(i,3)-A(j,3))^2);

23. if A(j,4)==A(i,4) && L < 3

24. A(i,5)=NaN;

25. A(i,6)=NaN;

26. end

27. end

28. end

Chapter 5 74


29. for i=1:n

30. C(i,1) = A(i,4);

31. C(i,2) = T(A(i,4));

32. L = 100;

33. for j=1:m

34. if B(j,4)==A(i,4)

35. LL = sqrt((A(i,2)-B(j,2))^2 + (A(i,3)-B(j,3))^2);

36. if LL < L

37. L = LL;

38. C(i,3) = A(i,5)-B(j,5);

39. C(i,4) = A(i,6)-B(j,6);

40. end

41. end

42. end

43. end

44. % Measure the velosity in X & Y direction for each bead [um/s]

45. Vx = nanmean(C(:,3));

46. Vy = nanmean(C(:,4));

47. D(:,1) = C(:,1);

48. D(:,2) = C(:,2);

49. Alpha = 0.05;

50. rep = 1;

51. [D(:,3), IDX, OUTLIERSX] = deleteoutliers(C(:,3),Alpha,rep);

52. [D(:,4), IDY, OUTLIERSY] = deleteoutliers(C(:,4),Alpha,rep);

53. Vx_Grubbs = nanmean(D(:,3));

54. Vy_Grubbs = nanmean(D(:,4));

55. STx = nanstd(D(:,3));

56. STy = nanstd(D(:,4));

57. Z = [Vx,Vy;Vx_Grubbs,Vy_Grubbs;STx,STy];

58. Res = 'Result.xlsx';

59. xlswrite(Res,Z,'B2:C4')

Chapter 5 75


60. R = (2.8/2)*10^-6; % Magnetic Bead's Radius [m]

61. m = 4*pi*10^-7; % The Permibility of the vacuum [T.m/A]

62. x = 0.51; % Bead's magnetic susceptibility

63. b = 593e-4; % Magnetic Field [T]

64. dB = 2.953; % Magnetic Field's Gradient [T/m]

65. vis = 1000*(2/9/(Vx_Grubbs*10^-6)*R^2*x/m*b*dB); % Bead's Velocity cp]

66. xlswrite(Res,vis,'B5:B5')

Chapter 6 76


6 REFERENCES

1. Kadler, K.E., et al., Collagen fibril formation. Biochem J, 1996. 316 ( Pt 1): p. 1-

11.

2. Canty, E.G. and K.E. Kadler, Procollagen trafficking, processing and

fibrillogenesis. J Cell Sci, 2005. 118(Pt 7): p. 1341-53.

3. Kadler, K.E., et al., Collagens at a glance. Journal of Cell Science, 2007.

120(12): p. 1955-1958.

4. Friess, W., Collagen--biomaterial for drug delivery. Eur J Pharm Biopharm,

1998. 45(2): p. 113-36.

5. Kadler, K.E., A. Hill, and E.G. Canty-Laird, Collagen fibrillogenesis: fibronectin,

integrins, and minor collagens as organizers and nucleators. Curr Opin Cell Biol,

2008. 20(5): p. 495-501.

6. Grinnell, F., Fibroblast biology in three-dimensional collagen matrices. Trends

Cell Biol, 2003. 13(5): p. 264-9.

7. Kannus, P., Structure of the tendon connective tissue. Scand J Med Sci Sports,

2000. 10(6): p. 312-20.

8. Birk, D.E. and R.L. Trelstad, Extracellular compartments in tendon

morphogenesis: collagen fibril, bundle, and macroaggregate formation. J Cell

Biol, 1986. 103(1): p. 231-40.

9. Birk, D.E., et al., Collagen fibril bundles: a branching assembly unit in tendon

morphogenesis. Development, 1989. 107(3): p. 437-43.

10. Birk, D.E., M.V. Nurminskaya, and E.I. Zycband, Collagen fibrillogenesis in situ:

fibril segments undergo post-depositional modifications resulting in linear and

lateral growth during matrix development. Dev Dyn, 1995. 202(3): p. 229-43.

11. Birk, D.E., et al., Collagen fibrillogenesis in situ: fibril segments become long

fibrils as the developing tendon matures. Dev Dyn, 1997. 208(3): p. 291-8.

12. Kadler, K.E., Y. Hojima, and D.J. Prockop, Assembly of collagen fibrils de novo

by cleavage of the type I pC-collagen with procollagen C-proteinase. Assay of

critical concentration demonstrates that collagen self-assembly is a classical

example of an entropy-driven process. J Biol Chem, 1987. 262(32): p. 15696-701.

Chapter 6 77


13. Nurminskaya, M.V. and D.E. Birk, Differential expression of genes associated

with collagen fibril growth in the chicken tendon: identification of structural and

regulatory genes by subtractive hybridization. Arch Biochem Biophys, 1998.

350(1): p. 1-9.

14. Kadler, K.E., Y. Hojima, and D.J. Prockop, Collagen fibrils in vitro grow from

pointed tips in the C- to N-terminal direction. Biochem J, 1990. 268(2): p. 339-43.

15. Fratzl, P., Collagen : structure and mechanics. 2008, New York: Springer. xviii,

506 p.

16. van der Rest, M. and R. Garrone, Collagen family of proteins. FASEB J, 1991.

5(13): p. 2814-23.

17. Boot-Handford, R.P. and D.S. Tuckwell, Fibrillar collagen: the key to vertebrate

evolution? A tale of molecular incest. Bioessays, 2003. 25(2): p. 142-51.

18. Hendershot, L.M. and N.J. Bulleid, Protein-specific chaperones: the role of hsp47

begins to gel. Curr Biol, 2000. 10(24): p. R912-5.

19. Porter, S., et al., The ADAMTS metalloproteinases. Biochem J, 2005. 386(Pt 1):

p. 15-27.

20. Risteli, J., et al., Radioimmunoassay for the Pyridinoline Cross-Linked Carboxy-

Terminal Telopeptide of Type-I Collagen - a New Serum Marker of Bone-

Collagen Degradation. Clinical Chemistry, 1993. 39(4): p. 635-640.

21. Akeson, W.H., et al., Collagen cross-linking alterations in joint contractures:

changes in the reducible cross-links in periarticular connective tissue collagen

after nine weeks of immobilization. Connect Tissue Res, 1977. 5(1): p. 15-9.

22. Eyre, D.R., M.A. Paz, and P.M. Gallop, Cross-linking in collagen and elastin.

Annu Rev Biochem, 1984. 53: p. 717-48.

23. Reiser, K., R.J. McCormick, and R.B. Rucker, Enzymatic and nonenzymatic cross-

linking of collagen and elastin. FASEB J, 1992. 6(7): p. 2439-49.

24. Kadler, K., Matrix loading: assembly of extracellular matrix collagen fibrils

during embryogenesis. Birth Defects Res C Embryo Today, 2004. 72(1): p. 1-11.

25. Hulmes, D.J.S., Building collagen molecules, fibrils, and suprafibrillar structures.

Journal of Structural Biology, 2002. 137(1-2): p. 2-10.

26. Paten, J., Investigation into the mechano-chemistry of de novo collagen assembly.

2014.

27. Gupta, S.V., Viscometry for liquids : calibration of viscometers. Cham : Springer,

2014. Vol. Springer series in materials science ; v.194. 2014.

28. White, F.M., Fluid mechanics. 6th ed. McGraw-Hill series in mechanical

engineering. 2008, New York: McGraw-Hill. xiii, 864 p. ill.

29. Viswanath, D.S., Viscosity of liquids : theory, estimation, experiment, and data.

2007, Dordrecht: Springer. xiii, 660 p.

30. Daniels, F., Experimental physical chemistry. 5th ed. 1956, New York,: McGraw-

Hill. 482 p.

31. Bingham, E.C., Fluidity and plasticity. 1st ed. 1922, New York etc.: McGraw-Hill

book company, inc. xi, 440 p.

32. Barr, G., Errors in viscometry due to surface tension. National Physical

Laboratory, Teddington, 1946.

33. Dinsdale, A. and F. Moore, Viscosity and its measurement. Monographs for

students. 1962, London,

Chapter 6 78


New York,: Published on behalf of the Institute of Physics and the Physical Society by

Chapman and Hall;

Reinhold. 67 p.

34. Wakeham, W.A., et al., Measurement of the transport properties of fluids.

Experimental thermodynamics. 1991, Oxford England ; Boston

Boca Raton, Fla.: Blackwell Scientific Publications ;

Distributed in the USA and North America by CRC Press. xx, 459 p.

35. Dontula, P., C.W. Macosko, and L.E. Scriven, Origins of concentric cylinders

viscometry. Journal of Rheology, 2005. 49(4): p. 807-818.

36. McKennell, R., Cone-Plate Viscometer. Analytical Chemistry, 1956. 28(11): p.

1710-1714.

37. Yabuno, H., et al., Self-excited vibrational viscometer for high-viscosity sensing.

Journal of Applied Physics, 2014. 116(12).

38. Akpek, A., C. Youn, and T. Kagawa, Temperature Measurement Control Problem

of Vibrational Viscometers Considering Heat Generation and Heat Transfer

Effect of Oscillators. 2013 9th Asian Control Conference (Ascc), 2013.

39. Osborne, M.F.M., The theory of the measurement of surface viscosity. Kolloid-

Zeitschrift und Zeitschrift für Polymere, 1968. 224(2): p. 150-161.

40. Alonso, C. and J.A. Zasadzinski, Linear dependence of surface drag on surface

viscosity. Physical Review E, 2004. 69(2).

41. Ghaskadvi, R.S., et al., Shear response of Langmuir monolayers of heneicosanoic

(C-21) acid studied using a torsion pendulum. Physical Review E, 1996. 54(2): p.

1770-1773.

42. Petkov, J.T., et al., Precise method for measuring the shear surface viscosity of

surfactant monolayers. Langmuir, 1996. 12(11): p. 2650-2653.

43. Waugh, R.E., Surface Viscosity Measurements from Large Bilayer Vesicle Tether

Formation .1. Analysis. Biophysical Journal, 1982. 38(1): p. 19-27.

44. Waugh, R.E., Surface Viscosity Measurements from Large Bilayer Vesicle Tether

Formation .2. Experiments. Biophysical Journal, 1982. 38(1): p. 29-37.

45. Ziemann, F., J. Radler, and E. Sackmann, Local Measurements of Viscoelastic

Moduli of Entangled Actin Networks Using an Oscillating Magnetic Bead Micro-

Rheometer. Biophysical Journal, 1994. 66(6): p. 2210-2216.

46. Bausch, A.R., et al., Local measurements of viscoelastic parameters of adherent

cell surfaces by magnetic bead microrheometry. Biophysical Journal, 1998.

75(4): p. 2038-2049.

47. Evans, E. and E. Sackmann, Translational and Rotational Drag Coefficients for a

Disk Moving in a Liquid Membrane-Associated with a Rigid Substrate. Journal of

Fluid Mechanics, 1988. 194: p. 553-561.

48. Rotkiewicz, K., K.H. Grellmann, and Z.R. Grabowski, Reinterpretation of the

anomalous fluorescense of p-n,n-dimethylamino-benzonitrile. Chemical Physics

Letters, 1973. 19(3): p. 315-318.

49. Haidekker, M.A. and E.A. Theodorakis, Environment-sensitive behavior of

fluorescent molecular rotors. J Biol Eng, 2010. 4: p. 11.

Chapter 6 79


50. Goh, W.L., et al., Molecular rotors as conditionally fluorescent labels for rapid

detection of biomolecular interactions. J Am Chem Soc, 2014. 136(17): p. 6159-

62.

51. Haidekker, M.A., et al., Effects of solvent polarity and solvent viscosity on the

fluorescent properties of molecular rotors and related probes. Bioorg Chem,

2005. 33(6): p. 415-25.

52. Gavvala, K., et al., Modulation of excimer formation of 9-(dicyano-

vinyl)julolidine by the macrocyclic hosts. Phys Chem Chem Phys, 2013. 15(1): p.

330-40.

53. Iwaki, T., et al., Antibodies for fluorescent molecular rotors. Biochemistry, 1993.

32(29): p. 7589-92.

54. Sawada, S., et al., Fluorescent rotors and their applications to the study of G-F

transformation of actin. Anal Biochem, 1992. 204(1): p. 110-7.

55. Kung, C.E. and J.K. Reed, Fluorescent molecular rotors: a new class of probes

for tubulin structure and assembly. Biochemistry, 1989. 28(16): p. 6678-86.

56. Vaccaro, G., et al., Direct monitoring of self-assembly of copolymeric micelles by

a luminescent molecular rotor. Chem Commun (Camb), 2013. 49(76): p. 8474-6.

57. Förster, T.H., G., Effect of viscosity on the fluorescence quantum yield of some

dye systems.

. Zeitschrift für Physikalische Chemie, 1971. 75: p. 63–76.

58. Kung, C.E. and J.K. Reed, Microviscosity Measurements of Phospholipid-

Bilayers Using Fluorescent Dyes That Undergo Torsional Relaxation.

Biochemistry, 1986. 25(20): p. 6114-6121.

59. Howell, S., et al., Intrinsic and extrinsic temperature-dependency of viscosity-

sensitive fluorescent molecular rotors. J Fluoresc, 2012. 22(1): p. 457-65.

60. Haidekker, M.A., et al., A novel approach to blood plasma viscosity measurement

using fluorescent molecular rotors. Am J Physiol Heart Circ Physiol, 2002.

282(5): p. H1609-14.

61. Kuimova, M.K., et al., Molecular rotor measures viscosity of live cells via

fluorescence lifetime imaging. J Am Chem Soc, 2008. 130(21): p. 6672-3.

62. Levitt, J.A., et al., Membrane-Bound Molecular Rotors Measure Viscosity in Live

Cells via Fluorescence Lifetime Imaging. Journal of Physical Chemistry C, 2009.

113(27): p. 11634-11642.

63. Hungerford, G., et al., Monitoring sol-to-gel transitions via fluorescence lifetime

determination using viscosity sensitive fluorescent probes. J Phys Chem B, 2009.

113(35): p. 12067-74.

64. Brown, R., A brief account of microscopical observations made in the months of

June, July and August, 1827, on the particles contained in the pollen of plants;

and on the general existence of active molecules in organic and inorganic bodies.

1828. p. 161-173.

65. Einstein, A., Über die von der molekularkinetischen Theorie der Wärme

geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen.

Annalen der Physik, 1905. 322(8): p. 549-560.

66. Crocker, J.C. and D.G. Grier, Methods of digital video microscopy for colloidal

studies. Journal of Colloid and Interface Science, 1996. 179(1): p. 298-310.

Chapter 6 80


67. Gao, Y.X. and M.L. Kilfoil, Accurate detection and complete tracking of large

populations of features in three dimensions. Optics Express, 2009. 17(6): p. 4685-

4704.

68. Cheng, N.S., Formula for the viscosity of a glycerol-water mixture. Industrial &

Engineering Chemistry Research, 2008. 47(9): p. 3285-3288.

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