Extensional Viscosity of Complex Fluids and the Effects of Pre-Shear
by
Anna E. Park
B.S., Mechanical Engineering
Massachusetts Institute of Technology, 2001
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
MASSACHUSETTS iNSTITUTE
February 2003
JUL08 2003
OF TECHNOLOGY
LIBRARIES
@ 2003 Massachusetts Institute of Technology
All rights reserved
Signature of Author..............................
........... .
.. .. . . .
Department of Mechanical Engineering
January 17, 2003
C ertified by.................................
Gareth H. kckinley
Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by......................................
Ain A. Sonin
Chairman, Department Committee on Graduate Students
BARKER
Extensional Viscosity of Complex Fluids and the Effects of Pre-Shear
by
Anna E. Park
Submitted to the Department of Mechanical Engineering
on January 17, 2003 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
A study is performed to compute the transient extensional viscosity of a number of
complex fluids and examine the effects of shearing the fluids before extending them. The
test fluids are glycerol, an aqueous solution of 2wt% polyethylene oxide (PEO), a
polystyrene Boger fluid (containing 0.025wt%) with and without a dispersion of 200nm
clay particles, yogurt, and acrylic paint. The extensional flow is created using a Capillary
Breakup Rheometer (Caber). The main parts of the instrument are two coaxial cylinders
6mm in diameter and a laser micrometer that measures the midpoint diameter of the fluid
filament as it thins under the action of capillary forces. The test fluid is loaded between
the cylinders and a step extensional strain is applied to the fluid by raising the upper
cylinder. To compare the material properties measured using the Caber, measurements
are also made using other instruments such as a cone and plate rheometer and a
tensiometer.
The bottom portion of the device is modified to enable steady shearing of the samples
prior to testing. As expected, the extensional flow properties of a Newtonian fluid,
glycerol, are not affected by the pre-shear. Pre-shearing PEO, PS025, and PS025 with
3wt% clay particles over a range of shear strains from 12.57 rad to 37.70 rad at rates
ranging from 1.88 s-1 to 18.84 s- resulted in lower extensional viscosities due to the
shear-induced alignment of polymers. The PS solution containing 1 Owt% clay did not
show significant changes after being pre-sheared over the ranges of strains and rates
specified above. Pre-shearing yogurt caused the breakup time of the filament to decrease
with both increasing shear strain and shear rate due to disruption of the natural gel
structure. For materials such as paint samples with volatile solvents, moderate amounts
of pre-shearing modified the transient behavior of the filament and the ultimate time to
breakup. Increasing the total strain imposed from 37.70 rad at 6.28 s-1 to 125.66 rad at
6.28 s-1 did not change the resulting fluid behavior. From the research, it was found that
Caber provides a fast and simple way to test the extensional flow behaviors of a wide
range of fluids.
Thesis Supervisor: Gareth H. McKinley
Title: Professor of Mechanical Engineering
2
TABLE OF CONTENTS
Chapter 1
Introduction .......................................................
Page:
7
1.1
Purpose ......................................................................
7
1.2
Background .................................................................
8
1.2.1
Newtonian Fluid ..................................................
8
1.2.2
Non-Newtonian Fluid .............................................
8
1.2.3
Extensional Flow ..................................................
9
1.2.4
Preshear ............................................................
10
1.3
General Approach ............................................................
Chapter 2
2.1
2.2
Previous Works ..................................................
12
13
Liquid Filament ............................................................
13
2.1.1
Newtonian Fluid ..................................................
15
2.1.2
Viscoelastic Fluid ..................................................
16
2.1.3
Generalized Fit ....................................................
19
2.1.4
Power Law .........................................................
19
Extension Viscosity .......................................................
21
2.2.1
Theoretical Extensional Viscosity ...............................
21
2.2.2
Extensional Viscosity from Radius verses Time Data .......... 22
2.2.3
Five-point Centered Difference Formula .........................
23
2.2.4
Slope Formula for Unequally Spaced Points ..................
23
2.3
Pre-Shear .....................................................................
24
2.4
Non-dimensional Numbers ..................................................
24
2.4.1
Deborah Number ..................................................
24
2.4.2
Trouton Ratio .....................................................
25
2.4.3
Bond Number .........................................................
25
2.4.4
Weissenberg Number .............................................
26
3
Chapter 3
3.1
3.2
Obtaining Material Properties ...............................................
4.2
27
27
3.1.1
Tensiom eter ...........................................................
27
3.1.2
Shear Rheometer ..................................................
28
T est F luids .....................................................................
30
3.2.1
Glycerol ............................................................ 30
3.2.2
Polyethylene Oxide ................................................ 32
3.2.3
Polystyrene Boger Fluid ......................................... 33
3.2.4
Y ogurt ...............................................................
35
3.2.5
Paint ...............................................................
40
Experiment ..........................................................
42
Chapter 4
4.1
Materials ..........................................................
Apparatus .................................................................... 42
4.1.1
Capillary Breakup Extensional Rheometer (Caber) ............ 42
4.1.2
Pre-Shear ............................................................45
Procedures .....................................................................48
4.2.1
Running Caber ..................................................... 48
4.2.2
Optical Imaging ...................................................... 50
Results and Discussion .........................................
52
5.1
Newtonian Fluid ............................................................
52
5.2
Viscoelastic Fluid ..........................................................
56
5.2.1
PE O .................................................................
56
5.2.2
PS025 ............................................................... 60
Chapter 5
5.3
Power Law Fluid ..............................................................
73
5.4
Fluid with Volatile Solvent ................................................
80
Chapter 6
Conclusion and Future Work .................................
86
4
6.1
C onclusion .....................................................................
86
6.2
Future Work ................................................................
89
Bibliography ...........................................................................
90
5
ACKNOWLEDGEMENTS
I would like to thank my family and friends for giving me lots of support. I'm very
appreciative of the times when Dan made my late nights in lab more pleasant by eating
dinner with me and by keeping me company as I worked. I'm also appreciative of my
roommates, who helped me to stay grounded throughout the whole thesis writing process.
I send my appreciation to my advisor, Gareth McKinley, for advising me through my
graduate years at MIT. I'm thankful for the opportunity I was given to do research with
his group and have so many great resources available to me.
I want to thank my lab mates (especially Jose, Pirouz, Suraj, and Hojun) in the NonNewtonian Fluids group for showing me how to use the equipment in lab and for all their
assistance. I thank Christian for answering my questions about LabVIEW. To the people
who were part of the lunchtime discussions on world history and culture, I thank you for
your interesting inputs
6
Introduction
Chapter 1
1.1
Purpose
Non-Newtonian fluids are found in many aspects of everyday life and in order to process
them or control their properties, they must be understood. For example, toothpaste must
flow easily enough to be squeezed out of the tube, and yet have a high enough viscosity
to stay on the toothbrush once it's been squeezed out (Prencipe et al. 1995).
(8mdn
(SWll
r/m
(a)
Figure 1.1:
(b)
Some examples of flows where shear
important: (a) spreading, (b) swallowing, and (c) nozzle
(c)
extensional properties are
sion (Padmanabhan 1995).
Other examples of combined shear and extensional flow are spreading and nozzle
extrusion. Spreading can be important for food products such as butter and cream cheese.
It is desirable for them to be easily spread on soft bread while not being watery (figure
1.1 a). The shear and extensional properties can also affect the feel of the food when it's
being swallowed or chewed (figure 1.1b). For the case of nozzle extrusion, the NonNewtonian behaviors can greatly affect the processing of the product (figure 1.1c). For
7
example, the normal stresses in the fluid can cause die swell, making the fluid stream
thicker near the exit of the flow as seen in figure 1.2 (Barnes et al. 1989).
Figure 1.2: Die swell occurring at the exit of a nozzle due to the normal stresses (Boger
and Walters 1993).
1.2
Background
1.2.1
Newtonian Fluid
Newtonian fluids have the following characteristics at constant temperature and pressure:
shear viscosity is constant, yield stress equals zero, normal stress differences equal zero,
and the fluid has no hysteresis. Most simple fluids such as water and glycerol are
Newtonian in standard settings. Any fluid that does not meet the requirements listed
above is a Non-Newtonian fluid (Barnes et al. 1989).
1.2.2 Non-Newtonian Fluid
Complex fluids such as dispersions, emulsions, and polymer solutions are usually NonNewtonian. When the viscosity of the fluid decreases with increasing shear rate, the fluid
is described as shear-thinning or pseudoplastic. When the behavior is the opposite and
the viscosity increases with shear rate, then it is described as shear-thickening or dilatant.
8
Another Non-Newtonian behavior is the presence of a yield stress. A Bingham plastic
does not flow until a shear stress greater than the yield stress is applied.
1.2.3
Extensional Flow
There are several different methods of studying extensional behavior of fluids that have
been explored by other researchers. In Barnes et al. (1989) and Padmanabhan (1995), the
method of using translating clamps is discussed. Two fluid reservoirs are held in clamps
as seen in figure 1.3. The clamps are moved in opposite directions to stretch the sample
and the force at one of the clamps is measured. The difficulty rises in clamping the liquid
and exponentially increasing the length of the sample.
Figure 1.3
Stretching liquid using moving clamps (Padmanabhan 1995).
A second method uses commercial rotational rheometers to measure the extensional
viscosity.
The experiments are run by clamping one end of the sample fluid and
attaching the other end to a drum rotating at a constant speed (Padmanabhan 1996).
Torque transducers are used at the clamp end to calculate the extensional viscosity. The
rheometers were found to be good tools for fluids with high viscosity.
9
Third method uses pressure drop measurements to compute the extensional viscosity. In
the paper by Padmanabhan and Macosko (1997), several other researchers such as Boger
(1987) and White et al. (1987) are mentioned to have also studied this type of flow. The
fluid sample is passed through an abrupt contraction and the pressure dissipated by the
fluid is measured for the calculation.
Another method of measurement generates uniaxial extensional flow by pulling a pool of
sample liquid from two opposing sides as seen in figure 1.4. This method is used for low
viscosity fluids (Padmanabhan 1995; Braithwaite 2000).
Figure 1.4: Diagram of opposed jets rheometer (Padmanabhan 1995).
Rheometrics Inc. made commercial rheometers called RFXTM employing this design, but
they are no longer manufactured.
1.2.4
Preshear
To understand how shearing affects the extensional properties of a fluid, the sample fluid
can be sheared prior to the extensional experiment.
This area of study is explored in
James et al. (1987). James and his colleagues discuss the effects of shearing fluid during
10
channel flow prior to measuring the apparent extensional viscosity using converging
channel flow. The fluids tested are dilute polyethylene oxide (PEO) and polyacrylamide
(PAM) solutions. The study showed that the PEO solutions were largely affected by the
pre-shear, whereas the PAM solutions were not.
Another method of pre-shearing is discussed in Rios et al. (2002). The fluid is sheared in
a concentric cylinders arrangement and the elongational flow of the fluid is examined as
it exits from the lower part of the cylinders. The paper discusses the preliminary tests
that were performed on glycerol using the apparatus.
Ferguson and co-workers (1998) and Schmidt and Munstedt (2002) also discuss the
affects of pre-shear on material properties.
The paper by Ferguson and colleagues
examines whether a polymer's 3D plot of the transient extensional viscosity (TEV) is
unique for that particular fluid and also whether fluids with different strain histories
always fall back on to the same surface. Using an Instron tester, it is found that there are
surfaces that are applicable to the TEV of a variety of fluids. In dealing with the question
regarding strain history, it is found that strain history moves the curve but the curve
returns to the original surface once the fluid has fully relaxed. It is concluded that the
time for the return is directly related to the relaxation time and the retardation time of the
fluid.
11
1.3
General Approach
To examine the extensional viscosity of complex fluids and study the effects of pre-shear,
various fluids are tested using a Capillary Breakup Extensional Rheometer (Caber) made
by Cambridge Polymer Group (www.campoly.com).
The fluids tested are glycerol,
polyethylene oxide, PS025 and PS025 with clay particles, yogurt, and acrylic paint. To
compare the material properties measured using the Caber, measurements are also made
using other instruments such as a cone and plate rheometer and a tensiometer. The
bottom portion of the Caber is modified to enable the pre-shearing of the samples prior to
testing.
12
Chapter 2
2.1
Previous Works
Liquid Filament
A liquid bridge is formed by loading a fluid sample between two cylinders and then
applying a step strain to the top cylinder. This can be seen in figure 2.1.
Figure 2.1: Liquid filament bridge is stretched between two cylinders.
Once the strain is applied, the liquid selects the dynamics such that the viscous, elastic,
gravitational, and capillary forces balance each other (McKinley and Tripathi 2000).
Since the pressure in the droplets at the ends of the filament is lower than that within the
filament, the fluid flows from the filament to the droplets as seen in figure 2.2
(Bazilevskii et al. 2001).
Figure 2.2: Filament thins under capillary pressure.
13
A simple surface energy analysis shows that for a fixed volume of liquid, two droplets of
equal diameter take up less surface area than a cylinder of smaller diameter. Therefore,
in accordance with the Rayleigh instability, the cylinder is not stable.
For slender
filaments, the middle portion can be modeled as a cylinder. The thinning of the filament
is modeled as steady simple elongational flow seen in figure 2.3.
Figure 2.3: Filament model under simple elongational flow.
With z-axis along the length of the filament,
1.
2
2.1
V, -- 6r,
VO =
where
0
is the elongational strain rate, and r is the radius (Tirtaatmadja and Sridhar
1993; Renardy 1994). The nonzero velocity gradient tensor components are
-av
az
{Vv}
=
{Vv}O
=-
'
.
ar
1 avo
r aO
2.2
v
+-
r
14
The rate of strain tensor is equal to
2U
0
0
0
0
f=Vv+(Vv)'=
0
0
2.3
-,
where for the midpoint of the filament
1
2
R
2.4
dRmid
dt
(Bird et al. 1987). From equation 2.4, the Hencky strain is found to be
c = 21n (
,
2.5
where RO is the filament midpoint radius before the stretch and Rmid is the filament
midpoint radius at a given time after the stretch (Malkin and Petrie 1997). Some papers
discussing this type of flow are Doyle et al. (1998), Entov (1999), Rasmussen and
Hassager (1999), Li and Larson (2000), Cruz-Mena et al. (2002), and Stelter et al. (2002).
2.1.1 Newtonian Fluid
From Liang and Mackley (1994), the total stress in the axial direction (czz)
direction (rz)
and radial
for the filament is given by
r. = -pO + r,=
r,. = --PO +
where po is the atmospheric pressure,
Ti
0,
r
.P
o-2.6
,.R
= -md
is the viscosity, and a is the surface tension. The
total stress along the z-axis is equal to zero because each end of the filament are attached
to relatively large quasistatic droplets that are in turn attached to the endplates (McKinley
and Tripathi 2000).
15
Using equations 2.3 and 2.6, the elongational stress is founds to be
0rE
Rmid
z
2.7
To obtain the equation of radius as it decreases in time, equations 2.1 and 2.7 are
combined and integrated resulting in:
oat
6q,
2.8
Rmig(t)=R ----
(Kolte and Szabo 1999). The R, is the radius at time equal to zero. From the equation, it
is predicted that the radius of a Newtonian liquid filament decreases linearly in time.
McKinley and Tripathi (2000) discuss how this theory does not correlate well with
the data and a correction factor must be used. The error results from the assumption that
the stress along the z-axis is equal to zero. The discrepancy is resolved when
F, = 2XrcRm ,
2.9
where Xis the correction factor. The equation now becomes
Rmid (t) =
R, -
(2X -1)
6
c
-t.
q,
2.10
Various authors discuss this factor (Eggers 1993; Papageorgiou 1995; Renardy 1995;
Eggers 1997).
For viscous Newtonian fluid, the value used for X is 0.7127
(Papageorgiou 1995).
2.1.2
Viscoelastic Fluid
A viscoelastic fluid is a material that exhibits both viscous and elastic properties (Reiner
1971). When studying dilute polymer solutions, the macromolecules can be modeled as
finitely extendable nonlinear elastic (FENE) dumbbells (de Gennes 1997).
The
16
dumbbells are two beads attached by a spring that is linear for small extensions, but
become increasingly stiffer for larger deformations until the ultimate extension limit, L, is
reached (Bird et al. 1977). The FENE-P model as seen in Anna 2001 is used to model the
viscoelastic samples. The upper convected derivative of the conformation tensor A is
defined as the following:
A,() =f(fA,
-I),
2.11
where I is the unit tensor and Ai is the relaxation time. The subscript "i" represents the
given mode. The FENE factors,fi, is defined by
1
1
- trA.
'
=2.12
The tensile stresses due to polymeric contributions is computed using
N.
Alp
[Izz Trr]=
Gf (Ai
- Arr),
2.13
where G is the elastic modulus.
The balance of viscous, elastic, and capillary forces in the filament gives
3r,.=L
Arj.
2.14
Rearranging equation 2.4 leads to
d ln
Z = -2
Rmid
dt
'.
2.15
17
After the fluid sample is stretched, the filament stretches at a rate that is just enough to
overcome the longest relaxation mode.
Since only the first term in the modes is
important, A,,, ~ A,,,i for all other modes. The equation for A,() and equation 2.15 are
combined and then integrated to yield,
R,)
A.0t)
exp
(AZ
2.16
,9
where A, is the value of the polymeric stretch right after the step strain. The balance of
elastic forces and capillary forces show that
2
2.17
3Az
Plugging this back into equation 2.14 and combining it with equation 2.16 gives
2)7, Rmid(t)
GAz R,
_-_
R-
Gr
Ri W
)r3
exp
J .
2.18
3 Az
As the filament thins after the applied step strain, the terms in equation 2.18 balance each
other with initial value of the axial stretch being
(GR1 )
2.19
GA1z
The conformation tensor grows according to the following equation:
( t ).
2.20
Incorporating equations 2.20 into 2.16 gives.
Rmid(t)
R, e xpK
.
3-
2.21
)
18
This agrees with the equation for dilute viscoelastic fluids found in Liang and Mackley
(1994), Bazilevskii et al. (1997), Stelter et al. (2000), Bazilevskii et. al (2001), Anna and
McKinley (2001), Stelter et al. (2000), and Anna et al. (2001).
2.1.3
GeneralizedFit
According to equation 2.21, a viscoelastic filament decreases in radius over time but does
not break. In reality, the filament does indeed break in finite time. Once the extension of
polymer molecules reaches a limit, the extensional viscosity curve plateaus and the fluid
breaks like a linearly viscous liquid. The third way of fitting the radius evolution data is
to use a derived model interpretation. This equation has the limiting expressions for
Newtonian, Oldroyd, and Bingham fluids. The expression from the notes by McKinley
(2000) is as follows:
=
Rmidft)
Rmida
rRl
iW{-jRij
I
)1/3
0
+
4ct
RG1/3
6?7,
RIGJ),(0 e -22
}.j42Ut+Ri
+
pg
tA
2.22
Replacing constant terms with coefficients, the following form is obtained:
Rmid(t) = A - Bt + Ce(-Dt).
2.23
Fitting this equation to experimental data gives an expression that represents the decrease
in radius over time (Anna and McKinley 2001).
2.1.4 Power Law
For certain shear thinning or shear thickening fluids, the shear viscosity verses shear rate
can be expressed using
7 (f)=
n-"1
2.24
19
where m and n are fitted parameters (Bird et al. 1987). The fluid is shear thinning for n<1
and shear thickening for n>1.
The following derivation is found in the notes by McKinley (2001). Using the second
invariant, the following equation for the shear rate is found:
.~
1
-(II)
2
Z_)2 + _.
(
2
)2
.
+(24)=
2.25
Balancing viscous and capillary forces,
2.26
Rid
Equations 2.24 and 2.25 are combined with equation 2.26 to obtain,
6 -
3 m ( N' )
2.27
-id
Rmi
Rearranging the above equation and using equation 2.4 for strain rate, it becomes
3m(3)(n-1) 2 2"n
1
dRmid
_R dt
1
2.28
Rn"
Equation 2.28 is then integrated from t-0 to t. The resulting expression is
+t
mid
n
2
F3m(3)("n-1) 2"
11n
2.29
where R, is the radius of the filament immediately after the application of the step strain.
The equation is rearranged to find the expression for Rmid,
6 o-
mdJ6n m
i1n 3
t2n0T
( tcrit -t)
,
2.30
20
where terit
=
R'"n
[3m(3)(n-1)/22n
I
To model the thinning filament of a power law fluid, the following equation is used:
2.31
Rmid(t) = A (te,, -t)".
2.2
Extension Viscosity
The stress tensor for a shear free flow is
;r= p+
=
0
P+,I
0
0
0
2.32
P + -Z
where p is pressure. The normal stress differences for the flows are
ZZ
rr
TOO -
For steady simple elongational flow,
viscosity, g
2is
1
2.33
=r 17246
equal to zero and ij is the apparent extensional
(Bird et al. 1987).
2.2.1 TheoreticalExtensional Viscosity
Using equation 3, the stress tensor of a Newtonian fluid is found to be
0
'2)7,
T=L0
0
-7
0
0J.
0
2.34
-77-Il
The extensional viscosity is equal to the first normal stress difference divided by the
extension rate,
qe,
= 37,.
2.35
21
For a Non-Newtonian fluid, the value of [r, -trr] also has to be taken into account (Bird
et al. 1987). This is set to equal the polymeric apparent extensional viscosity times the
elongational strain rate to give
-r,] = i, .
[rzz
2.36
As seen in notes by McKinley (2000), the total apparent extensional viscosity becomes:
2.37
iex, = 3q, + q,.
2.2.2 Extensional Viscosity from Radius verses Time Data
Stress balance is performed on the filament to obtain:
S2dRmid
2.38
F
Ri
rcR2
dt
mdmid
Rmid
(McKinley 2000). The above equation takes into account the viscous stress, tensile force
per area, elastic stress, and the capillary pressure (Anna et al. 2001). After combining
equations 2.9, 2.37, and 2.38, the extensional viscosity is calculated as
(2X -1)
17ext =
'
=Rmid
2 dRid
Rmid
2.39
dt
This method of measuring the extensional viscosity does not require a transducer since
the capillary force, R,
Rmid
acts as the force transducer. Equation 2.39 simplifies to
_ (2X-1)c
_7exd ( t2Rm
2.40
dt
22
Using equation 2.40, the extensional viscosity can be calculated using the data of the
midpoint radius as it decreases over time.
Though the equation is originally for
Newtonian fluids, the equation is also valid for fluids that form filaments that agree with
slender body approximations (McKinley 2002).
2.2.3 Five-point Centered Difference Formula
To calculate
dR
mid from data of radius over time, the following equation is used:
dt
_
dRmid
dt
2
Rmid(i+2) + Rmid(i+l) -R
)
d(_jl)-
2
24
Rmid(i2)
At
where the index i indicates the order of the data point (Becker and McKinley 1994). The
equation assumes equally spaced Rmid values. The derivation of the equation is found in
the journal article by Whitaker and Pigford (1960).
2.2.4 Slope Formulafor Unequally Spaced Points
For data that is not equally spaced in time, the following equation from McKinley (2002)
dRmi
mid
can be used to compute
dt
_C
mid (ti
l
P
SRmid(i+l)
pi)Rn~j - piRmid(j-l)
-(
dt
,
2.42
hi + h;_
where
p = ti+1 -t
ti
2.43
i-I
and
A. = tI.+- t .
2.44
23
2.3
Pre-Shear
The fluid sample between the two cylinders can be sheared before the top cylinder is
raised. The bottom cylinder rotates to apply the pre-shear orthogonal to the direction of
the step strain. The velocity profile of this flow is
v,
=0
Qrz
vO = Qz ,2.45
H
vZ = 0
where Q is the rotation rate of the bottom cylinder, r is the radius along the cylinders,
and H is the initial gap size (Anna 2000). The rate of strain tensor is given by
0
0
0
frlRoj-
frl/Ro
0_
=0
0
0
The shear rate at the edge of the system is calculated using
7R
=fR2
n
0
2.46
H
where jR is the shear rate at the edge of the cylinders, and R0 is the radius of the fluid
sample before the stretch (Bird et al. 1987).
2.4
Non-dimensional Numbers
2.4.1 DeborahNumber
The Deborah number was first defined by Marcus Reiner in 1969.
He states that
everything flows as long as one makes the observation on the correct relative time scale.
The Deborah number is the dimensionless deformation rate computed as the product of
the relaxation time of the fluid and the characteristic time of the experiment.
24
De = A / tflOW,
2.47
where A is the relaxation time and tflow is the time scale of the flow system (Bird et al.
1987). For steady simple elongational flow step strain experiments, 1/tflo, is the stretch
rate of the step strain. Deborah number becomes,
De = A4 .
2.48
A viscoelastic fluid behaves more like an elastic solid for De>1 and more like a viscous
fluid for De<1.
2.4.2 Trouton Ratio
Trouton ratio scales the extensional viscosity to the shear viscosity:
Tr = 77et
2.49
77S
(Tirtaatmadja and Sridhar 1993; Sizaire and Legat 1997). For Newtonian fluids, this
ratio is 3.
2.4.3 Bond Number
This dimensionless number gives the relative effects of gravity verses surface tension.
The equation for the number is as follows:
Bo = pgf Rm.d(t)
C-
2.50
Gravity has negligible effect when
Rmid(t) s
0.1cr
pg
2.51
25
(McKinley 2002).
2.4.4 WeissenbergNumber
The Weissenberg number, We, is the dimensionless shear rate.
It is computed by
multiplying the relaxation time of the fluid by the characteristic strain rate:
We = AK,
where
flow,
K
K
2.52
is the characteristic strain rate (Bird et al. 1987; Anna 2000). For the pre-shear
is the rotation rate, Q.
26
Chapter 3
Materials
3.1
Obtaining Material Properties
3.1.1
Tensiometer
KIOST tensiometer by Kruss is used to measure the surface tension of various liquids.
The device uses the Wilhelmy plate method. The testing plate, made of platinum, is
rough to improve wetting. It is 40.0 mm in wetting length, 19.9 mm in height, and 0.1
mm in thickness.
The plate is lowered to a sample vessel in the testing chamber
containing about 20 mL of test fluid. The measuring plate is lowered to the surface of the
liquid until it barely makes a contact. The liquid pulls the plate in when it makes the
contact. The force required to pull the plate back to the surface level is the Wihelmy
force (Digital Tensiometer Manual 1995). Figure 3.1 shows this occurrence.
The plate contacts the fluid
surfice.
The fluid pulls the plate into
the fluid.
The plate is lifted up to the
surace leveL
Figure 3.1: Diagram of the tensiometer plate and test fluid vessel (Digital Tensiometer
Manual 1995).
27
The tensiometer displays the surface tension calculated using the following equation:
-
F
1cos9
(3.1)
where F, is Wilhelmy force, / is the wetting length of the plate, and 0 is the contact
angle between the wetting line and the plate surface (Digital Tensiometer Manual 1995).
The calculation is made with the liquid wetting the entire wetting length and the contact
angle being 0. The actual angle is brought very close to this by using a clean platinum
plate with a rough surface.
3.1.1
Shear Rheometer
AR1000 by TA Instruments is used to measure shear viscosity and the relaxation time of
some of the test fluids. AR1000 is a controlled stress rheometer with the following
ranges:
Table 3.1: Ranges for AR1000
Torque
0.1 - 101jNm
Frequency
0.0001 - 100 Hz
Normal Force
0.01 - 50 N
Temperature
-150 - 4000 C
All of the tests are run using cone fixtures with the tip truncated. The truncation is small
enough that its effects on the fluid dynamics are ignored. With this fixture, the AR1000
is used as a cone and plate rheometer (see figure 3.2).
28
Fluid
(r
R)
0
Figure 3.2:
Diagram of the cone and plate rheometer.
The shear rate is constant
throughout the fluid (Welsh 2001).
The sample is loaded onto the plate and then the cone is lowered to where the tip would
touch the plate were it not truncated.
A steady flow experiment is chosen with a
specified range of stress. The cone rotates to achieve the given shear stress.
The
rheometer software computes the shear stress using the following equation:
=-7 3z-
(3.2)
where 7, is shear viscosity, z is torque on the plate, and R is the radius of the cone
(Macosko 1994; Rheometer Manual 1996).
To calculate the relaxation time of the fluid, various tests are performed also using the
cone and plate setup on the AR1000. First, three strain sweeps are done at various
frequencies.
The strain sweeps are set from the minimum % strain possible for the
machine (.028749) to a high % strain (3000). The sweeps are run at 0.1 Hz, 1 Hz, and 10
29
Hz. At each frequency, a plot of storage modulus (G') and loss modulus (G") verses %
strain on log-log scale is obtained. Looking at all of the plots, the maximum % strain
before G' and G" dips down is found. At this % strain, a frequency sweep is performed
from minimum frequency (1x10- 6 Hz) to a high frequency (100 Hz). On a log-log plot of
G'and G" verses co, the inverse of the frequency where G'and G" intersect is the
calculated value of the crossover relaxation time.
The longest relaxation time for a
polymer fluid is found by computing
A= G
G"co
(3.3)
for each point and finding the largest value of ) (McKinley 2002).
3.2
Test Fluids
3.2.1
Glycerol
The shear viscosity of 100% glycerol sample is obtained using the cone-plate rheometer.
The resulting output is seen in figure 3.3.
30
1.2 + '
''
'
''
0
1.00
0
0
0
0
0
0
0
0
0.80
.L
0.6
.L"
0.4--
0.2
0.0
0
2
4
6
8
10
Shear rate [s~1]
Figure 3.3: Using the cone and plate rheometer, the shear viscosity of pure glycerol at
23*C is found to be 1.04 Pa.s.
Since glycerol is a Newtonian fluid, the shear viscosity is not expected to change with
0
changing shear rate. The above data, from experiment run at 23 C, shows that the shear
viscosity is constant with some noise. The average shear viscosity from the data is 1.04
Pa.s. This value is lower than the listed value of 1.49 Pa.s at atmospheric pressure and
20*C. The difference is due to the experiment being run at a higher temperature and the
glycerol having aged. Over time, the fluid has absorbed some moisture from the air.
The surface tension of the glycerol sample measured by the tensiometer is 0.0648
N/m at 23 0 C.
31
Polyethylene Oxide
3.2.2
The polyethylene oxide (PEO) test fluid is 2wt% PEO in water. The PEO has an average
molecular weight of 2x106 g/mol. The mixture is stirred for three to five days with an
electric stirrer before being tested to ensure that the polymer is well mixed in the water.
The AR1000 is used to obtain the relaxation time of this elastic liquid at 20'C. The
following is a plot of the data from the frequency sweep:
I 1111
IIIIj
I
I
I
11151
I1111111
1
tjf!2
-
1.0
-
0.8
-
0.6
-
0.4
-
0.2
- EN 0000
10
momE00
r--
Co
0
ME
-
CL
0000
AEMO
1
AE0
A
00
A
E
AA
0
0.1
0
00t\
A
-0
0.01
L
:0
~0
L 0.0
0.1
1
10
100
Angular Frequency [s~1]
Figure 3.4: Data from oscillatory experiments on the cone and plate rheometer. Test
fluid is 2wt% PEO in water at 20'C.
The G' and the G" intersect at the angular frequency of 157.9 s4, resulting in a
crossover relaxation time of 6.33x10 3 s. The longest relaxation time is computed to be
1.06s. The surface tension of the PEO test fluid at 23*C is 0.0234 N/m.
32
3.2.3
Polystyrene Boger Fluid
The polystyrene test fluid, PS025, is a Boger fluid consisting of high molecular weight
polystyrene and styrene oil.
A Boger fluid is a dilute solution composed of low
concentrations of high molecular weight polymer dissolved in highly viscous Newtonian
liquid (Larson 1999). Polystyrene Boger fluid is non-toxic and non-volatile. It is fairly
stable even when exposed to radiation or higher temperatures, though it can still
experience some degradation. The material properties of PS025 are discussed more in
depth in Anna (2000).
The solvent used for the test fluid is oligomeric styrene
(Piccolastic A5 Resin) from Hercules. At 250 C, the material has a density of 1026 kg/m3
and surface tension of 0.0378 N/m.
To form PS025, 0.025 wt.% of high molecular
weight polystyrene (catalog no. 829, lot no. 03) from Scientific Polymer Products is
dissolved in the styrene oil. At this concentration, the polystyrene chains do not overlap
when at rest. The diluteness of the fluid is determined by examining the coil overlap
concentration, c *, given by
C=
(3.4)
M
/3
9R
where Rg is the radius of gyration and NA is Avogadro's number (Graessley 1980). For
PS025, the ratio of the weight concentration and the coil overlap concentration 0.33
(Anna 2000). The ratio is less than 1, indicating that the fluid is dilute.
Since kinetic theories on polymer solutions usually assume that the chains all have the
same molecular weight, it is necessary for the material used in the experiments to have a
narrow molecular weight distribution.
This distribution is measured using the
polydispersity index (PDI), which is the ratio of the weight-averaged molecular weight,
33
M., and the number-averaged molecular weight, M, (Flory 1953). For the polystyrene
chain, the PDI is 1.02 and the weight-averaged molecular weight is 2.32x10 6 g/mol.
Various material properties were obtained from a lab mate, Hojun Lee, who is also
working with the fluid.
The shear viscosity measured by running a constant shear
experiment on the cone and plate rheometer is 36.7 Pa.s. Computation of the longest
relaxation time, A%, for PS025 are found in both Anna (2000) and Welsh (2001). The
value was computed using the Zimm model fit on the frequency sweep data. The two
values differ by a small amount of 0.2 s. The value of 3.9 s found in Anna (2000) is
chosen to be used for calculations in chapter 5.
Two test fluids with different amounts of clay particles suspended in PS025 are also
tested. The clay is Cloisite 20A from Southern Clay Products. The particles are flat
slates with length of 200 nm and width of 100 nm. One of the test fluids has 3wt% of
clay and the other has 1 Owt% clay. The yield stress, r, of the fluid with 1 Owt% clay is
23.7 Pa. The relaxation time of the two test fluids is estimated to equal the time it takes
for the disks to relax in the fluid after being disturbed. The relaxation time of the 3wt%
and 1 Owt% clay polystyrene are calculated using the following formula, which is derived
using the rotary diffusivity of the particle:
=49,"
i
18kBT
where 7,
(3.5)
is the shear viscosity of the solvent, d is the diameter of the disk, kB is the
Boltzmann's constant (1.38x10- 23 J/K), and T is the temperature (Larson 1999). The
34
slates are approximated as disks for the calculation and the hydraulic diameter is used as
the diameter of the disk.
-4A
Dh
=
P
133.33 nm
(3.6)
where Dh is hydraulic diameter, A is area of the surface, and P is the perimeter (Fay
1998; White 1999).
For PS025 with clay particles, the relaxation time at 23*C is
calculated as 4.7 s.
3.2.4
Yogurt
According to Tamime and Robinson (1991), the main ingredient of yogurt is milk. Milk
is mostly water but it is made complex by its other components such as proteins,
carbohydrate, fats, minerals, and vitamins. The solid content in milk varies depending on
the time of the year and also the cow. To give the reader an idea of the variation in the
solid content in milk, a graph of the fat content and protein content per month in United
Kingdom is shown in figure 3.5.
35
Fat
4.24.1
4.3
4.0-
3.8
32-
A
M
J
A
0
S
N
D
J
F
M
Month
Figure 3.5: The fat and protein content variation in milk between months. The data is
from United Kingdom and the five graphs for each month represent different parts of the
U.K. (Tamime and Robinson 2000).
The fat content of the milk is standardized and is controlled by adding either skimmed
milk or cream. Whole milk is separated in order to get cream and skimmed milk. The
solids-not-fat (e.g. lactose and protein) content is also standardized.
The level can be
controlled by various methods. Application of heat, addition of milk powder, addition of
36
buttermilk powder, addition of whey powder, addition of casein powder, evaporation by
vacuum, and filtration through a membrane are all possible methods.
To make yogurt, the total solids in milk is raised to around 15g/100g. Then, it is
heated to a high temperature for 5-30 minutes. Next, the fluid is inoculated with bacterial
culture and incubated until a smooth viscous texture and desired flavor is achieved. The
yogurt is cooled and further processed if necessary. (Tamime and Robinson 2000)
The experiments are conducted using Dannon Natural Plain Yogurt and Dannon
Fat Free Plain Nonfat yogurt. The plain yogurt has 8g/227g total fat and 9g/227g protein.
The ingredients are cultured grade A milk and active yogurt cultures. The nonfat yogurt
has Og of total fat and 12g/227g protein. The ingredients are cultured grade A nonfat
milk, pectin, and active yogurt cultures. Both are stored in a refrigerator after being
bought from the store. Yogurt material properties are also examined in van Marle et al.
(1999) and Yu and Gunasekaran (2001).
For both of the yogurts, shear stress ramp up test and ramp down test are
performed at 23"C. The resulting plot is seen in figure 3.6.
37
2
+ Regular: Stress Ramp down
*- Regular: Stress Ramp up
A Nonfat: Stress Ramp down
2
Cn
100
A
Nonfat: Stress Ramp up
6
4
0
l
2
1042
5 16
.1
8
1.1.itlIll..
10
I.
h1h..
12
14
16
18
20
22
24
Shear Stress [Pa]
Figure 3.6: The stress ramp down and stress ramp up at constant temperature of 23"C for
both yogurts.
The yogurts have almost the same shear viscosity profiles. When stress ramp up test is
conducted, the yogurts show that a certain shear stress must be reached to break the gel
structure. During ramp down, the change in viscosity is more gradual.
Shear stress ramp down experiments are run to compare newly opened regular
yogurt (with expiration date of Dec. 4th) and an already opened regular yogurt (with
expiration date of Nov. 2nd). Both of the experiments are at 23*C. The 4 cm 1 degree
cone fixture is used. These tests are done to find out how much the yogurt changes after
it has been opened. It is later realized that the yogurts being from different containers
could have also affected the data. The resulting plot is seen in figure 3.7.
38
-- -- -- - -- -- --- - -- -
-- - -- - -- -- -
2
C',
+
100 -I
6
x
+
New
Old
x
4
2
0
0
U')
10
*
A
X +
x
6-
+
+
x
x
4
x
+
++
++
X
0
8
10
12
14
16
1.Ex
+ + ++
.
18
K....
20
+
"<
22
+
I
24
Shear Stress [Pa]
Figure 3.7: The stress ramp down of newly opened regular yogurt and already opened
regular yogurt. The experiment is conducted at 23"C.
Tests are run on both regular and nonfat yogurt to determine how the shear viscosity is
affected by temperature. Constant shear stress experiments at 30 Pa are conducted with
temperature increasing from 3"C to 25"C. The runs are each 12 minutes long. The results
are plotted in figure 3.8.
39
10
1
1 11 1
'
8+A
8
Regular
A Nonfat
, 6
+
4 -
+
5
10
15
20
25
Temperature [0C]
Figure 3.8:
Constant shear stress experiment at 30 Pa with increasing temperature.
Experiments are run using newly opened regular and nonfat yogurt.
The plot shows that the shear viscosity decreases with increasing temperature
for both
yogurts. The shapes of the curves are similar, but with nonfat yogurt having
a higher
viscosity at all temperatures.
At 23*C, the difference in viscosity between the nonfat
yogurt and regular yogurt is .3 Pa.s.
Using the tensiometer, the surface tension of regular and nonfat yogurt are found
to be 0.0435 N/m and 0.0418 N/m, respectively. The experiments are run with
the water
bath at room temperature after taking the yogurt out of the refrigerator. At the time
of the
experiment, the temperature of the yogurt is 14'C. The surface tension of regular
yogurt
re-measured in December at 21.4*C is 0.0411 N/m.
3.2.5
Paint
40
Delta Ceramcoat acrylic paint by Delta Technical Coatings, Inc. is examined as one of
the test fluids. It is a non-toxic, water based material with a surface tension of 0.0282
N/m at 20'C.
41
Chapter 4
4.1
Experiment
Apparatus
4.1.1 CapillaryBreakup ExtensionalRheometer (Caber)
The experiments are run using the Capillary Breakup Extensional Rheometer made by the
Cambridge Polymer Group.
A picture of the setup can be seen in figure 4.1.
equipment characterizes the flow of a test liquid in extension.
The
Similar setup is also
studied in Bazilevsky (8th International Congress), Spiegelberg et al. (1996), Chang et al.
(1999), Iza and Bousimina (2000), Tripathi et al. (2000), and Montanero et al. (2002).
r-
Figure 4.1: SolidWorks drawing of the Caber.
There are two main coaxial cylinders with one on top of the other. The bottom cylinder
is held stationary and the top cylinder is movable along the vertical axis. A portion of the
rod attached to the top cylinder is wrapped with a spring under compression. To achieve
42
a certain gap size, the linear motor presses down on the system to lower the upper
cylinder and compress the spring further (seen in figure 4.2). When the motor releases,
the spring pushes the upper cylinder up to the new location of the motor arm.
Linear
Linear
JMotor
Motor
II
F
Figure 4.2: The height of the upper cylinder is controlled by a linear motor. A spring
pushes the upper cylinder up when the motor releases.
The surfaces of the cylinders that come in contact with the liquid are both 6 mm in
diameter. At the beginning of an experimental run, the top cylinder is held 3 mm above
the bottom cylinder. A sample of around 100 pL is loaded in the gap. Then, a step strain
is applied to the liquid, stretching it to a final height of 13 mm. The applied strain, which
takes 50 ms, is at a rate as close to a step strain as possible with the current setup. On
opposing sides of the cylinders are the laser micrometer and the receptor, seen in figure
4.3. After a liquid filament is stretched, the laser micrometer measures the radius of the
filament over time as it breaks under capillary force. The laser sends a beam to the
43
receptor on the other side of the filament. The receptor measures how much of the beam
is received and the radius of the filament is calculated from this information. The laser
has accuracy of 5-10 pm.
Laser
Receptor
Figure 4.3: A laser micrometer measures the thickness of the liquid filament.
The entire experimental setup is controlled using the Caber software version 3, written in
LabVIEW.
The equipment and the computer communicate through a National
Instrument 1200 DAQ card.
Figure 4.4 is the layout of the experiment setup. The data travels to and from the
Caber and the DAQ card through the Caber control box. The positioning data for the
linear motor goes from the DAQ card to the Caber control box, then to the motor control
box, and finally to the motor. The data from the linear motor reaches the computer by
going through the motor control box.
44
Powr
Powr
Caber Control Box
Motor Control Box
DAQ 1200
Card
Caber
Computer
Linear
Motor
Figure 4.4: Diagram of the Caber experiment setup.
4.1.2 Pre-Shear
Modifications to the original Caber setup are made to enable pre-shearing of the fluid
samples. A new bottom cylinder is machined using 0.875 inch diameter steel rod. The
middle part of the cylinder is made to fit into a ball bearing and the bottom part attaches
to a rubber connecter. The plate that the bottom cylinder attaches to is modified so that
the ball bearing can sit in it.
The whole Caber is raised 6 inches above the table by four linch diameter rods.
The plate the Caber sits on is machined so that a Pittman D.C. geared motor with an
encoder (part no. GM9236C534-R2) can be mounted. Specifications for the motor and
the encoder are listed in table 4.1.
45
Table 4.1: Motor and Encoder Specifications
Motor
No Load Speed
800 rpm
Gear Ratio
5.9: 1
Encoder
Resolution
512ppr (before gear reduction)
The motor connects to the bottom cylinder through a rubber connector. SolidWorks
drawings of the Caber with the modifications is seen in figure 4.5. Figure 4.6 is a photo
of the setup.
7
nT
M
I
L
Figure 4.5: A new bottom cylinder is placed on a ball bearing and attached to a motor to
allow pre-shearing of fluid samples.
46
Figure 4.6:
Photo of the Caber with modifications to allow pre-shearing of fluid
samples.
BE12A6 by Advanced Motion Controls is added to the system to control the motor and
interpret the output from the encoder. The encoder outputs a frequency according to the
number of holes in the counter that are passing per seconds. The amplifier takes the
frequency data from the encoder and converts it to a voltage that is proportional to the
rotation rate of the motor. The input and the output of the amplifier are connected to the
DAQ pins through the Caber control box. The encoder is powered by one of the outputs
of the DAQ card. The diagram is seen in figure 4.7.
47
Power
Caber Control Box
Power
Amplifier
DAQ 1200
Card
Cornputer
CompterD.C.
Motor
Encoder
Figure 4.7: Pre-shear addition to the Caber experiment setup.
The system is controlled and monitored using LabVIEW.
4.2
Procedures
4.2.1
Running Caber
Turn on the equipment and run the Caber program. The program automatically checks to
make sure that the laser receptor is outputting the correct minimum voltage. The user
then commands the program to use the linear drive to determine variables for the cylinder
geometry. The motor takes several minutes to calibrate to the given geometry. Next, the
value for the desired initial gap size in mm is entered in. For all of the experiments, 3
mm is used. LabVIEW then displays the geometry data such as the relevant aspect ratios,
initial and final gap size, Henky strain, etc.
48
Various types of tests are available. The tests used for the experiments are trigger
test and batch test. For the triggered test, length of experiment and the sample rate can be
chosen. The batch test allows the user to define the options listed in table 4.2.
Table 4.2: User Defined Values for Batch Test
Option
Description
Length of relaxation [s]
Time in between cycles
Length of run [s]
Length of data capturing period
Total cycle [s]
Length of one cycle (sum of the
above two)
Number of cycles
Number
of
times
repeated
is
experiment
Total experiment [s]
Number of cycles times the
length of each cycle
Sample rate during run
Chosen sample rate [Hz]
the
After the desired test is chosen, the program asks the user whether or not all of the points
should be kept. If 'strip all points' is chosen, the program takes out points from the
regions in the data plot where the change of diameter over time is small.
It is
recommended that 'leave all data' is selected. Following this window is a new window
asking if the data should be permanently cropped or not. The 'permanently crop' option
deletes the last section of the data plot that consists of points taken after the fluid filament
has broken.
The user now can choose the strike time. This is the length of time the linear
motor takes to raise the top cylinder. To perform a step strain experiment, 50 ms is
inputted. Fifty milliseconds is the fastest the cylinder can open. Clicking on 'proceed'
49
from this window lowers the top cylinder to the initial height. About 100pL of sample
fluid is loaded onto the cylinders at this step using a 3 mL syringe. The fluid should form
the shape seen in figure 4.8.
Figure 4.8: Drawing of the cylinders after a fluid sample has been loaded.
The next window that comes up asks whether or not the sample should be pre-sheared. If
the sample is not pre-sheared, the experiment runs at this point and the top cylinder is
raised. If the pre-shearing option is chosen, the user inputs the number of revolutions of
the bottom cylinder and the rotation rate in revolutions per second. The Caber pre-shears
the sample and then the top cylinder rises to the final height.
4.2.2
Optical Imaging
The video images of the experiments are recorded using a Cohu camera, model MS12.
The camera records using 798 x 494 pixels. The longer length of the screen is aligned
with the axis of the Caber cylinders to maximize the usage of the viewing window. A
double-coated green lens filter by Hoya is used to block out the glare from the laser
micrometer. The experiments are lit from the back using either a sheet of light or light
diffused through a diffusing lens. Lighting is best when using a Fiber-Lite PL-800 as the
50
light source and Opal diffusing glass with thickness of 5.0-6.0 mm from Edmund
Industrial Optics. The video is recorded on digital videotapes.
51
Chapter 5
Results and Discussion
The extensional viscosity of each fluid is computed using Equation 2.40. Matlab is used
to compute the extensional viscosity of each test fluid from the Caber data. All of the
Caber experiments are run open to the lab environment. The temperature of the test
fluids is 21C ±2*C.
5.1
Newtonian Fluid
Pure glycerol at room temperature is tested using the Caber.
Pictures from the
experiment can be seen in figure 5.1.
t= 0.Ols
t=0.06s
t=0.lls
t=O.18s
Figure 5.1: Pictures of glycerol being tested on the Caber.
The Caber data for glycerol is plotted in figure 5.2. Glycerol is a Newtonian fluid and the
rate at which the diameter decreases with time does not change, as seen by the constant
slope of the plot. The data can be fit using a line with a slope of -5.1 lx10-3 m/s and a yintercept of 9.33x 104 m.
52
7
800x10
E
600
C,)
CU
400
200
01
0.05
0.10
0.15
Time [s]
Figure 5.2: Glycerol tested on the Caber plotted with the curve fit.
The extensional viscosity of glycerol is computed using equation 2.40 with surface
tension of 0.0648 Nm 1 and is plotted verses Hencky strain (figure 5.3). The extensional
viscosity is also computed using the curve fit to get a constant value of 2.70 Pa.s.
53
00
3.0 -00
0
00
0
0
2.5 70
2.0-
o
1.5
-3
.0
C)
F~
0
c
1.
1.0- -
-
Data
Curve fft
c
x
0.5
3
4
5
6
7
8
Hencky Strain
Figure 5.3: The extensional viscosity of glycerol computed from the Caber data. The
value from the curve fit is also plotted (m=-5.1 lx10-3 m/s; b= 9.33xl0~4 m).
The extensional viscosity computed using equation 2.35 with shear viscosity of 1.04 Pa.s
results in 3.12 Pa.s. The curve fit value is lower than both this value and the latter half of
the data points. This may be a result of the fluid filament being affected by gravity when
it is first formed. A closer look at the affects of gravity can be obtained by examining the
Bond number. It is estimated that gravity has negligible effects when the Bond number is
less than .1. For the glycerol sample, this occurs when
Rmid(t)
= 7.24* 10-4 m.
To determine the extensional viscosity after this point, a new curve fit is performed
(figure 5.4).
54
800x10 6
E
Data
Curve fitl
600
C,,
CU
0
--
-
~
I
I
I
I
i
i
i
i
I
i
i
i
*
I
400
200
0.05
0.10
0.15
Time [s]
Figure 5.4: A new curve is fit to the data starting from the point where effects from
gravity become negligible.
3
The slope of the new fit is -4.91x10- m, which results in extensional viscosity of 2.81
Pa.s. This value is still lower than the value calculated using the shear viscosity value,
but it is within 10 percent.
As expected, there is no effect of pre-shear on glycerol (figure 5.5). The sample is tested
after 12.57rad of shear strain at 1.88s-1 and 37.70 rad of shear strain at 18.84s~1.
55
0 y=Orad 4=0 s
y=12.57 rad, f=1.88 s-
800x10
A y =37.70 rad, '=18.84s'
600
,
E,
400
200
0.00
0.05
0.10
0.15
Time [s]
Figure 5.5: Glycerol is tested on the Caber with pre-shear at two different shear rates.
As expected, pre-shearing the sample does not affect the data.
5.2
Viscoelastic Fluid
5.2.1 PEO
The polyethylene oxide (PEO) test fluid is 2 weight % PEO in water. The PEO has an
average molecular weight of 2x10 6 g/mol. The pictures of the filament formed during the
Caber experiment are in figure 5.6.
t= 0.01s
t=0.IIs
t=0.21s
t=0.34s
Figure 5.6: Pictures of PEO being tested on Caber.
56
The fluid filament quickly thins to a long thin strand and the radius decreases more
slowly. The Caber experiment results are plotted in figure 5.7. The experiments are run
at room temperature and new sample is loaded onto the rheometer for each experiment.
The effects of pre-shear are explored by shearing the sample at two different shear rates
(6.28 s4 and 18.84 s-) for two different shear strains (12.57 rad and 37.70 rad).
8-
y=0 rad,=
6_
12.57 rad,
4-
y = 37.70 rad,
0 s= 6.28 s-
j
= 6.28 s-'
o
=12.57 rad, '=18.84 s-
E
1
-
= 37.70 rad. i=18.84 s-
C/)
10
-
8-6--
I
-44
,
0.00
,
,
I
ii,
0.05
,
Ii
0.10
,
i
0.15
I
I
0.20
i i
P
i
0.25
0.30
Time [sec]
Figure 5.7: Results from experiments on PEO at room temperature.
The plots are fairly exponential, deviating only near the breakup time. The pre-shearing
of the fluid sample results in the filament breaking sooner than without the pre-shear. To
take a closer look at whether or not the fluid should be affected by the shear rates, one
can calculate the Weissenberg number, We.
In order to compute this number, the
relaxation time must be determined. The results from the oscillatory tests conducted
using the cone and plate rheometer shows that the longest relaxation time is 1.06s.
57
Details on relaxation time spectrum can be found in Entov and Hinch (1997). Using the
measurement from the cone and plate rheometer, We = 0.04 for the slower shear rate and
0.12 for the faster shear rate. At this relaxation time, the fluid is not affected by the
shearing. For the longest relaxation time, the pre-shear is expected to make a difference.
The We numbers for the experiments are We=6.65 for the slower rate and We=19.97 for
the faster rate. Since We number is greater than 1, the pre-shear is expected to affect the
fluid.
Next, three of the curves are chosen as representatives and curve fits are
performed on them. The resulting plot is seen in figure 5.8.
8
6
E
2
..
M
Curve fit
86
0=s-'
rad,
C> y0
-
10-4
=37.7O0rad,=6.28s
-
Curve fit
-
4
-
I
Sy= 37.70 rad,
Curve fit
I
I
0.05
=18.84 s'
I
I
0.10
I
I
0.15
0.20
i iiI
0.25
4i
I i
0.30
Time [s]
Figure 5.8: Three of the curves with theoretical fits.
The coefficients from the curve fits are found on table 5.1.
58
Table 5.1: Fit Coefficients for PEO
R1 [mm]
1
[s- ]
3A
k [s]
y=0 rad, f= 0 s-1
.94
9.20
0.036
y= 37.70 rad, y= 6.28 s-I
.95
10.68
0.031
y=37.70 rad, j=18.84 s~'
.91
11.20
0.030
The Weissenberg number for the pre-shear experiments computed using the relaxation
time from the curve fit are as follows: We= 0.23 for the slower shear rate and We= 0.68
for the faster shear rate. In both of these cases, the We number predicts negligible effects
of pre-shear on the sample at these shear rates.
But the data shows that there is some
influence of pre-shear on the time of breakup.
The extensional viscosity for the above three curves with their curve fits are
plotted on figure 5.9.
59
11'
16
1'
'
'
l i'
'
=Orad,
S0
i
''i
=
'
''
'
'
'i I~ '
'
'
i
'
' '
|
s-'
Curve fit
C,)
y
C6)
12
= 37.70 rad,
j=6.28
s
Curve fit
y = 37.70 rad,
j
= 18.84 s-
Curve fit
0
8
->C
0,
MI
4
X~
1 1 1 i | 1 i i 1~~C C>
1I 1 1 1
W
2
4
6
11|
|
11I i
8
10
Hencky Strain
Figure 5.9: The extensional viscosity of PEO with varying amounts of pre-shear and
their respective curve fits. Block each with 3
The extensional viscosity of PEO is not significantly affected by the pre-shear. But as
expected, the extensional viscosity increases with Hencky strain for this exponential
fluid. The three curves collapse onto each other. It determined that applying pre-shear
did have some affect on the break time of the filament, but not on the shape.
5.2.2 PSO25
The pictures from the Caber experiments at 23*C for PS025, PS025 with 3wt% clay, and
PS025 with lOwt% clay are found in figure 5.10.
60
t= 0.01 s
t= 22.70 s
t= 11.71 s
t= 34.00s
(a)
VT
VT
Ai
t= 0.01 S
t= 11.30 s
t= 5.13 s
t=16.39 s
(b)
V
t= 0.01 s
t= 30.39 s
t= 15.20 s
t=45.61s
(c)
Figue 5.10: Pictures from Caber experiment. The materials tested are: (a) PS025, (b)
PS025 with 3wt% clay, and (c) PS025 with IOwt% clay.
61
The three test fluids are tested on the Caber and the resulting data can be seen in figure
5.11.
The curve for PS025 is fairly straight on the semi-log plot indicating that it is
nearly exponential.
0-38
6
4
C')
I5
(U-
0%
10-4 8
_
Fit for 0%
3%
Fit for 3%
10%
-
Fitforl10%
-
6
-
4
2,
0
10
20
30
40
Time [s]
Figure 5.11: Resulting data plots with generalized curve fits for Caber experiments of
PS025, PS025 with 3wt% clay, and PS025 with lOwt% clay.
Curve fits are performed on the data using the equation for generalized fit found in
chapter 2.
This equation has a constant, a linear, and an exponential term.
The
coefficients for each of the curves are listed in Table 5.2.
62
Table 5.2: Fit coefficients for PS025 and PS025- clay mixtures.
0%
3%
10%
A [in]
1.6919x10- 4
4.7720x10-4
1.059x10 3
B [m/s]
4.4123x10- 6
3.1952x10-5
2.1925x10-5
C [m]
1.0186 x10-3
9.3502x10- 4
2.4993x10-4
D [s ']
1.4754 x10-
1.9546x10-
1.5152x10-'
The coefficients from the fit can be used to approximate relaxation time, elastic modulus,
Newtonian viscosity, and the yield stress of the material according to the following
formulas:
1
3D
0.0709ry
(A+C - R)pg
First, the coefficients of the PS025 are examined. The relaxation time obtained using the
generalized curve fit is 2.26 s. Compared with the value of 3.9s found using Zimm fit to
frequency sweep data, the two values are fairly close since they differ by less than a
factor of two. The value of the Newtonian viscosity computed using the coefficients
from the fit is 607.40 Pa.s whereas the value measured from the cone and plate rheometer
is 36.7 Pa.s. The shear viscosity of PS025 is estimated to be the Newtonian viscosity of
the fluid since the polymer concentration is so low that it does not greatly affect the
Newtonian viscosity of the styrene oil base.
63
Next, the coefficients of the PS025 with 3wt% clay are evaluated. The relaxation time
obtained using the generalized curve fit is 1.71 s. Compared with the value of 4.7 s found
using rotary diffusivity, the two values differ only by about a factor of three. The
difference could have resulted from the estimations made about the shape and dimensions
of the clay particles, which in turn would affect the computed rotary diffusivity. A small
difference in the value of the diameter used in the computation can largely affect the
rotary diffusivity since this parameter is raised to the third power in the equation. The
value of the Newtonian viscosity computed using the coefficients from the fit is 83.88
Pa.s, whereas the value measured from the cone and plate rheometer is around 36.7 Pa.s.
The shear viscosity of PS025 is used in the comparison since the Newtonian portion of
the viscosity is being compared. There is a difference of about a factor of two between
the values computed using the different methods.
Lastly, the coefficients of the PS025 with 1Owt% are examined. The relaxation time
obtained using the generalized curve fit is 2.20 s. Compared with the value of 4.7 s
computed using rotary diffusivity of the clay particles, the two values differ by 2.5 s.
Since the same method and values are used to compute the relaxation time of both of the
PS025 samples with clay particles, the rotary diffusivity method estimates them to have
the same relaxation time. The values computed from the generalized curve fits for the
two fluids also show that the relaxation time of the two fluids are fairly close to each
other, differing by 0.41 s. The value of the Newtonian viscosity computed using the
coefficients from the fit is 122.24 Pa.s whereas the value measured from the cone and
plate rheometer is 36.7 Pa.s. There is a difference of about a factor of 3. Lastly, the yield
64
stress measured using the cone and plate rheometer is compared with the computed value
using the generalized fit. The value from the rheometer is 23.7 Pa and the one from the
generalized fit is 0.19 Pa.
Figure 5.12 is the plot of the computed extensional viscosity of each fluid using the Caber
data and the generalized curve fits.
0-6
1600
0 0%
-
0
Fit for 0%
0
.3%
00
o
1200
-
800
T)
Fit for 3%
10%
Fit for 10%
CO
-
00
00
400
2
4
6
8
Hencky Strain
Figure 5.12: Plots of computed extensional viscosity of polystyrene and polystyreneclay mixture Caber data and generalized curve fits.
The extensional viscosity of PS025 reaches the highest value at the larger Hencky strains
since the slope of the radius verses time curve decreases near then end. All three of the
fluids having the ability to form strands that thin slowly contributes to the high
extensional viscosities. The shape of the extensional viscosity curves is determined by
the change in slope of the Caber data curves. The curve fit for the PS025 with 3wt% clay
is straight and diverges from the data for Hencky strain greater than 5, because the curve
65
fit on the Caber data maintains the same slope as the radius goes to zero, ending more
abruptly. The curve fit for the PS025 with 10% clay diverges from the data also because
the fit maintains the same slope through most of the Caber data. The fit curve decreases
in radius more gradually then the data especially during the last third of the experiment.
Each of the three curves has distinct shapes.
The effects of pre-shearing PS025 are examined by keeping the shear rate constant at
1.88 s- while varying the shear strain from 0 to 37.70 rad (figure 5.13).
200
0
6
y =
+
4
0 rad,
s
=12.57 rad,
=
=1.88 s"
y = 25.13 rad,
j=1.88 s'
V y = 37.70 rad,
=1.88 s-
- - Curve fit
CU 104
0
5
10
15
20
25
30
Time [s]
Figure 5.13: The Caber experiment results of varying the amount of shear strain applied
to the sample of PS025.
The shear rate of 1.88 s1 is the slowest shear rate that the system could do reliably, yet
12.57 rad of shear strain at this rate caused the fluid to collapse onto the same curve as
the other amounts of shear strain at the same rate. Application of the pre-shear shortened
66
the breakup time by about 10 s.
The comparison of the curve shape can be better
examined by looking at the plot of the extensional viscosity (figure 5.14).
1600
0
1
y=Orad,y=0s I
-Curve
1200
-
fit
y=37.7Orad,
y
11.88 S
Curve fit
0
C'
C
0
800
--
1
400
C
U)
3
4
5
6
7
8
9
Hencky Strain
Figure 5.14:
Computed extensional viscosity of PS025 with and without pre-shear.
Since applying varying amounts of shear strain at constant shear rate had similar effects,
only one of the plots from the pre-shear experiments at 1.88 s-1 is plotted.
The extensional viscosity curves were computed only for the no pre-shear and the 37.70
rad of shear strain experiments since applying varying amounts of shear strain at constant
shear rate had similar effects. The 37.70 rad shear strain case is taken as a representative
case. The shapes of the extensional viscosity curves for the two cases similar, but the no
pre-shear has higher extensional viscosity values.
This indicates that the two fluids
thinned in the same manner, but the speed of the no pre-shear case was slower.
67
The effects of pre-shear on PS025 were also examined by varying the shear rate while
holding the shear strain constant at 12.57 rad (figure 5.15).
-108
e
y=0
5
2rad,
=
0
6 -+
y
12.57
rad,
p 1.88 s~
4-7
y = 12.57
12.57
rad,
f=6.28
r
=
rad,
s-
c18.84 s
2.
108
6 --
0
5
10
15
20
25
30
Time [s]
Figure 5.15: Caber results of PS025 with pre-shear at varying shear rates.
All of the pre-shear experiments resulted in similar data plots. Therefore, applying at
least 12.57 rad of pre-shear at a rate equal to or great than 1.88 s-1 caused the PS025 to
maintain the same thinning behavior but to break faster. This resulted in the extensional
viscosity curve to stay the same shape while have lower values.
Same types of experiments were repeated for PS025 with 3wt% clay. Figure 5.16 has the
plot of the Caber data of holding the shear rate constant while varying the shear strain.
68
1-3
6
4
r-9
10
a,=Orad,
0-
=Osrad, =6.28 s-
6-
y=12.57
4 -
y =37.70 rad,
Curve fit
-
0
2
4
=6.28 s
6
8
10
12
14
16
Time [s]
Figure 5.16:
Caber data of PS025 with 3wt% clay.
The results from pre-shear
experiments at different shear strain amounts are plotted against the no pre-shear data.
At this shear rate, applying 12.57 rad of shear strain did not have an affect, but 37.70 rad
of shear strain caused the fluid filament to break sooner. Looking at the Caber data, the
shape of the curve for the larger amount of shear strain still looks similar to that of the no
pre-shear experiment. The fit for both curves diverges from the data near the critical time
because the slope of the data decreases more than that of the fit.
The extensional
viscosity for the no pre-shear and the 37.70 rad shear strain cases are examined in figure
5.17.
The extensional viscosity plot also shows how the slope of the Caber curve
changes as the filament thins.
69
Cn
400
=
-
o
300
0
200 -
rad,
-1
Curve fit
-
y
-
= 37.70 rad,
=6.28 s
urv e fit.
N
-
200 0
:100 -
2
4
6
8
Hencky Strain
Figure 5.17: The computed extensional viscosity plots of PS025 with 3wt% of clay.
Similar to the results seen for PS025, the effect of pre-shearing the test fluid on the
extensional viscosity is that the curve is shifted down for the later portion of the
experiment. For PS025 with 3wt% clay, the shift down begins around Hencky strain of
4.8.
The effects of pre-shear on PS025 with 3wt% clay were also examined by varying the
shear rate while holding the shear strain constant at 37.70 rad (figure 5.18).
70
10-3
6
10
y=0 rad,= 0 s 1
-4
T3
-=37.70
rad,
6
3 7.7 0 rad,
Y=
4
0
37.70 rad,
2
4
=1.88 s~
= 6.2 8 s
j=18.84
6
s-
8
10
12
14
16
Time [s]
Figure 5.18: Caber data of PS025 with 3wt% clay with pre-shear at different shear rates.
Increasing the shear rate caused the fluid filament to break sooner, but the general shapes
of the curves are again similar.
Applying pre-shear to the PS025 with lOwt% clay at the rates that the current motor
could do did not significantly affect the results. The Caber data of pre-shear are seen in
figure 5.19. Though the break time did decrease by 2.96 s for the 37.70 rad shear strain
at 18.85 s~1 experiment compared to that of the no pre-shear, the change is not as big
of a
decrease as those seen for the other two PS025 fluids.
71
0-3
r-
6
4
E
CO,
--
2
0-4
=Orad,
A
r
6
4
-
0
y
=0s-1
j=18.84
37.70 rad, j -18.84
12.57 rad,
=
ss
30
20
10
40
Time [s]
(a)
10-3
6
4
CD)
:3
Sy=Orad,
10-4
rad,
j
y = 37.70 rad,
j
y = 37.70 rad,
j
y = 37.70
6
0
=0s
10
= 1.88 s
6.28 s-18.84
s
20
30
40
Time [s]
(b)
Figure 5.19:
The Caber data of PS025 with lOwt% clay.
(a) The results from
experiments with different amounts of pre-shear are plotted with the no pre-shear data.
(b) The same amount of pre-shear is applied at three different shear rates.
72
5.3
Power Law Fluid
Regular and nonfat plain Dannon yogurts are tested to study the behavior of power law
fluids. The pictures from the Caber experiments are seen in figure 5.20.
t= 0.01
t= 0.07
t=O.15
t=0.21
Figure 5.20: Pictures of regular yogurt from Caber experiment.
Though the yogurt does not form long cylindrical filaments, the same equations for
extensional viscosity are used since the filaments still fit under the slender body
approximations.
The regular and nonfat yogurts are experimented on the Caber and the resulting
plots are seen with their respective curve fits in figure 5.21. The yogurt containers are
taken out of the refrigerator during the experiments. The temperature of the yogurt when
placed on the Caber cylinders is around 19C.
73
2-
10-
E
C4)
0 Reg
V Nonfat
-0 -4
O10
--
4
0.00
Fit for Reg
F- for Nonfat
Fit
0.05
0.15
0.10
Time [s]
Figure 5.21: Yogurt data from Caber experiments. Power law curves are fit to the data.
The curves are fit to the following power law equation from chapter 2:
Rmi,,(t) = A (trit -t)
.
The coefficients for the yogurts are A= 0.0015 m, tcrit= 0.193 s, and n= 0.23 for regular
yogurt and A= 0.0017 m, tcrit= 0.172 s, and n= 0.30 for the nonfat yogurt.
It is
determined that there isn't much difference between the regular and nonfat yogurt. This
is not a surprise since it is seen in chapter 3 that the shear properties of the yogurt are
similar.
To compare the coefficients from the fit to the material property values obtained
using shear rheology, the viscosity verses shear rate data is plotted on a log-log scale
(figure 5.22).
74
2
100
rn
n= .10
8
--
66
EL
o Data
Curve fit: m= 11.71
464-
0
> 10
60
3
4 56
2
2
3 4 56
3
4 56
1
0.1
[s~1]
Shear rate
Figure 5.22: The shear viscosity verses shear rate date obtained from stress sweep test
on the cone and plate rheometer.
The above figure is the result from regular yogurt. This plot is used to compare with the
extensional data for both the regular and nonfat yogurt since the shear material properties
are almost identical for the two yogurts. First off, the values of n from the Caber curve
fits are compared with the cone and plate results. The former is 0.23 for regular yogurt
and 0.30 for the nonfat yogurt. This is in the same order of magnitude as the 0.1 obtained
from the shear tests.
The value of m can also be obtained from the Caber data using the following
equation from chapter 2:
1 o- Fi
1
75
The values of A from the curve fit to the Caber plots are used. Using A= 0.0015 m for
regular yogurt, m computed using the above equation is 16.67 Pa.s". Using A= 0.0017 m
for nonfat yogurt, m= 13.80 Pa.s". Both values are the same order of magnitude the value
of 11.71 Pa.s" obtained from fitting the shear rheology data.
The computed extensional viscosity data is plotted in figure 5.23.
r-
0n
5
o Regular
-
4
V
Curve fit
Nonfat
-- Curve fit
0
C
0
-
3
-V
-',
Cn
0
2
0
VV
0 Of
-_
C'
Cj
V
1
-
0
V
-W
10
v
~
1 1
1.5
2.0
2.5
1 1 1 1 1 1 1
3.0
3.5
4.0
Hencky Strain
Figure 5.23: Computed extensional viscosity plot of regular and nonfat yogurt.
The extensional viscosities of the two yogurts are similar. The shape of the curve is as
expected. The slope of the radius verses time curve is high at the beginning and the end
but lower in the middle. Since the extensional viscosity is indirectly proportional to the
slope, the opposite trend is seen in this curve.
76
Next, the effects of pre-shear are examined. Regular yogurt is loaded onto the Caber and
allowed to relax for 15s so that the yogurt. To ensure greater repeatability, the yogurt is
not stirred when the container is first opened. Instead, the mixing that occurs during
injection in and out of the syringe is expected to suffice. On figure 5.24 are the plots
from pre-shear experiments on the Caber where the amount of shear strain is varied.
6
4
r""
E
^V
2
0.1
6,
EU
4. -
2, . A
0.01
-
q
6
y=Orad,f=0s
y = 12.57 rad,f =1.88
y=25.13 rad,= 1.88
y = 37.70 rad, =1.88
yy=50.27rad,=1.88
-
s
s
1
s
s
y = 62.83 rad, j=1.88 s
0
10
1
20
30
40
50
60 70x10 3
Time [s]
Figure 5.24: The results from Caber experiments. Regular yogurt tested with varying
amounts of shear strain.
From the above plots, it is determined that adding pre-shear to the yogurt sample further
breaks the gel structure. But for strain greater than 37.70 rad, the yogurt reaches a limit
where increasing the amount of shear strain no longer has an effect.
The effects of varying the rate of the pre-shear are also examined. Figure 5.25 plots the
results from the Caber experiments.
77
6
4-
E
E
2-
r__9
<
0.1 -
-
Vn
0
-o
40
r=ra,=O
y= 37.70 rad,
2
S
--
=1.88 s~1
y = 37.70 rad, =3.77s
y= 37.70 rad, Y = 6.28 s
M y=37.70 rad,
0.01
-1
1
10
A
V
=12.57s
I
0
N
20
1
30
40
50
60 70x10
-3
Time [s]
Figure 5.25: The results from Caber experiments. Regular yogurt tested with varying
the shear rate of the pre-shear.
Similar to increasing the amount of shear strain, increasing the shear rate shortens the
critical time of the breakup. But increasing the shear rate causes a sharper drop from the
no pre-shear breakup time than the drop seen in figure 5.24 caused by increasing the
shear strain. This time decreases until a limit is reached. To further examine the effects
of varying the shear rate and shear strain on the critical time, a Pipkin diagram is
constructed (figure 5.26).
78
0.08
007
0.06
0.041
03
001
040
Shear rate [s
030
7
BD
Shear strain [rad]
Figure 5.26: Mesh plot of the critical break times for regular yogurt at various shear
rates and shear strains.
The highest time occurs when no pre-shear has been applied and the time decreases as
shear rate and shear strain increases. The values plotted in figure 5.26 are tabulated in
table 5.3 and 5.4.
79
Table 5.3: Critical time for Caber experiments with rate of pre-shear held constant for
each run.
1.88 s-
0 rad
12.57 rad
25.13 rad
37.70 rad
50.27 rad
62.83 rad
0.072 s
0.072 s
0.064s
0.042s
0.027s
0.028 s
'
Table 5.4: Critical time for Caber experiments with amount of pre-shear held constant
for each run.
37.70 rad
5.4
O s4
1.88 s'
3.77 s-
6.28 s1
12.57 s
0.072s
0.042s
0.033s
0.027 s
0.027 s
I
Fluid with Volatile Solvent
Acrylic paint is tested to see how it's extensional material properties change as the
solvent evaporates. The following are pictures of paint being tested on the Caber (figure
5.27).
t= 0.01
t= 0.20
t= 0.29
t= 0.36
80
every 60 s to
Figure 5.27: Caber experiment on acrylic paint. The sample is stretched
are from the
monitor the change in the fluid as the solvent evaporates. The pictures
second stretch where t= 120 s.
paint
The above pictures are from experiment that was conducted to examine how the
stretched
changes over time. A sample is placed between the Caber cylinders and is
s. The
every 60s. Figure 5.27 displays the pictures from the second stretch where t= 120
resulting plots of the runs are seen in figure 5.28.
6
4
2
E
W
ca
o
t= 60
S
0 t= 120 s
104
8
6
4
*S
Curve fit
t= 300 s
*1
-- Curve fit
St= 360
St= 420
2
.
Cure fit
t= 180s
17 t= 240 s
--
s
s
Curve fitO -IA
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
s run time.
Figure 5.28: Batch test on red paint using Caber with 60 s intervals and 1
the power
The runs are performed on the same sample. Curve fits are performed using
law equation.
81
Initially as the paint dries, the breakup time increases. After 240 s, the breakup time
jumps down and then fluctuates.
After looking at pictures of the experiment, it is
conjectured that this jump is a result a thin film of dried paint forming around the sample.
The film closes around the bottom reservoir of fluid when the cylinders are extended
decreasing the amount of fluid that can form the fluid filament. The film affects the
availability of the fluid differently on different runs because its collapse depends on its
shape, which can change during a run. Figure 5.29 is a set of pictures explaining this
occurrence.
Dried film
(a)
(b)
(c)
Figure 5.29: As the paint dries, a film of dry paint forms around the sample decreasing
the amount of fluid that is stretched. (a) The sample during a stretch. (b) After the
stretch, the top cylinder is brought back to the initial height. (c) Closed position.
Power law fits are performed on the curves for times 120s, 240s, 300s, and 420s and the
result are used to compute curve fit plots of extensional viscosity (figure 5.30).
82
30
-"
l
25
Curve fit
0Cn
40
v t= 240 s
Curve fit
t= 300s -:
Curve fit
- t= 420s -
V
S7
20
0
15
a,
0
Cn
C
w
t= 120s
V
Curve fit
-
10
5
0
.
I
I
I
1.. 5. 1.
4.0
.
4.5
5.0
.
.
5.5
Hencky Strain
Figure 5.30: The computed extensional viscosities using the Caber data for paint.
The extensional viscosity also fluctuates with increasing time. The shapes of the curves
are similar to that of yogurt in that they increase and then decrease.
Next the effects of pre-shear are examined. A shear strain of 37.70 rad at 6.28 s1
is applied to the sample before each stretch. There is a 30 s interval between each run so
that the paint can dry. The data are plotted in figure 5.31.
83
6
E
E
0.1
-V
8m
6a4-
8
V
<>
2[
t=- 180 s
K
0.00
t=- 30s
t=- 60 s
t=- 90 S
t=- 120 s
t>150
s
L
U
,e
UL
V
V
X
+o"
i
/
t--210 s
0.05
0.10
0.15
0.20
0.25
Time [s]
Figure 5.31: Paint tested on Caber. The sample is pre-sheared 37.70 rad at 6.28 s-1 for
each stretch. There is a 30 s interval between each run.
The break time increases for t= 60s and t= 90s, but decreases for longer times. The time
to break decreases repeatedly until it hits a plateau.
It is possible that as the solvent
begins to evaporate, the paint becomes stringier causing it to break later. But as more of
the solvent evaporates, the material changes so that the pre-shearing causes the particles
to orient themselves in directions of the pre-shear. This could be the reason why it breaks
earlier under extension. Another hypothesis is that as the paint reaches a certain solvent
level, the pre-shear breaks the polymer chains that formed as the solvent evaporated.
Also the thin film that forms on the outside of the sample could add to the decreasing of
the breakup time. In order prevent the forming of the outside film, experiments were run
without time delays in between the runs. Each run consisted of 125.66 rad at 6.28 s~ .
84
Fifteen runs were done on the same sample. Again the thin film formed on the outside of
the sample around 140 s and the resulting plots showed the same pattern as before (figure
5.32).
The paint's critical time increases with time for waits up to 100s and then
decreases starting that time, reaching a plateau starting at 160s.
0.24
A.,
-4
-
0.28:
-411
-
#41
.
1
0.20
14
1#
0.16
50
100
150
200
250
300
Time [s]
Figure 5.32: Plot of breakup times for paint filament. Fifteen Caber runs are performed
on the paint sample with 125.66 rad of pre-shear at 6.28 s-1 being applied for each run.
85
Chapter 6 Conclusion and Future Work
6.1
Conclusion
Glycerol, PEO, PS025 (with and without clay particles), yogurt, and paint were studied in
the current research. For glycerol, the predicted data was obtained. The radius decreased
in time linearly and applying pre-shear did not have an effect on the data.
The
extensional viscosity calculated using Caber data was 2.81 Pa.s. This value is lower than
the value calculated using the shear viscosity value, but it is within 10 percent
For the PEO samples, the critical time was lowered the most at the larger shear strain
applied, 37.70 rad, at the higher shear rate, 18.84 s-.
Between the two rates, the faster
rate decreased the break time more when same amount of shear strain was applied.
Similarly, the break time decreased more for the higher amount of shear strain when the
shear rate was held constant. The extensional viscosity was not significantly affected by
the pre-shear applied at these rates and amounts. This could be a result of the pre-shear
occurring near We of 1. When the number is calculated using the longest relaxation time
found from oscillatory experiments, X= 1.06 s, it is greater than 1 for both shear rates.
But when the relaxation time found using curve fits on the Caber data, X= 0.036 s, was
used to compute We, the value was less than one for both shear rates. It is predicted that
for higher shear rates, the critical time would be further lowered and the extensional
viscosity would decrease.
The extensional viscosity plot of PS025 showed that the extensional viscosity increases
with increasing Hencky strain, though the curve began to decrease in slope starting at
86
F=7. The extensional viscosity of PS025 was much larger than that of glycerol or PEO
with the value increasing to 1510 Pa.s at s= 8.95. Applying pre-shear of 37.70 rad at 1.88
s1 decreased the critical break time from 34.03 s to 24.31 s. The less shear strain at this
shear rate decreased the break time to time same time range ± 2 s. Appling 12.57 rad of
pre-shear at three different rates (1.88 s-1, 6.28 s-1 , and 18.84 s-1), also brought the break
time to the same region. The break time for the three runs ranged from 24.79 s to 26.47
s. The extensional viscosity curve for the pre-sheared sample was similar in shape to that
of the no pre-shear case, but the value for the extensional viscosity was increasingly less
than that of the no pre-shear run as Hencky strain increased. This decrease in extensional
viscosity was caused by the lining up of the polymers in the direction of the pre-shear.
The PS025 with 3wt% clay filament broke 17.61 s faster than the PS025 filament. The
shape of the curve is also different from that of PS025, the difference also showing up on
the extensional viscosity plot.
The increase in extensional viscosity with increasing
Hencky strain is not as drastic and only reaches up to 484 Pa.s at 6= 9.52. The clay infers
with the polymers in such a way that the filament breaks sooner and has lower
extensional viscosity at all values of F. Applying pre-shear to this test fluid had the same
effects that of PSO25.
The filament formed by PS025 with 1 Owt% clay broke later than that of the other PS025
test fluids at 45.88 s. The extensional viscosity plot shows relative speed at which the
filament broke. The filament for the sample with 1Owt% clay thinned slower than the
other two fluids for s<5.02, indicated by the extensional viscosity for this fluid being
87
greater than the other two in this range. Then, the curve dips further with increasing 6,
reaching extensional viscosity of 161 Pa.s at E= 9.22.
This amount of clay content
slowed the thinning down when the filament first formed. But once the thinning caused
the filament to reach a critical surface curve, the thinning quickly increased in speed.
Pre-shearing the fluid at the ranges that were possible with the current setup did not
significantly affect the break time or the extensional viscosity.
Both nonfat and regular plain yogurts are tested on the Caber. The data was curve fit
using a power law fit. The extensional viscosities of the two fluids were found to be
similar. The curve increases until it reaches c=2.5 ± 0.2, and then decreases back down.
The pre-shear test was performed only on the regular yogurt. The yogurt samples bought
at different times of the year showed significant breakup times (0.19 s for yogurt
purchased in July verses 0.07 s for yogurt purchased in December).
Therefore,
experiment sets for comparing pre-shear effects were performed on yogurt from the same
container. Applying pre-shear of different shear strains at the same rate and applying the
same amount of shear strain at various rates were examined.
Increasing the rate or
increasing the amount of strain decreased the critical break time until it reached a limit of
0.027 s ± 0.001 s. The pre-shearing of the fluid caused the gel structure of the yogurt to
break.
From the Caber experiments on acrylic paint, it is shown that the extensional properties
change as the paint dries. The break up time increased for the first few runs of all
experiment sets, but decreased during the rest of the set for some and jumped around
88
unpredictably for others. The shape of the curve also changed as the solvent evaporated,
indicated by the extensional viscosity plots. Although all of the extensional viscosity
curves increased with increasing Hencky strain until a maximum value and then
decreased for larger Hencky strain, the steepness of the curve changed as the paint dried.
Pre-shear experiments were also run on the paint samples with shear strain being applied
before each stretch. The same trends in the curves were found for these experiments
6.2
Future Work
For further research on the effects of pre-shear on the Caber data, a new motor allowing a
wider range of shear rates could be installed to replace the current motor. It would be
interesting to see the results of applying shear rates two or three times faster than the
current fastest shear rate of 18.84 s-1. Also, more in depth studies could be done on any
one of the fluids such as PEO. The affects of fluid buckling during pre-shear could be
examined to understand how the non-homogeneous application of pre-shear affects the
results. Dyes could be used on the outer rim of the loaded fluid sample to enhance the
visualization of the pre-shear.
89
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94