Appendix A Derivation of Creeping Flow and the …978-90-481-2845...A Appendix A Derivation of...

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.


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.


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


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.


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


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.


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).


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


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


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



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


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


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

Appendix A Derivation of Creeping Flow and the …978-90-481-2845...A Appendix A Derivation of...

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