Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65

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Journal of Non-Newtonian Fluid Mechanics

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Capillary flow of milk chocolate

http://dx.doi.org/10.1016/j.jnnfm.2014.06.0010377-0257/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +1 604 822 3107; fax: +1 604 822 6003.E-mail address: savvas.hatzi@ubc.ca (S.G. Hatzikiriakos).

Hesam Anvari Ardakani a, Evan Mitsoulis b, Savvas G. Hatzikiriakos a,⇑a Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canadab School of Mining Engineering and Metallurgy, National Technical University of Athens, Athens, Greece

a r t i c l e i n f o

Article history:Received 16 December 2013Received in revised form 31 May 2014Accepted 4 June 2014Available online 12 June 2014

Keywords:Chocolate rheologyCapillary extrusionThixotropyStructural parameterYield stressSemi-solid extrusion

a b s t r a c t

The rheological behaviour of three different commercially manufactured chocolates are studied as amodel paste system in order to understand their processing characteristics at different temperaturesby using parallel-plate and capillary flow. The rheological behaviour is distinctly different at tempera-tures below and above 30 �C. At temperatures below 30 �C it behaves as a soft solid. A previously devel-oped phenomenological mathematical model for polytetrafluoroethylene (PTFE) paste flow was used tomodel its rheology and capillary extrusion. The model takes into account the elastic–plastic (strain hard-ening) and viscous nature of the material in its non-melt state. The predictive capability of this model istested against capillary data for a variety of capillary dies having different length-to-diameter ratios (L/D)and contraction angles (2a). At temperatures above 30 �C, chocolate behaves as a visco-elasto-plasticliquid. Its behaviour has been determined as a Herschel–Bulkley viscoplastic material with a small degreeof thixotropy. The rheological data obtained from parallel plates and bob–cup rheometer were used toformulate a thixotropic model using a structural parameter. The capillary data were found to be welldescribed with the proposed models.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

A paste can be defined as a mixture of a solid and a liquid phaseor as a dense suspension, which is shown to demonstrate proper-ties between liquid and solid [1]. Many pastes are able to retaintheir shapes against gravity like solids [2]. At high shear stressesthey start to flow, which implies the existence of a yield stress, thatis, a critical stress for transition from solid-like to fluid-like behav-iour [1,3–5]. Moreover most of these pasty materials have shownto exhibit thixotropic behaviour, i.e., their viscosity decreases withtime, which implies structure breakup [1].

Chocolate is rheologically complex both above and below itsmelting range. It shows semi-solid behaviour at room temperature(20–25 �C). Chocolate has a broad melting range. It melts down andturns into liquid form (in reality a dense suspension of non-colloi-dal particles) at temperatures very close to oral temperature that isabout 30–32 �C. At room temperature the material typically con-tains about 10% liquid cocoa butter and this increases to 100%when the chocolate is fully molten above 35 �C. Generally, choco-late contains about 70% of solid sugar, some cocoa solids and crys-talline cocoa butter, which are dispersed in a continuous fat-phasecocoa butter. Different commercial chocolates can be found and are

categorized into three primary groups, namely, dark chocolate,milk chocolate, and white chocolate. They differ in content of cocoasolids, milk, and cocoa butter. Cocoa butter itself can be extractedfrom cocoa mass (ground cocoa beans) by pressing them duringprocessing [6–9]. Cocoa butter triglyceride is mainly formed fromPalmitic (P), Stearic (S), and Oleic (O) fatty acids [7,10]. Due tothe presence of these triglycerides, cocoa butter is able to formsix different crystal structures with different melting behaviours.Chocolate crystallinity is greatly influenced by temperaturetreatment during the process, fat content, and triglycerides type[11–14].

Usually, chocolates are made by pouring or extruding meltchocolate into a mould at temperature around 30 �C and cool downto retain the desired shape [15]. More recently, a new method hasbeen used, which involves cold extrusion of chocolate in tempera-ture between 5 �C and 25 �C [9]. During extrusion, chocolate isforced through the die by a pressure difference. As chocolate comesout of the die land it takes the shape of the die [16,17]. Althoughcold extrusion is an isothermal process, an increase in liquid fatcontent has been observed due to shear forces. Therefore, struc-tural decomposition occurs during the process that results in alower extrusion pressure [18].

Rheologically, ‘‘liquid’’ chocolates have shown non-Newtonianbehaviour with a yield stress and plastic viscosity (stress to keepfluid in motion) with mild shear-thinning characteristics. Differentparameters affect the rheological behaviour of chocolate such as fat

H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65 57

content, emulsifier content, water content, conching time, crystal-lization, particle size, and temperature [19–21]. Knowing the yieldstress and the thixotropic behaviour of pastes is extremely impor-tant from the rheological point of view in order to develop arheological constitutive equation capable of predicting correctlytheir flow behaviour in well-defined flows [1].

Chocolate fat content has a significant effect on yield stress.Generally, a lower amount of fat results in higher yield stress val-ues [22]. Particle size distribution is another important parameter,which plays a role in chocolate rheological behaviour [15]. Cocoaparticle size varies from 15 to 30 lm [23]. It has been reported thata bimodal distribution with a small amount of fine and largeamount of coarse particles reduces the apparent viscosity [24].Surfactant is another determining parameter in chocolate rheol-ogy. Addition of lecithin at low concentration (below 3%) reducesboth yield stress and viscosity. After around 5%, addition of morelecithin increases the yield stress while the shear viscosity of themelt continues to drop [15,22].

Different important rheological models have been used tocharacterize the rheological behaviour of chocolate melts includingthe Herschel–Bulkley, Casson, Bingham, and Carreau models[15,9,25,26]. Although the Casson is the recommended model byIOCCC (International Office of Cocoa, Chocolate, and Confection-ery), it has been reported that it is not able to accurately character-ize chocolate melt behaviour at low shear rates [26,27].

In the present paper, we are interested in studying first experi-mentally the rheological behaviour of a commercially manufac-tured chocolate at both semi-solid and melt forms using simplerheological flows. Based on these experimental results, a rheologi-cal constitutive equation is developed for ‘‘liquid’’ chocolate(T > 30 �C). Furthermore, capillary extrusion experiments are per-formed using capillary dies of different geometrical characteristics,such as die diameter, D, length-to-diameter ratio, L/D, and contrac-tion angle, 2a. Using the developed constitutive equation, flowsimulations in capillary flow are performed to predict the pressuredrop in the various geometries and examine in detail the flowbehaviour inside the dies with emphasis on the structure evolutionof chocolate. Since reliable rheological data for semi-solid statesare not available, a simple analytical phenomenological model isused to capture effects of different geometrical parameters of dieson this process at temperatures less than 30 �C.

Table 2List of capillary dies used in the present work. The barrel diameter, Db, is 15 mm.

Die number RR (–) L/D (–) 2a (degrees) D (mm)

1 277 18 15 0.92 311 20 30 0.853 311 20 45 0.854 311 20 60 0.855 311 20 90 0.856 311 20 180 0.857 45 20 45 2.2258 139 20 45 1.279 311 0 45 0.85

2. Experimental

Three commercially manufactured chocolates namely white,milk, and dark chocolate have been chosen for this study andobtained from Purdy’s chocolatiers (Vancouver, BC, Canada).Table 1 lists fat, sugar, and solid particles content of these threechocolates. Aside from fat, sugar, and solid particles, chocolatehas protein, soy lecithin (emulsifier), and other additives likevanilla. Note that the reported value for the fat content representsboth cocoa and milk fat. We assume that chocolate will behave in ahomogeneous way (at a length scale of 100 lm). Then it isexpected that liquid migration becomes negligible compared tothe case of granular materials with a liquid medium [28,29].

The thermal properties of these chocolates have been analyzedwith a Shimadzu DSC-60 calorimeter with air as heating flow. A

Table 1Fat, sugar, and solid particles content of three different chocolate types used in this work

Chocolate type Fat content (w%) Sugar content (w

White chocolate 34 54Milk chocolate 32 54Dark chocolate 32 46

sample was placed in a sealed aluminium pan and heated at rateof 5 �C/min from 20 �C to 50 �C.

The rheological properties of chocolate, such as the flow curve,the yield stress, and thixotropy, have been studied using a rota-tional rheometer (Kinexus, Malvern) equipped with a cup-and-bob geometry. In order to obtain consistent and reproducibleresults, the samples are pre-sheared at the experiment tempera-ture (a common practice for thixotropic fluids) for a certain periodof time (15 min) under a certain shear rate (50 s�1) and left to restfor 2 h thereafter in order to recover their structure. Initial struc-ture of these commercially manufactured samples is unknown.Shearing at moderately high shear rate for 15 min fully decom-poses the structure. This pre-shearing stage is followed by a restingstage in which structure starts to re-build itself. Pursuing the samepre-shearing protocol for all samples guarantees same structureand initial condition. Experiments have been carried out at differ-ent temperatures to study the effect of temperature on the extru-sion process with emphasis on two temperatures 30 �C and 35 �C.The same rheometer has been used to determine the complex vis-cosity of the material from 25 �C (solid state) to 35 �C (liquid state).

Capillary extrusion experiments were carried out using a pis-ton-driven constant-speed capillary rheometer (Bohlin RH 2000).Capillary dies of various geometrical parameters were used tostudy the processability of chocolate in capillary flow. All the cap-illary dies were made from stainless steel with electrical dischargemachining. Machined surfaces with this method are smooth andhave average roughness of less than 1 lm. Obtained data from cap-illary rheometer were also used to check the consistency of the for-mulated constitutive equations, which were based on rheologicaldata from the parallel-plate rheometer. Important parametersinclude the die diameter, D, the length-to-diameter ratio of thedie, L/D, the reduction ratio, RR � ðD2

b=D2Þ, where Db is the barreldiameter, and the entrance contraction angle, 2a. Table 2 lists allcapillary dies used in this study along with their geometrical char-acteristics. The barrel diameter, Db, is 15 mm. Experiments wererepeated at several different temperatures (25 �C to 40 �C) in orderto study effects of temperature on the extrusion process.

3. The effect of temperature and ingredients

3.1. DSC analysis

As mentioned before, six different polymorphic terms of cocoabutter can exist in the semi-solid state [11–14]. In general the

.

%) Solid particles content (w%) Density (kg/m3)

0 �13002 �13008 �1300

DSC Analysis

Temperature, T (OC)

10 20 30 40 50

Hea

t Flo

w, (

m W

/g)

-800

-600

-400

-200

0

200

400White ChocolateMilk ChocolateDark Chocolate

Fig. 1. DSC heating thermograms of three chocolates studied in this work.

58 H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65

thermal behaviour of chocolate is influenced by the existence andcontent of each of these forms in the final product. The main differ-ences between these forms are the distance between fatty acids,angle of tilt, and the way that triglycerides are packed in crystalli-zation [30]. The most stable form of chocolate has a melting pointof around 37 �C, while the less stable form has a melting point of17 �C [30]. Fig. 1 depicts the heating thermograph of the threechocolate samples. The melt peak in this thermograph is equal to30.2 �C, 30.8 �C, 32.2 �C for white, milk, and dark chocolate respec-tively. Melting peak separates solid from liquid behaviour. Above29 �C, all four first unstable polymorphic forms of chocolate buttercrystal melt down and only two stable forms remain in chocolatepaste [30]. The area beneath the DSC thermograph indicates theamount of energy that is needed to melt down the chocolate struc-ture. Therefore, it can be concluded that dark chocolate has thestrongest structure. While, white chocolate has shown to havethe weakest structure. Although DSC analysis would not give anyexact information about the distribution of different cocoa buttercrystal types, it can be seen that at higher temperatures more heathas transferred to the dark chocolate compared to milk and whitechocolate. This implements the existence of higher content of morestable crystal types in the dark chocolate.

3.2. The effect of temperature on rheology

Processing temperature is a significant factor in chocolateextrusion pressure. At low temperatures below the melting point,

Temperature, T ( oC)

15 20 25 30 35 40 45 50 55102

103

104

105

106

107

108

109

Com

plex

Vis

cosi

ty,

| η∗ (ω

)| (

Pa.

s) ω = 0.1 Hz

ω = 1 Hz

ω = 10 Hz

(a)

Fig. 2. (a) The complex viscosity of the milk chocolate as a function of temperature at t30 �C, which marks the transition from solid-like to liquid-like behaviour. (b) The phase a

chocolate behaves as semi-solid material [31], and a much higherpressure is needed for extrusion. Increasing the extrusion temper-ature melts down the fat structure and reduces the chocolate solidfat content. The melting process of more stable fat structure startsbelow 20 �C, which was verified by DSC [30]. However, liquid fatcontent still remains below 50% for temperatures lower than30 �C [32]. Note that different heating history dramatically altersthe results of rheological tests. Therefore, for all the experimentsin this work solid chocolate is grinded and then heated to thedesired temperature (similar to the DSC test).

Simple temperature sweep oscillatory test was performed inorder to study effects of temperature on the milk chocolaterheology. Serrated parallel-plate geometry was used in order tominimize slip effects particularly at low temperatures. For allmeasurements the strain amplitude was held constant at 10�4

and the complex viscosity measured at three different frequenciesas a function of temperature. Fig. 2a and b summarize the results,i.e., the effects of temperature on complex viscosity and phaseangle at different frequencies. Note that there is a gradual dropin the complex viscosity before 30 �C which is followed by a sud-den drop at about 30 �C that marks the transition from a solidliketo a liquidlike behaviour (also seen from the phase angle results ofFig. 2b). The gradual decrease of chocolate complex can beexplained by an increase of the liquid-fat content due to partialmelting of the cocoa butter. These findings are in agreement withthe DSC results depicted in Fig. 1. For example De Graef et al.[22] studied the effects of fat content on chocolate yield stressand viscosity. Based on their work, the average value of complexmodulus at 40 �C for 29% fat content is above 104 Pa. Consideringthat the milk chocolate of the present work has 32% fat content,our findings are consistent with previous studies.

Capillary extrusion experiments were carried out at four differ-ent apparent shear rates over a broad temperature range to inves-tigate the effects of temperature on the extrusion pressure of milkchocolate. Fig. 3 shows that at all investigated apparent shear rates,the extrusion pressure decreases rapidly with temperature atabout 30 �C. These results are consistent with DSC analysis(Fig. 1) and complex viscosity measurements (Fig. 2a and b).

3.3. The effect of chocolate composition on rheology

Different chocolate types can be categorized based on theircocoa butter, cocoa particle, sugar, and milk content. Dark choco-late is produced by adding sugar and cocoa butter to cocoa solids.Generally, there is no milk involved in dark chocolate processing.

Temperature, T ( oC)

15 20 25 30 35 40 45 50 55

Pha

se A

ngle

, φ

10

20

30

40

50

60

ω = 0.1 Hz

ω = 1 Hz

ω = 10 Hz(b)

hree different frequencies. Note the gradual drop of the complex viscosity at aboutngle of the milk chocolate as a function of temperature at three different frequencies.

Temperature, T (oC)

24 26 28 30 32 34 36 38 40 42

Ext

rusi

on P

ress

ure,

P (

kPa)

100

101

102

103

104

105

γA = 25 s-1

γA = 100

γA = 400

γA = 1600

RR=311, L/D=20, 2α=45

.

.

..

Fig. 3. The capillary extrusion pressure of the milk chocolate as a function oftemperature at several values of the apparent shear rate using a capillary die havinga length-to-diameter ratio of 20, reduction ratio (RR) of 311, and contraction angleof 45�. Note the sudden pressure drop at about 30 �C corresponding to the suddendrop of complex viscosity at about 30 �C.

H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65 59

The difference between milk and dark chocolate processing is add-ing milk to the chocolate mixture in the form of milk powder orliquid milk. Meanwhile, white chocolate is produced in a slightlydifferent manner, where, cocoa butter, sugar and milk are mixedtogether in absence of cocoa solid particles [6]. Fig. 4a and b showsthe extrusion pressure versus apparent shear rates for three differ-ent chocolates at two different temperatures, namely 25 �C and35 �C respectively. According to DSC analysis all three kinds ofchocolate are in solid state at 25 �C. The chocolate structure comesfrom two contributions. First it is the liquid/melted fat content.Any small changes in the temperature can easily alter the balanceof liquid/melted fat content. Therefore, this contribution is highlysensitive to changes in temperature. The other contribution tochocolate structure is arising from solid links and protein emulsi-fier linkages. It has been claimed that during the semi-solid choc-olate extrusion, the cocoa butter crystals structure breaks down byshear forces and the material starts to yield. Furthermore, thereleased energy by viscous dissipation melts down the fat andweakens the structure by increasing the liquid fat content[31,32]. Dark chocolate has stronger structure due to both liquid/melted fat content (Fig. 1) and solid links. Therefore, a higher shearforce is needed as seen in Fig. 4a. Note that for all three samples thefat content is about 32%. However, the cocoa butter crystal formwith no milk fat is more stable for the dark chocolate. Furthermore,

Apparent Shear Rate, γ (s-1)

Ext

rusi

on P

ress

ure,

P(k

Pa)

10 3

10 4

White ChocolateMilk ChocolateDark Chocolate

RR= 340, L/D=20, 2α=45

.

T = 25 oC×4 10 4

(a)

10 100 1000 5000

Fig. 4. The capillary extrusion pressure of three different chocolates as a function ofreduction ratio (RR) of 311, and contraction angle of 45�: (a) T = 25 �C, (b) T = 35 �C.

dark chocolate has the highest solid particles content (chocolateparticles) among the others. Existence of solid particles in semi-solid chocolate may increase the strength of structure directly.Besides, it can indirectly enhance the structure by helping hetero-geneous nucleation of cocoa fat crystal. Fig. 4b shows the corre-sponding extrusion pressures versus apparent shear rates for thedifferent chocolates at 35 �C where the chocolates are in the meltstate. All three samples show a similar behaviour and a small dif-ference in the solid particles content amongst the three samplesseems to play a minor effect.

4. Rheology of semi-solid chocolate (T < 30 �C)

Below 30 �C chocolate behaves as a semi-solid as was discussedabove based on the DSC thermogram. Therefore, rheological mea-surements of the apparent viscosity are not possible with the exist-ing instruments, and thus a viscous constitutive model cannot beformulated. Instead, a semi-solid phenomenological constitutivemodel for paste extrusion can be used [28,33]. This semi-solidpaste extrusion model assumes that the material slips on the wallwith Coulombic friction. Phenomenological analytical flow modelsfor calculating the extrusion pressure as a function of the operatingconditions and geometrical characteristics of dies are based on the‘‘radial-flow’’ hypothesis [33–37]. This hypothesis assumes that theflow is along the radial direction in the die (assuming a sphericalcoordinate system as in Fig. 5), and points located on virtual spher-ical surfaces of a constant radius r from the die apex have the sameradial velocity.

The mathematical form of the ‘‘radial-flow’’ hypothesis for acylindrical die (Fig. 5) can be written as:

drdt¼ � Q

2pð1� cos aÞr2 : ð1Þ

Based on this hypothesis, the kinematics of semi-solid chocolatepaste flow can be calculated independent of any rheological consti-tutive measurements at a given volumetric flow rate Q. The consti-tutive rheological behaviour of paste is modelled by elasto-plasticand viscous contribution to stress, essentially a modified Kelvinstress–strain relation with an added power-law viscous term[38–40]. The stress expression can be written as:

r ¼ Ccn þ K _cm; ð2Þ

where r is the shear stress, C is a consistency constant, c is theshear strain, _c is the shear rate, K is the consistency index, and nand m are power-law exponents.

Apparent Shear Rate, γ (s-1)

10 100 1000

Ext

rusi

on P

ress

ure,

P(k

Pa)

100

101

102

103

White ChocolateMilk ChocolateDark Chocolate

RR= 340, L/D=20, 2α=45

.

T = 35 oC

5000

(b)

apparent shear rate using a capillary die having a length-to-diameter ratio of 20,

Fig. 5. Schematic of a typical capillary die used in paste extrusion showing the various characteristic dimensions, Db, D, 2a and reduction ratio RR = (Db/D)2.

Milk Chocolate

Apparent Shear Rate, γA

(s-1)

10 100 1000

Ext

rusi

on P

ress

ure,

P (

kPa)

1000

10000

RR = 311RR = 139RR = 45Eq. 3

L/D=20, 2α=45

.

T = 25oC20000

5000

Fig. 6. The effect of apparent shear rate and reduction ratio on the steady-stateextrusion pressure of chocolate at 25 �C using capillary dies having length-to-diameter ratio of 20 and contraction angle of 45�. Solid lines are model predictions(Eq. (3)) with fitted parameters listed in Table 3.

60 H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65

Based on this hypothesis (‘‘radial flow’’) and the given constitu-tive equation, an analytical expression for extrusion pressure wasderived by [36], which can be written as follows:

P ¼ rraðRRÞB þ 2 1þ Bð Þ CDb

2 sina

� �2B Z rb¼Db

2 sin a

ra¼ D2ffiffiffiRRp

tan a

ð3 ln rb=rð ÞÞn

r2Bþ1 dr

(

þ Kð3mþ 2BÞ

12Q tan3 apð1� cos aÞD3

b

!m

ðRRÞBþ3m=2 � 1� �)

; ð3Þ

where

rra ¼ roaðe�4fL=Da � 1Þ þ rzLe�4fL=Da ð4Þ

and

�roa ¼ C32

lnðRRÞ� �n

þ K12Q sin3 aðRRÞ3=2

pð1� cos aÞD3b

!m

; ð5Þ

where P is the extrusion pressure, f is a Coulombic friction coeffi-cient, B = f sina/[2(1 � cosa)], rzL is the shear stress imposed atthe die exit, which is negligible or zero, and K and C are materialconstants, which can be evaluated by fitting extrusion pressuredata. Note that the first term in RHS of Eq. (3) represents pressuredrop in the die land and the second term represents the pressuredrop in the conical part of the die (more details in [36]). Table 3 liststhe optimized values for this model that best fit the capillary exper-imental data presented below.

All the reported extrusion pressure data refer to steady-statevalues. Depending on the piston speed and die diameter, an appar-ent shear rate can be defined by _cA � 32Q=pD3, where Q is the vol-umetric flow rate defined as Q � 1=4VpðpD2

bÞ, where Vp is thepiston velocity.

Fig. 6 shows the effect of apparent shear rate and reductionratio on the extrusion pressure. The extrusion pressure generallyincreases with an increase in the apparent shear rate (hence volu-metric flow rate), due to the increase in the viscous resistance ofthe paste. In addition it can be seen that the extrusion pressureincreases slightly with increasing reduction ratio. Generally, a

Table 3Fitted parameters for the semi-solid paste extrusion model (Eq. (3)).

C (Pa) n K (Pa sm) m f

134 � 103 4 � 10�10 313 � 103 0.31 0.024

higher extrusion pressure is needed to yield chocolate throughtapered dies with a higher reduction ratio. Model has shown tohave a better fit for higher reduction ratios. This might be a resultof having more dies with different contraction angles and L/D ratioat the higher reduction ratio (RR = 311).

Fig. 7 summarizes the effect of contraction angle on the extru-sion pressure at several apparent shear rates. The extrusion pres-sure initially decreases, goes through a minimum and thenslightly increases, with increasing die entrance angle. The extru-sion pressure is higher at low contraction angles as a result of Cou-lombic friction (larger area of die at smaller contraction angles).

The effect of L/D on the extrusion pressure is illustrated in Fig. 8.Due to the semi-solid-like behaviour of chocolate paste at 25 �C,the pressure drop in the conical part of the die is considerable. Itcan be seen that at the lower shear rates, about half of pressuredrop occurs in the conical part, i.e., comparing the data forL/D = 0 with that for L/D = 20. At low apparent shear rates, about30% of pressure drop occurs in the die land. Based on the proposedmodel this value represents the pressure drop due to the frictionbetween the semi-solid chocolate and the die land wall. Althoughthe model has been able to predict this value at low shear rates, itunderestimates the pressure drop in the die land at higher shearrates. As a result of viscous dissipation, chocolate liquid fat content

Milk Chocolate

Contraction Angle, 2 α0 20 40 60 80 100 120 140

Ext

rusi

on P

ress

ure,

P (

kPa)

10000γ

A = 25 s-1

γA = 50

γA = 100

γA = 200

γA = 400

γA = 800

γA = 1600

Eq. 3

.

.

.

.

.

.

.

T= 25 oC

RR=310, L/D=20

20000

2000

Fig. 7. The effect of entrance angle on the steady-state extrusion pressure ofchocolate at 25 �C using capillary dies having length-to-diameter ratio of 20 andreduction ratio (RR) of 311. Solid lines are model predictions (Eq. (3)) with fittedparameters listed in Table 3.

Milk Chocolate

Apparent Shear Rate, γA (s-1)

10 100 1000 10000

Ext

rusi

on P

ress

ure,

P (

kPa)

1000

10000

L/D=20L/D=0Eq. 3

RR =311, 2 α=45

.

T = 25 oC20000

Fig. 8. The effect of L/D on the steady-state extrusion chocolate at 25 �C usingcapillary dies having reduction ratio (RR) of 311 and contraction angle of 45�. Solidlines are model predictions (Eq. (3)) with fitted parameters listed in Table 3.

Milk Chocolate

Shear Rate, γ (s -1 )

0.1 1 10 100 1000

She

ar S

tres

s, σ

(Pa)

101

102

103

T=30oCT=32.5 oC T=35 oC, smallT=35 oC, largeHB modelThixotropic model

.

Fig. 9. The flow curve of chocolate using bob-and-cup geometries of variousdimensions to check for slip effects. The flow curve of milk chocolate at 35 �C isindependent of gap size showing that the effects of slip are negligible. Continuouslines represent the fits to the data of various constitutive models (see text foradditional explanations).

H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65 61

increases at higher shear rates. The existence of higher liquid fatcontent decreases the pressure drop in the conical part of thedie and increases the pressure drop in the die land as a result ofthe extra viscous dissipation. This effect was not considered inthe present model. Overall the model fit to the available experi-mental data has shown a good consistency. Optimized values formodel parameters reveal an important understanding about therheological properties of paste. The parameter n describes the sig-nificance of the strain hardening of material. When n is greaterthan 1, the material shows significant strain-hardening behaviour.If n is equal to zero, the material shows simple plastic deformation.Since in this case the value of n is very close to zero, it can be con-cluded that chocolate is a semi-solid paste with plastic deforma-tion (yield stress), which is consistent with previous findings[16,18,31].

5. Rheology of melt chocolate (T > 30�C)

5.1. Equilibrium flow curve

Steady shear experiments were first performed using differentgaps in order to examine possible effects of slip [40,41]. Fig. 9 plotsthe steady shear results for the cup-and-bob geometry using asmall and large gap (in the small geometry the bob diameter is14 mm and the cup diameter is 15.5 mm, while in the large

geometry, the diameters are 25 mm and 27.5 mm, respectively)at 35 �C. It is noted that the results within the accuracy of the exist-ing rheometer are independent of the gap. A small gap dependenceat small shear rates where yield stress effects prevail, althoughconsistent with the assumption of slip, is within experimentalerror. Therefore, this set of data provides evidence that slip inthe flow of chocolate is negligible at temperatures above themelting peak.

Steady shear experiments were performed for chocolate sampleat 30 �C, 32.5 �C and 35 �C in order to determine the flow curve ofthis material as a basis for the development of an appropriate con-stitutive equation (Fig. 9). As seen the temperature has a significanteffect on the rheology of the material at temperatures close to30 �C where it dramatically drops. However, the rheological behav-iour of chocolate melt at 32.5 �C and 35 �C is almost identical.These results are consistent with the previous results for complexviscosity measurements (Fig. 3). They will be used to formulate aconstitutive equation for the capillary simulations.

5.2. Time dependency – Thixotropy

For a complete rheological characterization it is also necessaryto identify the time-dependent effects, in particular the thixotropicbehaviour of the chocolate [25]. Various start-up experiments werecarried out. In each experiment, pre-shearing and rest to recoverthe structure as described above was first used. The shear rate issuddenly increased from zero to a certain value. As a result theshear stress increases from zero to a high value rapidly and subse-quently gradually decreases with time until it reaches its steadyvalue. Essentially a shear stress overshoot is obtained in each case,also indicating time dependency. This initial overshoot is related tothe viscoelastic response of the material. After this viscoelasticresponse, thixotropic behaviour can be observed. Fig. 10a and bshow both viscoelastic and thixotropic effects. The viscoelasticeffect can only be tracked at low shear rates where the thixotropiceffect is limited. Therefore for model development, the elasticresponse is neglected and the material is assumed to be purelythixotropic.

Fig. 10a and b summarize the thixotropic results for milk choc-olate at 30 �C and 35 �C. Thixotropic effects are more significant at30 �C as chocolate has a higher content of solid cocoa butter at alower temperature [30]. During flow, shear forces cause a phasetransition of cocoa butter and increase the overall liquid fat con-tent of the material. Increase of the liquid fat content causes a dropin viscosity over time until it reaches a steady-state value. As can

Time, t (s)0 10 20 30 40 50 60

She

ar S

tres

s, σ

(P

a)

0

1000

2000

3000

4000 experiments

γ = 50 s-1

γ = 10 s-1

γ = 5 s-1

γ = 1 s-1

T=30oC, Milk Chocolate.

.

.

.

(a)

Time, t (s)

0 10 20 30 40 50 60

She

ar S

tres

s, σ

(Pa)

0

100

200

300

400

500

600

experiments

γ = 100 s-1

γ = 50 s-1

γ = 25 s-1

γ = 5 s-1

T=35oC, Milk Chocolate.

.

.

.

(b)

Fig. 10. Thixotropic behaviour of milk chocolate in three start-up tests at different shear rates: (a) T = 30 �C and (b) T = 35 �C. The continuous lines present predictions of thethixotropic model (Eq. (7)) as explained in the text.

62 H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65

be seen the overall description of the changes is satisfactory.Deviations at short times are related to the viscoelastic nature ofthe material, which is not taken into account.

5.3. Constitutive modelling of melt chocolate

As discussed above and based on the experimental data,chocolate behaves as a viscoplastic melt at temperatures greaterthan 30 �C. The rheological data obtained by using the rotationalrheometer (Fig. 9) can be used to model fluid behaviour. TheHerschel–Bulkley model has been widely used to model the rheo-logical behaviours of pastes, suspensions, and emulsions in whichmaterial shows power-law behaviour after yielding. Thereforethe stress can be written as

s ¼ sy þ k _cn; ð6Þ

where sy is the yield stress, k is the consistency index, and n is thepower-law index (note that this n is not the same as the n of Eq. (2);it is used here for convenience). The equilibrium flow curve datawas used in order to determine the parameters of the model at30 �C and 35 �C. Table 4 shows the optimized values for theHerschel–Bulkley model, and predictions are illustrated as dashedlines in Fig. 9. De Graef et al. [22] studied the effects of fat contenton chocolate yield stress and viscosity. Based on their work, theaverage value of yield stress at 40 �C for 29%, and 33% added fat is23 Pa, and 11 Pa, while the viscosity values are 3 Pa s, and1.25 Pa s, respectively. Considering that the milk chocolate of thepresent work has 32% fat content, our findings are consistent withthese studies. A more comprehensive summary of previousattempts to model the molten chocolate behaviour with theHerschel–Bulkley and the Casson models can be found in the workof Taylor et al. [26].

Although the Herschel–Bulkley model predicts equilibrium datavery well, this model is not able to capture thixotropic behaviouras observed above (Fig. 10a and b with time-dependence). Aconstitutive model for a typical paste should possess at least twoelements. First, a yield stress which is the stress below no flow isobserved. Then, thixotropy should be included through an

Table 4Herschel–Bulkley model parameters (Eq. (6)).

Temperature (�C) sy (Pa) k (Pa sn) n

30 1.2 � 103 29 0.8935 30 2.8 0.98

appropriate kinetic equation involving a structural parameter[42]. It has been claimed by many authors that the existence ofyield stress indicates the existence of structure that requires initialenergy to decay and let the material flow, which implies that flowmight alter the structure and thus thixotropy is expected [43,44].

To represent chocolate rheology, the proposed model includestwo terms and can be written as follows:

s ¼ ð1þ nÞsy;th þ ð1þ nÞg1 _cn: ð7Þ

The first term represents the yield stress behaviour of the mate-rial. It is often assumed that the yield stress is related to the struc-ture of the material through a structural parameter, n [42,45]. Thesecond term includes the viscous part of the response, and it is afunction of the structural parameter, n, which can be describedby a kinetic equation. The rate of change in structure can bedescribed by [46]:

dndt¼ �k1 _cnþ k2ð1� nÞ: ð8Þ

This way the structural parameter n is a normalized quantitythat varies between 0 and 1 and indicates the integrity of the net-work (n = 0: no network or structure; n = 1: fully developed net-work or structure). The first term on the RHS of Eq. (8) indicatesbreakdown of the network due to material deformation; the sec-ond term is responsible for build-up of the network with a timeconstant 1/k2 associated to it.

Structure formation occurs due to Brownian motion and is par-tially due to imposition of shear rate [43]. According to the aboveEq. (8), the shear contribution in structure build-up is neglected,and the rate of formation is set proportional to (1 � n) [47]. It hasbeen assumed that the shear rate may break down structure. Therate of structural break-down is proportional to the shear rateand also to the degree of structure.

According to this kinetic equation, the structural parameterapproaches a steady-state value, neq, at a given value of shear rate,_c, that is:

neq ¼k2

k1 _cþ k2: ð9Þ

Therefore the equilibrium flow curve is given by:

seq ¼ ð1þ neqÞsy;th þ ð1þ neqÞg1 _cn: ð10Þ

Then the equilibrium apparent viscosity geq is given by:

geq ¼ð1þ neqÞsy;th

_cþ ð1þ neqÞg1 _cðn�1Þ: ð11Þ

Table 5Fitted parameters for the structural model.

Temperature (�C) sy,th (Pa) g1 (Pa s) n k1 k2 (s�1)

30 615 29.3 0.89 0.001 0.032535 15 2.84 0.98 0.018 0.18

H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65 63

It is noted that g1 and n were set equal to consistency index (kin Eq. (6)) and power-law index in the Herschel–Bulkley model forthe sake of convenience. At high shear rates, the structural param-eter (neq) approaches asymptotically zero (Eq. (9)). Therefore, bothmodels result in the same apparent viscosity. At low shear rates,the structural parameter (neq) approaches the value of 1 (Eq. (9))and for consistency the thixotropic yield stress (sy,th) has beenset half of the yield stress in the Herschel–Bulkley model. Thus,both models predict the same value for yield stress in the lowershear rates. The ratio k1/k2 was calculated by fitting Eq. (11) tothe data of Fig. 9 by writing neq ¼ 1=½1þ ðk1=k2Þ _c� from Eq. (9).Subsequently, using the data of Fig. 9a and b, the individual valuesof k1 and k2 were calculated. Optimized values for the proposedmodels are summarized in Table 5 and the model predictions forthe equilibrium flow curves are illustrated as dotted line in Fig. 9.

5.4. Capillary melt flow simulations

We consider the conservation equations of mass and momen-tum for incompressible fluids under isothermal, creeping, steadyflow conditions. These are written as [48]:

r � �u ¼ 0; ð12Þ

0 ¼ �rpþr � ��s; ð13Þ

where �u is the velocity vector, p is the pressure and ��s is the extrastress tensor.

The viscous stresses are given for inelastic incompressible fluidsby the relation [48]:

��s ¼ gðj _cjÞ��_c; ð14Þ

where gðj _cjÞ is the apparent viscosity of Eq. (6), in which the shearrate _c is replaced by the magnitude j _cj of the rate-of-strain tensor��_c ¼ r�uþr�uT , which is given by:

j _cj ¼ffiffiffiffiffiffiffiffiffi12

II _c

r¼ 1

2ð _c : _cÞ

� �1=2

; ð15Þ

where II _c is the second invariant of ��_c

Milk Chocolate

Apparent Shear Rate, γA (s-1)

101 102 103

Ext

rusi

on P

ress

ure,

P (

kPa)

10 2

10 3

10 4

RR = 311RR = 139RR = 45HB modelThixotropic model

L/D=20, 2 α=45

.

T = 30oC

(a)

Fig. 11. The effect of apparent shear rate and reduction ratio on the steady-state extrusio20, and contraction angle of 45�: (a) T = 30 �C, (b) T = 35 �C. Symbols are experimental d

II _c ¼ ð _c : _cÞ ¼X

i

Xj

_cij _cij; ð16Þ

Thus, the apparent viscosity is written as:

gðj _cj; nÞ ¼ ð1þ nÞsy;th

j _cj þ ð1þ nÞg1 _cðn�1Þ: ð17Þ

In the above, the apparent viscosity is a function of n, whichobeys the kinetic Eq. (8). In general flows, Eq. (8) becomes the con-vective-transport equation [49,50]:

@n@tþ �u � rn ¼ �k1j _cjnþ k2ð1� nÞ: ð18Þ

For steady-state conditions, on/ot = 0.The above rheological model (Eqs. (14) and (17)) is introduced

into the conservation of momentum (Eq. (13)) and closes the sys-tem of equations. The boundary conditions applied, the solutiondomain, as well as the method used to solve the equations, havebeen explained in detail previously [49–51].

5.5. Results and discussion

5.5.1. Effect of apparent shear rateFig. 10a and b depict the steady-state extrusion pressure as a

function of apparent shear rate for several reduction ratios at30 �C and 35 �C, respectively. The extrusion pressure generallyincreases with an increase in the apparent shear rate (hence volu-metric flow rate), due to the increase in the viscous resistance ofthe paste.

At 35 �C almost all of cocoa fat butter is converted into liquid. Itis expected that chocolate has the lowest crystallized structure andsolid content at 35 �C. Fig. 11b shows that at 35 �C both the Her-schel–Bulkley and the thixotropic model capture the extrusionpressure well and result in very similar values. However, at thetemperature of 30 �C (Fig. 11a) chocolate is very close to meltingand therefore a higher level of structure is expected. In this case,the thixotropic effects become very important. In other words,the apparent viscosity of chocolate paste during the process ishigher than that predicted for the equilibrium flow curve. It takesa longer time for structure to break down. Thus, the Herschel–Bulkley model underpredicts the extrusion pressure. In general,the thixotropic model gives better predictions compared to theHerschel–Bulkley model. However, it can be seen that even thismodel has some difficulties capturing the flow behaviour at lowershear rates.

Milk Chocolate

Apparent Shear Rate, γA

(s-1)

101 102 103

Ext

rusi

on P

ress

ure,

P (

kPa)

100

101

102

103

104

RR = 311RR = 139RR = 45HB modelThixotropic model

L/D=20, 2α=45

.

T = 35oC

(b)

n pressure of chocolate paste using capillary dies having length-to-diameter ratio ofata and lines are the model predictions.

Milk Chocolate

Contraction Angle, 2 α0 20 40 60 80 100 120 140 160

Ext

rusi

on P

ress

ure,

P (

kPa)

100

1000

γA = 25 s-1

γA = 50

γA = 100

γA = 200

γA = 400

γA = 800

γA = 1600

HB thixotropic

.

.

.

.

.

.

.

T= 30oC6000

(a)

RR=311, L/D=20

Milk Chocolate

Contraction Angle, 2α0 20 40 60 80 100 120 140 160

Ext

rusi

on P

ress

ure,

P (

kPa)

10

100

1000

γA = 25 s-1

γA = 50

γA = 100

γA = 200

γA = 400

γA = 800

γA = 1600

HB Thixotropic

.

.

.

.

.

.

.T= 35oC(b)

RR=311, L/D=20

Fig. 12. The effect of contraction angle on the steady-state extrusion pressure of chocolate paste using capillary dies having length-to-diameter ratio of 20 and reduction ratio(RR) of 311: (a) T = 30 �C, (b) T = 35 �C. Symbols are experimental data and lines are the model predictions.

Apparent Shear Rate, γΑ

(s-1)

10 100 1000

Ext

rusi

on P

ress

ure,

P (

kPa)

101

102

103

104

L/D= 20L/D= 0HB modelthixotropic model

RR=311, 2α=45

.

T = 30oC Milk Chocolate

(a)

Milk Chocolate

Apparent Shear Rate, γΑ

(s-1)

10 100 1000

Ext

rusi

on P

ress

ure,

P (

kPa)

100

101

102

103

L/D=20L/D=0HB modelThixotropic model

RR=311, 2α=45

.

T = 35oC

(b)

Fig. 13. The effect of L/D on the steady-state extrusion pressure of chocolate paste using capillary dies having reduction ratio (RR) of 311 and contraction angle of 45�: (a)T = 30 �C, (b) T = 35 �C. Symbols are experimental data and lines are the model predictions.

64 H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 210 (2014) 56–65

5.5.2. Effect of die entrance angleFig. 12a and b depict the effect of contraction angle on the

extrusion pressure at several apparent shear rates and tempera-tures, namely 30 �C and 35 �C, respectively. The effects of contrac-tion angle on extrusion pressure are negligible in both cases. It isworth mentioning that most of the pressure drop occurs in thedie land region, and therefore the effect of die entry design onthe extrusion pressure is much less compared to the case ofsemi-solid chocolate. It is noted that at 25 �C, a significant portionof the pressure drop occurs in the conical part of the die.

5.5.3. Effect of length-to-diameter ratioThe effect of L/D ratio on the extrusion pressure at different

temperatures is shown in Fig. 13a and b. Again, it can be seen thatat the lower temperature, the Herschel–Bulkley model underpre-dicts the extrusion pressure. Note that close to 30 �C, the flowcurve of chocolate changes more non-linearly. The extrusion pres-sure for L/D = 0 is underpredicted as viscoelastic effects might beimportant here and are not taken into account in the modelling.

6. Conclusions

Rheological experiments and simulations in capillary flow wereperformed for a commercial chocolate. It was found that itsrheological behaviour is distinctly different at temperaturesbelow and above 30 �C. At temperatures below 30 �C chocolatebehaves as a soft solid. A previously developed phenomenological

mathematical model for polytetrafluoroethylene (PTFE) paste flowwas used to model its rheology and capillary extrusion. The modeltakes into account the elastic–plastic (strain hardening) and vis-cous nature of the material in its non-melt state. The predictivecapability of this model is tested against capillary data for a varietyof capillary dies having different length-to-diameter ratios (L/D)and contraction angles (2a). It was found that it can represent itsbehaviour in capillary flow very well.

At temperatures above 30 �C, chocolate behaves as a visco-elasto-plastic liquid. Its behaviour has been determined as aHerschel–Bulkley viscoplastic material with a small degree ofthixotropy. The rheological data obtained from parallel-platesand bob–cup rheometers were used to formulate a thixotropicmodel using a structural parameter. The capillary data were foundto be well described with the proposed model.

Acknowledgements

Financial assistance from the Natural Sciences and EngineeringResearch Council (NSERC) of Canada and the THALES program(2012–2015) of the Ministry of Education of Greece are gratefullyacknowledged.

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