structure and mechanical properties of fat crystal...

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



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.

16

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

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

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


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'


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!

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


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


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


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


2.372.063D Db average


High SFC range(>10%)

Low SFC range(<10%)

SFC

93

Rheology and microscopy fractal dimensions of HMF

2.743D Dr 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!

structure and mechanical properties of fat crystal...

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