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A
Appendix A
Derivation of Creeping Flow and the Result for Low
Reynolds Number Flow Around a Sphere
A.1 Derivation of Creeping Flow
The Navier Stokes equation is
uuuu 2∇+−∇=
∇•+∂∂
µρ pt
(A.1)
where u is the vector of fluid velocities, p is the fluid pressure, µ is the fluid
viscosity and ρ is the fluid density. The left-hand side of Eq. (A.1) represents the
inertia of the system, or the acceleration of the fluid. On the right-hand side, the
pressure gradient represents a force acting on the fluid because of a non-uniform
pressure, while the viscous term represents a shear stress on the fluid opposing any
motion that is occurring. The volume of liquid is conserved, and this condition is
stated as
0=•∇ u (A.2)
Equation (A.1) is a vector equation, comprising three separate equations. To-
gether with (A.2), the Navier Stokes equation comprises four simultaneous
equations for the four unknown variables in the system, which are the three
components of the velocity (u) and the fluid pressure (p).
The physical problem under consideration will dictate the natural coordinate
system to use for the equations. For instance, when considering flow past a
spherical particle, it is conventional to use spherical coordinates with the origin at
the centre of the particle. Fig. A.1 uses spherical coordinates for such a case.
276 Appendix A: Derivation of Creeping Flow and the Result for Low Reynolds Number Flow
A.2 Scaling of the Navier-Stokes Equation
The Navier-Stokes equations are non-linear, and no complete analytical solution
exists. For complex flows, it is common to resort to numerical integration. In
certain circumstances, however, it is possible to make analytic progress. The case
of flows in colloidal systems is one such example, where the smallness of the
colloidal particles and the slowness of the flows results in the non-linear terms in
(A.1) becoming negligibly small. This condition is derived below with the result
that the inertial terms in colloidal flows (left-hand side of (A.1)) are irrelevant.
The relative magnitude of the different terms in (A.1) may be estimated by scal-
ing them. The characteristic length scale is the particle radius R. The characteristic
velocity is taken as u*. Hence, the characteristic time is expressed as R/u*. The
t∂
∂u
term in Eq. A.1 may be written as
tuR
u
∂∂u
*/
*
where the overbars indicate dimensionless quantities that have a magnitude
between zero and one. Performing the same analysis on each term in (A.1) results
in a scaled equation:
uuuu 2
2
2 ***∇+∇−=
∇•+∂∂
R
up
R
P
tR
u µρ (A.3)
It is conventional to divide through by the viscous term and to choose the char-
acteristic pressure to be µu*/R. The result is
uuuu 2*
∇+∇−=
∇•+∂∂
pt
Ru
µρ
(A.4)
where the dimensionless group
µ
ρ Ru *
is called the ‘Reynolds number’. It is a measure of the relative magnitude of the
inertial terms in relation to the viscous terms. For a 1 µm particle in water (µ =
10–3 Pa s) with a characteristic velocity of 1 µm/s, the Reynolds number is 10
–6.
Because this is so tiny, the inertial terms on the left-hand side of (A.4) may be
A.3 Stokes Flow 277
safely ignored. In dimensionless terms, the governing equations – in the low
Reynolds number limit – become
02 =∇+∇− up (A.5)
0∇ • =u (A.6)
Equation (A.5) tells us that the flow is independent of time. The viscous drag,
u2∇ balances the applied pressure gradient p∇ . The condition described by these
equations is called ‘creeping flow’, because the equations apply for very low
velocities. The equations may be solved analytically for flow around a colloidal
particle. This solution is called ‘Stokes flow’.
A.3 Stokes Flow
For creeping flow around a spherical particle, an analytic solution is available, if
the flow at a great distance from the particle is at a uniform velocity in a single
direction, as shown in Fig. A.1. The boundary condition assumes no flow ( 0=u )
at the particle surface. From the derived flow pattern, the shear stress on a
colloidal particle may be calculated. Integration of the stress over the particle
surface gives the drag force Fdrag on a particle. Physically, this is the force that
resists flow of a particle through a fluid and that arises from the fluid’s viscosity.
For a particle of radius R in a uniform flow of velocity U, the drag is given by
URFdrag πµ6= (A.7)
In this limit, the drag resistance experienced by a colloidal particle moving in a
viscous fluid is proportional to its size.
A.4 Sedimentation
Particles will settle if their density is greater than that of the solvent surrounding
them. If the particles are less dense than the solvent, they rise in a process called
‘creaming’. Thus, the direction of particle motion (up or down) depends on
whether the difference in density between the particles and solvent (∆ρ) is positive
or negative. Sedimentation and creaming are the result of the gravitational force
associated with the acceleration due to gravity, g. The gravitational force acting on
a single particle is given by Fgrav = 4/3 π R3 ∆ρ g. In this expression, a force is
obtained by multiplying g by the difference in mass between a particle and the
fluid it occupies. At equilibrium, this gravitational force exactly balances the
278 Appendix A: Derivation of Creeping Flow and the Result for Low Reynolds Number Flow
Stokes drag force (A.7). Setting the two forces equal, the sedimentation velocity is
then found to be
µ
ρ gRUsed
2
9
2 ∆= (A.8)
The strong dependence of Used on R provides a means to separate particles by
size. Larger particles sediment to the bottom of a container faster than smaller
ones. Similarly, mono-sized particles may be separated by their density differ-
ences. Stable colloidal particles, such as in a latex, resist sedimentation and
creaming.
U∞ Fdrag = 6 π µ U R r
θ
Stokes Flow Solution
θµ
θ
θ
θ
CosR
U
r
RPP
Sinr
R
r
RUU
Cosr
R
r
RUUr
∞
∞
∞
−=
++−=
+−=
2
2
0
3
3
2
2
2
3
44
31
22
31
R
Fig. A.1 Flow of a Newtonian fluid past a sphere in the limit of low Reynolds number results in
Stokes flow and a drag on the sphere that is linear in the flow velocity.
Appendix B
Appendix B
GARField Profiling Techniques and Experimental
Parameters
In an MR profiling experiment, the pulses of RF radiation are used to rotate the
magnetisation of the aligned nuclear spins. An excitation pulse that lasts for a
sufficient time to rotate the magnetisation, from the longitudinal axis into the
transverse plane, is referred to as a 90° pulse. In the GARField design, the time
duration of a 90° pulse is about tpd = 1 µs (Glover et al. 1999). A 180° pulse
logically lasts twice as long as a 90° pulse, and it inverts the direction of the
magnetisation along the longitudinal axis. An excitation pulse of RF radiation is
followed by an RF pulse sequence that is used to create a series of echoes by
refocusing the nuclear spins that are out of synchronisation.
In GARField profiling, NMR signals are obtained using a quadrature spin-echo
sequence (McDonald and Newling 1998, Mitchell et al. 2006), which is described as:
(90°x – τ – (90°y – τ – echo – τ)N – τR) .
After the first excitation pulse, there are pulses to create N echoes separated in
time by 2τ, where τ is called the ‘pulse gap’. There is repetition delay time of τR
before the sequence is repeated again for NS times to enable many averages. In a
typical experiment, N = 32 and τ = 95.0 µs (Mallégol et al. 2006). In order to
enable the nuclei to relax fully before repeating an acquisition, τR should be
chosen so that it is at least five times greater than the spin-lattice relaxation time,
T1, of the sample.
The same RF coil that is used to create the RF pulse is used to detect the echo
signal. As the time between RF pulses is 2τ, and as the pulse itself requires a short
amount of time, then the time to acquire the echo signal is slightly less than 2τ.
The echo is recorded as a fixed number of points (or acquisitions), Nacq, acquired
at regular intervals in time, called the ‘dwell time’, tD. As explained, these two
parameters must be chosen so that NacqtD < 2τ.
280 Appendix B: GARField Profiling Techniques and Experimental Parameters
Echoes are acquired as a signal varying in time. A Fourier transform of an echo
takes the signal from the time domain to the frequency domain, so that the
frequency can be correlated with spatial position through Eq. (2.13). To obtain a
GARField profile, each of the N echoes in a train is Fourier-transformed and then
summed for each of the NS scans. Profiles are normalised by an elastomer standard
to correct for sensitivity decline over film thickness.
It is helpful to note that when the spacing in time between the measured points in
an echo is tD seconds, then the total range in the frequency domain after a Fourier
transform will be 1/tD Hz. This range of frequency converts to the length range that
can be viewed in a profile, which will contain Nacq points spaced evenly apart.
The thickness of a wet latex film determines the minimum field-of-view (FOV)
that is required. The quality of the data increases with an increase in the signal
intensity over the background noise level, or the so-called ‘signal-to-noise ratio’
(SNR). Almost always, there is a desire to reduce the resolution (i.e., distance
between points in a profile, ∆y), to obtain information on non-uniformities at short
length scales. The resolution may be adjusted through the spacing of points in the
profile in the frequency domain (1/NacqtD). If the echo train decays quickly in time,
as caused by a short T2 relaxation time, then this time sets the resolution rather
than the echo acquisition time (NacqtD). The mathematical relationships presented
in Table B.1 show how these profile characteristics are determined. For instance,
in GARField profiles when Nacq = 256, tD = 0.4 µs (Glover et al. 1999), the pixel
resolution is
6 1 6
16.7
(42.58 10 )(17 / )(0.4 10 )(512)y m
HzT T m sµ
− −∆ = =
⋅ ⋅
It is apparent that a stronger gradient, Gy, will decrease the resolution, ∆y, but
the field of view will likewise be decreased. Also, increasing NacqtD will improve
the resolution but require longer pulse gaps and hence longer total profiling times,
possibly prohibiting the study of fast drying. Taking more scans, NS, will increase
the signal-to-noise ratio, but at the expense of more time per profile.
References 281
Table B.1. Interrelationship between NMR Profile Characteristics and Experimental Parameters
(McDonald and Newling 1998, Mitchell et al. 2006)
Parameter Relevant Meaning Equation
Field-of-view,
FOV
Thickness of excited
volume; Maximum film
thickness that can be
analysed pdyy tGG
FOVγγ
1∝
∆Ω=
Relaxation-limited
resolution, ∆y
Distance between data
points in a profile in
more solid-like samples 2
2
TGy
yγ=∆
Recording-window-
limited resolution, ∆y
Distance between data
points in a profile acqDyacqy NtGtG
y⋅
==∆γ
π
γ
π 22
Signal-to-noise ratio,
SNR
Indicates the extent to
which the signal is
above the noise level
and hence the data
quality
y
NSNR S
∆∝
∆Ω = pulse bandwidth; tpd = time duration of RF pulse; τ = pulse gap or delay time between RF
pulses; tD = dwell time between acquisitions of an echo; Nacq = number of acquisitions in a single
echo; tacq = tDNacq = acquisition time; NS = the number of scans (i.e., repetitions of the pulse
programme)
References
Glover P.M., Aptaker P.S., Bowler J.R., Ciampi E., McDonald P.J. (1999) A
novel high-gradient permanent magnet for the profiling of planar films and
coatings. J Magn Reson 139: 90-97
Mallégol J., Bennett G., Dupont O., McDonald PJ, Keddie JL (2006) Skin
development during the film formation of waterborne acrylic pressure-sensitive
adhesives containing tackifying resin. J Adhesion 82: 217-238
McDonald P.J. and Newling B. (1998) Stray field magnetic resonance imaging.
Rep Prog Phys 61: 1441-1493
Mitchell J., Blümler P., McDonald P.J. (2006) Spatially resolved nuclear magnetic
resonance studies of planar samples. Progr Nucl Magn Reson Spectrosc 48:
161-181
Appendix C
Appendix C
Terminology of Humidity and an Expression for
Evaporation Rate
The first part of this appendix defines a number of terms relating to humidity. Our
experience from numerous undergraduate lectures is that the definitions are
difficult to understand upon first reading, but illustration with relevant examples
makes them clear. The definitions of relevant terms are given immediately
below in Sections C.1 through C.9, and this is followed by a number of worked
examples.
C.1 Humidity
Consider the atmosphere you are in. The air is at a certain average temperature,
and it has some water dissolved in it, making it somewhat moist. The amount of
water dissolved in a sample of air, called the ‘humidity’, is given the symbol H
and is measured in the SI units of kg of water per kg of dry air. For example, the
air in a vessel may have a mass of 100 kg, and it may contain 1 kg of moisture,
making the total mass 101 kg. The humidity in this example is H = 1/100 = 0.01
kg water per kg of dry air (or 0.01 kg/kg).
C.2 Relative Humidity
The air will be able to support only a certain maximum amount of water. At
loadings above this level, the air will be saturated and liquid water will condense
from the gas phase. The maximum amount of water that can be supported is a
function of temperature and pressure. For instance, warmer air can hold more
284 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
moisture. A common outcome of this is dew forming on grass in the early
morning. At the lowest temperature of the day, the atmosphere cannot hold as
much moisture as it could when warm. As a result, water condenses.
The ratio of the air’s humidity to the maximum possible humidity at the same
pressure and temperature is called the ‘relative humidity’ and is given the symbol
H0. As an example, at 25oC and at a pressure P of one standard atmosphere (1.02
x 105 Pa), the saturation humidity is 0.02 kg of water per kg of dry air. For the case
considered in Section C.1 with a humidity of 0.01 kg/kg, the relative humidity will
be H0 = 0.01/0.02 = 0.5 or 50%. Relative humidity is conventionally expressed as
a percentage.
C.3 Dry Bulb Temperature
The temperature of a sample of air is termed the dry bulb temperature. It is measured
by placing a thermometer into the air sample while ensuring that no liquid water is
on the thermometer, i.e., the ‘bulb’ is dry. The significance of dry bulb temperature,
and the reason to define what seems like such a mundane quantity, will be clear after
defining the counterpart called the ‘wet bulb temperature’.
C.4 Wet Bulb Temperature
When liquid water is in contact with water in the air at a pressure less than the
vapour pressure of the liquid, there will be evaporation. The process requires
energy to overcome the latent heat of evaporation of water, Λl. Hence, during
evaporation, there is a heat flux from the air into the liquid water. The result is that
liquid water is at a lower temperature than the air in contact with it. The tempera-
ture of the water is called the ‘wet bulb temperature’. It is measured with a ‘wet
bulb’ by either placing a thermometer into liquid water or by covering it with a
wet cloth. The wet bulb temperature is determined by how fast the water evapo-
rates, and so it is a function of the dry bulb temperature and the relative humidity.
C.5 Specific Volume
Air is a compressible gas that will change its volume, V, with a dry bulb temperature,
T, as described by the well-known ideal gas equation, PV = nRT, where n represents
the number of moles of the gas, and R is the gas constant equal to 8.314 J mol–1 K
–1.
The specific volume of air is measured in the SI units of m3 per kg of dry air at the
specified pressure, P. Humid air will have a higher specific volume than dry air at the
same temperature, because of the volume contribution of the water vapour.
C.6 Enthalpy of Air 285
C.6 Enthalpy of Air
As water evaporates, the latent heat of the water requires a large energy input from
the air to achieve the phase change. This is the equivalent to stating that the
enthalpy of air increases with humidity. The enthalpy of air should not be con-
fused with the enthalpy of vaporisation, because the former depends on how much
water is in the air and is not a function of the water itself. The magnitude of this
enthalpy allows for the design of heating duties in industrial dryers and for the
estimates of the time needed to allow for drying.
C.7 Psychrometric Chart
The quantities defined in the previous sections can all be found on a psychromet-
ric chart, an example of which is given in Fig. C.1. The horizontal axis is the dry
bulb temperature, measured in Celsius. The vertical axis, on the right-hand side of
the chart, presents the humidity H in units of kJ/kg. Note that the axis is labelled
on its left-hand side. With the dry bulb temperature and humidity specified, every
other variable may be determined from the chart. The relative humidity runs
diagonally on the chart on lines with a slight curvature and with a positive slope;
higher values are on the upper left side. As an example, the 40% relative humidity
line is labelled.
The specific volume is presented as straight lines running diagonally from the
upper left to the lower right. The values are identified above the lines. The line of
specific volume of 0.875 m3 per kg of dry air is shown in Fig. C.1 as an example.
The wet bulb temperature is shown on the curved left hand axis that corresponds
to the 100% relative humidity line. The lines run from left to right with a down-
ward slope. The labels for the temperature are written on the lines. The line
corresponding to a wet bulb temperature of 20°C is identified in the example.
Lines of constant enthalpy of air are also shown on the psychrometric chart.
They are presented as the solid lines running nearly parallel to the lines of constant
wet bulb temperature. The values are labelled on the vertical axis on the right-
hand side of the chart and on the diagonal axis on the left-hand side. The line of
40 kJ/kg dry air is identified in Figure C.1.
The use of the chart is given in the following example. At a dry bulb tempera-
ture of 30°C, and with a humidity of H = 0.02 kg/kg, the chart tells us that the
relative humidity is about 75%. The wet bulb temperature is approximately 26°C,
showing that evaporative cooling has reduced the water temperature by 4°C. The
specific volume of the air is 0.885 m3/kg. Finally, the enthalpy of air for this
example is 80 kJ/kg.
286 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
C.8 Dew Point
If a sample of humid air is cooled, at some point liquid water will condense. The
temperature at which liquid water first appears is called the ‘dew point’. Because
the cooling of an air sample is a constant humidity operation, the dew point can be
found by moving horizontally on the psychrometric chart (along a line of constant
H) to the 100% relative humidity line and then reading off the temperature. The
dry bulb and wet bulb temperatures are identical at 100% relative humidity.
C.9 Relating Humidity to Partial Pressure
Recall that humidity is defined as the mass of water per mass of dry air. Another
way of describing the water vapour content is to use the water’s partial pressure.
In an air sample at total pressure P, the partial pressure of water, Pw leaves a
pressure P-Pw remaining for the dry air. Assuming an ideal gas relation, the
number of moles of water in the sample is n = PwV/RT using the symbols defined
in Section C.5. With 18 being the molar mass of water, 18 Pw V/RT follows as the
mass of water in the air. Assuming that the dry air consists of nitrogen gas, its
mass is given by 29 (P-Pw)V/RT. Hence, the humidity is related to the partial
pressure of water by
( )
18
29
w
w
P
P P=
−H (C.1)
This equation may be inverted to give the partial pressure of water as
29
2918 ~29 18
118
wP
P=
+
H
H
H
(C.2)
where the last approximation is valid if 29H/18 <<1, which is typically the case.
Hence, humidity is simply a measure of the water partial pressure.
Example 1
A sample of air is at a dry bulb temperature of 20°C and has a relative humidity of
H0 = 40 %. What is its humidity, specific volume, enthalpy, wet bulb temperature
and dew point?
C.9 Relating Humidity to Partial Pressure 287
Solution
The point of 20°C and 40% RH locates the point marked A on Fig. C.1. The
humidity can be read as 6 g H2O per kg of dry air. The specific volume is 0.837
m3 per kg of dry air, the enthalpy is 35 kJ per kg of dry air, the wet bulb tempera-
ture is 12.5°C, and the dew point is 6.5°C.
Fig. C.1 Psychrometric chart for air and water at 1 atm. Reproduced with permission from
Nedderman and Blackadder (1971).
Example 2
Air at a dry bulb temperature of 25°C and H0 = 50% comes into contact with latex
at a moisture content of 200 mg per kg of dry latex. The air is used to dry the
latex. What is the minimum amount of dry air required to produce one kg of dry
latex? Assume that the enthalpy of the air remains unaltered.
Solution
The amount of air must be sufficient to transport the water as it evaporates. To
answer the question, the amount of water that the air can hold under the stated
conditions needs to be calculated.
The inlet air is at point E2.1 on Fig. C.2.
288 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
Fig. C.2 Psychrometric chart for examples 2 and 3. Reproduced with permission from Nedder-
man and Blackadder (1971).
The humidity at this point is 0.010 kg water per kg of dry air.
The air comes into contact with the latex and becomes laden with more water.
Following the constant enthalpy line, from point E2.1 to the 100% relative
humidity line, determines point E2.2.
The humidity of this point is 0.013 kg water per kg of dry air.
The maximum amount of moisture that can be added to the air stream is 0.013
– 0.010 = 0.003 kg water per kg of dry air.
The latex contains 0.2 kg of moisture per kg of dry latex.
The minimum mass of air required is 0.2/0.003 = 66 kg dry air per kg of latex.
Note the assumption of constant enthalpy is common when contacting air with
a material to be dried. An enthalpy exchange is minimal compared with the latent
heat of the water, and hence error is small.
Example 3
We wish to dry cloth from a moisture content of 0.1 kg of water per kg of dry
two separate driers. An air stream at 40°C and 20% RH is contacted with the wet
cloth in drier 1. The air leaves the drier at a relative humidity of 60%. This air is
then heated at constant humidity to a temperature of 45°C. This hot stream is then
contacted with wet cloth in drier 2 and leaves the drier at 80% RH.
cloth to a bone dry composition. The wet cloth is split into two streams and fed to
C.9 Relating Humidity to Partial Pressure 289
1. Sketch the flow diagram and indicate the dry bulb temperature and humidity at
every stage.
2. How much moisture is removed per kg of dry air in the first drier?
3. How much moisture is removed per kg of dry air in the second drier?
4. The first drier processes 100 kg per hour of dry cloth. How much dry cloth is
processed in the second drier?
5. What is the volumetric flow rate of air into the first drier?
Solution
1. The flowsheet is sketched below. Each point is located on the psychrometric
chart (Fig. C.2) following the procedure outlined below.
Drier 1 Heater Drier 2
40 oC
RH 20%
Point E3.1
RH 60%
Point E3.2
45 oC
Point E3.3
RH 80%
Point E3.4
Wet cloth
0.1 kg water per kg dry cloth
Bone dry cloth
Wet cloth
0.1 kg water per kg dry cloth
Bone dry cloth
Point E3.1 is located on chart C.2 as 40°C and 20% RH. The humidity at this
point is 0.00925 kg water per kg of dry air.
Following the constant enthalpy line, from point E3.1 to the 60% relative hu-
midity line, locates point E3.2. The humidity is 0.01425 kg water per kg of dry air,
and the dry bulb temperature is 28°C.
Following a horizontal line (constant humidity) from point E3.2 to the 45°C
dry bulb temperature locates point E3.3 with a humidity of 0.01425 kg/kg.
Following a constant enthalpy line from point E3.3 to a relative humidity of
80% locates point E3.4 at a dry bulb temperature of 29°C and a humidity of
0.02075 kg/kg.
2. In the first drier, the air removes 0.01425 – 0.00925 = 0.005 kg water per kg of
dry air,
3. In the second drier, the air removes 0.02075 – 0.01425 = 0.0065 kg of water
per kg of dry air.
4. We process 100 kg per hour of dry cloth in drier 1. With 0.1 kg of water per kg
of dry cloth, we remove 10kg per hour of water.
290 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
We have an air flowrate of 10/0.005 kg per hour of dry air = 2000 kg per hour
of dry air.
The moisture removed in drier 2 is 0.0065 x 2000 = 13 kg of moisture per hour.
The flow rate of cloth to drier 2 is 130 kg of dry cloth per hour.
5. The specific volume at point E3.1 is 0.9 m3 per kg of dry air. Hence, the
volumetric flow rate of air is 0.9 x 2000 = 1800 m3 per hour.
Example 4
Air (dry bulb temperature 40°C, wet bulb temperature 27°C) is scrubbed with
water, which is maintained at 24°C. Assume that equilibrium is reached between
the air and water. The air is heated to 50°C by passing over steam coils. It is then
used in an adiabatic rotary drier from which it issues at 45°C. The drier produces
110 kg per hr of dry product, and the material loses 0.1 kg H2O per kg of dry
solid.
1. What is the humidity of the air
a. initially?
b. leaving the scrubber?
c. after reheating?
d. leaving the drier?
2. What is the total weight of dry air used per hour?
3. What is the total volume of air leaving the drier?
4. How much heat is supplied by the steam coils?
Solution
A flowsheet is shown below and the points are readily found on the psychrometric
chart (Fig. C.3)
Scrubber Steam
coils
Drier
40 oC d.b
27 oC wb
Point A
24 oC
100% RH
Point B
50 oC
Point C
45 oC
Point D
C.9 Relating Humidity to Partial Pressure 291
Fig. C.3 Psychrometric chart for examples 4 and 5. Reproduced with permission from Nedder-
man and Blackadder (1971).
1. From the chart:
Humidity at point A 0.017 kg/kg
Humidity at point B 0.019 kg/kg
Humidity at point C 0.019 kg/kg
Humidity at point D 0.0215 kg/kg
2. We remove 110 x 0.1 = 11 kg/hr of H2O.
Remove 0.0215–0.019 = 0.0025 kg H2O per kg of dry air.
11/0.0025 = 4400 kg dry air per hour
3. Specific volume of air is 0.93 m3/kg dry air.
0.93 x 4400 = 4092 m3/hr
4. Enthalpy after heating at point C is 100 kJ/kg dry air.
Enthalpy before heating at point B is 72 kJ/kg dry air.
Total heating is 28 x 4400 kJ/hr = 123.2 MJ/hr.
Example 5
A drier uses 25 kg of dry air per hour at 50°C and 10% relative humidity to dry
biscuits. At full throughput, 5 kg per hour of bone dry biscuit is made. The air
leaves completely saturated.
292 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
1. How much water is present in the biscuits originally?
2. Assuming adiabatic operation, what is the temperature of the air leaving the drier?
3. How much energy is required to heat the inlet air to its initial state in the drier?
Assume the air is heated from 20°C and that the humidity remains constant
during the heating up.
4. How much energy is required to dry a kg of dry biscuit?
Solution
1. Inlet air in at point E in Fig. C.3. It has a humidity of 0.0075 kg/kg. The
enthalpy at this point is 70 kJ/kg of dry air.
Following a constant enthalpy line to 100% humidity leads to point F.
The humidity at this point is 0.0185 kg/kg.
We remove 0.011 kg of water per kg of dry air.
We use 25 kg per hour of air, so we remove 0.275 kg per hour of water.
We process 5 kg per hour of biscuits, so we remove 0.055 kg water per kg of
dry biscuit.
2. The temperature at point F is 23.5°C.
3. The air is initially at point G. The enthalpy is 40 kJ/kg of dry air.
We add 70–40 = 30 kJ/kg of dry air.
Since we use 25 kg of dry air per hour, we use 30 x 25 = 750 kJ/hour.
4. We dry 5 kg per hour of biscuits, so the energy usage is 750/5 = 150 kJ per kg
of dry biscuit.
A blank chart is provided in Figure C.4 for future reference.
C.10 Evaporation Rate
The rate of mass loss per unit area from liquid water, E, is determined by the
difference in chemical potential of the water just above the liquid and in the bulk
of the liquid. It is commonly called the ‘evaporation rate’, E, and has the SI units
of kg m–2s–1.
It is easier to think of the problem in terms of the partial pressure created by the
liquid water acting as a driving force. The partial pressure of the water vapour
immediately above the liquid water is the saturated vapour pressure, Pvap*. Far
away from the liquid, the partial pressure of water in the atmosphere is determined
by the humidity and is given the symbol Pw. The saturated vapour pressure, Pvap*,
of water at atmospheric pressure is plotted as a function of the liquid temperature
in Fig. C.5. Hence, the evaporation rate is expressed as
( )* w
m vap w
ME k P P
RT= − (C.3)
C.10 Evaporation Rate 293
Fig. C.4 A blank chart is provided for the use of readers. Psychrometric chart for air and water.
Reproduced with permission from Nedderman and Blackadder (1971).
0
0.5
1
1.5
2
0 20 40 60 80 100 120
Temperature (oC)
Fig. C.5 Saturated vapour pressure of water as a function of temperature at atmospheric pressure.
Note the pressure of one standard atmosphere at a temperature of 100°C. Data obtained from
Haywood (1968).
294 Appendix C: Terminology of Humidity and an Expression for Evaporation Rate
where the constant of proportionality is the mass transfer coefficient km; the molar
mass of the water is given as Mw = 18 g mol–1. It is apparent that evaporation will
occur when Pvap* > Pw, whereas there will be condensation when Pvap* < Pw. It is
not usually easy to adjust Pvap* without also changing Pw. To achieve a fast
evaporation rate, however, their difference should be maximised.
References
Haywood R.W. (1968) Thermodynamic Tables in SI (metric) units, Cambridge
University Press
Nedderman R.N. and Blackadder D.A. (1971) A Handbook of Unit Operations,
Academic Press
Appendix D
Appendix D
Fracture Mechanics: Terminology and Tests
Chapter 5 provides an overview of the historical development of the understand-
ing of how interdiffusion and mechanical properties of latex films are related. This
understanding has emerged from studies of the fracture at polymer and polymer
interfaces as a function of annealing times and temperatures. This Appendix
explains the meaning of the some of the relevant terms from the techniques of
fracture mechanics that were used in such experiments. It also describes the
experimental set-up for fracture mechanics tests.
D.1 Fracture Toughness, KIC
Brittle fracture is when a material breaks by having a crack travel across it.
Fracture toughness is an indication of how resistant a material is to brittle fracture.
It provides the condition for brittle fracture in terms of the applied stress, σ, which is acting on a pre-existing central crack of length 2a. The parameter can be
measured when the crack is in a state of plane stress or plane strain. Most of the
experiments to measure the fracture toughness at polymer and polymer interfaces
are in the condition of plane strain. The stress distribution around the tip of a crack
as a function of position in a solid is described by using the stress intensity factors,
KI and KII. By convention, subscript I refers to plane strain; a subscript of II is
used to indicate measurements in plane stress (Williams 1977, 1978).
A crack will propagate in a material above a certain value of the stress at the
crack tip set by the critical stress intensity factor, KIC. This critical value is
commonly called the ‘fracture toughness’. It is a material property that is inde-
pendent of the test geometry. The fracture toughness is given as
KIC2 = Y
2σ 2a (D.1)
296 Appendix D: Fracture Mechanics: Terminology and Tests
Y 2 is a geometric factor that depends on the sample size and shape (Williams
1977). When the fracture toughness and typical flaw size in a material are both
known, then (D.1) can be used to calculate the maximum stress it can bear without
failure. At a higher stress, the crack will propagate.
Equation (D.1) can also be used to measure the fracture toughness. Typically, a
crack of known length, a, is initiated in a specimen. A load is applied to impose a
stress on the crack, and the load P is recorded when the crack begins to propagate
(Williams 1977). A typical experimental set-up for this test is the double cantile-
ver beam (DCB) geometry (Fig. D.1). KIC is then found from:
hb
aPYKIC = (D.2)
where h and b represent the dimension of the specimen illustrated in Fig. D.1. KIC
has the rather peculiar units of N m–3/2. In studies of crack healing, the surfaces of
two beams (either fracture surfaces or polished surfaces) are fused together for
known times and temperatures. Then, the fracture toughness of the healed
interface is measured in a test in the DCB geometry.
In the test geometry shown in Fig. D.2, a wedge of thickness ∆ is used. Typi-cally, the wedge is the sharp side of a razor blade (Schnell et al. 1999). As the
wedge travels along the interface at a slow and constant velocity, the crack
extends ahead of it by a distance a. With this set-up, in the limit where a > 2h, Gc
is then calculated as (Kanninen 1973):
2
3
2
2
12
3
c
IC
Eh
aK
α
∆= (D.3)
where αc is a correction factor. The geometry presented in Fig. D.2 is for the
symmetric base in which the same polymer is on either side of the interface and
has the same beam dimensions on both sides. The equation must be modified
when applied to the interface between different polymers (with different elastic
moduli and dimensions) in an asymmetric DCB test. The calculation of KIC for
various other test geometries is presented elsewhere (Williams 1978).
D.2 Plastic Zone Size at the Crack Tip, ry 297
h
bHealed interface
a
Side
view
P
h
bHealed interface
a
Side
view
P
Fig. D.1 Double cantilever beam geometry used to measure fracture toughness in plane strain, KIC.
Healed interface
h
a
∆
Healed interface
h
a
∆
h
a
∆
Fig. D.2 A symmetric double cantilever beam test geometry used to measure the fracture
toughness KI at a healed interface by driving a wedge at a constant velocity.
D.2 Plastic Zone Size at the Crack Tip, ry
The fracture mechanics equation presented in the previous section applies to
brittle fracture, which means that failure is by crack propagation. If the stress on a
polymer exceeds its yield stress, σy, then there is plastic deformation. Beyond the yield stress, deformation is not fully reversible when the load is removed. Below
the yield stress, deformation is elastic and the material returns to its original
dimensions after the load is removed.
When there is plastic deformation in the zone of maximum stress at the crack
tip, the crack tip will become rounded out or blunted. Plastic deformation at the
crack tip leads to ductile failure. The likelihood of ductile failure may be assessed
by estimating the radius at the crack tip, ry, using this expression:
2
2
2
1
y
cy
Kr
σπ= (D.4)
298 Appendix D: Fracture Mechanics: Terminology and Tests
For a plate thickness of h, brittle fracture will occur when ry < h/4. When the ry > h/4,
there will be blunting of the crack tip and a transition to ductile failure. Glassy
polymers might be expected to undergo brittle fracture, but experiments on
poly(styrene) (Tg ≈100 °C) at room temperature have found evidence for deformation at the crack tip (Schnell et al. 1998). In latex films with a glass transition temperature
far below the temperature of testing, brittle fracture is clearly not expected.
D.3 Critical Energy Release Rate, Gc
The energy release rate is defined as the amount of energy that is released per unit
area as a crack grows. The critical energy release rate required to initiate crack
growth, Gc is an additional parameter measured in fracture mechanics experi-
ments. The units for Gc are rather intuitive and easy to understand, as they are
given in terms of energy per unit area (J m–2 in SI units). Thus, one can think of Gc
as the energy required per unit area of crack for stable crack growth. At equilib-
rium, this energy will equal the energy released in the fracture process.
Gc can be found from KIC (or KIIC as appropriate, depending on the experiment)
through the elastic modulus, E, of the specimen (Williams 1977, 1978) as
E
KG ICc
2
= (D.5)
Young’s modulus E, (in units of force per unit area), is determined by the slope of
the linear relation between stress and strain in the elastic region. Combining (1)
and (D.5) shows that
aYE
G c2
2
22
=
σ (D.6)
In fracture mechanics experiments, a crack at an interface can be driven by a
wedge travelling along an interface at a constant slow velocity. Low velocities,
where there is slow, stable crack growth, are used.
D.4 Fracture Strength
In a related experiment, called the ‘notched beam test’, a wedge with a 45° angle is cut out of a specimen, as shown in Fig. D.3a. The specimens are strained at a constant
rate until they fracture, and the stress at the point of fracture, σf, is recorded. In studies of latex film strength, a common approach is to test specimens under
a tensile load (Kim et al. 1994). The specimens are strained at a constant rate. The
maximum stress that can be applied before fracture of the specimen is recorded as
D.5 Plastic Zone Size at the Crack Tip, ry 299
the fracture strength, σT. The test is illustrated in Fig. D.3, and an example of how to find σT from a stress-strain curve is also given.
D.5 Fracture Energy
Another way to analyse the stress-strain data is to find the total energy required for
a specimen to break, which is called the ‘fracture energy’, WB (Zosel and Ley
1993). It can be expressed in units of energy per unit volume (J/m3), which is
equivalent the units of stress (N/m2 = Pa). It can be calculated from the area under
a stress-strain curve, as shown in Figure D.3c. WB is a function not only of the
elastic modulus (indicated by the slope of the linear region at low strain), but is
also affected by plastic deformation or flow of the specimen.
σ = σTσ = 0
σ = σTσ = 0
(b) (c)Stress (Pa)
σT
εf
Fracture
Strain
WB
(a)
45°
σf
45°
σf
Fig. D.3 a An illustration of the notched beam test geometry. b An illustration of the deforma-
tion of a tensile specimen until failure at a stress of σT. c An example of a stress-strain curve for a brittle specimen, showing the meaning of σT and WB.
References
Kanninen M.F. (1973) Augmented double cantilever beam model for studying
crack propagation and arrest. Int. J Fract. 9: 83-92
Kim K.D., Sperling L.H., Klein A., Hammouda B. (1994) Reptation time,
temperature, and cosurfactant effects on the molecular interdiffusion rate dur-
ing polystyrene latex film formation. Macromolecules 27: 6841-6850
Schnell R., Stamm M., Creton C. (1998) Direct correlation between interfacial
width and adhesion in glassy polymers. Macromolecules 31: 2284-2292
Williams, J.G. (1977) Fracture mechanics of polymers. Polym. Engin. Sci. 17:
144-149
300 Fracture Mechanics: Terminology and Tests
Williams J.G. (1978) Applications of linear fracture mechanics. Failure in
Polymers. Springer-Verlag, Berlin.. Advances in Polymer Science Series. Vol
27, 67-120
Zozel A, Ley G. (1993) Influence of crosslinking on structure, mechanical
properties and strength of latex films. Macromolecules 26: 2222-2227
Index
acomustic waves 34
acrylates 2
acrylic acid groups 174
acrylic copolymers 3
adhesion, effect of surfactants 190
adhesion energy 159
adsorption isotherms 192–3
AFM see atomic force microscopy
aggregation, definition of 20
alkyd film 74
anisotropic particles 259–61
anisotropy 259
anthracene 77, 81
Arrhenius equation 166
aspect ratio 240, 246
atomic force microscopy 62–8, 144
cantilever 62, 68
experimental parameters 65
height artefacts 64
indentation depth 64
intermittent contact 63
microtomed cross-sections 67
particle deformation 67
phase imaging 66
contrast in 67
set point ratio 65
TappingModeTM 63
tip 69
contamination 68
atom transfer radical
polymerization 220
autocorrelation 45
autohesion 151
barrier resistance
effect of surfmers 206
in nanocomposites 216
beam bending 32–34
blocking 159, 169, 216, 245
boundary layer 96
Bragg’s law 232
brittle fracture 293
brittleness 215
Brown, Robert 1
Brownian dynamics simulations
of drying 106
Brownian motion 1, 44
applications of 50
Brown mechanism 125
capillary deformation 124–5, 135
experimental evidence 142
capillary length 110, 111
capillary pressure 111–3, 124, 230
effect on cracking 116
capillary waves 157
carbon nanotube 221, 234, 246,
263
carboxylic acid groups 173
carpet backings 6
302 Index
chain
branching 164–5
entanglement 159
length 249
pull-out 158
scission 159
chalking 245
chemical patterning 231
Clausius-Mossotti equation 51
clay
exfoliation 221
intercalation 221
close packing, random 10, 23, 100
cloudy-clear transition 29, 143
coalescent reduction 268
coalescing aid 174–5
effect on Tg 175
selection of 175
coffee rings 110
Col.9® 245
colloidal crystal 23, 232
classification 238
growth 231
colloidal stability, effect on drying
114
colloid dispersion 1
colloid science 17–23
complex longitudinal modulus 35
confocal microscopy 49–50
laser scanning 50
confocal Raman microscopy 52, 74
construction materials 6
convection of surfactant 194
core-shell particle see particle
crack healing 152, 294
cracking 116–7
in nanocomposites 235
relaxation mechanism 117
crack point 29
crack spacing 117
creaming 275
creeping flow 22, 273
critical coagulation concentration
115
critical energy release rate 295–6
critical micelle concentration 191
critical stress intensity factor 293
critical volume fraction 234
crosslinking 58, 73–4, 175
autoxidative 74
control parameter 179
molecular weight effects 178
two-pack 175
two-pack in one pot 175
cryogenic electron microscopy see
electron microscopy
currant-bun particle 221
dangling chains 178
Darcy flow 112
Darcy’s law 104
Debye length 18, 114
deformation map 133–4, 139
depletion interactions 17, 20
Designed DiffusionTM 269
desorption of surfactant 199
deuterium 44
dew point 283
dialysed latex 189
diffraction limit 49
diffusing wave spectroscopy 46, 263
diffusion 10, 151
activation energy for 166
competition with crosslinking
175
effect of chain branching 164
effect of coalescing aids 174
effect of membranes 173
effect of molecular weight 164–5
effect of particle size 172
effect of reduced mobility 171
effect of temperature 165
in gel 177
near Tg 167
of core shell particles 172
particle shell effects 164
scaling prediction 165
scaling relations 157
shift factor 168
surfactant 195
tortuosity effects 169
Index 303
diffusion coefficient 22, 153, 166
dirt pick-up 189, 245
DLVO theory 17, 19
double cantilever beam 294
drag coefficient 22
dry bulb temperature 282
drying 10, 95–117
effect of Peclet number 104, 106
effect of salt 106, 115
effect of surfactant 114
horizontal 107–114
factors that affect 112
fronts 108, 109
MRI of 113
importance of 95
particle distribution during 99
three-stage process 98
two-stage process 98
vertical 99–107
factors affecting 102
drying fronts 15
dry sintering see sintering
dwell time in MR profiling 277
dynamic speckle 48
elastic particles 127
elastic spheres 128
electrical conductivity 36, 216
electrical impedance 36
electric force microscopy 69–70
electron beam damage 40
electron microscopy 36–42
cryogenic scanning 37, 104, 108
cryogenic transmission 125
dark field 41
environmental, pump down 41
environmental scanning 36,
37–40, 145
design 39
scanning 36, 72–3
backscattering electron images
73
scanning transmission 36
transmission 41, 71–2
freeze-fracture 72
staining 72
wet STEM 41–2
electron paramagnetic resonance 60
electron scattering 40
electrostatic repulsion 17, 18–19
ellipsometry 50, 52, 143
emulsion polymerisation 2
emulsion polymers, market for 9
encapsulated particle 221
entanglement molecular weight
155, 160, 178
enthalpy of air 283
environmental (gaseous) detector
38
environmental legislation 15–16
environmental scanning electron
microscopy see electron microscopy
EU Directive 2004/42/EC 15
evanescent wave 49
evaporation
effects on 97
rate 96, 296
evaporative cooling 32, 96
evaporative lithography 267
face-centred cubic 225
Fickian diffusion 153
filler particles 168, 171
effect on diffusion 81, 170
film formation
mechanical probe 32
stages of 10, 11
film formation paradox 174
film scratching 32
film topography 267
flame retardancy 214
flammability 214
flocculation, definition of 20
flow, particle in Newtonian fluid
276
flow instabilities 266
fluorescence decay curves 80, 81
fluorescence resonance energy
transfer 61, 76
simulations 79
forced Rayleigh scattering 58, 59
Forster radius 77
304 Index
Forster relation 76
fraction of mixing after
interdiffusion 79
fracture energy 159, 296
effect of diffusion 160
time dependence 162
fracture strength 159, 296
fracture toughness 159, 293–4
free radicals 40
Frenkel theory 128
FTIR spectroscopy 73
further gradual coalescence 151
GARField 56–58, 277–9
experimental design 57
experimental profiles 105
gel point 35
glass transition temperature,
definition of 2
gloss, effect of surfactant 188
Graham, Thomas 1
gravimetry 32
Guinier plot 75
Halpin-Tsai equations 214
Hamaker constant 18
Hertz theory 127
hetero-flocculation 223–4
homogeneous particles 213–4
honeycomb 13
horizontal drying see drying
humidity 281–92
definition of 95, 281
relative, definition of 281–2
hybrid 213, 224
types of 217–25
hydrophobicity 245
ideal gas equation 282
industrial coater 6
infrared microscopy 53, 146
infrared spectroscopy 52
inisurfs 205, 207
inks 6
inorganic nanocomposite particles
219
inorganic nanoparticles 245
Institute Laue Langevin 43
interaction potentials 17
interdiffusion 152
effects on 80
techniques to study 74
interdiffusion distance 162
interfacial chain density 162
interfacial strength 247
interfacial width 75, 152
interparticle interference 51
interpenetration distance 75, 157
interphase 215
interstitial space between latex
particles 169
inverse micro-Raman spectroscopy
53, 263
iridescence 232
Janus particles 260
Johnson, Kendall and Roberts 127
Kelvin probe force microscopy
69–70
knife point 29
Krieger-Dougherty expression 23
Langmuir isotherm 193
laponite 264
laponite clay 228
lapping time 111
latex
blends 213
definition of 1
dialysed 189
gloves 8
market for 9
natural 8
sensitisation 8
latex film formation 10
publications on 16
latex foam structures 173
light scattering 44, 83
dynamic 45
in nanocomposites 234
magnetic resonance imaging 55
magnetic resonance profiling and
particle deformation 140
magnetogyric ration 54
Marangoni flows 199, 202
Index 305
Marangoni instabilities 200
mass transfer coefficient 97
mass transfer resistance 98
melt compression 232
membrane bending 34
membranes 172–3
meniscus 124, 125, 230
MFFT see minimum film formation
temperature
micelle 191
microrheology 45
miniemulsion polymerisation 217
minimum film formation
temperature
and particle size ratio 236
definition of 14
effect of particle size 30
effect of surfactant 191
interpretation 30
MFFT bar 29–31, 139, 143
for studying deformation 138
standard for 29
time effects 30
modern art 189
moist sintering see sintering
molecular mobility 171
molecular weight 165
Monte Carlo simulation of drying
106
MRI see magnetic resonance
imaging
multispeckle 46
nanocomposites 213–49
classification 213
conductivity 216
cracking in 235
failure mechanism 247
in paints 216
light scattering in 234
properties 214
silica 227
soft-soft 242
stiffness of 214
toughness of 215
viscoelasticity 215
nanoparticle
dispersion 233–4
encapsulated 222
hybrid 224
Navier-Stokes equation 22, 273–5
Newtonian fluid 22
NMR see nuclear magnetic
resonance
non-adsorbing polymer 20
non-radiative energy transfer 58, 61
nuclear magnetic resonance 54
MOUSE 56
spectroscopy 74, 202
occupational exposure limits 15
oligomers 268
opal structure 232
double-inverse 262
inverse 262
open time 107, 111
optical cantilever see beam bending
optical clarity front 113
optical stethoscopy 70
optical transmission 143
optical transparency 14
packing, face-centered cubic 12
paints, formulation of 4
paper coatings 6
parameter map 131
partial pressure 296
particle
core-shell 218–10, 226
film formation 227, 264
half moon 218
lobed 218
occluded structures 218
particle assembly 225, 261
particle blends
advantages of 233
film formation 234
hard-soft 243
particle compressibility 102
particle deformation 10, 12
atomic force microscopy 144
driving forces 121, 122
effect of particle size 139
306 Index
effect of temperature 137–9
MFFT bar 143
scaling argument 135
particle deposition methods 230
particle interfacial area 122
particle packing 12, 260
effect of surfactant 191
front 109
size ratio effects 235–6
particle spacing 51
patterned substrate 231
peak-to-valley height 144, 145
Peclet number 101, 195
effect on drying 104, 106
peel strength 190
pendular rings 126
percolation 238–42
effect on properties 241
model 239
of rods 240
thresholds 239
phase separation 234
in particles 261
phenanthrene 77, 81
photoacoustic spectroscopy 73
photon correlation spectroscopy 45
photonic crystals 262
Pickering emulsion polymerisation
222
plane strain 293
plane stress 293
plasticisation 16, 81
by surfactant 187
plasticisers 174
plastic zone 295
Plateau borders 146
Poisson’s ratio 127
poly-condensation 217
poly(dimethyl siloxane) 245
Porod law 75–6
pressure-limiting apertures 38
pressure sensitive adhesives 190
application of 5
psychrometric chart 283
pulse gap in MR profiling 279
quadrature spin-echo sequence 277
quality factor 64
quantum efficiency of energy
transfer 78
quartz crystal microbalance 73
radiolysis 40
radius of gyration 157, 170, 172
compared to diffusion distance
170
Raman spectroscopy 52
surfactant analysis 202
random coil 172
raspberry particles 222
Rayleigh theory 51
reactive surfactant 205
refractive index 143
measurement of 52
replicas, transmission electron
microscopy 71
reptation 14, 152, 154
reptation time 156
Reynolds number 274
rheology modifiers 4
rhombic dodecahedron 13, 122
root mean square displacement of
chains 156
Rouse entanglement time 156
Rouse relaxation time 156
Routh and Russel film deformation
model 130
Rutherford backscattering
spectrometry, surfactant analysis
202
saturated vapour pressure 296
scanning electric potential
microscopy 69–70
scanning electron microscopy see
electron microscopy
scanning near-field optical
microscopy 50, 70–1
scanning transmission electron
microscopy see electron
microscopy
scattering angle 43
scattering techniques 42–52
Index 307
scratch resistance 216
secondary ion mass spectrometry
201, 204
sedimentation 275
sedimentation coefficient 102
sedimentation velocity 276
seeded emulsion polymerisation
219
shear force microscopy 70–1
shear modulus 158
Sheetz deformation 126, 136
silica
nanocomposites 227
nanoparticles 244
particles 169, 172
sintering
dry 123–4, 136
theory 129
moist 126
wet 123, 135
skin formation 58, 107, 141, 146
experimental evidence 142
study of 59
skin layer 55, 115, 146
small-angle neutron scattering
42–4, 145
parameters for 43
surfactant analysis 202
to study interdiffusion 75
small-angle X-ray scattering
42–4
sodium dodecyl sulphate 187
soft-soft nanocomposites 242
sorptive capacity 107
specific volume 282
speckle
commercial instrument 49
interferometry 48
spectrophotometry 83
specular reflection 188
spin-casting 228
spin-spin relaxation time 55, 58, 74
star polymer 165
steric stabilisation 234
stick-slip 116
Stokes-Einstein diffusion coefficient
22, 44, 101
Stokes flow 22, 275
stray-field imaging 55
strength 214
stress relaxation modulus 131
styrene-acrylic copolymers 3
surface patterns 230
surface roughness 144
surfactant 185–207
anionic 185
cationic 185
classification of 185
convection of 194
desorption 187, 199
exudation 187
cause of 192
effect of surfmers 206
effect of Tg 199
fate of 186–7
gloss effect 188
non-ionic 185
plasticisation by 187
segregation 198
solubility in polymer 187
surfactant-free emulsion
polymerisation 185
surfactant-induced flow 267
surfmer 205–7
temperature, effect on particle
deformation 137–9
templates for drying 231
tensile strength 160
TexanolTM 174
textile backings 6
thermal conductivity 216
thermoelectric applications 263–4
thin film analyser 146
time-temperature superposition 167
tortuosity 169, 172
toughness 215
transmission electron microscopy
see electron microscopy
transmission spectrophotometry 50
transport coefficient 104
308 Index
transurfs 205, 207
tube model 155
turbidity 83
ultramicroscopy 50
ultrasonic reflection 34–35, 73
van der Waals attraction 17, 128
van der Waals forces 115
varnishes, formulation of 4
vertical deposition 228–9
vertical drying profiles see drying
viscoelastic particles 122, 130
viscosity
dependence on volume
fraction 23
measurement of 32
viscous flow of particles 128
VOC see volatile organic
compounds
volatile organic compounds 15, 138
water
adsorption 190
diffusion coefficient of
vapour 96
distribution profiles 141
surface tension 125
water whitening 191
wavevector 43
wet bulb temperature 282
wet sintering see sintering
wet STEM see electron microscopy
wetting 152, 157
Williams-Landel-Ferry equation
167
Winnik, M.A., 76
X-ray photoelectron spectroscopy
201
X-ray scattering 44
Young’s modulus 241