Post on 16-Apr-2018
1
Structure and Mechanical Properties of Fat Crystal
Networks
Alejandro G. Marangoni
University of Guelph
Guelph, Ontario, Canada
What is this? How is it different from this?
margarineolive oil
2
So, what is a fat after all?• To the chemist and chemical engineer: a complex mixture of high melting
point triacylglycerols in low melting point triacylglyceorls with complex phase behavior – all can be explained from knowledge of molecular composition and phase behavior (SFC and phase diagrams are king…)
• To the crystallographer: a metastable polycrystalline material prone to recrystallization and fractionation – all can be explained from knowledge of polymorphism (XRD is king and DSC is queen…)
• To the colloidal scientist: a colloidal gel (or colloidal crystal) composed of a network of polycrystalline fat particles which trap liquid oil within – all can be explained from knowledge of the mesoscale structure, i.e., colloidal sizes, interactions, distribution (rheometer is king and microscope is queen…)
• To the consumer: something that makes you fat, but tastes good (cookies and chocolate rule…)
• All agree on the last point only
Factors affecting the textureof fats and fat-structured foods
1. Solid fat content
2. Primary crystal habit (polymorphism)
3. Nano and Microstructurea. crystallite morphology and sizeb. spatial distribution of network mass
4. Interparticle interaction forces
3
96b344b60.0c4.08E+07d24
90b312b80.5b1.63E+07c20
223a593a90.5a2.64E+07b15
231a602a90.5a2.19E+07a5
K (N/mm)
Fy (N)SFC (%)G´ (Pa)Temperature (ºC)
Means with a common superscript within a column are not significantly different (P>0.05).
What controls hardness?
dD
Bulk fat Network
Floc
Cluster
Single crystallite
Domain Lamella
4
Triacylglycerol Molecules
Crystals
Crystal Clusters
Crystal Network
Fat
Nanostructure0.4–100nm
Microstructure0.1-200μm
Macroscopic World>0.2mm
Polymorphism
RheologyMechanical StrengthSensory Impressions
Heat, mass, momentum transfer
Heat, mass, momentum transfer
Heat, mass, momentum transfer
Molecular Structure
Solid Fat Content
Triacylglycerols– Molecular Structure
H2C-O-C-CH2-(CH2)n-CH3
H2C-O-C-CH2-(CH2)n-CH3
|CH3-(CH2)n-CH2-C-O-C-H
|
O
O
O
Triacylglycerol = glycerol + 3 fatty acids
GlycerolBackbone
Fatty Acyl Chain
5
Fatty Acids and Triacylglycerol Types
Fatty Acids Long-Chain Saturated 14:0, 16:0, 18:0Medium-Chain Saturated 8:0, 10:0, 12:0Short-Chain Saturated 4:0, 6:0
Mono-Unsaturated 18:1Poly-Unsaturated 18:2, 18:3, 22:6
Geometric Isomers (cis vs. trans)Positional Isomers (ω-3,6,9)
Triacylglycerols Simple HomogeneousMixed
Complex and varied positional distribution of fatty acids on TAG molecule.
Triacylglycerol Crystallization Process
Isotropic Melt
Crystalline Solid
Decrease temperature below melting point of solid into metastable region (undercooled or supersaturated)
T<Tm
T>Tm
6
Crystallization Process – more to the story?
Isotropic Melt Structured Liquid?
Crystalline SolidAmorphous Solid?
Gibbs-Thompson Equation
ΔGn=Anγ-Vn(Δμ/Vm)
1.0×10-08 1.0×10-07 1.0×10-06 0.00001-1.0×10-13
-5.0×10-14
0.0×10-00
5.0×10-14
rc
σ=30 mJ m-2
ΔGm=150 J mol-1
Vms=8.5⋅10-4 m3 mol-1
ΔGn=1.46⋅10-14J/nrc=3.4⋅10-7m
r (m)
ΔG
n(J
/nuc
leus
)
7
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
Long Spacings(001)
2L
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
Long Spacings(001)
3L
8
Crystalline structure of fats
2L 3L
Modified from Garti & Sato: Crystallization Processes in Fats and Lipid Systems, 2nd ed.
Polymorphism→Same chemical composition but different solid-state structure
α→β’→β
melt
Lower melting pointLower densityLower stability
Higher melting pointHigher densityHigher stability
Melt-mediate and/or solid-state phase transitions (polymorphic transformations)
9
The Crystalline State – The Subcell(Polymorphism)
Characteristic short spacings:
β: 4.6Å (Triclinic)
β’: 3.8Å and 4.2Å
(orthorhombic ⊥)
α: 4.15Å (Hexagonal)
Stability, melting point and packing density:
α < β’ < β
Also: γ (orthorhombic ⊥), δand several β’ polymorphs
Subforms possible
Triclinic
Orthrhombic
Hexagonal
10
Rigaku Multiflex
11
d
2
2x x
2
2
X-rays
Crystal Planes
2 sinn dλ θ=
BURN-THROUGH
X-RAY
REFLECTION
SAMPLE
DETECTOR
α
rl
r2
12
γ α
β’ β
0
500
1000
1500
2000
2500
3000
0 5 10 15 20 25 30 35 40
65.71
46.34
32.65
16.0112.87 8.08
5.435.18
4.60
4.23
3.99 3.883.77
3.71
Cocoa Butter Powder XRD
001
002
004005
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
O
O
OH
O
O
O
2θ
Inte
nsity
SAXSWAXS
13
Differential Scanning CalorimeterTA Q1000
0 10 20 30 40 50 60
Tcrystalization
Tmelting
ΔH
ΔH
Tonset crystalization
Tonset melting
Temperature (°C)
Hea
t Flo
w
14
-20 -10 0 10 20 30 40 50 60-2.0
-1.5
-1.0
-0.5
0.0
β'
β
α
Temperature (oC)
Hea
t Flo
w (W
/g)
A
-10 0 10 20 30 40 50 60-1.00
-0.75
-0.50
-0.25
0.00
α β'
Temperature (oC)
Hea
t Flo
w (W
/g)
B
0 10 20 30 40 50 60-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
β '
β
Temperature (oC)
Hea
t Flo
w (W
/g)
C
-30 -20 -10 0 10 20 30 40 50-1.00
-0.75
-0.50
-0.25
0.00
γ
α
Temperature (oC)
Hea
t Flo
w (W
/g)
D
R2 C 0
0
CH
CH2 0 C R1
0
CH2 0 C R3
0Triacylglycerol (TAG) Molecule
Double- and triple-chain-length structures
Cocoa Butter
Predominant fatty acids include: palmitic acid (16:0), stearic acid (18:0) and oleic acid (18:1)
Mainly composed of symmetrical triacylglycerolsPOP, SOS and POS
15
Table 3: Fatty Acid Composition of Cocoa Butter Fatty Acid Cocoa Butter (wt %) 16:0 28.1 ± 0.45a
18:0 34.4 ± 0.47a
18:1 32.6 ± 0.28a
18:2 2.81 ± 0.06a
18:3 and 20:0 1.95 ± 0.21a
aData represents the average of seven replicates ± standard error.
Table 4: Triacylglycerol Composition of Cocoa Butter Triacylglycerol (carbon number) Experimental Results (wt%) 50 19.8 ± 0.27b 52 47.6 ± 0.21b
54 31.4 ± 0.33b
56 1.11 ± 0.13b
bData represents the average of four replicates ± standard error.
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Table 2 : Characteristic short spacings as determined by XRD for the various polymorphic forms of cocoa butter (Larsson, 1994).
Polymorphic form
Short Spacings (Å)
γ (sub -α) (I) 3.87(m), 4 .17(s)α (II) 4.20(vs)
β 2’ (III) 3.87(vw), 4.20(vs)β 1’ (IV) 3.75(m), 3.88(w), 4.13(s), 4.32(s)β 2 (V) 3.65(s), 3.73(s), 3.87(w), 3.98(s), 4.22(w), 4.58(vs), 5.13(w), 5.38(m)
β 1 (VI) 3.67(s), 3.84(m), 4.01(w), 4.21(vw), 4.53(vs), 5.09(vw), 5.37(m)The relative intensity: very strong (vs), strong (s), medium (m), weak (w) or very weak (vw).
Table 1: Polymorphic forms of cocoa butter as determined by various research groups Wille and Lutton (1966) Merken and Vaeck (1980) as reported by van Malssen
et al. (1996a) Polymorphic Form
Tm (°C) Polymorphic Form
Tm (°C) Polymorphic Form
Tm (°C)
I 17.3 γ 16-18 γ -5-+5 II 23.3 α 20.7-24.2 α 17-22 III 25.5 IV 27.5 β’ 26-28 β’ 20-27 V 33.8 β 33.7-34.9 β 29.34 VI 36.3
29-34
17
0
500
1000
1500
2000
2500
3000
0 5 10 15 20 25 30 35 40
65.71
46.34
32.65
16.0112.87 8.08
5.435.18
4.60
4.23
3.99 3.883.77
3.71
18
19
20
21
Phase Behavior
Fats are mixtures of lipids, predominantly triacylglycerols, but also contain minor polar components such as diacyl-glycerols, monacylglycerols, free fatty acids, phospho-lipids, sterols, etc.
Continuous solid solution (mixed crystals), eutectic, monotectic (partial solid solution), compound formation phase behaviors are commonly observed
Conditions for fat compatibility
1. Equivalent thermal propertiesMelting pointsMelting and solidification rangesSFC
2. Similar molecular size, shape and packingTo allow isomorphous replacement orformation of a single lattice unit in mixtures
3. Similar polymorphismTransformation from stable to unstable formsshould occur as readily for binary mixtures as with individual components
22
Continuous solid solution – compatible
Eutectic – incompatible (minima)
Monotectic – significant amounts of lower melting component solubilized by higher melting component. Eutectic point close to m.p. of lower melting component; partial solid solution
Compound formation (maxima)
Liquidus lines predicted by Hildebrand equation
ln(xi)=ΔHi/R (1/Tm-1/T)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0330
335
340
345
350
PPPSSST=[(R*ln(x)/ΔH)-1/Tm]-1
mol fraction
End
of m
elt (
K)
23
Continuous solid solution: SSS/ESS, POS/SOS
Eutectic behavior: PPP/SSS, EEE/SOS, POS/PSO,PPP/LLL, PPP/SOS
Compound formation: SSO/SOS, POP/OPO
Monotectic behavior: SSS/OOO, SSS/LLL, PPP/POP,SSS/SOS
Bruker Minispec pNMR
24
Mag
netiz
atio
n Si
gnal
Inte
nsity
PulseTime (μs)
Solid – liquid components (total hydrogen content)
Liquidcomponent only
tB=10 tA=70
Mag
netiz
atio
n Si
gnal
Inte
nsity
PulseTime (μs)
Solid – liquid components (total hydrogen content)
Liquidcomponent only
tB=10 tA=70
SFC vs. temperature profile forAMF/Cocoa Butter
0 5 10 15 20 25 30 35 40 45 500
102030405060708090
1000% AMF10% AMF20% AMF30% AMF40% AMF50% AMF60% AMF70% AMF80% AMF90% AMF100% AMF
Temperature (oC)
Sol
id F
at C
onte
nt (%
)
Isosolid Diagrams (Phase Diagrams?)
25
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
35
40
Composition (%w/w MMF)
Tem
pera
ture
(o C)
0 10 20 30 40 50 60 70 80 90 10010
20
30
40
50
60
Composition (% HMF)
Tem
pera
ture
(o C)
26
0 10 20 30 40 50 60 70 80 90 10020
25
30
35
40
45
50
55
60
AMFMMFHMF
Composition (%w/w milkfat fraction)
Dro
ppin
g Po
int (
o C)
SFC influences hardness…
24 hr Tempering 48 hr Tempering0
153045607590
105120135150
Cocoa ButterMMFAMF40% HMF40% AMF40% MMF10% MMF10% HMF10% AMF
* * *
*<150
Tempering Time (hours)
Pen
etra
tion
Dep
th(1
/10
mm
)
27
Ideal Solubility – Fact or fiction?
ln(xi)=ΔHi/R (1/Tm-1/T)
Polymorphism and Microstructure
28
Time –Temperature State Diagram for the Polymorphism of Statically Crystallized Cocoa
Butterβ’ βγ SOS
1 10 100 1000 10000 100000-30-25-20-15-10-505
1015202530
Time (min)
Tem
pera
ture
(o C)
γ + α
α
β’
α+β’
β’+ β
The α formA
B
C
D
29
The β’ formA
B
C
D
The β form A
B
C
D
E
F
30
Cocoa butter crystallized at 0ºCA D
B E
C F
α
α →β’
α→ β’
β’
β’
β’
(1 day)
(5 days)
(7 days)
(14 days)
(21 days)
(28 days)
Cocoa butter crystallized at 20°CA
B
C
D
E
F
G
β’
β
β
β
(1 day)
(7 days)
(21 days)
(21 days)
(35 days)
(35 days)
(35 days)
β’→β
β’→β
β’→β
31
Cocoa butter crystallized at 26°C
(1 day)
A
B
C
D
E
F
γ SOS
β’
β
β
β
β
(3 days)
(7 days)
(7 days)
(28 days)
(28 days)
Growth of a PolycrystallineFat Crystal Network
32
Microscope
Camera
Heating Freezing Stage
33
Growth of a fat crystal network
34
Effect of cooling rate
0.4oC/min
1.2oC/min
6.0oC/min
Effect of Surfactant Addition
0.1% vs. 0.5% Tween 60
35
Temperature Effect
5oC 25oC
Effect of film size on slide
170:m 20:m
36
Cocoa Butter Fat Crystal NetworkVisualized using AFM
Cryo-TEM Isobutanol-extracted SSS
37
38
Relationship between structure and crystallization behavior
nkte1)(SFC)t(SFC −−=
∞
1. Growth mode (Avrami model)k - rate constant of crystallizationn - dimensionality of growth and
the type of nucleation
2. Nucleation (τ, ΔGc- Fisher-Turnbull model)
( )kTdGcG
eJ hNkT
Δ+Δ−=ΔGc - activation free energy of nucleationΔ Gd - activation free energy of diffusion
Table 6: Avrami exponent values for the different types of growth and nucleation (Sharples, 1966)
Avrami Exponent Various types of growth and nucleation 3 + 1 = 4 Spherulitic growth from sporadic nuclei 3 + 0 = 3 Spherulitic growth from instantaneous nuclei 2 + 1 = 3 Disc-like growth from sporadic nuclei 2 + 0 = 2 Disc-like growth from instantaneous nuclei 1 + 1 = 2 Rod-like growth from sporadic nuclei 1 + 0 = 1 Rod-like growth from sporadic nuclei
39
n=4 n=1
0 50 100 150 200 250 300 3500
25
50
75
k=0.00103min-1.5
SFC∞=67.7%n=1.50
Time (min)
SFC
(%)
40
Avrami Exponent vs. Crystallization Temperature
-30 -20 -10 0 10 20 300
1
2
3
4
5
Crystallization Temperature (oC)
Avra
mi E
xpon
ent
Phase Technology Cloud Point Analyzer
41
0 100 200 300 4000
25
50
75
τnucleation-A τnucleation-B
Time (min)
Sign
al (a
.u.)
0.00000 0.00001 0.00002 0.000037.5
10.0
12.5
15.0
Slope = 221100R2=0.99
A
1/T(ΔT)2
ln( τ
T)
0 10 20 300.0
2.5
5.0
7.5B
Crystallization Temperature ( °C)
ΔG
c (kJ
/mol
)
42
Avrami Exponent vs. Crystallization Temperature
-30 -20 -10 0 10 20 300
1
2
3
4
5
Crystallization Temperature (oC)
Avra
mi E
xpon
ent
Induction Times as Determined from Crystallization Curves at Various
Temperatures
-30 -20 -10 0 10 20 30-5.0
-2.5
0.0
2.5
5.0
7.5
10.0α β' βα +β'
Crystallization Temperature (oC)
log
τ
43
Rheology of Fats
Small DeformationDynamic ControlledStress Rheometer
TA AR2000
44
Viscoelasticity of Fats
G”/G’ ~ 0.1
Shear strains of ~ 0.1% or less
G’ frequency-independent
45
0 2000 4000 6000 8000 100000
5000000
10000000
15000000
20000000
25000000
30000000
0.00
0.01
0.02
0.03
0.04
0.05
Stress (Pa)
Stor
age
Mod
ulus
(Pa)
Strain (%)
0 2 4 6 8 100
4000000
8000000
12000000
16000000
20000000A
Frequency (Hz)
Shea
r M
odul
us (P
a)
0 2 4 6 8 100
400000
800000
1200000
1600000B
Frequency (Hz)
Shea
r M
odul
us (P
a)
0 2 4 6 8 100
2000000
4000000
6000000
8000000
10000000C
Frequency (Hz)
Shea
r M
odul
us (P
a)
0 2 4 6 8 100
15000000
30000000
45000000
60000000D
Frequency (Hz)
Shea
r M
odul
us (P
a)
46
0 5000 10000 15000 20000 25000 30000 350001000000
10000000
100000000
0.0
0.1
0.2
0.3
0.4
0.5A
Stress (Pa)
She
ar M
odul
us (P
a)S
train (%)
Large Deformation Mechanical Tests
47
0 .5 1 .0 1 .5 2 .00 .0
5 .0
2 .5
7 .5
1 2 .5
1 0 .0
1 5 .0
1 7 .5
2 0 .0
2 2 .5
2 5 .0
2 7 .5
0 .0
D is p la c e m e n t (m m )
Forc
e (N
)
Y ie ld F o rc e (N )
Y ie ld d e fo rm a tio n (m m )
Y ie ld W o rk (N m m )
0 .5 1 .0 1 .5 2 .00 .0
5 .0
2 .5
7 .5
1 2 .5
1 0 .0
1 5 .0
1 7 .5
2 0 .0
2 2 .5
2 5 .0
2 7 .5
0 .0
D is p la c e m e n t (m m )
Forc
e (N
)
Y ie ld F o rc e (N )
Y ie ld d e fo rm a tio n (m m )
Y ie ld W o rk (N m m )
Modeling the microstructure of a fat
48
Φ= 5.3
5.0
245
oHADGπ
Nederveen, C.J. Dynamic mechanical behavior of suspensions of fat particles in oil. J. Colloid Science 18: 276 (1963)
Φ= 32 odAE
π
For small deformations only
There also exists a modification of van den Tempel’s model by Sherman (1968)
49
G~Φ
30 35 40 45 50 55 600.0
2.5
5.0
7.5
10.0A
Φ (%)
G' (
MP
a)
40 50 60 70 805
10
15
20
25B
Φ (%)
G' (
MP
a)
20 30 40 50 60 700
5
10
15C
Φ (%)
G' (
MP
a)
20 25 30 35 40 450.0
0.5
1.0
1.5D
Φ (%)
G' (
MP
a)
milkfat
palm oil
lard
tallow
Network model needs to takeinto account the presence of particle (crystal) aggregates
50
G~Φ for single particle network doesnot provide satisfactory description ofsystem’s behavior
Rather: G~Φμ
Several parameters that determinethe state of aggregation are lumpedtogether into one dimensionless‘scaling factor’
He also suggests that μ~N
Good agreement with experiments
About 106 particles peraggregate
More agitation leads to less aggregation
51
μΦ~'G
D
Ddxd
−≈
−+
=
33.4μ
μ
Heertje, I. (1993)
Microstructural Studies in Fat Research
Food Structure 12, 77-94
52
Support for the view of fats as crystallite fractal gels grows
Johansson also shows that:
ξ~Φ1/(D-d)
53
Weak-link vs. Strong Link
High SFC vs. Low SFC
Fractal analysis explains the softening observedin milkfat upon chemical interesterification
Narine, S.S. and Marangoni, A.G. (1999) Fractal nature of fat crystal networks Physical Review E 59: 1908
54
Structural view – a colloidal gel?
Fractal Floc
Particle
Link
Fractality and the Fractal Dimension
55
Characterization of the SpatialDistribution of Mass within the
Fat Crystal Network
Fractal Geometry
Fat crystal networks display statistical self-similarity
and scaling behavior characteristic of stochastic
fractal systems
56
Statistical Self similarity – you guess!
57
58
Db=1.60
4x
10x
40x
100x
Self-similarity?
Scaling Behavior and Fractional Dimensionality
3.50 3.75 4.00 4.25 4.5014
15
16
17
18
19
20
D=2.45r2=0.88
ln Φ
ln G
'
Scaling behavior G~Φμ
3.5 4.0 4.5 5.0 5.5 6.02
3
4
5
6
D=2.01 (0.06)r2=0.994
log (length)
log
(N)
Scaling behavior N~LD
59
~ μξ Φ
Marangoni and Ollivon, 2007, Chemical Physics Letters 442: 360-364
60
2.00 2.25 2.50 2.75 3.00 3.251.25
1.50
1.75
2.00
2.25
slope = 0.67 ± 0.009A
D = 1.49 ± 0.019
SFC linear with time
Log10 Time (seconds)
Log 1
0 D
iam
eter
(μm
)
2.00 2.25 2.50 2.75 3.00 3.25
slope = 0.72 ± 0.055
B
SFC linear with time
D = 1.40 ± 0.106
Log10 Time (seconds)
ξ~t1/D
Milkfat Spherulites at 20oCfor M~t
Batte and Marangoni, 2005, Crystal Growth and Design 5: 1703-1705
Fracal Structural-mechanical modelThe view
61
Structure-mechanical modelThermodynamic arguments
Dd
o
D
daA
aE −− Φ=Φ=
1
2*3
1
* 26
επεδ
EG 31=
Marangoni, 2000, Phys. Rev. B 62, 13951Marangoni and Rogers, 2003, Appl. Phys. Lett. 82, 3239
D
ocadAG −Φ= 3
1
36π
Narine and Marangoni, 1999, Phys. Rev. E 60, 6991
Structure-mechanical modelParticle-spring arguments
62
The Yield Stress (Compression)
D
a
E
−Φ=
⋅=
31
*
**
6δσ
εσ
Marangoni and Rogers, 2003, Appl. Phys. Lett. 82, 3239
Model simulations
0.15 0.20 0.25 0.30 0.350
1000
2000
3000a=200 nmD=2.8δ (J m-2)→var
(a)
0.005
0.03
0.01
σ* (P
a)
0.15 0.20 0.25 0.30 0.350
1000
2000
3000(b)δ=0.01J m-2
D=2.8a (nm)→var
100
500
σ* (P
a)
200
0.15 0.20 0.25 0.30 0.350
1000
2000
3000(c)δ=0.01J m-2
a=200 nmD→var
2.75
2.7
2.8
Φ
σ*(P
a)
63
0.1 0.2 0.3 0.4 0.5 0.60
5000
10000
15000
Φσ*
(Pa)
δ=0.01J m-2
a=740 nmD=2.69
(c)
0.15 0.20 0.25 0.30 0.350
1000
2000
3000δ=0.01J m-2
a=130 nmD=2.79
(a)
σ*(P
a)
0.1 0.2 0.3 0.4 0.50
2000
4000
6000
8000
(b)
δ=0.01J m-2
a=810 nmD=2.67
σ*(P
a)
Methods of Determination
64
Processing procedure for calculation of scaling fractal dimensions of fat crystal
network
Original polarized light microscopy image
Adjust contrast of the images
Threshold it to binary image
Calculate Db by Benoit 1.3, Df by a particleCounting.m,and DFT by Photoshop
OriginalPolarized light microscopyimage
Thresholdedimage
Fractal Dimension by Microscopy - particle counting
• Df is related to the spatial distribution of mass
• higher Dfmore orderly distribution
PLM image Threshold Particle Count
65
Mass Fractal Dimension (D) by particle counting (sensitive to order)
N = number of particlesL = length of ROI c = constant of proportionality
Log N = D Log R + Log cY = m X + b
N cLD=
Narine and Marangoni, 1999, PRE 59:1908
Fractal Dimension Determination by Particle Counting
3.5 4.0 4.5 5.0 5.5 6.02
3
4
5
6
D=2.01 (0.06)r2=0.994
log (length)
log
(N)
66
Artifacts!
-0.25 0.00 0.25 0.50 0.75 1.000
1
2
3
4
Dexc=2.18
Dinc=2.05
Dexc=2.49Dinc=2.17
r2>0.99
log L
log
N
Mass Fractal Dimension by Box Counting (sensitive to degree of occupancy of embedding space)
Benoit 1.3 (Truesoft Int’l Inc.)
67
Low (1.5) High (1.9)
Effect of crystallization temperature on cocoa butter microstructure
68
0 5 10 15 20 25 301.4
1.5
1.6
1.7
1.8
1.9
2.0
Temperature (oC)
Dbo
x
Effect of crystallization temperature on milkfatmicrostructure
0 1 2 3 4 5 61.65
1.70
1.75
1.80
1.85
1.90
1.95
0 days1 day7 days14 days
Cooling rate (oC/min)
D (d
=2)
Cooling rate effects on milkfat microstructure
69
Fractal Dimension by Microscopy: box counting
• Db is sensitive to the degree of fill in the network:higher Db higher degree of fill
Φ = NpVp/NV ~ (>/a)D/(>/a)d
Φ ~ (>/a)D/(> /a)d = (> /a)(D-d) > ~ aΦ1/(D-d)
N~>d Np~>D
> >
What is the fractal dimension?
70
Fractal dimension from the characteristic slope of FFT power spectrum
0.0 0.5 1.0 1.5 2.0 2.5-3
-2
-1
0
1S= -0.811
S= -2.14Lo
g (M
agni
tude
2 )
-0.5 0.0 0.5 1.0 1.5 2.0 2.5-3
-2
-1
0
1
S= -0.76
S= -2.11
S= 0
Log
(Mag
nitu
de2 )
0.0 0.5 1.0 1.5 2.0 2.5-4
-3
-2
-1
0
1
S= 0S= -1.16
S= -3.42
Log (Frequency)
Log
(Mag
nitu
de2 )
(A)
(C)
(B)
| |22
slopeD = +
Fractal dimension - rheology
μ
λ ⎟⎠⎞
⎜⎝⎛=
100' SFCG
⎟⎠⎞
⎜⎝⎛+=
100lnln'ln SFCG μλ
71
Weak-link Theory - Fat Crystal Networks
Developed for colloidal gels and adapted to fat crystal networks:
G’ = dynamic shear elastic modulus (storage modulus)λ = constant which depends on particle properties and
particle-particle interactionsΦ = solids’ volume fraction of fat sample
G’ ~ Φμ ~ λΦ1/(d-D)
Narine and Marangoni, 1999, PRE 59:1908
Using the Weak-Link Theory to Calculate Fractal Dimensions Rheologically
Log G’ = [1/(3-D)] log Φ + logfcλ
G d D' ~ λΦ1−
72
30 35 40 45 50 55 600.0
2.5
5.0
7.5
10.0A
Φ (%)
G' (
MP
a)
40 50 60 70 805
10
15
20
25B
Φ (%)
G' (
MP
a)20 30 40 50 60 70
0
5
10
15C
Φ (%)
G' (
MP
a)
20 25 30 35 40 450.0
0.5
1.0
1.5D
Φ (%)
G' (
MP
a)
15 20 25 30 35 400
2
4
6
8A
Solid Fat Content (%)
Stor
age
Mod
ulus
(MPa
)
20 25 30 35 40 45 50 55 600
2
4
6
8
10
12B
Solid Fat Content (%)
Stor
age
Mod
ulus
(MPa
)
3.00 3.25 3.50 3.75 4.00 4.25-6
-4
-2
0
2
4
6A
ln φ
ln G
'
3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.42.0
2.5
3.0
3.5
4.0
4.5
5.0B
ln φ
ln G
'
73
0 20 40 60 80 1000
20
40
60
80
100AMFCBPO
Fraction of Fat in Oil (% w/w)
SFC
(%)
Dilute fat with vegetable oil under conditions where fat is not dissolved significantly in the oil?
Milkfat
-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.68
10
12
14
16
18
S=8.40I=25.9r2=0.93
S=6.01I=22.3r2=0.88 S=2.20
I=18.1r2=0.79
ln (SFC/100)
ln G
'
74
Palm Oil
-2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.012
13
14
15
16
17
18
19
S=3.77I=21.1r2=0.98
S=1.80I=18.2r2=0.88
S=3.36I=18.3r2=0.96
ln (SFC/100)
ln G
'
Cocoa Butter
-2.4 -2.0 -1.6 -1.2 -0.8 -0.4 -0.010
12
14
16
18
20
ln (SFC/100)
ln G
'
S=3.98I=I=20.3r2=0.96
S=1.56I=I=17.2r2=0.39
75
-2.0 -1.5 -1.0 -0.52
4
6
8
10(A)
ln ( σ
o)
S=4.9r2=0.89
S=1.97r2=0.83
S=8.83r2=0.96
-2.5 -2.0 -1.5 -1.0 -0.5 0.04
5
6
7
8
9
10(B)
ln ( σ
o)
S=2.054r2=0.93
-2.5 -2.0 -1.5 -1.0 -0.5 0.04
5
6
7
8
9
10(C)
ln (SFC/100)
ln ( σ
o)
S=2.96r2=0.84
Strain at the limit of linearity alwaysIncreases with SFC, thus fats are Always in the weak-link rheological Regime and thus:
:=1/(3-D)
Awad, Rogers, Marangoni, 2004, J. Phys.Chem. B 108, 171
1.8 2.0 2.2 2.4 2.6 2.8 3.00.0
0.2
0.4
0.6
Df
G'/ λ
( φ=0
.5)
Effect of D on the Elastic Modulus
76
Relationship to Large DeformationRheological Behavior?
4 6 8 10 120
1000
2000
3000
4000
r2=0.80
E' (MPa)
Yie
ld F
orce
(g fo
rce)
0 100 200 300 4002.0
2.2
2.4
2.6
2.8
3.0
21oC26.5oC
Shear Rate (RPM)
Dr
Effects of shear on Dr (AMF)
VANHOUTTE, 2002 (Ph.D. thesis, Gent University)
77
0 100 200 300 400-10
-5
0
5
10
21oC26.5oC
lnλ=:5.5-0.018⋅SR
Shear Rate (RPM)
lnλ
lnλ=:8.1-0.045⋅SR
SRDD β
=−max)(ln SR
o
αλλ
−=
0.000 0.005 0.010 0.015 0.020 0.0252.0
2.2
2.4
2.6
2.8
3.0
21oC
26.5oC
D=3.0-50⋅SR-1
D=2.8-29⋅SR-1
1/Shear Rate (RPM-1)
Dr
-10 -8 -6 -4 -2 0 2 4 6 81.5
1.6
1.7
1.8
1.9
2.0D=0.023⋅ln(τ-1) -1.81r2=0.91
A
ln (τ-1)
D
0 1 2 3 41.5
1.6
1.7
1.8
1.9
2.0
D=1.92-0.08⋅nr2=0.93
B
n
D
Crystallization behavior and structure
JDD ln* β+=
Marangoni and McGauley, 2003,Crystal Growth and Design 3, 95
78
JDdDd
aaln*
11* 66 βδδσ −−− Φ=Φ=
Thermomechanical Method for the Determination of the Rheology
Fractal Dimension
79
AMF SFC vs Temperature
0
10
20
30
40
50
60
0 5 10 15 20 25 30 35 40
Temperature (deg Celcius)
SFC
(%)
AMF G' vs Temperature
0.00E+00
2.00E+06
4.00E+06
6.00E+06
8.00E+06
1.00E+07
1.20E+07
1.40E+07
1.60E+07
1.80E+07
2.00E+07
4 7 10 13 16 19 22 25
Temperature (deg Celcius)
G' (
Pa)
80
AMF G' vs SFC
0.00E+00
5.00E+06
1.00E+07
1.50E+07
2.00E+07
0 10 20 30 40 50 60
SFC (%)
G' (
Pa)
-1.4 -1.2 -1.0 -0.8 -0.613
14
15
16
17
18
Thermor2=0.996D=2.68
Dilutionr2=0.938D=2.70
Φ (SFC/100)
ln G
'
81
CB SFC vs. Temperature vs.
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35 40
Temperature (deg Celcius)
SFC
(%)
CB G' vs. Temperature
0.00E+005.00E+061.00E+071.50E+072.00E+072.50E+073.00E+073.50E+074.00E+074.50E+07
15 17 19 21 23 25 27 29
Temperature (deg Celcius)
G' (
Pa)
82
CB G' vs. SFC
0.00E+005.00E+061.00E+071.50E+072.00E+072.50E+073.00E+073.50E+074.00E+074.50E+07
0 10 20 30 40 50 60 70 80 90
SFC (%)
G' (
Pa)
-2.0 -1.5 -1.0 -0.5 0.013
14
15
16
17
18
19Dilutionr2=0.92D=2.43
Thermor2=0.98D=2.40
Φ (SFC/100)
ln G
'
83
Dongming Tang’s project
• Determine the physical basis of the scaling fractal dimensions used to quantify the microstructure of fat crystal networks in 2D and 3D
• Compare the scaling fractal dimensions of fat crystal networks in 2D and 3D space
• Use fractal dimensions in 3-dimensional space to explain the rheological properties of fat crystal networks
Common methods to determine fractal dimensions
N ~ R-Db
N ~ RDf
• Df
• Db
DFT in 2D = (4+slope)/2
• DFT
84
2D simulation images of fat crystal networks
Sample images generated by computer simulation. The shape, crystal size, area fraction and the distribution orderliness for the images are (A) Line, 13 pixels, AF=10%, 100% randomly distributed; (B) disk, 29 pixels, AF=10%, 100% evenly distributed; (C) square, 49 pixels, AF=15%, 100% randomly distributed; (D) diamond, 25 pixels, AF=8%, 30% evenly distributed; (E) diamond, 25 pixels, AF=10%, 50% evenly distributed; (F) diamond, 25 pixels, AF=8%, 70% evenly distributed.
Tang D. and A.G. Marangoni (2006) Journal of American Oil Chemists Society 83: 309-314.
2D simulation on Db
0 5 10 15 20 25 30 35 401.0
1.2
1.4
1.6
1.8
2.0
R=2R=6R=14R=32R=42
Area fraction of crystals (%)
Db
The effects of area fraction on Db for different crystal sizes. R is the radius of fat crystals
0 10 20 30 40 500.8
1.0
1.2
1.4
1.6
1.8AF=20%
AF=10%
AF=2%
Crystal radius
Db
The effects of crystal size on Db. Solid symbols are for 100% randomly distributed crystals, and open symbols are for 100% evenly distributed crystals.
85
2D simulation on Df
Radial distribution pattern of simulation images with different Df. (A) Df = 0.998; (B) Df = 3.052.
Tang and Marangoni (2006) Journal of American Oil Chemists Society 83: 309-314.
2D simulation on DFT
1 2 3 4 5 6 71.3
1.4
1.5
1.6
1.7
1.8
2%
6%
10%20%
30%
Radius of crystals
DFT
Area fractionof crystals
AF2% AF4% AF6% AF8%AF10%1.58
1.60
1.62
1.64
1.66
1.68
0%30%50%70%90%
Distribution order of crystals
Area fraction of particles
DFT
86
2D simulation summary
X
√
√
√
Db
X
X
√
√
DFT
√?Distribution pattern
XXArea fraction of crystals
X?Crystal size
X?Crystal shape
DfRheology DAffecting Factors
Different measures of fractalitycannot agree since they are sensitive to different structural features within
the network.
Cannot use the argument that all values of the fractal dimension must
be the same for the system to be fractal…….
87
2D fractal dimension vs 3D fractal dimension
Tang D. and A.G. Marangoni (2006) Chemical Physics Letters 433: 248-252.
Acquiring images of fat samples in 2-and 3- dimensional space
2-3
um
Glass slide
200
umCover slip
Fat sample3D image taking method
20 u
m
2D image taking method
88
3-Dimensional Imaging of Bulk Fats by Wide-field DeconvolutionPolarized Light Microscopy
Novel Approach: Wide-field Deconvolution
• traditional uses (confocal, fluorescence)– adapted to TLB (Holmes & O’Conner)
• allows for thicker samples (true volumes)• 3-dimensional data sets (x, y, z)• employs a deconvolution algorithm
(removes haze from unfocussed portions of the specimen)
• 2D projection, 3D renderings
89
bottom
z
x
ytop
Z-series • image acquisition
• 50x LWD plano• N.A. = 0.55
Deconvolution Algorithm• Openlab: Volume Deconvolution
– theoretical point spread function– uses 8 nearest neighbors (NN)– removes unfocussed haze from each plane
• Autoquant: Autodeblur– Estimates the point spread function– Uses the image data (Blind)– Removes unfocussed haze from each plane
• sample: high-melting fraction of milk fat (HMF) diluted in triolein(70:30), crystallized isothermally at 28ºC for 15 minutes
90
Comparisons:2D Projections
Raw
NN
Blind
Raw
NN
Blind
Comparisons:XZ and YZ
91
3D-FD
The main window of 3D-FD after the calculation of particle-counting dimension, Df. Note that the first data points on the left calculation graph was manually checked off.
© Marangoni, A.G., Tang, D. and TruSoft Inc., 2006
3D simulation study on Db
0 5 10 15 20 251.7
1.9
2.1
2.3
2.5
2.7
r=3r=4r=5
r=6
r=10r=14r=18
r=22
Area fraction of crystals (%)
Db
The effects of AF on Db for different crystal sizes. r is the radius of fat crystals
0 5 10 15 20 25 301.7
1.9
2.1
2.3
2.5
2.7
AF=2%AF=10%AF=20%
Crystal radius (pixels)
Db
The effects of crystal size on Db of fat crystal networks with evenly distributed crystals. AF is area fraction of crystals.
92
2D vs 3D fractal dimension
(a) Db of 2D slices at different depths of a thick sample of the high melting fraction of milk fat in canola oil at a 7% solids’ mass fraction (b) Comparison between the 2D and 3D box countingdimensions of samples of high melting fraction of milk fat in canola oil at 20°C at different mass fractions of solids (Φ).
Tang and Marangoni, 2006, Chemical Physics Letters 433: 248-252.
Microscopy fractal dimensions of HMF
2.672.763D Df average
1.901.992D Df average
2.372.063D Db average
1.651.342D Db average
High SFC range(>10%)
Low SFC range(<10%)
SFC
93
Rheology and microscopy fractal dimensions of HMF
2.743D Dr average
2.672.763D Df average
2.372.063D Db average
High SFC range(>10%)
Low SFC range(<10%)
SFC
Modified fractal model
Heterogeneous Stress DistributionPrinciple
94
Homogeneous factor and dynamic microstructure
Spatial distribution of strain energy in a perfectly connected DLCA network
Ma, H.S. et al. (2000). Journal of Non-Crystal Solids, 277, 127-141.
Fat crystal network at high SFC
Force
95
Dk b
ecG −Φ−−∗= 31
)1('
Tang and Marangoni, 2008, JCIS 318: 202
96
Modified fractal model of colloidal fat crystal networks
What about intermolecular interactions?
Van der Waals are the end-all?
97
φπγ
φ D31
D-31
da 6Aλ G'
20
−==
Fat Crystal Network Model
Variables Explanation
λ Experimental value, obtained from a ln G’ vs. ln Φ plot
a diameter of particles
γ Strain at the limit of linearity
d0 separation between the “flocs”
Hamaker Constant
SFC
Hamaker constant
• McLachlan and Lifshitz approach
• Semi-classical approach
• Fat Crystal Network Model
Material:Dispersed phase: Fully hydrogenated canola oil
Continuous phase: High oleic sunflower oil
98
McLachlan and Lifshitzapproach
J.Israelachvili –Intermolecular & Surface Forces, 2002 page 184
k Boltzmann’s constant 1.38 10-23 J/K
h Planck’s constant 6.626 x 10-34 J s T Temperature (K )ε dielectric permittivity
n index of refractionelectronic absorption frequency in the UV region 3 x 1015 s-1
Index 1 and 2 are for dispersed and continuous phases
( )( ) 2
322
21
222
21
2
22
2100 216
343
nnnnhkTAAA e
vv+
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−
=+= >=ν
εεεε
eν
Permanent dipole-permanent dipole interaction are very important in fat crystal
interactions!
Av=o % Av=o Av>o % Av>o Total A
Svaikauskas 3.5 10-22 J 67 % 1.70 10-22J 33% 5.2x10-22 J (30°C)
Svaikauskas 9.1 10-22 J 82% 1.77 10-22 J 18% 1.1x10-21 J (20°C)
Johannson(JAOC, Vol.69, 8 1992)
3.8 10-23 J 12% 1.77 10-22 J 88% 2.0x10-22 J
Kloek(Ph.D Thesis, 1998)
3.8 10-23 J 8.5% 1.83 10-21 J 91.5% 2.0x10-21 J
99
Hamaker from nSSS nOOO εSSS εOOO A
Svaikauskas(tristearin in sunflower
oil)
1.503867
1.51299
1.469903
1.466420
1.68*
1.926
2.7
3.17
5.2x10-22 J(30°C)
1.1x10-21 J (20°C)
Johannson (JAOC, Vol.69,no8,1992)
(tristearin in soybean oil)
1.44711 1.47351 23 2.53 2x10-22 J
Kloek (Ph.DThesis,1998) (tristearin
in soybean oil)
1.562 1.47351 23 2.53 2x10-21 J
1- Bailey’s Industrial Oil and fats Products (measured at 60°C) * Estimated from 20 °C data.2- Meeussen, Personal communication 3 Handbook of Chemistry and Physics, R.C East, 56th edn.
Semiclassical approach
crystal-melt interfacial tension (30°C)equilibrium space between molecules
Hiemenz and Rajagopalan, Principles of Colloid and Surface Chemistry, 1997
20A 24π dδ=
δd 0
* L.W.Phipps, Trans. Faraday Soc. 60, 1873 (1964).
* * Latifeh Ahmadi’s work
d0 (nm) δ (J/m2 ) A (J)0.2 0.01 * 3.0x10-20
0.2 0.005** 1.5x10-20
100
Fat Crystal Network Model
φπγ
φ D31
D-31
da 6Aλ G'
20
−==da 6 λ A 20πγ=
Variables Explanation Experimental values
λ Experimental value, obtained from the ln G’ vs. ln Φ graph
1.17 107 Pa
a diameter of particles 2µm,1 µm, 0.4µm,0.2 µm
γ Strain at the limit of linearity 1.9 10-4
d0 separation between the “flocs” 0.2 nm
A= 3.35 10-21 J when a = 2 μm A= 1.68 10-21 J when a = 1 μm A= 6.7 10-22 J when a = 0.4 μm A= 3.35 10-22 J when a = 0.2 μm
Summary of Hamaker values
101
Thank You!